Decomposable coherence and quantum fluctuation relations
Erick Hinds Mingo
1
and David Jennings
1,2,3
1
Controlled Quantum Dynamics Theory Group, Imperial College London, Prince Consort Road, London SW7 2BW, UK
2
Department of Physics, University of Oxford, Oxford, OX1 3PU, UK
3
School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, UK.
November 11, 2019
In Newtonian mechanics, any closed-system
dynamics of a composite system in a microstate
will leave all its individual subsystems in dis-
tinct microstates, however this fails dramatically
in quantum mechanics due to the existence of
quantum entanglement. Here we introduce the
notion of a ‘coherent work process’, and show
that it is the direct extension of a work process
in classical mechanics into quantum theory. This
leads to the notion of ‘decomposable’ and ‘non-
decomposable’ quantum coherence and gives a
new perspective on recent results in the theory
of asymmetry as well as early analysis in the
theory of classical random variables. Within the
context of recent fluctuation relations, originally
framed in terms of quantum channels, we show
that coherent work processes play the same role
as their classical counterparts, and so provide
a simple physical primitive for quantum coher-
ence in such systems. We also introduce a pure
state effective potential as a tool with which to
analyze the coherent component of these fluc-
tuation relations, and which leads to a notion
of temperature-dependent mean coherence, pro-
vides connections with multi-partite entangle-
ment, and gives a hierarchy of quantum correc-
tions to the classical Crooks relation in powers
of inverse temperature.
1 Introduction
The superposition principle is at the core of what makes
quantum mechanics so special [1–4]. Classically a par-
ticle might be located in a microstate at any one of a
number of spatial sites with sharp momentum, however
quantum mechanics allows a fundamentally new kind
of state – the particle being a superposition of multiple
different locations x
1
, x
2
, . . . , x
n
. For this, the particle
is in a state |ψi =
P
k
e
iθ
k
√
p
k
|x
k
i where θ
k
are phase
angles, and p
k
is the probability of a measurement of
position returning the classical outcome x
k
. While the
measurement outcomes are random, the state |ψi is a
perfectly sharp state (a pure state, or state of maximal
knowledge) and so should be viewed exactly on a par
with a classical state of well-defined location |x
i
i, say.
More precisely, the structure of the state space in any
physical theory determines many of the distinct char-
acteristics of the particular theory. State spaces are
always convex sets determined entirely by the extremal
points of the set – the pure states of the physical the-
ory. The admissible measurements that the theory al-
lows are defined in relation to this state space (see [5] for
more details) and so are secondary theoretical ingredi-
ents. Therefore a comparison of classical and quantum
mechanics at a fundamental level should equate sharp
microstates with pure quantum states |ψi, as opposed
to say comparing measurement statistics.
In classical mechanics a system with initial position
and conjugate momentum (x
0
, p
0
) evolves in time along
a well-defined trajectory (x(t), p(t)) in phase space de-
termined by the Hamiltonian of the system. This evo-
lution extends into quantum mechanics where a quan-
tum state evolves under a unitary transformation. The
unitary dynamics of such a system S can also be un-
derstood in terms of paths, however it is now de-
scribed in terms of a path integral [6] where the tran-
sition amplitudes are given by integrating the func-
tional exp[
i
~
R
dt(p(t) ˙x(t)−H
S
[x(t), p(t)])] over all paths
(x(t), p(t)) in phase space consistent with the bound-
ary conditions, and where H
S
[x(t), p(t)] is the classi-
cal Hamiltonian for the system S. At the operator
level this dynamics is described a unitary transforma-
tion on states |ψi → U(t)|ψi, for some unitary operator
U(t). Thus, in both classical and quantum mechanics
the Hamiltonian plays a key role in the time evolution
of a system. Moreover in quantum mechanics, if the
system is energetically closed then we have [U, H
S
] = 0
for the unitary evolution of the system. In both classi-
cal and quantum mechanics an initial sharp (pure) state
is evolved to a final sharp state.
The path integral perspective describes this unitary
evolution as a sum over all consistent paths, and so re-
ceives contributions from trajectories that respect nei-
Accepted in Quantum 2019-10-24, click title to verify 1
arXiv:1812.08159v3 [quant-ph] 8 Nov 2019
ther energy conservation nor the classical equations of
motion. However, in the “limit of ~ → 0” a stationary
phase argument tells us that the dominant contributions
to the evolution come from those trajectories around
the classical phase space trajectory (x
class
(t), p
class
(t)),
namely the one that obeys the Hamiltonian equations
of motion [7] given by ˙p = −∂
x
H
S
, ˙x = ∂
p
H
S
. Thus we
recover classical mechanics in the limit.
In the case of two quantum systems S and A, exactly
the same formalism applies, and coherent dynamics of
the composite quantum system can be analysed without
any further fuss – governed by a total Hamiltonian H
SA
for the joint system SA. However, we can now ask the
following question:
What unitary transformations of a composite quantum
system SA in a state |ψi
SA
are possible that (a) obey
energy conservation over the system SA, but (b) give
rise to a transformation of the system S from an
initial pure state and into a final pure state?
It is clear that in the classical limit the answer to this
trivial with sharp microstates – all classical dynamics
that conserve the energy H
SA
give rise to deterministic
transformations of S. However for coherent quantum
systems, which can become entangled with each other,
the answer is less obvious. Indeed, as we shall see, the
above is directly linked to central results from the re-
source theory of asymmetry [8–12] and foundational re-
sults from the 1930s on classical random variables [13–
16].
The physical motivation for this is given by consider-
ing the elementary notion of a mechanical work process
on a classical system along some trajectory. For this,
a mechanical system S initially in some definite phase
space state (x
0
, p
0
) is evolved deterministically to some
final state (x
1
, p
1
) and the mechanical work w for the
process is computed from w =
R
path
F ·dx, where F (t)
is the force exerted on the system along the particular
path [7]. However, the system in this case is not en-
ergetically isolated, and so the quantity of energy w is
only meaningful because it corresponds to an equal and
opposite change in energy of some external classical de-
gree of freedom A that is used to induce the process.
For example, we could imagine some external weight of
mass m that is initialised at some definite height h
0
= 0
and finishes deterministically in some final height h
1
.
In order to associate the height value h
1
for A to w the
work done in the process on S (via w = mgh
1
) it is nec-
essary that energy is exactly conserved between the two
systems S and A. However, if we now wish to consider
superpositions of work processes on S then this account
becomes non-trivial, and so provides the starting point
for our analysis.
In this paper we shall address the above question
and show that the classical notion of deterministic me-
chanical work can be extended into quantum mechan-
ics in a way that exactly parallels the classical case,
and gives rise to connections with topics in contem-
porary quantum physics. Firstly, it leads to a notion
of decomposable and non-decomposable coherence and
has an immediate description in terms of recent frame-
works for quantum coherence [8–11, 17, 18]. We then
extend these considerations in noisy quantum environ-
ments and show that this notion of a coherent work pro-
cess connects naturally with fluctuation relations [19–
26]. The coherent fluctuation theorems we develop go
beyond traditional relations, such as the Crooks rela-
tion, and allow the analysis of quantum coherence in
a noisy thermal environment. Because quantum coher-
ence is handled explicitly this also leads to connections
to many-body entanglement and our results provide a
clean explanatory framework for recent experimental
proposals in trapped ion systems [25].
1.1 Summary of the core results
This paper’s aim is to extend classical notions of work to
quantum systems and provide novel tools for the anal-
ysis of quantum thermodynamics and the role that co-
herence plays. To a large extent, quantum thermody-
namics has taken inspiration from statistical mechanics
when studying notions of ‘work’ [27–30]. In the present
analysis, we instead revisit the deterministic Newto-
nian concept of work and propose a natural coherent
extension into quantum mechanics. We then show how
this coherent notion of work automatically arises in the
non-deterministic context of recent fully quantum fluc-
tuation theorems [23] and provides a fully quantum-
mechanical account of thermodynamic processes.
In Section 2 we begin by defining a coherent work
process
|ψ
0
i
S
ω
−→ |ψ
1
i
S
, (1)
which is a deterministic primitive to describe coherent
energy exchanges. The evolution is constrained to con-
serve energy microscopically and preserve the statistical
independence of the systems. In particular, the exclu-
sion of entanglement generating processes in this defi-
nition is a choice required in order to be the quantum-
mechanical equivalent of Newtonian work involving a
deterministic transition of a ‘weight’ system. Coher-
ent work processes are found to separate into coher-
ently trivial and coherently non-trivial types, with the
distinction appearing as a consequence of the notion
of ‘decomposability of random variables’ in probabil-
ity theory [14, 16]. This connection enables us to show
that coherent states form a closed subset under coherent
work processes. As a result, we find the classical limit of
this fragment of quantum theory is equivalent to New-
Accepted in Quantum 2019-10-24, click title to verify 2
tonian work processes under a conservative force. After
this, we explicitly provide a complete characterization
of coherent work processes for quantum systems with
equal level spacings, and also prove that measurement
statistics on such systems always become less noisy un-
der coherent work processes.
In Section 3 we move to a thermodynamic regime in
which we now have a thermal mixed state described by
an inverse temperature β = (kT )
−1
. We derive a quan-
tum fluctuation theorem using an inclusive picture, and
frame the result on the athermal system. Under the as-
sumptions of time-reversal invariant dynamics and mi-
croscopic energy conservation, we derive a fluctuation
theorem in terms of cumulant generating functions that
has the classical Crooks result as its high-temperature
limit. However we find that the quantum fluctuation
theorem admits an infinite series of corrections to the
Crooks form in powers of β. The higher order terms
account for the coherent properties of the initial pure
states of the system and gives an operational perspec-
tive on related works [23, 26]. There exists another nat-
ural decomposition of the fluctuation theorem in terms
of ‘average change in energy’ and ‘average change in
coherence’ with the caveat that the average coherence
must be carefully interpreted. We derive a closed ex-
pression for the ‘average coherence’ using readily inter-
pretable quantities.
A key result of the work is in Section 4 where we show
that the quantum fluctuation theorem becomes a func-
tion of the coherent work state in a precise sense, and
so further justifies the choice of coherent work processes
as a coherent primitive.
1.2 Relation to no-go results
In the literature, one finds a range of different notions of
work. For example, given some closed system in a state
ρ, described by a time-dependent Hamiltonian H(t) and
evolving under a unitary U, the average work is fre-
quently expressed as W = tr(H(t
f
)UρU
†
) −tr(H(t
i
)ρ).
Two-point measurement schemes (TPM) are common
ways of defining the work done in a process. In such a
scheme, one might define
W =
X
w
X
i,j
0
:
w=E
j
0
−E
i
w p
j
0
|i
p
i
(2)
where p
i
is the probability of measuring |iihi| when the
system is in the state ρ, and p
j
0
|i
is the transition prob-
ability of going from |ii → |j
0
i given |j
0
i is an eigenstate
of H(t
f
). Within a TPM definition of work, one would
wish to recover the traditional results from classical sta-
tistical mechanics when ρ is diagonal (fully incoherent)
state. However, a no-go result was obtained in [27] that
shows that these two requirements are not compatible
for general coherent states. In particular, given any tu-
ple (H(t
i
), H(t
f
), U), it is impossible to always assign
POVM operators for the TPM scheme
M
(w)
T P M
=
X
i,j
0
:
w=E
j
0
−E
i
p
j
0
|i
p
i
|iihi| (3)
that provide an average work from the expectation value
of X =
P
w
wM
(w)
T P M
that is equal for the expectation
value of the operator X = H(t
i
) − U
†
H(t
f
)U.
Our results include the TPM scheme as a special case,
but circumvent no-go results by distinguishing between
a random variable obtained by a POVM measurement
on a quantum state with coherence, and the quantum
state itself as a complete description of a physical sys-
tem
1
. For us, ‘coherent work’ is a fully quantum me-
chanical property and not a classical statistical feature.
This stance is further supported by the fact that our
coherent work processes play precisely the same role in
the coherent fluctuation theorem, as do Newtonian work
processes within the classical Crooks relation.
2 Coherent superpositions of classical
processes and decomposable quantum co-
herence.
We now go into more detail on what is required in order
to have superpositions of processes that overall conserve
energy and give rise to deterministic pure state trans-
formations on a subsystem. The criteria are the same
as in the case of mechanical work discussed in the in-
troduction. We assume a process in which the following
hold:
1. (Sharp initial state) The quantum system S begins
in a definite initial pure state |ψ
0
i.
2. (Deterministic dynamics) The system undergoes a
deterministic quantum evolution, given by some
unitary U (t).
3. (Conservation) Energy conservation holds micro-
scopically and is accounted for with any auxiliary
‘weight’ system A initialised in a default energy
eigenstate |0i.
4. (Sharp final state) The system S finishes in some
final pure state |ψ
1
i.
1
In the sense that quantum states in general do not admit
satisfactory hidden variable theories.
Accepted in Quantum 2019-10-24, click title to verify 3
Since the dynamics is unitary, and S finishes in a pure
state, this implies the system A must also finish in some
pure state |ωi. In the classical case this change in state
of the ‘weight’ exactly encodes the work done on the
system S, and so by direct extension we refer to |ωi
as the coherent work output of the process on S. For
simplicity in what follows, we shall use the notation
|ψ
0
i
S
ω
−→ |ψ
1
i
S
, (4)
to denote the coherent work process on a system S in
which |ψ
0
i
S
⊗|0i
A
→ U (t)[|ψ
0
i
S
⊗|0i
A
] = |ψ
1
i
S
⊗|ωi
A
under a energy conserving interaction. If separate co-
herent work processes |ψ
0
i
S
ω
−→ |ψ
1
i
S
and |ψ
1
i
S
ω
0
−→
|ψ
0
i
S
are possible, the transformation is termed re-
versible and denoted by |ψ
0
i
ω
←→ |ψ
1
i. We are able to
label the two processes by ω instead of (ω, ω
0
) because
the states |ωi and |ω
0
i are related in a very simple way,
as we prove in Appendix A.
The coherent work output is in general a quantum
state with coherences between energy eigenspaces, how-
ever its form is essentially unique for a given initial state
|ψ
0
i and given final state |ψ
1
i for the system S. This
is discussed more in Appendix A. Finally, we note the
assumption that the output system is the same as the
input system S can be easily dropped, where we require
that SA → S
0
A
0
under an isometry, with energy conser-
vation defined over the composite input/output systems
involved.
We can now provide some concrete examples of co-
herent work processes.
Example 2.1. Consider some finite quantum system
S with Hamiltonian H
S
. We assume for simplicity that
the energy spectrum of H
S
is non-degenerate and given
by {0, 1, 2, . . . }. We write |ki for the energy eigenstate
of H
S
with energy k.
Consider the following two incoherent transitions of
a system S between energy states. The first is |0i → |5i
and the second is |1i → |6i. In both cases the process
can be done by supplying 5 units of energy to the sys-
tem from an external source A, and thus |ωi = | − 5i.
Moreover there is a unitary V that can do both processes
coherently in superposition:
V [(a|0i + b|1i) ⊗ |0i] = (a|5i + b|6i) ⊗ | − 5i,
where a, b are arbitrary complex amplitudes for the
quantum state. Note also, that for the case of |a| =
|b| = 1/
√
2 this coherent transition could equally arise
as a superposition of |0i → |6i and |1i → |5i. For this
realisation, a different energy cost occurs for each indi-
vidual transition, however the net effect also gives rise
to | − 5i on the external source system. This process is
reversible.
Figure 1: Coherent superpositions of mechanical processes.
What unitary transformations are possible on a joint system
that conserves total energy, but gives rise to a deterministic
transformation of a pure quantum state |ψ
0
i into some final
pure quantum state |ψ
1
i? Conservation laws place non-trivial
obstacles on quantum systems, and not all such processes can
be superposed. This corresponds to the existence of quantum
states that have non-decomposable coherence. States in the
set C of semi-classical states, defined in the main text, are
infinitely divisible and in the ~ → 0 limit recover the classical
regime.
Beyond this simple example subtleties emerge and
one can obtain non-trivial interference effects in the
work processes, as the following example demonstrates.
Example 2.2. Consider again, a quantum system S
as in the previous example. One can coherently and
deterministically do a work process where
1
2
(|0i + |1i + |2i + |3i)
ω
−→
1
√
2
(|5i + |6i), (5)
which can be viewed as a merging of classical work tra-
jectories. The coherent work output for this process is
|ωi =
1
√
2
(| − 3i + | − 5i). This process is irreversible.
The proof that this transition is possible determin-
istically and irreversibly follows directly from Theorem
2.6 below.
However quantum mechanics also has prohibitions
that give rise to highly non-trivial constraints that rule
out certain processes, as the following illustrates.
Example 2.3. Let S be the same quantum system as
in the above examples. It is impossible to superpose the
two transitions |0i → |5i and |1i → |7i in a coherent
work process, even if we allow A to finish in some super-
posed state |ωi with amplitudes over different energies.
This result again follows from Theorem 2.6 below, but
one way to see that this is impossible is that the swap-
ping of the initial states |0i and |1i and the swapping of
Accepted in Quantum 2019-10-24, click title to verify 4
the two final states |5i and |7i do not correspond to the
same change in energy, and so the “which way” infor-
mation for the process cannot be erased in a way that
does not leave an external energy signature (as occurs
in the first example for |a| = |b| = 1/
√
2).
2.1 “Which-way” information for processes and
energy conservation constraints
We can make more precise what we mean by the en-
ergy conservation condition providing a restriction what
classical work processes can be superposed when the
quantum state has some permutation symmetry. For
simplicity we can consider the same quantum system
S as in the above examples. Suppose it is in some
initial state |ψi
S
that is invariant under swapping of
energy eigenstates |ai and |bi. We can let F
a,b
:=
(|aihb| + |biha| +
P
k6=a,b
|kihk|) be the operator that
swaps these levels, and thus F
a,b
⊗
A
|ψi⊗|0i = |ψi⊗|0i.
This must map to a corresponding operation on the out-
put state |φi⊗|ωi, which we denote
˜
F , and which must
obey V F
a,b
⊗
A
=
˜
F V , where V is the interaction uni-
tary that conserves energy and performs the transfor-
mation. Therefore we have that
˜
F = V (F
a,b
⊗
A
)V
†
.
However because of energy conservation we must also
have that
V |ai ⊗ |0i = |xi ⊗ |a − xi (6)
V |bi ⊗ |0i = |yi ⊗ |b − yi, (7)
for some integers x, y, and thus,
˜
F = |x, a − xihy, b − y| + |y, b − yihx, a − x|
+ other terms.
However in order for such a which-way symmetry to
relate solely to the system S, we must have that
˜
F fac-
tors into a product operator of the form X ⊗
A
so
that the invariance on the initial superposition is associ-
ated with a corresponding transformation on the output
state. Restricting just to the space spanned by |x, a−xi
and |y, b −yi we see that this implies a −x = b −y, and
so y − x = b − a and the invariance is under the swap-
ping operation X = F
x,y
. Thus, the invariance under
permutations in a superposition together with energy
conservation implies that a transformation such as the
one in example 2.3 is forbidden.
2.2 Superposition of classical processes in a
semi-classical regime
Having provided some initial examples for coherent
work processes, we next link with the concept of work
in classical mechanics through the following result. For
this we consider a mechanical coordinate x together
with its conjugate momentum p, which are quantised
in the usual way with commutation relation [x, p] =
i~ . We define a coherent state for the system via
|αi = D(α)|0i for any α ∈ C and where a|0i = 0, with
a := (x + ip)/
√
2 and with the displacement operator
D(α) := exp[αa
†
−α
∗
a]. This defines the set of coherent
states {|αi = D(α)|0i : α ∈ C} for the system.
We now note that we always have the freedom to
rigidly translate a coherent state in energy |ψi →
∆
k
|ψi, where ∆ :=
P
n≥0
|n + 1ihn| for some non-
negative integer k (∆ is occasionally referred to as the
phase operator [31]). We will also allow arbitrary phase
shifts in energy |ni → e
iθ
n
|ni. We now define the states
|α, ki := ∆
k
|αi, (8)
where |αi is a coherent state [32–36]. We denote by C
the set of quantum states obtained from any coherent
state via rigid translations by a finite amount together
with arbitrary phase shifts in energy. Note that arbi-
trary energy eigenstates are included in the set C as a
rigid-translation of the vacuum |α = 0i state. For os-
cillator systems the energy eigenstates are not usually
considered as classical states, however their inclusion in
the set C is required for consistency in the ~ → 0 limit.
Explicitly, the states take the form
C :=
|ψi = e
−iLt
|α, ki : k ∈ N
0
, α ∈ C, t ∈ R, [L, H] = 0
where we allow the use of an arbitrary Hermitian ob-
servable L that commutes with H to generate arbitrary
relative phase shifts in energy.
We now consider the harmonic oscillator, but it is
expected that a similar statement for the classical ~ → 0
limit can be made for more general coherent states.
Theorem 2.4 (Semi-classical regime). Let S be a
harmonic oscillator system, with Hamiltonian H
S
=
hνa
†
a. Let C be the set of quantum states for S as
defined above. Then:
1. The set of quantum states C is closed under all pos-
sible coherent work processes from S to S with fixed
Hamiltonian H
S
for both the input and output.
2. Given any quantum state |ψi ∈ C there is a unique,
canonical state |α, ki ∈ C such that we have a re-
versible transformation
|ψi
ω
c
←→ |α, ki, (9)
with |ω
c
i = |0i and α = |α|. Moreover, the only
coherent work processes possible between canonical
states are
|α, ki
ω
−→ |α
0
, ki, (10)
such that |α
0
| ≤ |α|. Modulo phases, the coherent
work output can be taken to be of the canonical form
Accepted in Quantum 2019-10-24, click title to verify 5
|ωi
A
= |λ, ni
A
, where λ =
p
|α|
2
− |α
0
|
2
and n, k
0
are any integers that obey n + k
0
= k.
3. In the classical limit of large displacements |α| 1,
from coherent processes on C we recover all classical
work processes on the system S under a conserva-
tive force.
Note that we can weaken the assumption that the
output system is S and the Hamiltonian H
S
is the same
at the start and at the end. It is readily seen that
any admissible output system S
0
must have within its
energy spectrum infinitely many discrete energies with
separations being multiples of hν, and the output state
distribution is always Poissonian on a subset of these
discrete eigenstates. A similar degree of freedom ex-
ists for the auxiliary system A, which also must have
a Poissonian distribution on a subset of discrete energy
levels. The output parameters α, α
0
, λ must obey the
same conditions as in Theorem 2.4.
The proof of the above theorem is given in Appendix
A.3, and provides a simple correspondence between the
coherent work processes and work processes in classical
Newtonian mechanics. The main aspect of this result
that is non-trivial is to show that C is closed under work
processes, but follows from a non-trivial result in prob-
ability theory.
Care must be taken when defining the classical limit
as we do in Theorem 2.4. Conservative forces are char-
acterised by their ‘path-independence’, and so the en-
ergy change is dependent only upon the initial and final
phase space co-ordinates (x
0
, p
0
) and (x
1
, p
1
) respec-
tively. But the distinction between quantum and clas-
sical is that in classical physics we can assign a well-
defined pair of co-ordinates to our system, labelling the
position and momentum. In addition to this, the energy
is also as sharp as instruments allow. We therefore use
the fact that for |α| large, the standard deviation of the
position, momentum and energy of our system grow in-
creasingly negligible compared to their expected values.
Furthermore, the energy change of the two phase space
points entirely captures the work done in the process.
The coherent state result shows that in a semi-
classical limit we recover the familiar work behaviour.
However, the earlier examples show that once we de-
viate from the semi-classical regime the structure of
coherence in such processes becomes highly non-trivial
and certain transformations are impossible. This is cap-
tured by the notion of decomposable coherence and non-
decomposable coherence, which we now describe.
2.3 Decomposable and non-decomposable quan-
tum coherence
The topic of coherent work is directly related to whether
a random variable in classical probability theory is ‘de-
composable’ or not. In classical theory a random vari-
able
ˆ
X is said to be decomposable if it can be written as
ˆ
X =
ˆ
Y +
ˆ
Z for two independent, non-constant random
variables
ˆ
Y and
ˆ
Z [13, 16]. This turns out to provide
a different characterization of coherent work processes.
For compactness, given a system X with Hamiltonian
H
X
, denote by
ˆ
X the random variable obtained from
the measurement of H
X
in some state |ψi. The proba-
bility distribution is given by p(E
k
) = hψ|Π
k
|ψi, where
Π
k
is the projector onto the energy eigenspace of H
X
with energy E
k
.
Within our quantum-mechanical setting we now say
that a state |ψi has non-decomposable coherence if given
a coherent work process |ψi
ω
−→ |φi this implies that
either |φi or |ωi are energy eigenstates of their respec-
tive Hamiltonians, and otherwise the state is said to
have decomposable coherence. We also refer to such
non-decomposable transformations as “trivial” coherent
processes. With this notion in place, we obtain the fol-
lowing result.
Theorem 2.5. Given a quantum system X with Hamil-
tonian H
X
with a discrete spectrum, a state |ψi admits
a non-trivial coherent work process if and only if the
associated classical random variable
ˆ
X is a decompos-
able random variable. Furthermore, for a coherent work
process
|ψi
X
ω
−→ |φi
Y
, (11)
the associated classical random variables are given by
ˆ
X =
ˆ
Y +
ˆ
W , where
ˆ
Y and
ˆ
W correspond to the measure-
ments of H
Y
and H
W
in |φi
Y
and |ωi
W
respectively.
We see that the states in C have infinitely divisible
coherence, and thus admit an infinite sequence of non-
trivial coherent work processes. A random variable
ˆ
Y is
infinitely divisible if for any positive integer n, we can
find n independent and identically distributed random
variables
ˆ
X
n
that sum to
ˆ
Y [14]. The proof of this is
provided in Appendix A.2.
2.4 General coherence decomposition and disor-
der in measurement statistics
We now specify exactly what coherent work processes
are possible for a quantum system S with discrete,
equally-spaced energy levels. To avoid boundary effects,
we assume for mathematical convenience that both S
Accepted in Quantum 2019-10-24, click title to verify 6
Figure 2: Decomposability of coherence. Shown is a
schematic of decomposable coherence. A state |ψ
0
i for a quan-
tum system S can be split into pure state coherence |ψ
1
i on
S
0
and a coherent work output |ωi on A under an energetically
closed process. The probability distributions for the energy
measurements on both |ψ
1
i and |ωi always majorize the distri-
bution for |ψ
0
i and so the energy measurement disorder on S
is always non-increasing.
and A are doubly infinite ladders and make use of core
results from asymmetry theory [10].
Note that for a finite d-dimensional system S with
constant energy gaps, essentially the same analysis ap-
plies since one can always embed the finite system into d
levels of such a ladder, apply the below theorem and en-
sure that the support of the output state does not stray
outside the d levels of the embedding. For systems with
general energy spacings, again a similar analysis can
be applied, since the required dynamics will not cou-
ple different Fourier modes in a quantum state and so
will only transform coherences between subsets of levels
with equal level spacings.
For the core model of a ladder system with equally
spaced levels we have the following statement for coher-
ent work processes, with the proof provided in Appendix
A.4.
Theorem 2.6. Let S be a quantum system, with energy
eigenbasis {|ni : n ∈ Z} with fixed Hamiltonian H
S
=
P
n∈Z
n|nihn|. Given an initial quantum state |ψi
S
=
P
i
√
p
i
|ii
S
, a coherent work process
|ψi
S
ω
−→ |φi
S
, (12)
is possible with
|φi
S
=
X
j
e
iθ
j
√
q
j
|ji
S
(13)
|ωi
A
=
X
k
e
iϕ
k
√
r
k
|ki
A
(14)
for arbitrary phases {θ
j
} and {ϕ
k
}, with output Hamil-
tonian H
S
as above and H
A
=
P
n∈Z
n|ni
A
hn|, if and
only the distribution (p
n
) over Z can be written as
p
n
=
X
j∈Z
r
j
(∆
j
q)
n
.
=
X
j∈Z
r
j
q
n−j
(15)
for distributions (q
m
) and (r
j
) over Z.
Again, note that we may weaken the assumption that
the Hamiltonian of S is the same at the start and at the
end with essentially the same conclusion. For example,
one may introduce unoccupied energy levels in the out-
put system S. Moreover if we drop the assumption of
the precise form of the Hamiltonian H
A
, it is readily
checked that the output Hamiltonians of S and A could
be shifted by equal and opposite constant amounts while
respecting the conservation of energy condition. In each
of these variants, however, the core structure of the out-
put distributions is essentially the same.
Now the Birkhoff-von Neumann theorem states that
every bistochastic matrix is a convex combination of
permutations [37, 38], and thus since equation (15) is
a convex combinations of translations (which are them-
selves permutations) this means the two distributions
are related by a bistochastic mapping q → p = Aq,
where A is bistochastic. However this implies that q
majorizes p, written p ≺ q, and which establishes a
very useful relation between the input distribution on
S and output distributions on S and A. Specifically
it implies that in any such coherent work process we
must have both p ≺ q and p ≺ r. Now if f is any con-
cave real-valued function on probability distributions,
then a standard theorem [38] tells us that f(p) ≥ f(q)
whenever p ≺ q. This leads us to the following result.
Corollary 2.6.1 (Disorder in coherent processes
never increases). Let f
dis
(ψ) be some real-valued con-
cave function of the distribution (p
k
) over energy in
|ψi =
P
k
e
iθ
k
√
p
k
|ki (such as the Shannon entropy
function), then in any coherent work process |ψi
ω
−→ |φi
we have that
max{f
dis
(φ), f
dis
(ω)} ≤ f
dis
(ψ). (16)
The significance of this is that every measure of “dis-
order” f
dis
for the energy statistics in a pure quantum
Accepted in Quantum 2019-10-24, click title to verify 7
state |φi will be a convex function of this form. The
corollary thus shows that the energy statistics of both
the coherent work output and the final state of S are
always less disordered than that of the initial state of S
with respect to all measures. This also provides an in-
tuitive perspective on the decomposability of coherence
in a such a process.
Under repeated coherent work processes on the final
state of S and the work output state, the disorder will
become diluted but for a general scenario will stop when
one reaches non-decomposable components. However
for the class of states in C there is no such obstacle
and the disorder can be separated indefinitely. Note
also that if in addition the function f
dis
is additive over
quantum systems, namely f
dis
(φ⊗ψ) = f
dis
(φ)+f
dis
(ψ),
then the total disorder over all systems as measured by
this function will remain constant throughout.
2.5 Multiple processes and coherently connected
quantum states
The framework we have laid out involves an auxil-
iary system initialised in an energy eigenstate. It is
for this reason that the disorder dilutes among the
bipartite system, rather than being able to increase
in one subsystem. This has the advantage of isolat-
ing the coherent manipulations on the primary system.
Due to this, in a sequence of coherent work processes
|ψ
1
i
S
ω
1
−→ |ψ
2
i
S
···
ω
n
−→ |ψ
n+1
i
S
, there might only be a
finite number of non-trivial processes possible before the
system ends in a state with non-decomposable coher-
ence. However, a notable counter-example is the semi-
classical result for which this never happens.
The method we have chosen is the simplest concep-
tually, but it is not the most general procedure possi-
ble. One could imagine an auxiliary system prepared in
an arbitrary state |φi
A
. In such a case, the approach
involving random variables must be slightly modified
but the same tools can be applied. The most gen-
eral coherent work process possible, excluding entan-
glement generation, is a transformation of the form
N
n
i=1
|ψ
i
i −→
N
m
i=1
|ψ
0
i
i, with the only two constraints
that the dynamics conserve energy globally, and the
subsystems are left in a pure product state. In this mod-
ified framework, results such as Corollary 2.6.1 would no
longer hold, since coherence can be concentrated from
many subsystems to fewer subsystems. We leave this as
an open question for later study.
A special case that is relevant here is if we have two
quantum states |ψi
S
and |φi
S
and the auxiliary system
A begins in a state |ωi
A
with coherence and terminates
in some energy eigenstate |0i
A
. This can be denoted as
|φi
S
ω
−→ |ψi
S
, (17)
however, if we are concerned with transitioning from
|ψi
S
to |φi
S
we can use the fact that if V is an energy
conserving unitary realising the process (17) then its
inverse V
†
exists and is also energy conserving. This
implies that we can write the inverse transformation as
|ψi
S
−ω
−→ |φi
S
, (18)
which is realised through the unitary transformation
V
†
[|ψi
S
⊗ |ωi
A
] = |φi
S
⊗ |0i
A
. (19)
This extension of notation allows us to define the nega-
tive coherent work output −ω, which physically means
the coherent quantum state |ωi
A
required to realise the
transformation – in other words ω is the coherent work
input to the process. If a coherent work process exists
in either direction between |ψi
S
and |φi
S
then we say
the states |ψi
S
and |φi
S
are coherently connected. From
this definition we see that the whole set C is a coherently
connected set in the sense that any two states in C are
coherently connected. Note however that coherent con-
nectedness is not a transitive relation: if (|ψ
1
i, |ψ
2
i) and
(|ψ
2
i, |ψ
3
i) are each coherently connected pairs then
it does not imply that (|ψ
1
i, |ψ
3
i) are coherently con-
nected.
2.6 But shouldn’t ‘work’ be a number?
We have used the term ‘coherent work output’, however
the issue of what ‘work’ is in general, is subtle. Here
we do not wish to use the term without justification,
and so now expand on why this is not an unreasonable
use of language. In particular, we argue that the ap-
proach is fully consistent with operational physics and
is a natural generalisation of what happens in both clas-
sical mechanics and statistical mechanics. For ease of
analysis we list the main components of the argument:
• Work is done on a system S in a process when an
auxiliary physical system A is required to trans-
form, under energy conservation from a default
state, so as to realise the process on S.
• Work on a system S in Newtonian mechanics is
a number w ∈ R that operationally can be read
off from the transformed state of S or equivalently
from the transformed auxiliary system A’s state.
• Work on a system S in statistical mechanics is not a
number, but is described mathematically by a ran-
dom variable W : (dµ(x), x) on the real line. Op-
erationally, individual instances are read off from
measurement outcomes on S, and by energy con-
servation are in a one-to-one correspondence with
outcomes on A.
Accepted in Quantum 2019-10-24, click title to verify 8
Figure 3: When does “work” make sense? The two left-
hand cases concern deterministic processes, whereas the two
right-hand cases involve mixed states with unavoidable prob-
abilistic elements. In Newtonian mechanics, work is a deter-
ministic number w and a largely unambiguous concept. Ex-
tensions into statistical mechanics mean that work is no longer
described by a number, but instead is a classical random vari-
able W : (dµ(x), x). Classical mechanical work occurs when
this distribution is sharp, µ(x) = δ(x − w). However one may
extend the deterministic concept of work from Newtonian me-
chanics in a natural way into quantum mechanics. This leads
to a coherent form of work, that coincides with a superposition
of classical work processes and in the ~ → 0 limit recovers the
classical notion. It also it arises in fluctuation relations in the
same manner as its classical counterpart.
• Quantum mechanics is a non-commutative exten-
sion of statistical mechanics, in which classical dis-
tributions dµ(x) are replaced with density opera-
tors ρ on some Hilbert space H.
• Complementarity in quantum theory implies that
general POVM measurements on a state ρ are in-
compatible, and moreover “unperformed measure-
ments have no result” [2]. No underlying values are
assigned to a pure state |ωi in a superposition of
different classical states.
The last two points are vital. One can of course consider
random variables obtained from measurements on quan-
tum systems, but this is not the same thing and cannot
describe quantum physics in any sensible way. Trying
to use classical random variables to fully describe quan-
tum physics is essentially a “hidden variable” approach,
however we know from many results that it is impossi-
ble to define a hidden variable theory for quantum me-
chanics unless one is willing to make highly problematic
assumptions, such as violation of local causality [2].
In our approach, the coherent work process does not
associate a single, unique number to the coherent work
output, instead we describe the output of the process
in terms of a pair (|ωihω|, H) that generalizes the clas-
sical stochastic pair W : (dµ(x), x) into a fully coherent
setting.
Obviously one could measure the state |ωi in
the energy eigenbasis to get measurement statistics
(p(E
1
), p(E
1
), p(E
2
) . . . , ), and associate energy scales
to these observed outcomes, however this is a fundamen-
tally different scenario – the state |ωi has no random-
ness in itself, instead it has complementarity in observ-
ables which is physically different. The measurement
on |ωi, in contrast, introduces classical randomness and
so steps out of the deterministic mechanical domain.
Indeed, one can make the following comparison: the co-
herent work output |ωi is to the measurement statistics
p = (p(E
1
), p(E
1
), p(E
2
) . . . , ) what a laser is to inco-
herent thermal light. We can obtain interference effects,
entanglement and other non-classical phenomena from
a quantum state |ωi, but not from a probability distri-
bution p.
Note also, that essentially the same approach is taken
for quantum entanglement. There is no measurement
one can do corresponding to the “amount of entangle-
ment” in a state, despite it being a crucial physical
property of a state with dramatic effects. Instead, en-
tanglement is quantified in terms of pure state ‘units’,
for example the Bell state |ψ
−
i. In both the coherent
work case and the entanglement case, one ultimately
concerned with empirical statistics and not abstract
quantum states. The reconciliation of this is that quan-
tification in terms of the abstract quantum states will
determine the empirical statistics in a way that depends
on the particular context involved.
In the context of the next section, by adopting this
coherent description, we can evade recent no-go results
[27] on fluctuation relations for thermodynamic systems
and provide insight into recent developments in this field
[23, 25, 28, 39, 40] while also connecting naturally with
well-established tools for the study of quantum coher-
ence [8, 9, 41, 42].
3 Coherent Fluctuation Theorems
In this section we show that the concept of coherent
work processes discussed in the previous section nat-
urally occurs within a recent fluctuation theorem [23–
25] that explicitly handles all quantum coherences and
which is experimentally accessible in trapped ion sys-
tems [25]. We first introduce the core physical assump-
tions involved and then explain how these naturally re-
late to the concept of coherent work processes. But
first, we briefly re-cap on the classical Crooks fluctua-
tion relation, where phase space trajectories are center
stage.
3.1 Classical, stochastic Tasaki-Crooks relation
In the most basic setting one has a bath subsystem B,
initially in some thermal equilibrium state γ
0
=
e
−βH
0
Z
with respect to an initial Hamiltonian H
0
, which is
then subject to some varying potential that changes its
Accepted in Quantum 2019-10-24, click title to verify 9
Hamiltonian H
0
→ H(t) → H
1
and finishes in some fi-
nal Hamiltonian H
1
. Note that through an appropriate
choice of units we can always arrange that a protocol
starts at t = 0 and finishes at time t = 1.
By doing sharp, projective measurements of the en-
ergy at the start and end of the protocol one obtains
stochastic changes in energy on B for the protocol that
lead to a stochastic definition of work (see the review
[22] and the recent paper [29] by Talkner and Hänggi for
excellent discussions on the concept of work in statis-
tical mechanics). In this setting one can readily derive
a Tasaki-Crooks fluctuation relation that compares the
work done during a forward protocol P : H
0
→ H(t) →
H
1
with the work done in the time-reversed protocol
P
∗
: H
1
→ H(t)
rev
→ H
0
, obtained by reversing the
time-dependent variation of the system’s Hamiltonian
under P and beginning in equilibrium with respect to
H
1
. The Tasaki-Crooks relation compares the probabil-
ity of doing work W = w in the forward protocol with
the probability of doing work W = −w in the reverse
protocol, and is given explicitly by
P [w|γ
0
, P]
P [−w|γ
1
, P
∗
]
= e
−β(∆F −w)
(20)
where ∆F := F
1
−F
0
, and F
k
is the free energy of the
equilibrium state γ
k
=
e
−βH
k
Z
for k = 0, 1. Note that
there is no assumption that B finishes in the equilibrium
state γ
1
, and there is no assumption that the protocol
is in any way quasi-static. From this one can derive the
Jarynski relation and the Clausius relation hWi ≥ ∆F
from traditional thermodynamics.
3.2 From stochastic relations to genuinely quan-
tum fluctuation relations
While the formalism of the two-point measurement
scheme is applicable to both classical and quantum
systems alike, it must be emphasized that any gen-
uine coherent features are destroyed and the operational
physics is entirely equivalent to classical stochastic dy-
namics on a classical system (see [43] for a discussion
of why this stochasticity is essentially classical). Sev-
eral works [24, 27, 28, 39, 44–46] have proposed vari-
ants of the two-point scheme with the aim of providing
fluctuation relations that capture genuinely quantum-
mechanical features. Of note are the no-go results of
[27, 47] that prove that certain desirable features for a
measurement scheme are incompatible. One model that
attempts to circumvent these obstacles was proposed in
[28] where projective measurements were replaced by
weak measurements, and in which quasi-probability dis-
tributions arise. A related analysis in terms of quasi-
probabilities was provided in [45], while in [47] it was
Figure 4: Classical fluctuation setting. In the classical set-
up, any particular phase space point (x, p) is taken along a
unique trajectory (x(t), p(t)), and work is accumulated in the
process. Work originates as a change in Hamiltonian from a
time t
0
to a time t
1
. This change is induced via an interaction
with an implicit energy source, in which the total system can be
described by a time-independent Hamiltonian. Note that if one
follows a classical Crooks derivation but imposes a restriction to
coarse-grained resolutions of phase space (such as the shaded
regions shown) then it is expected that one will obtained similar
results to the coherent fluctuation relations for quantum theory.
shown that these “negative probabilities” are in fact
a witness of contextuality – a fundamental feature of
quantum mechanics that includes quantum entangle-
ment as a special case [48, 49] and is conjectured to
be the key ingredient providing speed-ups in quantum
computers [50, 51].
However weak measurements are a very particular
type of quantum measurement and so one can ask if an
over-arching framework exists that includes the classical
stochastic setting, weak measurements and fully gen-
eral POVM measurements that lead to coherent fluc-
tuation relations. In [23] a general framework was de-
veloped by Åberg. The framework developed can be
viewed as a fully coherent version of the “inclusive” de-
scription taken by Deffner and Jarzynski in [52] for the
thermodynamics of classical systems, and leads to an
extremely general fluctuation relation that handles co-
herence, and includes the classical fluctuation relations
as special cases. However the technical level of the anal-
ysis is formidable and quite different from existing ap-
proaches. The physical assumptions of the model are
natural, however it was not clear how one should inter-
pret the broad, abstract results in more familiar terms.
3.3 The coherent fluctuation framework
Before proceeding, we make a brief comment on our
notation. We shall use capital Roman letters (A, B, C)
Accepted in Quantum 2019-10-24, click title to verify 10
at the start of the alphabet to denote quantum systems
that are initialized in quantum states with no coherence
between energy eigenspaces. For example, for an auxil-
iary ‘weight’ systems initialized in |0i we use A. In this
analysis we are primarily concerned with coherent fea-
tures and so we use the notation S for a quantum system
that initially has coherence between energy eigenspaces.
For fluctuation relation contexts it is typically the
case that we have a large thermal bath that we do not
control, together with another quantum system that we
do control and which is initially in thermal equilibrium
with the bath. Following the above convention we label
these B
1
(e.g.for the uncontrolled system) and B
2
(e.g.
for the controlled thermal system), and reserve S for
additional coherent degrees of freedom that we can also
control and wish to study.
The approach taken in [23] can be viewed as a gen-
eralisation of [52] to a fully quantum setting in which
all energies and all coherences in energy are accounted
for explicitly. Moreover, it is exactly in the same spirit
as the coherent work processes as introduced earlier –
quantum coherences in energy are always relational de-
grees of freedom and so an inclusive approach is natu-
ral. Specifically, in addition to the primary system S
one introduces an additional bath subsystem B and the
composite system SB is energetically closed in terms
of energy flows and coherence flows, but not necessar-
ily dynamically closed
2
. Therefore we make use of an
inclusive “microscopic” description of the energetic de-
grees of freedom with the aim of arriving at a fluctuation
relation that handles arbitrary coherence.
A key thing to highlight is that every fluctuation rela-
tion (classical or quantum) involves such an additional
system S, however this system is generally left implicit
as the external physical degrees of freedom that pro-
vide energy and coherence so as to change the Hamilto-
nian of B in some time-dependent manner. Such time-
dependent variations of a Hamiltonian interact non-
trivially with coherent structures and so this is a pri-
mary reason why such an inclusive microscopic descrip-
tion is needed.
A crucial point is the following: since the system SB
is closed in terms of energy and coherence flows this
means that all energy/coherence changes in S corre-
spond in a one-to-one fashion to the energy/coherence
changes in B.
The assumptions of the framework are as follows:
1. A microscopic, inclusive description is taken for a
thermal bath and quantum system.
2
This does not mean that the system is “autonomous” – an ex-
perimenter can still implement general time-dependent protocols
by macroscopically varying internal parameters of the system.
2. The microscopic description is energetically closed,
but not dynamically closed.
3. Time-reversal symmetry holds for the microscopic
dynamics of the composite system.
4. The thermal system B is initially in some Gibbs
state with respect to an initial Hamiltonian and S
is in some arbitrary quantum state.
The model therefore involves the initialisation of SB
in some joint product state ρ⊗γ
0
where B is in thermal
equilibrium, as above, while S is allowed to be in an
arbitrary state ρ. The primary system is then subject
to open system dynamics that transforms the state as
ρ → E(ρ) where
E(ρ) = tr
B
V (ρ ⊗ γ
0
)V
†
, (21)
in terms of a microscopic unitary V on SB that may be
partially controlled by macroscopic parameters by an
experimenter. Note that any protocol that an experi-
menter implements will in general be through macro-
scopic parameters, however any such transformation
will admit such a description via a Stinespring dilation
[53]. We refer the reader to [23] for a discussion of this
point.
In the incoherent regime, the fluctuation theorem
compares transition probabilities of ρ → σ with transi-
tion probabilities of ΘσΘ
†
→ ΘρΘ
†
where Θ is a time-
reversal operator. We shall find that when coherence is
present, this must be generalised so that the core Crooks
construction can be implemented.
3.4 Time-dependent Hamiltonians within a mi-
croscopic, inclusive constraint–description
Since we must first account for microscopic degrees of
freedom, any protocol for the change in Hamiltonian
H
0
→ H(t) → H
1
should be handled with care. The
physical reason for this is that a general time-dependent
Hamiltonian will interact non-trivially with the quan-
tum coherence between energies and so to properly de-
scribe the latter one must be careful with the former.
At the microscopic level Hamiltonians are generators of
time-translation, whereas any time-dependent Hamil-
tonian is always an effective description that arises
through the interaction with some external system.
Consider any time-dependent Hamiltonian H(t) sce-
nario that evolves from some initial H
0
:= H(0) to some
final H
1
:= H(1). Since the ‘t’ in H(t) corresponds to
a physically discernible value this implies that it can
always be made explicit within a Hilbert space descrip-
tion via a degree of freedom of the composite system
(otherwise the parameter is physically meaningless!).
Accepted in Quantum 2019-10-24, click title to verify 11
Therefore, there always exists a Hilbert space descrip-
tion of a time-dependent Hamiltonian in terms of a com-
posite SB system with a Hilbert space that takes the
form
H
SB
= H
S
⊗ (H
0
B
⊕ H
1
B
⊕ H
other
B
), (22)
where H
0
B
is the span of the eigenstates {|E
0
k
i} of the
initial Hamiltonian H
0
, and similarly for H
1
B
and H
1
.
The subspace H
other
B
corresponds to any other physical
degrees of freedom that may be accessed at intermediate
times 0 < t < 1 of the protocol
3
. Therefore any protocol
H
0
→ H(t) → H
1
on a quantum system can be under-
stood at a microscopic level as the evolution of the bath
subsystem B from being constrained solely to the sub-
space H
0
B
, at the start of the protocol (t=0), to it being
constrained to the subspace H
1
B
at the end of the pro-
tocol (t=1). The underlying Hamiltonian for the com-
posite system is therefore H
SB
= H
S
⊗
B
+
S
⊗H
B
,
with
H
B
= H
0
⊕ H
1
⊕ H
other
, (23)
where we simply combine all possible intermediate
Hamiltonians H(t) into the term H
other
simply for com-
pactness. This underlying description in terms of con-
straints, while appearing strange from the traditional
H(t) formulation, is entirely consistent with the usual
story. More importantly, it turns out to provide a pow-
erful perspective in the context of fluctuation relations
with coherence.
3.5 Coherent thermalisation of a system with re-
spect to constraints
In our analysis, the constraint–description for the time-
dependent protocol turns out to connect with the tra-
ditional notion of thermodynamic constraints in phe-
nomenological settings (see for example Callen [54]).
Specifically, we can talk of a “coherent thermalisation”
of B with respect to a constraint C at inverse tempera-
ture β, as we now describe through examples.
Let C
0
be the constraint “B is constrained to H
0
B
”.
Mathematically this constraint is described by the pro-
jector Π
0
=
P
k
|E
0
k
ihE
0
k
| onto the subspace H
0
B
. Exper-
imentally this corresponds to a POVM measurement on
SB given by {
S
⊗Π
0
,
S
⊗Π
1
,
S
⊗Π
other
} that asks:
does B have a Hamiltonian H
0
, the Hamiltonian H
1
, or
some intermediate Hamiltonian H(t)? In actual experi-
ments it simply amounts to the experimenter looking at
what the classical dials of the apparatus are set to, or if
the time-varying protocol is fixed, looking at a clock to
3
The description of a continuous infinity of ‘t’ values in this
vein has technical subtleties, namely the mathematical awkward-
ness of non-separable Hilbert spaces. Experimentally however,
physics always involves a finite resolution scale and so this sub-
tlety has no physical content here.
determine the time t. In the event of the first outcome,
the system B is updated via the projector Π
0
that en-
sures it is entirely constrained to H
0
B
and so C
0
simply
amounts to the statement: the system B has Hamilto-
nian H
0
.
Thermalisation with respect to the constraint C
0
rep-
resented by a projector Π
0
is now defined [23] by the
transformation
Π
0
→ Γ(Π
0
) =
e
−βH
B
/2
Π
0
e
−βH
B
/2
tr(e
−βH
B
Π
0
)
(24)
=
e
−βH
0
Z
= γ
0
, (25)
and thus Γ(Π
0
) represents the statement that B has
Hamiltonian H
0
and is thermalised with respect to it,
at inverse temperature β. The mapping Γ should be
viewed as transforming a Hamiltonian constraint into
a thermodynamic constraint. In exactly the same way,
we also have that Γ(Π
1
) = γ
1
.
The particular form of the coherent thermalisation
transformation, which we shall simply call Gibbs rescal-
ing, is required for various reasons, however certain im-
portant cases should first be highlighted. If the con-
straint C is “The system B is in the energy eigenstate
|E
k
i of H
0
” then this is a much stronger constraint.
Again, this is represented via a projector |E
0
k
ihE
0
k
|, how-
ever now we find that
Γ(|E
0
k
ihE
0
k
|) =
e
−βH
B
/2
|E
0
k
ihE
0
k
|e
−βH
B
/2
tr(e
−βH
B
|E
0
k
ihE
0
k
|)
= |E
0
k
ihE
0
k
|.
(26)
Thus if B is constrained to be exactly in the sharp eigen-
state |E
0
k
i then there are no remaining degrees of free-
dom to thermalise, and the state remains the same.
A more interesting case is if the only thing we know
is that B is in pure state, with a uniform superposi-
tion over all the eigenstates of H
0
. This condition is
described by the projector |
0
ih
0
| where
|
0
i :=
1
√
d
0
X
k
|E
0
k
i, (27)
where d
0
is the dimension of H
0
. It is readily checked
that for this case
Γ(|
0
ih
0
|) = |γ
0
ihγ
0
|, (28)
where |γ
0
i =
1
√
Z
0
P
k
e
−βE
0
k
/2
|E
0
k
i is the coherent
Gibbs state with respect to the initial Hamiltonian
H
0
. Or, in terms of entanglement we can consider the
maximally entangled state |φ
+
i
AB
=
1
√
d
P
d−1
k=0
|E
k
i
A
⊗
|E
k
i
B
on two quantum systems of dimension d. In this
case, coherent thermalization of the maximally entan-
Accepted in Quantum 2019-10-24, click title to verify 12
gled state with respect to A leads to
Γ(|φ
+
ihφ
+
|) = |
˜
φih
˜
φ| (29)
|
˜
φi =
1
√
Z
X
k
√
e
−βE
k
|E
k
i
A
⊗ |E
k
i
B
, (30)
which is the thermofield double state [55] in high energy
physics and condensed matter.
These examples justify X 7→ Γ(X) as describing a
formal coherent thermalisation: if no coherences are
present Γ provides the thermal Gibbs state; pure states
are always sent to pure states; and coherent superpo-
sitions over energy are re-weighted with Gibbs factors
at inverse temperature β. However, it is important to
emphasise that Γ is not a physical transformation but
an invertible mapping on states that pairs (ρ, Γ(ρ)), as
being distinguished – in the same way as time-reversal
pairs (ρ, ΘρΘ
†
).
The time-reversal operator pairs a forward temporal-
direction with a backward temporal-direction, while Γ
pairs a Hamiltonian constraint with a thermodynamic
constraint. It might seem strange that we pair a me-
chanical constraint with one that has an explicit ther-
modynamic feature (namely a temperature), but the
need for this arises from not necessarily having sharp
microstate properties in quantum superpositions and it
is this indeterminacy that requires the Gibbs-rescaling.
Such a scenario would also arise in a classical exam-
ple where instead of doing a Crooks relation on mi-
crostates (x, p) one follows the same construction on
distinguished distributions {q
k
(x, p)}, e.g. if we took
coarse-grained resolutions of finite sized support on
phase space. The q
k
(x, p) would also have to be Gibbs-
rescaled in order to obtain a coarse-grained Crooks re-
lation (this is the classical analog of the above example
of projector Π that is not of rank-1).
For the case where the constraint is the above Π
0
,
the pairing (Π
0
, Γ(Π
0
)) could potentially be viewed as
a statement of the equivalence of the microcanonical
and canonical ensembles in a similar vein to recent clas-
sical results [56]. This perspective can be formulated
in terms of dualities between constrained and uncon-
strained optimization problems, and we conjecture that
the present Gibbs-rescaling could be naturally described
in such terms. We leave this to future analysis.
Beyond its role with time-reversal in identifying the
correct trajectories to pair, we note that the transforma-
tion Γ is not a linear physical map but instead amounts
to an change in our description of the system. In par-
ticular, one that is covariant under energy conserving
unitaries V , with V Γ(ρ)V
†
= Γ(V ρV
†
). This structure
arises naturally in fluctuation settings and scenarios in
which one reverses a general quantum operation. In
[57] Crooks considered the reversal of a Markov pro-
cess, and found the resulting transformation generates
factors of e
−βE
k
/2
/
√
Z that Gibbs re-scale all probabil-
ity distributions. The above mapping should be viewed
as the extension of this to a fully coherent setting. The
transformation Γ(X) also arises in the context of the
Petz recovery map [58] for quantum channels, and of
course in the paper [23] by Åberg, where it is dis-
cussed in more detail. Its form also arises in quantum-
mechanical settings for the transition between micro-
scopic and macroscopic descriptions of thermodynamic
systems [59], which is perhaps the most appropriate per-
spective in light of the above constraint discussion. It
would be of interest to obtain an independent opera-
tional analysis of this transformation (perhaps as a co-
herent form of a maximum entropy principle). We do
not expand any more on this here, but instead refer the
reader to [26] and [23] for more discussion.
3.6 A key relation between forwards and reverse
protocols for quantum systems
Given the preceding discussion we can derive the co-
herent fluctuation relation, however this follows from a
core structure that is describable on a single quantum
system. This was discussed in [23], but here we provide
a slightly modified approach that will prove useful for
what follows. Let S
tot
be any quantum system, with
Hamiltonian H
tot
. We consider the following sequence
that abstracts the classical notion of a ‘trajectory’ to a
coherent form for S
tot
:
1. A measurement is done on S
tot
with outcome given
by a (not necessarily rank-1) projector Π
0
.
2. Coherent thermalisation of Π
0
with respect to H
tot
and at inverse temperature β updates the system
to the state Γ(Π
0
).
3. The system S
tot
evolves unitarily under a unitary
V that commutes with H
tot
.
4. A final measurement is done on S
tot
with projective
outcome Π
1
(again not necessarily rank-1).
We denote the probability of this sequence as P [Π
1
|Π
0
],
and is given explicitly as
P [Π
1
|
˜
Π
0
] = tr[Π
1
V (Γ(Π
0
))V
†
]. (31)
The expression can be viewed as the probability of a
particular ‘trajectory’ under the above protocol.
The fluctuation setting also requires a notion of a
time-reversal of a trajectory. In quantum mechan-
ics, time-reversal is an anti-unitary transformation at
the level of the Hilbert space. A state transforms as
|ψi → Θ|ψi where Θ is both anti-linear (e.g. Θ(α|ψi) =
α
∗
Θ|ψi for any α ∈ C) and Θ
†
Θ = ΘΘ
†
= . How-
ever any anti-unitary Θ can be written as KU where
Accepted in Quantum 2019-10-24, click title to verify 13
U is some unitary and K is complex conjugation in
a preferred basis. At the level of Hermitian operators
(e.g. projectors, observables, quantum states...) for
which we have that X = X
†
, this complex conjuga-
tion X → K
†
XK = X
∗
is equivalent to simply tak-
ing a transpose of the operator X → X
T
. For the
case of time-reversal, we can identify the time-reversal
of a state ρ as ρ 7→ ρ
∗
= ρ
T
. An inspection of the
off-diagonal components of ρ in the energy eigenbasis
(which is the relevant basis for the harmonic oscillator
system below) makes this clearer: a typical component
ρ
ij
will oscillate in time as e
−iω
ij
t
and thus by taking
the complex conjugation one obtains an oscillation of
e
+iω
ij
t
for the matrix component – in other words one
swaps the positive and negative modes in the quantum
state.
Since tr[X] = tr[X
T
] for any Hilbert space operator
X, and (XY )
T
= Y
T
X
T
, we see that Equation (31)
can be re-written as
P [Π
1
|
˜
Π
0
)] = tr[(Π
1
V Γ(Π
0
)V
†
)
T
] (32)
= tr[(V
†
)
T
Γ(Π
0
)
T
V
T
Π
T
1
]. (33)
where we have introduced the short-hand notation
˜
X :=
Γ(X) for any operator X. We now make the final as-
sumption that V , in addition to [V, H
tot
] = 0, is also
invariant under the above time reversal transformation
– namely V = V
T
. Informally, this condition can be
interpreted as assuming that the unitary V does not
inject in any microscopic time-asymmetry into the sys-
tem, and thus any differences in probabilities between
forward and reverse trajectories are purely due to the
thermodynamic structure.
Using that V = V
T
, and writing X
∗
= X
T
for any
Hermitian X we see that
P [Π
1
|
˜
Π
0
] = tr[(V
†
)Γ(Π
0
)
T
V Π
∗
1
]
=
1
tr(e
−βH
tot
Π
0
)
tr[e
−βH
tot
/2
Π
∗
0
e
−βH
tot
/2
V Π
∗
1
V
†
]
=
tr(e
−βH
tot
Π
∗
1
)
tr(e
−βH
tot
Π
0
)
tr[Π
∗
0
V Γ(Π
∗
1
)V
†
]
=
tr(e
−βH
tot
Π
1
)
tr(e
−βH
tot
Π
0
)
P [Π
∗
0
|
˜
Π
∗
1
], (34)
where we used the fact that H
T
tot
= H
tot
to write
tr(e
−βH
tot
Π
∗
1
) = tr(e
−βH
tot
Π
1
). We can therefore ex-
press the ratio of an abstract ‘forward trajectory’ to its
‘reversed trajectory’ as
P [Π
1
|
˜
Π
0
]
P [Π
∗
0
|
˜
Π
∗
1
]
=
tr(e
−βH
tot
Π
1
)
tr(e
−βH
tot
Π
0
)
. (35)
This is the key relation that we will use to obtain the
fluctuation relations.
Figure 5: Global system schematic. A system S, initially
uncorrelated with a thermal system B, evolve under a micro-
scopically energy conserving unitary V , to produce open-system
dynamics on S.
3.7 Coherent Tasaki-Crooks relation.
As already mentioned, we adopt an inclusive descrip-
tion in which the total system S
tot
is actually composed
of two subsystems S and B, and the total system SB is
energetically closed, but not dynamically closed. As dis-
cussed earlier, one should view the initially incoherent B
system as sub-divided into two components B = B
1
B
2
where B
1
is a large, uncontrolled thermal bath while B
2
corresponds to a quantum system for which we can con-
trol its degrees of freedom – for example we can vary its
Hamiltonian in some manner. The total Hamiltonian
for SB is as before and has eigenspaces given by the
decomposition in Equation (22).
We now consider an initial constraint C
0
for t = 0 and
an final constraint C
1
for t = 1, given by
C
0
: B has Hamiltonian H
0
and S is in a state |ψ
0
i.
C
1
: B has Hamiltonian H
1
and S is in a state |ψ
1
i.
Mathematically, these are represented by Π
SB,0
=
|ψ
0
ihψ
0
| ⊗ Π
0
and Π
SB,1
= |ψ
1
ihψ
1
| ⊗ Π
1
respectively.
where Π
k
is now the projector onto H
k
B
for the system
B.
The coherent thermalisation for the initial set-up
gives Γ(Π
SB,0
) = |
˜
ψ
0
ih
˜
ψ
0
| ⊗ γ
0
where |
˜
ψ
0
ih
˜
ψ
0
| :=
Γ
S
(|ψ
0
ihψ
0
|) where Γ
S
is Gibbs-rescaling purely with
respect to the Hamiltonian H
S
. If we now substitute
these components into Equation (35) we obtain
P [Π
SB,1
|
˜
Π
SB,0
]
P [Π
∗
SB,0
|
˜
Π
∗
SB,1
]
=
tr(e
−βH
SB
|ψ
1
ihψ
1
| ⊗ Π
1
)
tr(e
−βH
SB
|ψ
0
ihψ
0
| ⊗ Π
0
)
=
tr(e
−βH
1
)
tr(e
−βH
0
)
tr(e
−βH
S
|ψ
1
ihψ
1
|)
tr(e
−βH
S
|ψ
0
ihψ
0
|)
= e
−β∆F −∆Λ
, (36)
Accepted in Quantum 2019-10-24, click title to verify 14
where βF
k
= −log tre
−βH
k
is the free energy of B at
time t = k, ∆F = F
1
− F
0
, and ∆Λ := Λ(β, ψ
1
) −
Λ(β, ψ
0
), where we define Λ(β, ρ) the effective potential
of a state ρ at inverse temperature β as
Λ(β, ρ) := −log tr(e
−βH
S
ρ). (37)
With the understanding that the Hamiltonian of B
changes from H
0
to H
1
we can simply write
P [ψ
1
|
˜
ψ
0
]
P [ψ
∗
0
|
˜
ψ
∗
1
]
=
−β∆F −∆Λ
, (38)
as the final coherent Crooks relation for the ratio of the
probabilities of a given forward coherent trajectory to
the probability of its reverse trajectory.
It is important to note that if one ranges over all pos-
sible POVMs on S then the above relation is equivalent
to the abstract channel relation that was first derived
by Åberg in [23]. The difference here is that we focus
on projective measurements that admit a simple con-
straint interpretation and introduce the effective poten-
tial. Both of these turn out to be key in our physical
analysis of the relation. We now state the core quantum
fluctuation theorem in terms of the effective potential
of the quantum states.
Theorem 3.1 (Effective potential form). Let H =
H
S
⊗
B
+
S
⊗ H
B
be a microscopic, time-reversal
invariant Hamiltonian and assume B begins in a ther-
mal state at inverse temperature β = 1/(kT ). Addition-
ally, assume the dynamics admit a microscopic, time-
reversal invariant unitary V such that [V, H] = 0. Then
with P [ψ
1
|
˜
ψ
0
] and P [ψ
∗
0
|
˜
ψ
∗
1
] defined as above, the tran-
sition probabilities satisfy:
P [ψ
1
|
˜
ψ
0
]
P [ψ
∗
0
|
˜
ψ
∗
1
]
= e
−β∆F −(Λ(β,ψ
1
)−Λ(β,ψ
0
))
, (39)
where ∆F = (F
1
−F
0
) is the difference in free energies
with respect to the final and initial Hamiltonians of B,
and Λ(β, ρ) is the effective potential for any state ρ on
S at inverse temperature β.
The effective potential Λ(β, ψ) can be viewed as a log-
arithm of the Laplace transform for the quantum state
|ψi of S with respect to H
S
, and as such it corresponds
to the cumulant generating function [60] for the mea-
surement statistics of energy in the quantum state |ψi.
Thus by expansion in terms of the cumulants, the fluc-
tuation relation can be re-expressed as
P [ψ
1
|
˜
ψ
0
]
P [ψ
∗
0
|
˜
ψ
∗
1
]
= e
−β∆F −
P
n≥1
(−1)
n
β
n
n!
∆κ
n
, (40)
where ∆κ
n
= κ
n
(p
1
) − κ
n
(p
0
), and κ
n
(p
k
) is the n
th
cumulant for the random variable obtained if one mea-
sured H
S
in the state |ψ
k
i, and which has probability
distribution p
k
over energy. Under the assumption that
all cumulants are finite, one immediately sees that in the
high temperature regime one recovers a classical form
for the coherent Crooks relation in which the first order
β term dominates.
Corollary 3.1.1. In the high temperature limit,
P [ψ
1
|
˜
ψ
0
]
P [ψ
∗
0
|
˜
ψ
∗
1
]
= e
β(W
B
−∆F )+O(β
2
)
(41)
where W
B
:= hH
B
i
t=1
− hH
B
i
t=0
= −[hH
S
i
ψ
1
−
hH
S
i
ψ
0
].
Note that since the composite system SB is energet-
ically closed a change in energy in B, as measured by
the first moment hH
B
i, is identified with a correspond-
ing change in energy in S.
A more interesting question is how to interpret the
higher order corrections
(−1)
n
β
n
∆κ
n
n!
for n > 1. These
vanish if the state of S is an energy eigenstate, and so
arise from the non-trivial coherent structure of the input
and output pure states on S. One might suspect that
they are measures of quantum coherence, but this turns
out to not be the case, for example κ
3
can both increase
and decrease under incoherent quantum operations and
therefore cannot be a genuine measure of coherence [41,
61].
Despite the fact that a general cumulant κ
n
is not
a measure of coherence it turns out that κ
2
is distin-
guished and admits such an interpretation. The second
cumulant κ
2
is the variance of energy in a quantum state
|ψi, and it has been shown that if one restricts to pure
quantum states then in the asymptotic regime of many
copies of a state |ψi there is an essentially unique way
to quantify coherence [9] between different eigenspaces,
and is given by χ(ψ) := 4πκ
2
(ψ). With this in mind,
we next provide a decomposition of the effective poten-
tial Λ(β, ψ) of a pure quantum state into energy and
coherence contributions.
3.8 Separation of the effective potential into en-
ergetic and coherent contributions
We can show that the general fluctuation relation takes
a particularly simple form by exploiting how Λ(β, ψ)
varies as a function of the inverse temperature β. We as-
sume that Λ(β, ρ) is differentiable to second-order with
respect to β. Firstly, it is clear that Λ(0, ρ) = 0 for any
quantum state ρ, and moreover that
∂
β
Λ(β, ρ)|
β=0
= hHi
ρ
. (42)
Looking at the second derivative ∂
2
β
Λ(β, ρ) we find that
−∂
2
β
Λ(β, ρ) =
tr[ρH
2
e
−βH
]
tr(e
−βH
ρ)
−
tr[ρHe
−βH
]
tr(e
−βH
ρ)
2
. (43)
Accepted in Quantum 2019-10-24, click title to verify 15
The right-hand side of this only depends on the dis-
tribution p = (p
k
) over energy eigenstates for ρ. We
therefore define p
k
= tr[Π
k
ρ] where Π
k
is the projector
onto the energy eigenspace with energy E
k
, and so
−∂
2
β
Λ(β, ρ) =
P
k
[p
k
E
2
k
e
−βE
k
]
P
j
(e
−βE
j
p
j
)
−
P
k
[p
k
E
k
e
−βE
k
]
P
j
(e
−βE
j
p
j
)
!
2
.
(44)
Now define the distribution
˜p
k
=
e
−βE
k
P
j
e
−βE
j
p
j
p
k
for all k. (45)
This is a Gibbs re-scaling of the distribution p
k
at
inverse temperature β. In particular, we see that
∂
2
β
Λ(β, ρ) = −κ
2
(
˜
p), the variance of energy under the
re-scaled distribution ˜p
k
.
We can now apply the second–order Mean Value The-
orem to Λ(β, ρ) to deduce that for some β
m
∈ [0, β] we
have
Λ(β, ρ) = Λ(0, ρ) + β∂
β
Λ(β, ρ)|
β=0
(46)
+
1
2
β
2
∂
2
β
Λ(β, ρ)|
β=β
m
= βhHi
ρ
+
1
2
β
2
∂
2
β
Λ(β, ρ)|
β=β
m
, (47)
and thus we have that in general
Λ(β, ρ) = βhHi
ρ
−
1
8π
β
2
χ
m
(˜ρ), (48)
where we have used that χ(ρ) = 4πκ
2
(p), with the
short-hand notation ˜ρ = Γ(ρ). Here χ
m
(˜ρ) is non-trivial
and should be interpreted with care. It involves com-
puting χ on a Gibbs re-scaling of the quantum state,
and then evaluated at an effective inverse temperature
β
m
≤ β that is determined by the statistics of the orig-
inal state.
The utility of this is that for a pure state |ψi, the
re-scaled state |
˜
ψi is also pure, and thus χ, the vari-
ance in energy, is always a genuine measure of coher-
ence when restricted to pure states [10]. Therefore, the
higher order cumulants for n > 1 arise from the co-
herent structure of |ψi and while individual cumulants
are not coherence measures, one can still deduce that
the sum of all higher order terms can be reduced to a
single coherence measure of a rescaled pure quantum
state. We refer to χ
m
(
˜
ψ) as the mean coherence at in-
verse temperature β of the pure quantum state |ψi, and
it allows the effective potential to split in a remarkably
simple way as
Λ(β, ψ) =
(mean energy)
kT
−
(mean coherence)
8π(kT )
2
, (49)
however the way in which χ
m
(
˜
ψ) corresponds to “mean
coherence” is subtle. When β
m
= β then χ
m
is precisely
the amount of coherence in the physically prepared ini-
tial state Γ(ψ) =
˜
ψ after Gibbs re-scaling. However
more generally, χ
m
is the amount of coherence in the
pure state ∝ exp[−(β
m
−β)H
S
]|
˜
ψi with β
m
determined
by the energy statistics of the quantum state. We go
into more detail on χ
m
in the next section.
In terms of the mean coherence the core fluctuation
relation can be re-stated as follows.
Theorem 3.2 (Mean coherence & mean energy
decomposition). Let the assumptions Theorem 3.1
hold. Then with χ
m
(˜ρ) defined as before at an inverse
temperature β, for any two pure states |ψ
0
i and |ψ
1
i,
the fluctuation theorem takes the form
P [ψ
1
|
˜
ψ
0
]
P [ψ
∗
0
|
˜
ψ
∗
1
]
= e
−β∆F −βW
S
+
β
2
8π
∆χ
, (50)
where W
S
= hH
S
i
ψ
1
− hH
S
i
ψ
0
and ∆χ = χ
m
(
˜
ψ
1
) −
χ
m
0
(
˜
ψ
0
).
This fully separates the change in the first moment
hH
S
i from the higher order corrections. Note that
χ
m
(
˜
ψ
1
) and χ
m
0
(
˜
ψ
0
) are in general evaluated at two
different temperatures β
m
and β
m
0
respectively. Such a
difference depends on β and also the original statistics
of each state |ψ
0
i and |ψ
1
i.
3.9 An explicit form for the mean coherence at
inverse temperature β
The above form of χ
m
, and its dependence on tempera-
ture is slightly opaque, however one can obtain a simple
expression for computing it that makes clear the differ-
ent contributions. To this end we can make use of the
following result.
Lemma 3.3. Given a quantum system S with Hamil-
tonian H
S
and {Π
k
} the projectors onto the energy
eigenspaces of H
S
, let us denote the de-phased form of
state in the energy basis by D(ρ) =
P
k
Π
k
ρΠ
k
. Then,
for any quantum state ρ, the effective potential Λ(β, ρ)
is given by
Λ(β, ρ) = min
σ
{βhH
S
i
σ
+ S(σ||D(ρ))}, (51)
where the minimization is taken over all quantum states
σ of S, and S(σ||ρ) = tr[σ log σ − σ log ρ] is the rel-
ative entropy function. Moreover, the minimization is
attained for the state σ = Γ(D(ρ)).
The proof of this is provided in the Appendix B.3.
If the probability distribution over energy of ρ is p =
(p
k
) then this is unaffected by the dephasing ρ 7→ D(ρ).
Denoting by
˜
p the Gibbs rescaling of p, we therefore
have that
Λ(β, ρ) = βhH
S
i
˜
p
+ S(
˜
p||p), (52)
Accepted in Quantum 2019-10-24, click title to verify 16
where the relative entropy term is now the classical rel-
ative entropy for distributions and hH
S
i
˜
p
:=
P
k
E
k
˜p
k
is the expectation value of energy for the classical dis-
tribution
˜
p. We thus have the following.
Theorem 3.4. Given a quantum system S with Hamil-
tonian H
S
, the mean coherence at inverse temperature
β of a quantum state |ψi is given by
β
2
8π
χ
m
(
˜
ψ) = β(hH
S
i
p
− hH
S
i
˜
p
) − S(
˜
p||p). (53)
where p = (p
k
) is the distribution over energy of ψ and
˜
p is the distribution over energy for |
˜
ψih
˜
ψ| = Γ(|ψihψ|).
Everything on the right-hand side of Equation (53)
is easily computable, and its form sheds light on what
the mean coherence at inverse temperature β actually
depends on. The first term quantifies the degree to
which the coherent thermalisation raises or lowers the
expected energy of |ψi in units of β, while the second
term quantifies the degree to which the coherent ther-
malisation makes the state |
˜
ψi distinguishable from the
state |ψi (in the sense of hypothesis testing optimized
over all possible quantum measurements [62]). Since the
left-hand side is always non-negative, we also see that
energy term is never smaller than the relative entropy
term.
Example 3.5. Consider the system S in the uni-
form superposition over d energy eigenstates |ψi =
1
√
d
P
d−1
n=0
|E
n
i. Then, as discussed earlier, this trans-
forms under Γ to the coherent Gibbs state |
˜
ψi = |γi.
Then by direct calculation with equation (53), we find
that
β
2
8π
χ
m
(|γihγ|) = β[F
1
d
1
d
− F (γ)], (54)
where F (ρ) := hHi
ρ
− k
B
T S(ρ) is the free-energy of
the state ρ and γ is the Gibbs state. Thus the mean
coherence is proportional to the difference between the
free energy of S with a trivial Hamiltonian H
S
= 0
and the free energy of S with non-trivial Hamiltonian
H
S
6= 0 at inverse temperature β.
Finally, note that while χ is a genuine measure of
coherence for any pure quantum state, this is different
from χ
m
being a monotone under the general dynamics
of the setting. While this is not immediately obvious
from the discussion, we highlight that the reduced dy-
namics for any state ρ of S is of the form
E(ρ) = tr
B
V (ρ ⊗ γ
0
)V
†
, (55)
and it is readily seen that this quantum operation is
time-translation covariant, and so the total coherence
between energy eigenspaces of S can never increase
[8, 42, 63]. In addition, given any elements M
0
, M
1
that occur in some POVM M = {M
k
}, one can always
include them in a covariant measurement on the sys-
tem, by simply considering the orbit of each under the
adjoint action X → U (t)XU (t)
†
for all t and if these
are not in the measurement set, expanding the set to
include them. Thus, a POVM M = {M
k
} exists that
contains both M
0
and M
1
and the total non-selective
evolution is time-translation covariant. Therefore there
is no coherence being exploited to perform the measure-
ment [64] and so over the whole procedure on S we have
that the coherence in S can never increase.
4 Coherent work processes, quantum
fluctuation relations and the semi-
classical limit.
We can now show that the earlier notion of a coherent
work process fits neatly into the inclusive quantum fluc-
tuation setting. Recall that the classical Crooks relation
takes the form
P [w|γ
0
, P]
P [−w|γ
1
, P
∗
]
= e
−β(∆F −w)
(56)
that relates the probability of an amount of work w for
a forward trajectory with −w for the reversed trajec-
tory. As already discussed this quantity w is always as-
sociated with some auxiliary system within an inclusive
description. It is readily seen that the previous quan-
tum fluctuation relations can be re-cast in the following
form that provides a natural extension of the classical
result.
Theorem 4.1 (Fluctuations and coherent work
processes). Let S be a quantum system with Hamilto-
nian H
S
and let |ψ
0
i
S
and |ψ
1
i
S
be two pure states of
S that have energy statistics with finite cumulants of all
orders. Then the following are equivalent:
1. The pure states |ψ
0
i
S
and |ψ
1
i
S
are coherently con-
nected with coherent work output/input ω on an
auxiliary system A.
2. Within the fluctuation relation context with a ther-
mal system B and quantum system S we have that
P [ψ
1
|
˜
ψ
0
]
P [ψ
∗
0
|
˜
ψ
∗
1
]
= e
−β(∆F ±kT Λ(β,ω))
, (57)
for all inverse temperatures β ≥ 0, for some state
|ωi
A
with finite cumulants on an auxiliary system
A and some choice of sign before Λ(β, ω).
We note that for |ψ
0
i
S
and |ψ
1
i
S
being energy eigen-
states we have that |ωi
A
is an energy eigenstate |wi
A
,
Accepted in Quantum 2019-10-24, click title to verify 17
Figure 6: Coherent work processes (CWP) and thermal in-
teractions (TI). We introduced coherent work processes which
are deterministic processes that transition a pure state to an-
other pure state. Thermal interactions are stochastic, hence
the output is a probabilistic mixture of pure states. In the
context of the fluctuation relation, we post-select a particular
pure state. The coherent work output/input |ωi associated to
the initial and final quantum states contributes Λ(β, ω) to the
fluctuation relation and generalises the classical term βw from
the standard Crooks relation.
for which Λ(β, ω) = βw for some sharp value w ∈ R, and
so the above expression reduces to a classical Crooks
relation form. The proof of the theorem is provided in
Appendix B.4.
4.1 Physical interpretation of the result
Interpreting this theorem requires some care. State-
ment 1 is simply a deterministic transformation between
pure states with coherence on a quantum system S.
In contrast, statement 2 involves a stochastic thermal
process in which initial coherence in S interacts with a
thermal environment B and projective measurements
at the start and at the end is assumed. Moreover this
coherence is not outputted to some ordered degree of
freedom but is dissipated into the large thermal envi-
ronment B. The appearance of an auxiliary system A
in statement 2 is independent of the actual thermody-
namic context and takes no part in the process.
This shows that coherent work processes and coher-
ently connectedness relate to the coherent fluctuation
in a simple way, via the coherent work output/input
ω
A
. Moreover, the numerical contribution of ω
A
to the
fluctuation relation exponent is given simply by the ef-
fective potential of ω
A
. From a structural perspective
we may therefore summarise this conclusion as follows:
Coherent work processes are to coherent fluctuation
relations what deterministic Newtonian work processes
are to classical fluctuation relations.
Moreover in the semi-classical regime, where coherence
becomes negligible, the two fluctuation relations con-
verge, and therefore we have that the coherent frame-
work is a faithful and consistent extension of the classi-
cal work relations into quantum mechanics.
4.2 Mixed state coherent work output/input
The connection with coherent work processes can be
made more explicit by observing that the Gibbs state
γ
0
is a probabilistic mixture of energy eigenstates and
therefore the unitary part of the dynamics can be writ-
ten as
V (ψ
0
⊗ γ
0
)V
†
=
X
k
e
−βE
0
k
Z
0
V (|ψ
0
ihψ
0
| ⊗ |E
0
k
ihE
0
k
|)V
†
.
(58)
In general |ϕ
k
i
SB
= V [|ψ
0
i⊗|E
0
k
i] is not a product state
over S and B, and so does not describe a coherent work
process in itself, however the projective measurement
on S will collapse |ϕ
k
i to a product state |ψ
1
i ⊗ |φ
1
k
i
and so the overall effect is the probabilistic transition
|ψ
0
i ⊗ |E
0
k
i → |ψ
1
i ⊗ |φ
1
k
i. Thus the particular energy
eigenstate |E
0
k
i in the Gibbs ensemble acts as a refer-
ence energy level and |E
0
k
i → |φ
1
k
i can be viewed as a
coherent work output conditioned on |E
0
k
i.
We can now link with more abstract formalism by
considering the evolution E of S and looking at what is
called the complementary channel [65]
¯
E from S into B
given by
¯
E(ρ) := tr
S
V (ρ ⊗ γ
0
)V
†
. (59)
This provides a stochastic mixture of coherent work out-
puts, from the system S into the bath system B.
In fact this perspective can be taken as a more gen-
eral formulation of coherent work processes in which a
general state ρ of a system S transforms through some
energy-conserving interaction with a second system A
in a default energy eigenstate |0ih0| as ρ → E(ρ) =
tr
A
V (ρ ⊗ |0ih0|)V
†
. Then we define a general coherent
work process as
ρ
ω
−→ E(ρ). (60)
where ω =
¯
E(ρ) is now the mixed state coherent output
for the process.
The set of such quantum transformations are simply
the time-translation covariant quantum operations [8,
42, 63], and so in the same way as before we can say that
two quantum states ρ and σ are coherently connected if
there exists a time-translation covariant channel E from
one state to the other. The coherent work output/input
is given via the complementary channel
¯
E acting on the
input system. We leave the study of such features to
future work, where coherent work processes could prove
useful tools in studying asymmetry theory.
In the next section, we return to more concrete sys-
tems, and briefly outline how coherent work processes
Accepted in Quantum 2019-10-24, click title to verify 18
are experimentally accessible in existing trapped ion
proposals.
4.3 Coherent fluctuation relations in the experi-
mental semi-classical regime
We have shown that our analysis of coherent work pro-
cesses, in which classical deterministic transitions are
superposed together while respecting energy conserva-
tion, appears naturally within the fluctuation relation.
We can now re-visit the case that the quantum system
S is a harmonic oscillator system, and restrict to states
in C from Section 2.2, which are closed under coherent
work processes. This is particularly of interest because a
fluctuation relation for this system can be demonstrated
within existing trapped ion systems [25], and therefore
shows that the theoretical analysis presented here is of
relevance to existing experimental work.
We can without loss of generality restrict to the
canonical states |α, ki in the set C, and to simplify
things further we consider the subset formed of pure
coherent states (with k = 0), although the fully gen-
eral case can also be easily computed. This gives rise
to the same fluctuation relation as in [25], which fol-
lowed from the analysis of [23]. For a coherent state
|αi of a quantum harmonic oscillator with Hamiltonian
H
S
= hν(a
†
a+
1
2
), the effective potential is readily com-
puted and takes the form
Λ(β, α) =
1
2
βhν +
1
hν
hH
S
i
α
−
1
2
hν
(1 − e
−βhν
),
(61)
where hH
S
i
α
:= hα|H
S
|αi = hν(|α|
2
+ 1/2).
A straightforward calculation leads to the follow-
ing quantum fluctuation relation restricted to coherent
states of an oscillator system S.
Theorem 4.2 (Semi-classical relation [25]). Let the
assumptions of Theorem 3.1 hold, and let S is a har-
monic oscillator with Hamiltonian H
S
= hν(a
†
a +
1
2
).
Then for two coherent states of the system |α
0
i and
|α
1
i, the following holds
P [α
1
|˜α
0
]
P [α
∗
0
|˜α
∗
1
]
= exp
−
∆F
kT
+
¯
W
B
hν
th
, (62)
where hν
th
= hH
S
i
γ
is the average energy of a Gibbs
state γ of a quantum harmonic oscillator, related to the
thermal de Broglie wavelength λ
dB
(T ) via
hν
th
=
h
2
mλ
dB
(T )
2
+
1
2
hν, (63)
and
¯
W
B
:= −
1
2
(W
S
+
˜
W
S
), W
S
= hH
S
i
α
1
− hH
S
i
α
0
,
˜
W
S
= hH
S
i
˜α
1
− hH
S
i
˜α
0
.
The interpretation of this is quite natural, with only
the term
¯
W needing care. We first note that
¯
W is the
average of two changes in pure state energy: the energy
change in |α
0
i → |α
1
i and its Gibbs rescaled version
|˜α
0
i → |˜α
1
i. Since Gibbs rescaling leaves energy eigen-
states unchanged this implies that
¯
W reduces to the
classical sharp energy transition in the case of zero co-
herences, otherwise the coherences provide a non-trivial
distortion for a semi-classical ‘work’ term
4
. By compar-
ing with the previous analysis one sees that
¯
W is not
simply a change in first moments of H, and thus incor-
porates part of the mean coherence contribution from
χ
m
.
The reason that this form of the fluctuation relation
is natural is that coherent states are the “most classi-
cal” states for the quantum system (they saturate the
Heisenberg bound), and so gives rise to a fluctuation
relation that is as close to the classical one as possible
while still having coherent structure. This is reinforced
by the fact that all the terms on the right-hand side,
except for the ‘work’ term
˜
W are simple equilibrium
properties of a quantum system – as in the standard
Crooks relation. The principle difference is that the
equilibrium temperature term kT for the work is re-
placed by hν
th
which is a quantum mechanical equilib-
rium property, related to the de Broglie thermal wave-
length. This shows that in the high temperature regime
where kT hν, the equilibrium thermal fluctuations
dominate the de Broglie thermal wavelength and we re-
cover the classical regime hν
th
→ kT . In an intermedi-
ate temperature regime we get a smooth temperature-
dependent distortion of the standard Crooks relation,
while for very low temperatures we have hν
th
→ 1/2hν
in a high coherence regime – namely it is now the vac-
uum fluctuations that dominate in the fluctuation rela-
tion.
One can also readily compute the mean coherence χ
m
for the oscillator system, and find that for any coherent
state |αi it takes the form
χ
m
= 8π|α|
2
(kT )
2
[βhν + e
−βhν
− 1]. (64)
Expansion of the exponential gives that χ
m
=
4π|α|
2
[(hν)
2
−
1
3
β(hν)
3
+ ···] and so we see the tem-
perature dependence only begins at cubic order in hν.
We also see that lim
h→0
χ
m
= 0, for all temperatures
T , and so the mean coherence χ
m
is a purely quantum-
mechanical feature that disappears in the classical limit.
Perhaps more interestingly, we see that χ
m
vanishes in
the T → 0 limit also. Therefore, since χ
m
only con-
tributes when we simultaneously have both a non-zero
4
We do not refer to this as “the work” since it is purely re-
stricted to the semi-classical case of coherent states, and there is
no reason to believe that a physically sensible and unique “work”
quantity should exist for general quantum systems.
Accepted in Quantum 2019-10-24, click title to verify 19
Planck’s constant h and non-zero temperature T , the
mean coherence is a genuinely quantum-thermodynamic
property of the system.
4.4 The macroscopic regime and multipartite
entanglement.
One method of characterising the classical regime is by
letting thermal fluctuations dominate quantum fluctu-
ations, as we showed. But one expects a notion ‘clas-
sicality’ to also arise in the large system limit, where
the state of the system is given in terms of a small set
of intrinsic variables. In this section we show that if
the system S is composed of many independent, iden-
tically distributed (IID) systems S
1
, ..., S
n
, all higher
order corrections bar the variance vanish in the limit
n → ∞, under an appropriate scaling of energy units.
Given a macroscopic regime in which a quantum sys-
tem with no correlations is well-described by a bounded
list of intrinsic parameters, the total quantum state can
be written as ρ
tot
≈ ρ
⊗n
, for some effective subsys-
tem state ρ that encodes the intrinsic parameters. We
can now consider the macroscopic states |ψ
n
i := |ψi
⊗n
and |φ
n
i := |φi
⊗n
on the reference system, for two pure
non-energy eigenstates |ψi and |φi and see how the fluc-
tuation relation behaves.
In particular, we consider a state |ψi =
P
d−1
m=0
√
p
m
|mi of a d-dimensional system, where
{|mi} are the eigenstates of the Hamiltonian
H
1
=
P
d−1
m=0
m|mihm|. Looking at n copies of
the system,
|ψi
⊗n
=
n(d−1)
X
j=0
√
c
j
|e
j
i
n
, (65)
where we have decomposed |ψi
⊗n
into a superposition
of energy eigenstates {|e
j
i
n
} of n copies of the system
[9], where |e
j
i
n
has total energy j and it is assumed H
tot
is the sum of the non-interacting individual Hamiltoni-
ans. The coefficients {c
j
} are then described by multi-
nomial coefficients
c
j
=
n
k
0
...k
d−1
p
k
0
0
p
k
1
1
...p
k
d−1
d−1
(66)
where j =
P
i
i k
i
and n =
P
i
k
i
. Furthermore, if the
spectrum {p
j
} is gapless, with p
j
6= 0 for 0 ≤ j ≤ d −1,
then in the asymptotic limit the distribution c
j
over
energies becomes [9]
c
j
=
1
p
2πσ
2
n
e
−
(j−µ
n
(ψ))
2
2σ
2
n
(ψ)
+ O
n
−1
, (67)
where µ
n
(ψ) = hψ
n
|H
tot
|ψ
n
i = nhψ|H
1
|ψi = nµ and
σ
2
n
(ψ) = nσ
2
is the variance of H
tot
evaluated on |ψ
n
i.
In the large n limit we know from the central limit
theorem that the distributions over energy will tend to
a Gaussian in the neighborhood around the mean en-
ergy. As we show in the Supplementary Material, the
fluctuation theorem takes the following form: for any
> 0 there is an M such that for all n > M we have
P [ψ
n
|
˜
φ
n
]
P [φ
∗
n
|
˜
ψ
∗
n
]
− e
−n(β∆f+β∆µ−
1
2
β
2
∆σ
2
)
≤ , (68)
however with the proviso that the scaling β =
β
0
√
nλ
is
adopted, where β
0
is a dimensionless unit and λ a char-
acteristic energy scale – which is technically required in
order to apply the central limit theorem rigorously, and
amounts to a choice of units for temperature for a fix
scale n. This captures the informal statement that for
the IID case the states “become more like Gaussians”
and
P [ψ
n
|
˜
φ
n
]
P [φ
∗
n
|
˜
ψ
∗
n
]
n→∞
∼ e
−n(β∆f+β∆µ−
1
2
β
2
∆σ
2
)
(69)
Thus in the macroscopic limit, the central limit theo-
rem can be used to truncate the exponent in the fluctu-
ation relation to the second-order cumulants, with the
usual provisos in this statement.
As in the asymptotic IID regime one reduces to only
the first two cumulants, we can use the second cumu-
lant to place a bound on many-body entangled systems.
Suppose we start with a state |φi
⊗n
and perform any
unitary U that is block diagonal in the total Hamilto-
nian H =
P
i
H
i
then in general the state U|φi
⊗n
will
be entangled between the n subsystems. But it is read-
ily seen that Λ(β, U φ
n
U
†
) = Λ(β, φ
n
) and thus (a) in
the large n limit will have a Gaussian profile in energy
and (b) will generically be entangled between subsys-
tems.
When the total system S is a large multi-partite sys-
tem composed of n spin–1/2 particles one can exploit
the fact that the variance V ar(ψ, σ
i
) of a Pauli operator
σ
i
in a pure quantum state |ψi is upper bounded when
limited to k–producible states. A state |ψi is called
k–producible if it can be written in the form:
|ψi = |φ
1
i ⊗ |φ
2
i ⊗ ... ⊗ |φ
m
i, (70)
where m ≥ n/k and each pure state |φ
j
i involves at
most k particles [66] (which captures the notion of mul-
tipartite entanglement). It is clear from the defini-
tion that 1–producible states are product states with
m = n, containing no entanglement. A pure state
|ψi has genuine k–partite entanglement if |ψi is k–
producible but not (k − 1)–producible. Now for such
a state, the variance of a many-body Pauli spin opera-
tor Σ
(n)
i
= σ
(1)
i
+ ··· + σ
(n)
i
, where i = x, y, z and the
superscript denotes the qubit label, obeys
4Var(ψ, Σ
(n)
i
) ≤ sk
2
+ (n − sk)
2
(71)
Accepted in Quantum 2019-10-24, click title to verify 20
on the set of k–producible states [67, 68]. Here s is the
integer part of n/k and the bound ranges from n up to
n
2
.
By using the above bound we see that the largest
possible changes in variance will be ±[sk
2
+ (n − sk)
2
].
Therefore for a system S composed of n particles, with
each subsystem having a Hamiltonian H
i
= σ
z
giving
a total Hamiltonian H
S
= Σ
(n)
z
,
P [ψ|
˜
φ
n
]
P [φ
∗
n
|
˜
ψ
∗
]
≤ e
β(W
B
−∆F )
e
(β)
2
8
(sk
2
+(n−sk)
2
)+O(1/
√
n)
.
(72)
where W
B
= −[hH
S
i
ψ
− hH
S
i
φ
n
] is the change in en-
ergy of the bath subsystem. This holds for any large n
IID product state |φ
n
i and any k–producible state |ψi
with genuine k–partite entanglement. While this bound
(which turns out to be tight [67]) grows weaker and
weaker as n increases, it is nonetheless striking that one
can apply the fluctuation relation to provide non-trivial
statements on transitions between quantum states with
different multipartite entanglement and it would be of
interest to further develop this connection further.
5 Outlook
We started by extending the core defining character-
istics of deterministic work in classical mechanics into
quantum mechanics, which lead to the notion of coher-
ent work processes. It turns out that the set of coher-
ent processes is non-trivial for quantum systems, and it
would be of interest to extend this analysis to determine,
for example, what quantum systems admit infinitely de-
composable quantum coherence. Because of Theorem
2.5 this can be determined from the large literature on
the decomposability of classical random variables.
Another open question is whether a mixed state ver-
sion of coherent work processes leads to interesting
physics, or could be related to on-going research in ther-
modynamics. The asymmetry theory perspective on
coherent work processes means that this is expected
to have non-trivial structure, and so applying exist-
ing asymmetry results within a thermodynamic context
would be of interest.
The fluctuation relations and the effective potentials
discussed here suggest that a wide range of tools (e.g.
from many-body physics [69]) could be brought to bear
on coherent fluctuation relations. It is essential to de-
velop greater connection with existing fluctuation rela-
tions, but it is conjectured that essentially any of the
existing extensions of the classical stochastic Crooks re-
lation could be faithfully represented in this setting.
6 Acknowledgements
We would like to thank Johan Åberg, Zoë Holmes,
Hyukjoon Kwon, Myungshik Kim, Jack Clarke and
Doug Plato for useful discussions. EHM is funded by
the EPSRC Centre for Doctoral Training in Controlled
Quantum Dynamics. DJ is supported by the Royal So-
ciety and also a University Academic Fellowship.
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Accepted in Quantum 2019-10-24, click title to verify 25
A Coherent work processes
Here we gather the proofs of the statements for coherent
work processes. First we show that for auxiliary ‘weight’
systems with non-degenerate spectra, the coherent work
output is effectively unique.
Proposition A.1. Let |ωi
A
=
P
k
e
iθ
k
√
p
k
|ki
A
be
the coherent work output from a coherent work process
|ψi
S
ω
−→ |φi
S
where {|ki
A
} is the eigenbasis of the non-
degenerate Hamiltonian H
A
. Then the distribution {p
k
}
is uniquely determined, while the phases {θ
k
} are arbi-
trary.
Proof. Assume |ωi
A
is the coherent work output of the
process |ψ
S
i
ω
−→ |φ
S
i, enacted by the unitary U . Define
the characteristic function ϕ
X,η
(t) := tr(e
iH
X
t
|ηihη|
X
).
Then under a coherent work process,
ϕ
S,ψ
(t)ϕ
A,0
(t) = tr(e
i(H
S
+H
A
)t
|ψihψ|
S
⊗ |0ih0|
A
)
(73)
= tr(e
i(H
S
+H
A
)t
U
†
U|ψihψ|
S
⊗ |0ih0|
A
)
(74)
= ϕ
S,φ
(t)ϕ
A,ω
(t), (75)
where U is a unitary and [U, H
S
+ H
A
] = 0 by the
construction of coherent work processes. Due to the
uniqueness property of characteristic functions [13],
ϕ
A,ω
(t) = ϕ
A,ω
0
(t) for all t ∈ R if and only if the distri-
bution for ω and ω
0
are equal. Therefore with ψ and φ
pre-determined, the distribution of ω is uniquely deter-
mined.
Since the only constraint on U is that it is en-
ergy conserving, then for any Hermitian operator L
A
such that [L
A
, H
A
] = 0, we can define a unitary
V = (1
S
⊗ e
−iL
A
θ
)U that also conserves energy, with
[V, H
S
+H
A
] = 0. Furthermore, with appropriate choice
of L
A
, any set of phases {θ
k
} can be selected.
It is necessary to constrain the spectrum of A to be
non-degenerate, as given a degenerate Hamiltonian H
A
0
for a system A
0
, one could enact a unitary T that lo-
cally conserves energy ([H
A
0
, T ] = 0) and can arbitrarily
change the state |ωi
A
in the degenerate subspace.
Though we describe the coherent work output as ef-
fectively unique, one is always free to enact evolution
locally on the auxiliary system that will induce a set
of relative phases. In this way, any phases gained from
the joint evolution between S and A can be arbitrarily
changed with local evolution. Since relative phases are
a signature of time-evolution, the result distinguishes
between global and local effects of evolution on states.
We can also prove that all reversible transformations
involve trivial coherent work processes.
Proposition A.2. A coherent work process is trivial if
and only if it is reversible.
Proof. Let V |ψ
0
i
S
⊗ |E
0
i
A
= |ψ
1
i
S
0
⊗ |ωi
A
be a co-
herent work process. In terms of random variables
with respect to Hamiltonians H
S
and H
A
, an equiv-
alent condition for the existence of this process is
ˆ
S =
ˆ
S
0
+
ˆ
A, where
ˆ
S,
ˆ
S
0
,
ˆ
A are the random variables for
|ψ
0
i
S
, |ψ
1
i
S
0
, |ωi
A
respectively. If it is a trivial coher-
ent work process, then at least one of
ˆ
S
0
or
ˆ
A is a con-
stant random variable. Without loss of generality, we
assume
ˆ
A = a is constant and so we must have that
ˆ
S
0
=
ˆ
S −a and
ˆ
S =
ˆ
S
0
+ a. This gives |ψ
0
i
S
a
−→ |ψ
1
i
S
0
and |ψ
1
i
S
0
−a
−→ |ψ
0
i
S
.
Conversely, assume we have a reversible coherent
work process
V |ψ
0
i
S
⊗ |E
0
i
A
1
= |ψ
1
i
S
0
⊗ |ω
1
i
A
1
, (76)
U|ψ
1
i
S
0
⊗ |E
0
0
i
A
2
= |ψ
0
i
S
⊗ |ω
2
i
A
2
, (77)
where we introduce a different auxiliary system for each
process. Then
ˆ
S =
ˆ
S
0
+
ˆ
A
1
and
ˆ
S
0
=
ˆ
S +
ˆ
A
2
. As
the random variables are all independent, we have both
Var(
ˆ
S) = Var(
ˆ
S
0
) + Var(
ˆ
A
1
) and Var(
ˆ
S
0
) = Var(
ˆ
S) +
Var(
ˆ
A
2
). Therefore
Var(
ˆ
A
1
) + Var(
ˆ
A
1
) = 0. (78)
Due to the non-negativity of the variance for real-
valued random variables this implies that Var(
ˆ
A
1
) =
Var(
ˆ
A
2
) = 0. This implies that both
ˆ
A
1
ˆ
A
2
are con-
stant random variables and so both processes are trivial
as required.
The consequence of this result is that any reversible
coherent work process can simply be labeled by the con-
stant value of
ˆ
A
1
. For example, if we have |ψ
0
i
S
ω
−→
|ψ
1
i
S
0
and |ψ
1
i
S
0
ω
0
−→ |ψ
0
i
S
then this implies |ωi = |E
k
i
and |ω
0
i = | − E
k
i for some energy E
k
(if E
0
= 0),
then we label the reversible process simply as |ψ
0
i
S
E
k
←→
|ψ
1
i
S
0
.
A.1 Decomposition of Poisson distributions
The main theorem used in this section is Theorem A.3
due to Raikov and later generalized by others, but we
first re-cap on some background details [13].
The cumulative distribution function (CDF) of a ran-
dom variable
ˆ
X on the real line is defined as F (x) :=
P rob[
ˆ
X ≤ x]. If
ˆ
X has CDF F
1
and
ˆ
Y has CDF F
2
, and
ˆ
X and
ˆ
Y are independent, then the CDF of the random
Accepted in Quantum 2019-10-24, click title to verify 26
variable
ˆ
Z :=
ˆ
X +
ˆ
Y has CDF given by the convolution
F
ˆ
Z
(x) =
Z
∞
−∞
F
1
(x − y)dF
2
(y)
=
Z
∞
−∞
F
1
(x − y)
dF
2
dy
(y)dy
=
Z
∞
−∞
F
1
(x − y)f
2
(y)dy, (79)
where f
2
(y) is the probability density for
ˆ
Y . For com-
pleteness of notation, we also define the step function
(x) = 1 if x ≥ 0 and zero otherwise.
With this notation in place, we can now state the
theorem by Raikov.
Theorem A.3 (Raikov, 1938). The Poisson law has
cumulative distribution function
F
x − α
σ
, λ
, σ > 0, α ∈ R, λ ≥ 0, (80)
where
F (x; λ) = e
−λ
X
k≤x
λ
k
k!
. (81)
(for x < 0 the sum is empty and F (x; λ) = 0). F(x; λ)
corresponds to the Poisson cumulative distribution for
a random variable that takes only integer values at non-
negative values k with probability
λ
k
k!
e
−λ
. When λ = 0
we obtain the singular distribution law (x − α).
The integral equation
Z
∞
−∞
F
1
(x − y)dF
2
(y) = F
x − α
σ
, λ
, (82)
where F
1
(x) and F
2
(x) are cumulative distribution func-
tions only has solutions
F
1
(x) = F
x − α
1
σ
, µ
(83)
F
2
(x) = F
x − α
2
σ
, ν
, (84)
where µ ≥ 0, ν ≥ 0, µ + ν = λ and α
1
+ α
2
= α, with
α
1
, α
2
∈ R.
We refer the reader to [15] for the proof.
We note that the original statement of Raikov’s the-
orem found in [15] has a typographical error to do with
the shifting parameters α and β. Namely, F
1
(x) =
F (
x−α−β
σ
, µ) and F
2
(x) = F (
x−α+β
σ
, ν), with β ∈ R.
This error can be seen in the following manner. Sup-
pose we have a Poisson distributed random variable
ˆ
Y
shifted by a ∈ R, such that
ˆ
Y
0
= (
ˆ
Y − a) ∼ Pois(µ),
where Pois(µ) denotes a Poisson distribution on the
non-negative integers. Similarly, let us define another
Poisson distributed random variable
ˆ
Z shifted by b ∈ R,
such that
ˆ
Z
0
= (
ˆ
Z −b) ∼ Pois(ν). If we assume
ˆ
Y and
ˆ
Z
are independent, then the sum of the random variables
is
ˆ
X =
ˆ
Y +
ˆ
Z (85)
= (
ˆ
Y − a) + (
ˆ
Z − b) + (a + b). (86)
Since a and b are constants, we are able to re-arrange
this to find
ˆ
X − (a + b) = (
ˆ
Y − a) + (
ˆ
Z − b). (87)
However, the right hand side of this equation corre-
sponds to the sum of two unshifted Poisson distributed
random variables and thus
ˆ
X − (a + b) =
ˆ
Y
0
+
ˆ
Z
0
∼
Pois(µ + ν), where we deduce this from Raikov’s theo-
rem. From this we see that
ˆ
X is a Poisson distributed
random variable shifted by an amount (a + b). Now let
us choose a = α+β and b = α−β. Thus a+b = 2α and
we expect the sum of the shifted random variables to
have a shifting of 2α, rather than the α that is claimed.
Another consistency check is to consider the limit of a
Poisson cumulative distribution function. We have that
lim
λ→0
F
x − α
σ
; λ
= lim
λ→0
X
k≤
x−α
σ
e
−λ
λ
k
k!
(88)
=
(
1, if x ≥ α
0, otherwise
(89)
where we make use of the limit of a Poisson probability
mass function [70]. Now suppose
ˆ
Y
0
∼ Pois(µ) and
ˆ
Y =
ˆ
Y
0
+ a. Then lim
µ→0
ˆ
Y = lim
µ→0
ˆ
Y
0
+ a = a.
If we similarly define
ˆ
Z
0
∼ Pois(ν), with
ˆ
Z =
ˆ
Z
0
+ b,
then lim
ν→0
ˆ
Z = b. Thus when we sum the two shifted
random variables, lim
µ,ν→0
ˆ
Y +
ˆ
Z = a + b, as expected.
A.2 Proof for decomposability theorem
Theorem A.4. Given a quantum system X with
Hamiltonian H
X
with a discrete spectrum, a state |ψi
X
admits a non-trivial coherent work process if and only if
the associated classical random variable
ˆ
X is a decom-
posable random variable. Furthermore, for a coherent
work process
|ψi
X
ω
−→ |φi
Z
, (90)
the associated classical random variables are given by
ˆ
X =
ˆ
Z+
ˆ
W , where
ˆ
Z and
ˆ
W correspond to the measure-
ments of H
Z
and H
W
in |φi
Z
and |ωi
W
respectively.
Proof. Given quantum systems X, W, Y, Z with corre-
sponding Hamiltonians, suppose first the state |ψi
X
has
decomposable coherence. This implies that there exists
Accepted in Quantum 2019-10-24, click title to verify 27
an auxiliary system Y with zero energy eigenstate |0i
Y
and an isometry V from H
X
⊗H
Y
into H
Z
⊗H
W
with
V (H
X
⊗
Y
+
X
⊗H
Y
) = (H
Z
⊗
W
+
Z
⊗H
W
)V
†
,
(91)
such that
V [|ψi
X
⊗ |0i
Y
] = |φi
Z
⊗ |ωi
W
, (92)
and where neither |φi
Z
nor |ωi
W
are energy eigenstates
of their respective Hamiltonians. The associated classi-
cal random variables for the input systems are
ˆ
X and
ˆ
Y , while the associated classical random variables on
the output systems are
ˆ
Z and
ˆ
W .
Now since
ˆ
Y = 0 the distribution for
ˆ
X coincides
with the distribution
ˆ
X +
ˆ
Y , which is simply the total
energy of the input systems. However equation (91)
implies that V is block diagonal in the total energy and
thus preserves the measurement statistics for the total
energy. Therefore
ˆ
X =
ˆ
Z +
ˆ
W . Since |φi
Z
and |ωi
W
are not energy eigenstates this means that both
ˆ
Z and
ˆ
W have non-trivial distributions and are independent
since the composite state on the output is a product
state, and thus
ˆ
X is a decomposable random variable.
Conversely, given an initial state |ψi
X
with associ-
ated classical random variable
ˆ
X, now suppose that
ˆ
X
is decomposable. This means there exist classical ran-
dom variables
ˆ
Z and
ˆ
W , both with independent, non-
trivial distributions, such that
ˆ
X =
ˆ
Z +
ˆ
W . Since
H
X
is assumed to have a discrete spectrum the clas-
sical random variable
ˆ
X only has support on a discrete
set of values, and thus
ˆ
Z and
ˆ
W only take on discrete
values too. Let E
1
, E
2
, . . . denote the values that the
random variables
ˆ
X,
ˆ
Z,
ˆ
W take on, and define the set
Ω = {E
1
, . . . , E
d
, . . . }. Denote by (r
k
) and (s
k
) respec-
tively the distributions of
ˆ
Z and
ˆ
W over Ω.
Given these distributions, we first introduce two
quantum systems Z, W , each of countable dimension
|Ω| with Hamiltonians H
Z
, H
W
and each with an en-
ergy spectrum lying in Ω. Define the states
|φi
Z
:=
|Ω|
X
k=1
√
r
k
|E
k
i
Z
(93)
|ωi
W
:=
|Ω|
X
k=1
√
s
k
|E
k
i
W
, (94)
on quantum systems Z and W where we label the eigen-
states of each Hamiltonian with their corresponding
eigenvalues. We also introduce a third system auxil-
iary Y of arbitrary dimension, but with a zero energy
eigenstate |0i
Y
for some Hamiltonian H
Y
. We now con-
struct an isometry V
tot
from H
X
⊗H
Y
to H
Z
⊗H
W
that
sends |ψi
X
⊗ |0i
Y
to |φi
Z
⊗ |ωi
W
.
First observe that
|φi
Z
⊗ |ωi
W
=
|Ω|
X
z,w=1
√
r
z
s
w
|E
z
i
Z
⊗ |E
w
i
W
. (95)
However
ˆ
X =
ˆ
Z +
ˆ
W , with
ˆ
Z and
ˆ
W independent ran-
dom variables. This implies that the distribution (p
x
)
of
ˆ
X over Ω is given by the convolution
p
x
=
X
z,w:
E
x
=E
z
+E
w
q
z
r
w
. (96)
Therefore we instead write the above expression for the
output states as
|φi
Z
⊗ |ωi
W
=
|Ω|
X
z,w=1
√
r
z
s
w
|E
z
i
Z
⊗ |E
w
i
W
=
X
E
x
X
z,w:
E
x
=E
z
+E
w
√
r
z
s
w
|E
z
i
Z
⊗ |E
w
i
W
.
(97)
We let Π
x
denote the projector onto the E
x
eigenspace
of H
X
, and so for those x for which p
x
6= 0 we have
Π
x
|ψi
X
=
√
p
x
|ψ
x
i
X
, (98)
which defines a normalised pure state |ψ
x
i
X
within the
E
x
eigenspace of H
X
. Thus
|ψi
X
=
|Ω|
X
x=1
√
p
x
|ψ
x
i
X
. (99)
We next define a linear mapping V from H
sup
:=
span{|ψ
x
i
X
⊗ |0i
Y
}
x
into H
Z
⊗ H
W
for which
V [|ψ
x
i
X
⊗ |0i
Y
] =
1
√
p
x
X
z,w:
E
x
=E
z
+E
w
√
r
z
s
w
|E
z
i
Z
⊗|E
w
i
W
,
(100)
for all x = 1, . . . |Ω| for which p
x
6= 0. If p
x
6= 0 for some
x then equation (96) implies that there is at least one
pair (z, w) such that q
z
and r
w
are non-zero and so for
such an x the image under V is a non-zero vector.
V is an isometry: It is readily checked, using the
orthogonality of the eigenstates of the output systems
and equation (96) that
hψ
x
|
X
⊗ h0|
Y
V
†
V |ψ
x
i
X
⊗ |0i
Y
= 1 (101)
for all x for which p
x
6= 0. Note that since different en-
ergy eigenspaces are orthogonal, we have both that the
input states {|ψ
x
i
X
⊗ |0i
Y
}
x
are an orthonormal basis
for H
sup
, and also the output states are orthonormal
hψ
x
|
X
⊗ h0|
Y
V
†
V |ψ
y
i
X
⊗ |0i
Y
= 0, (102)
Accepted in Quantum 2019-10-24, click title to verify 28
for any x 6= y. The above provide the matrix compo-
nents for V and imply that V is an isometry from H
sup
into H
Z
⊗ H
W
.
V is energy conserving: Let H
XY
:= H
X
⊗
Y
+
X
⊗
H
Y
be the initial Hamiltonian. Now since H
X
⊗H
Y
=
H
sup
⊕ H
⊥
sup
and since |ψ
x
i
X
⊗ |0i
Y
are eigenstates of
H
XY
we can write H
XY
= H
XY,s
⊕ H
XY,p
, where
H
XY,s
is simply the restriction of the total Hamil-
tonian to the subspace H
sup
and H
XY,p
is the com-
ponent on H
⊥
sup
. Now each pair (z, w) in equation
(100) contributes a vector
√
r
z
|E
z
i
Z
⊗
√
s
w
|E
w
i
W
that
is an E
x
energy eigenvector of the total Hamiltonian
H
ZW
:= H
Z
⊗
W
+
Z
⊗ H
W
, and so V |ψ
x
i
X
⊗ |0i
Y
is an E
x
energy eigenstate of H
ZW
for any x. Thus
V sends E
x
eigenstates of H
XY,s
to E
x
eigenstates of
H
ZW
as required by energy conservation.
Finally, by rearranging equation (100) and summing
over x we see that V [|ψi
X
⊗|0i
Y
] = |φi
Z
⊗|ωi
W
. To get
an isometry on the full space, we again write H
X
⊗H
Y
=
H
sup
⊕H
⊥
sup
, and extend the systems Z and W and their
Hamiltonians so that now
H
Z
⊗ H
W
= span{|E
z
i
Z
⊗ |E
w
i
W
}
z,w
⊕ H
other
(103)
with dim(H
other
) ≥ dim(H
⊥
sup
). The final isometry on
the input system is then V
tot
= V ⊕V
other
, where V
other
is
an arbitrary energy conserving isometry from H
⊥
sup
into
H
other
. This constructs a globally energy conserving
V
tot
such that
V
tot
|ψi
X
⊗ |0i
Y
= |φi
Z
⊗ |ωi
W
, (104)
and since both of
ˆ
Z and
ˆ
W are non-trivial random
variables we have that neither of the output states
|ωi
W
, |φi
Z
is an energy eigenstate of its respective
Hamiltonian. Thus the state |ψi
X
has decomposable
coherence as required.
A.3 Proof of semi-classical coherent work pro-
cesses theorem
The proof of the semi-classical theorem relies on
Raikov’s theorem, which is a result for classical proba-
bility distributions. The core idea behind the proof is
outlined in Fig. 7. For the non-trivial direction, one
starts with a tuple (S, |ψi
S
, H
S
) with S a quantum sys-
tem with Hamiltonian H
S
, and |ψi
S
a state of the sys-
tem that admits a non-trivial coherent work process.
Any state |ψi
S
uniquely determines a classical random
variable
ˆ
S, although the converse is not true since we
have freedom in, for example, choosing phases or the
dimension of S. If
ˆ
S is decomposable, then Raikov’s
theorem tells us the relationship between the compo-
nents of the decomposition
ˆ
S =
ˆ
S
0
+
ˆ
A. From this we
can construct (non-unique) tuples (S
0
, |φi
S
0
, H
S
0
) and
(A, |ωi
A
, H
A
) associated to the random variables
ˆ
S
0
and
ˆ
A respectively. On these quantum systems we then
show that a coherent work process can be constructed.
The precise statement and proof are as follows.
Figure 7: Raikov’s theorem and coherent work processes.
Given any quantum system S with Hamiltonian H
S
in a state
|ψi
X
there is a unique classical random variable
ˆ
X
S
corre-
sponding to the energy measurement on S. Conversely, given
a random variable
ˆ
X
S
we may construct a (non-unique) tuple
(S, |ψi
S
, H
S
) associated to this random variable.
Theorem A.5. Let S be a harmonic oscillator system,
with Hamiltonian H
S
= hνa
†
a. Let C be the set of
quantum states for S as defined above. Then:
1. The set of quantum states C is closed under all pos-
sible coherent work processes from S to S with fixed
Hamiltonian H
S
for both the input and output.
2. Given any quantum state |ψi
S
∈ C there is a
unique, canonical state |α, ki
S
∈ C such that we
have a reversible transformation
|ψi
S
ω
c
←→ |α, ki
S
, (105)
with |ω
c
i
A
= |0i
A
and α = |α|. Moreover, the only
coherent work processes possible between canonical
states are
|α, ki
S
ω
−→ |α
0
, ki
S
, (106)
such that |α
0
| ≤ |α|. Modulo phases, the coherent
work output is of the form |ωi
A
= |λ, ni
A
, where
λ =
p
|α|
2
− |α
0
|
2
and n, k
0
are any integers that
obey n + k
0
= k.
3. In the classical limit of large displacements |α| 1,
from coherent processes on C we recover all classical
work processes on the system S under a conserva-
tive force.
Proof. (Proof of 1.) Let |ψi
S
= e
iL
S
θ
|λ, mi
S
be a state
in C defined with respect to a Hamiltonian H
S
= hνa
†
a,
with [L
S
, H
S
] = 0, L
S
= L
†
S
and θ ∈ R. Denote by
ˆ
S
the classical random variable obtained from an energy
measurement of S, which takes values in {s
n
= nhν :
Accepted in Quantum 2019-10-24, click title to verify 29
n ∈ Z}, and with distribution p = (p
n
= |hψ|ni|
2
). We
can expand the state |ψi
S
in the energy basis as
|ψi = e
−λ/2
∞
X
n=0
e
iθ
n
r
λ
n
n!
|n + mi, (107)
for some arbitrary phases {θ
n
} and some m ∈ N. De-
note by σ = hν the energy spacing of H
S
. Then the en-
ergy measurement distribution is Poissonian and given
by
p
n
= P
ˆ
S = (n + m)hν
= e
−λ
λ
n
n!
, (108)
for n ≥ 0 and p
n
= 0 for n < 0. The associated cumu-
lative distribution function is given by
F
ˆ
S
(x) = F
x − mhν
σ
, λ
. (109)
The set C is clearly closed under trivial coherent work
processes. Now suppose that |ψi
S
admits some non-
trivial coherent work process given by
|ψi
S
ω
−→ |φi
S
0
. (110)
By Theorem 2.5 this implies that
ˆ
S =
ˆ
S
0
+
ˆ
A, where
ˆ
S
0
and
ˆ
A are non-trivial, independent classical ran-
dom variables. From these we can construct tuples
(S
0
, |φi
S
0
.H
S
0
) and (A, |ωi
A
, H
A
) that realise the asso-
ciated coherent work process, as detailed in the proof of
Theorem 2.5. For any |ψi
S
∈ C we have that
ˆ
S =
ˆ
S
0
+
ˆ
A
is a decomposable Poisson random variable, and thus by
Raikov’s theorem the cumulative distribution functions
for
ˆ
S
0
and
ˆ
A are also Poissonian, with
F
ˆ
S
0
(x) = F
x − α
1
σ
, µ
, (111)
F
ˆ
A
(x) = F
x − α
2
σ
, κ
, (112)
where ν, κ ≥ 0 and constrained as µ + κ = λ, α
1
+ α
2
=
mhν and α
1
, α
2
∈ R.
As σ appears identically in all cumulative distribution
functions this implies that both H
S
0
and H
A
must have
within their spectral decompositions, countably infinite
discrete energy eigenstates with equal spacings hν, cor-
responding to the outcomes of the energy measurements
for the constructed states |φi
S
0
and |ωi
A
. We write the
corresponding distributions for
ˆ
S
0
and
ˆ
A as
q
n
:= P [
ˆ
S
0
= nhν + c
S
0
] = e
−µ
µ
n
n!
r
a
:= P [
ˆ
A = ahν + c
A
] = e
−κ
κ
a
a!
, (113)
for integers n, a ≥ 0, and where c
S
0
and c
A
are any
constants that obey
c
S
0
+ c
A
= mhν. (114)
By suitable constant shifts of the Hamiltonians H
S
0
and H
A
, that shift the zero energy level, we can en-
sure that c
S
0
= m
0
hν for some integer m
0
, and thus
c
A
= (m −m
0
)hν. This implies that both (q
n
) and (r
a
)
are both Poissonian distributions on a discrete set of
energy eigenvalues with constant separation hν.
Let Π
S
0
n
and Π
A
a
denote the projectors for the discrete
energy eigenvalues in the energy measurements of S
0
and A with probabilities q
n
and r
a
on |φi
S
0
and |ωi
A
respectively. We now define the states |n + m
0
i
S
0
and
|a + m − m
0
i
A
via
√
q
n
|n + m
0
i
S
0
:= Π
S
0
n
|φi
S
0
(115)
√
r
a
|a + m − m
0
i
A
:= Π
A
a
|ωi
A
, (116)
with n, a ≥ 0 and where we have ensured trivial phases
via the definition of the states. Therefore we can write
the two output states as
|φi
S
0
= e
−µ/2
∞
X
n=0
r
µ
n
n!
|n + m
0
i
S
0
|ωi
A
= e
−κ/2
∞
X
a=0
r
κ
a
a!
|a + m − m
0
i
A
. (117)
Recall that the translated coherent states are given by
D(α)|0i = |αi = e
−
|α|
2
2
∞
X
n=0
α
n
n!
|ni (118)
|α, mi = e
−
|α|
2
2
∞
X
n=0
α
n
n!
|n + mi. (119)
and so we can define real constants β
1
=
√
µ and
β
2
=
√
κ that allow us to write |φi
S
0
= |β
1
, m
0
i
S
0
and
|ωi
A
= |β
2
, m − m
0
i
A
, in terms of the notation for C
states. Therefore |φi
S
0
, |ωi
A
∈ C and so C is closed
under coherent work processes.
(Proof of 2.) Any state in C can be expressed in the
form
|ψi
S
= e
−λ/2
∞
X
n=0
e
iθ
n
r
λ
n
n!
|n + mi, (120)
where m is some non-negative integer, λ ≥ 0 and {θ
n
}
are phases. We now define a unitary of the form V =
V
S
⊗ 1
A
where
V
S
=
∞
X
n=0
e
−iθ
n
|n + mihn + m| +
m−1
X
k=0
|kihk|, (121)
which realises the transformation V [|ψi
S
⊗ |0i
A
] =
|
√
λ, mi
S
⊗|0i
A
with λ ≥ 0. It is clear that both V and
its inverse V
†
are energy conserving and so we have a
reversible coherent work process with |ω
c
i
A
= |0i
A
as
claimed.
Accepted in Quantum 2019-10-24, click title to verify 30
As constructed in the proof of part 1, we saw that
modulo arbitrary phases between the energy eigen-
states, any coherent work process on a state |α, mi
S
takes the form
|α, mi
S
ω
−→ |β
1
, m
0
i
S
0
(122)
and with |ωi
A
= |β
2
, m − m
0
i
A
. With a freedom to
shift the energies of S
0
and A by a constant factor
±c respectively. Moreover, since the Poisson parame-
ters are given, as above, by λ = |α|
2
, µ = |β
1
|
2
and
κ = |β
2
|
2
, Raikov’s theorem tells us that we must have
|α|
2
= |β
1
|
2
+ |β
2
|
2
, and therefore |β
2
| =
p
|α|
2
− |β
1
|
2
as claimed.
(Proof of 3.) For a classical work process under a
conservative force, we need two conditions to hold: (i)
the position, momentum and energy are sharp and (ii)
the process depends only on the initial and final en-
ergies, not the path. For the first, consider the posi-
tion and momentum operators q =
q
~
2mω
(a
†
+ a) and
p = i
q
~mω
2
(a
†
− a) with the canonical commutation
relation [q, p] = i~, and ~ω = hν. It is straightfor-
ward to verify that the variances satisfy σ
2
q
=
~
2mω
and
σ
2
p
=
~mω
2
for any coherent state. Then coherent states
are minimal uncertainty states with Heisenberg uncer-
tainty relation σ
2
q
σ
2
p
=
~
2
4
. For a coherent state |αi,
the expectation value for the position and momentum
are hqi
α
=
q
~
2mω
(α
∗
+ α) and hpi
α
= i
q
~mω
2
(α
∗
−α).
Therefore
σ
q
hqi
α
=
1
(α
∗
+ α)
, (123)
σ
p
hpi
α
=
i
(α − α
∗
)
. (124)
In the limit of large displacement |α| 1, the values for
hqi
α
and hpi
α
become arbitrarily sharp with the ratio of
standard deviation to value tending to zero. Similarly,
for a Harmonic oscillator Hamiltonian H
S
= ~ωa
†
a, the
variance of energy is σ
2
H
= ~
2
ω
2
|α|
2
leading to the ratio
of the standard deviation to the mean
σ
H
hH
S
i
α
=
1
|α|
, (125)
which again becomes sharp in the limit in the same
manner. Thus, all energy transfers, positions and mo-
menta become sharp in the limit of large displacements
and the classical mechanics of a harmonic oscillator is
recovered.
For (ii), the energy transfer depends only on the ini-
tial and final states of the system |αi
S
and |α
0
i
S
. As
all coherent states are included in C, then coherent work
processes on C states with large displacement are classi-
cal work processes under a conservative force. Further-
more, for the transition |αi
S
ω
−→ |α
0
i
S
, we have that
|α| ≥ |α
0
| and therefore hH
S
i
α
≥ hH
S
i
α
0
– meaning the
process on S is work extracting only for the coherence
component of |α, ki. However variations in the k label
are always sharp and can involve either positive or neg-
ative work and it is clear that for large displacements
the fractional energy separation of levels tends to zero
and one has an effective continuum of energy values that
recovers the classical case.
A.4 Proof of Theorem II.6 (coherent work pro-
cesses on a ladder system)
Theorem A.6. Let S be a quantum system, with en-
ergy eigenbasis {|ni : n ∈ Z} with fixed Hamiltonian
H
S
=
P
n∈Z
n|nihn|. Given an initial quantum state
|ψi
S
=
P
i
√
p
i
|ii
S
, a coherent work process
|ψi
S
ω
−→ |φi
S
, (126)
is possible with
|φi
S
=
X
j
e
iθ
j
√
q
j
|ji
S
(127)
|ωi
A
=
X
k
e
iϕ
k
√
r
k
|ki
A
(128)
for arbitrary phases {θ
j
} and {ϕ
k
}, with output Hamil-
tonian H
S
as above and H
A
=
P
n∈Z
n|ni
A
hn|, if and
only the distribution (p
n
) over Z can be written as
p
n
=
X
j∈Z
r
j
(∆
j
q)
n
. (129)
for distributions (q
m
) and (r
j
) over Z.
Proof. Assume |ψi
S
ω
−→ |φi
S
. Then there exists a uni-
tary V such that V |ψi
S
⊗|0i
A
= |φi
S
⊗|ωi
A
. Further-
more, the unitary satisfies the global energy constraint
[V, H
S
+ H
A
] = 0, where H
S
and H
A
are the Hamilto-
nians of S and A respectively.
We expand |φi
S
and |ωi
A
in their energy eigenbases
as
|φi
S
=
X
j
e
iθ
j
√
q
j
|ji
S
|ωi
A
=
X
k
e
iϕ
k
√
r
k
|ki
A
.
If V provides a valid coherent work process then so too
does the unitary UV where U = e
iB
S
θ
⊗ e
iC
A
ϕ
for any
Hermitian B
S
and C
A
such that [B
S
, H
S
] = [C
A
, H
A
] =
0. By choosing the operators B
S
and C
A
appropriately
one can freely vary the phases θ
j
and ϕ
k
. We have
that A has the same Hamiltonian ladder structure as
Accepted in Quantum 2019-10-24, click title to verify 31
S. Energy conservation implies that for any n ∈ Z, the
action of V on the state |ni
S
⊗ |0i
A
gives
V |ni
S
⊗ |0i
A
=
X
j∈Z
e
iα
n,j
√
m
j|n
|n − ji
S
⊗ |ji
A
. (130)
for some distribution (m
j|n
)
j
over j, and phases (α
n,j
).
This implies that
|φi
S
⊗ |ωi
A
=
X
n,j∈Z
e
iα
n,j
√
m
j|n
p
n
|n − ji
S
⊗ |ji
A
,
(131)
=
X
k,l∈Z
e
i(θ
k
+ϕ
l
)
√
q
k
r
l
|ki
S
⊗ |li
A
. (132)
Let Π
n
be the projector onto the energy eigenspace of
total energy n ∈ Z for the joint system SA. We thus
have that
hφ|
S
⊗ hω|
A
Π
n
|φi
S
⊗ |ωi
A
=
X
j∈Z
m
j|n
p
n
, (133)
=
X
j∈Z
q
n−j
r
j
, (134)
However
P
j
m
j|n
= 1 and so we have that the distri-
bution (p
n
) decomposes as
p
n
=
X
j∈Z
r
j
q
n−j
=
X
j∈Z
r
j
(∆
j
q)
n
, (135)
as required. Conversely, suppose we have a distribution
(p
n
) over Z which can be decomposed as
p
n
=
X
j∈Z
r
j
q
n−j
=
X
j∈Z
r
j
(∆
j
q)
n
, (136)
for two distributions (r
j
) and (q
k
). From these we define
for ladder systems S and A the states
|ψi
S
=
X
i
√
p
i
|ii
S
, (137)
and
|φi
S
=
X
j
e
iθ
j
√
q
j
|ji
S
(138)
|ωi
A
=
X
k
e
iϕ
k
√
r
k
|ki
A
(139)
where θ
j
and ϕ
k
are arbitrary phases. The joint system
SA therefore has the two states
|ψi
S
⊗ |0i
A
=
X
i
√
p
i
|ii
S
⊗ |0i
A
|φi
S
⊗ |ωi
A
=
X
k,l∈Z
e
i(θ
k
+ϕ
l
)
√
q
k
r
l
|ki
S
⊗ |li
A
.
= U
†
X
k,l∈Z
√
q
k
r
l
|ki
S
⊗ |li
A
, (140)
where U
†
is a phase unitary that generates the phases
θ
k
, ϕ
j
and which is diagonal in the energy eigenba-
sis. Again, we let Π
n
denote the projector onto the
n–eigenspace of the system SA and so
Π
n
|ψi
S
⊗ |0i
A
=
√
p
n
|ii
S
⊗ |0i
A
Π
n
U|φi
S
⊗ |ωi
A
=
X
m∈Z
√
q
n−m
r
m
|n − mi
S
|mi
A
. (141)
Thus we have that
hφ|
S
⊗ hω|
A
U
†
Π
n
U|φi
S
⊗ |ωi
A
=
X
m
q
n−m
r
m
= p
n
,
(142)
and thus the vectors Π
n
|ψi
S
⊗|0i
A
and Π
n
U|φi
S
⊗|ωi
A
both lie in the n–eigenspace of the total hamiltonian and
have equal norms. Therefore, we may define a unitary
V
n
within each energy eigenspace such that
V
n
Π
n
|ψi
S
⊗ |0i
A
= Π
n
U|φi
S
⊗ |ωi
A
. (143)
For those n–eigenspaces outside the support of |ψi
S
⊗
|0i
A
we simply let V
n
be the identity unitary on that
subspace. We now write V := ⊕
n
V
n
to obtain an energy
conserving unitary on the full space, and using the fact
that U commutes with the energy projectors we have
that
U
†
V |ψi
S
⊗ |0i
A
= |φi
S
⊗ |ωi
A
, (144)
where U
†
V is an energy conserving unitary on the joint
system. Therefore the required coherent work process
exists.
A.5 Physical example of decomposable coher-
ence
In the main text, an example of infinitely divisible co-
herence was provided in the form of a coherent state.
Here we provide a physical system in which one might
encounter such examples of coherent state manipula-
tions. Consider a beam splitter set up, with a coherent
state of light injected as the input on one mode and the
vacuum on the other mode, such that the initial state
is |αi
S
⊗ |0i
A
. We use the beam splitter unitary
B(θ) = e
θ(ab
†
−a
†
b)
(145)
where a and b are the annihilation operators for S and A
respectively [71]. Under the action of a beam splitting,
which conserves photon number, we want to achieve
B(θ)|αi ⊗ |0i = |βi ⊗ |ωi, (146)
for some tunable parameter θ. It is well known that the
two mode annihilation operators satisfy the algebraic
Accepted in Quantum 2019-10-24, click title to verify 32
property:
B
†
(θ)aB(θ) = a cos θ + b sin θ (147)
B
†
(θ)bB(θ) = b cos θ − a sin θ. (148)
Since coherent states are displaced vacuum states, we
have that
B(θ)|αi ⊗ |0i = B(θ)e
−
1
2
|α|
2
e
αa
†
e
−α
∗
a
|0i ⊗ |0i (149)
= e
−
1
2
|α|
2
B(θ)e
αa
†
B
†
(θ)|0i ⊗ |0i (150)
= e
−
1
2
|α|
2
e
αB(θ)a
†
B
†
(θ)
|0i ⊗ |0i (151)
= e
−
1
2
|α|
2
e
α(a
†
cos θ−b
†
sin θ)
|0i ⊗ |0i
(152)
= |α cos θi ⊗ | − α sin θi (153)
where we used the fact that B(θ)|0i ⊗ |0i = |0i ⊗ |0i
and B(θ) is unitary. Here, r = cos θ and t = sin θ are
the transmission and reflection coefficients. The con-
sequence is that we have achieved the desired coherent
state splitting with a photon number conserving uni-
tary. The second mode that began in the vacuum there-
fore carries information of the coherent energy transfer.
As expected, the condition |α|
2
= |α cos θ|
2
+ |α sin θ|
2
is satisfied.
If one were to perform measurements on the ancilla sys-
tem, they would extract information on the energetics
and coherences. However, by performing the measure-
ment, the state would no longer be available for ma-
nipulations that involve interference effects. For this
reason, the coherent work output is the state |α sin θi
rather than a post-processed state.
B Fluctuation relations and the effective
potential
Here we gather together the proofs for the fluctuation
relation analysis.
B.1 Basic properties of the effective potential.
It is worth describing some properties of the function
Λ(β, ρ) to shed light on the fluctuation relations.
We first note that the effective potential has an in-
variance under the group generated by the observable
H. More precisely, let U(t) be the unitary repre-
sentation of time-translations generated by H. Then
Λ(β, U (t)ρU (t)
†
) = Λ(β, ρ) for any t ∈ R, and so we
have that Λ(β, ρ(t
1
)) = Λ(β, ρ(t
2
)) for any two times
t
1
, and t
2
. Thus, Λ is a constant of motion under the
free Hamiltonian evolution. For a pure quantum state
|ψi =
P
k
e
iθ
k
√
p
k
|E
k
i, Λ is purely a function of the
classical distribution p
k
over the energy eigenstates. For
a projective measurement of energy in this eigenbasis we
can denote by
ˆ
X the classical random variable for the
energy obtained in the measurement, then it is readily
seen that Λ(β, ψ) is essentially the negative cumulant
generating function for
ˆ
X in the inverse temperature β,
Λ(β, ψ) = −
X
n≥1
(−1)
n
β
n
n!
κ
n
(
ˆ
X). (154)
We shall make use of the following theorem for our
analysis.
Theorem B.1. If A is Hermitian and Y is strictly pos-
itive, then
ln tr(e
A+ln Y
) = max{tr(XA) − S(X||Y ) : X is positive
& trX = 1}.
Alternatively, if X is positive with Tr(X) = 1 and B is
Hermitian, then
S(X||e
B
) = max{tr(XA) − log tr(e
A+B
) : A = A
†
}.
The proof of this can be found in [72].
We now establish basic properties of the effective po-
tential Λ(β, ρ). As a function of the quantum state ρ,
the effective potential has the following properties.
Lemma B.2. Given a quantum system S with Hamil-
tonian H , and inverse temperature β > 0. We have
that
1. (Invariance) Let ρ(t) = e
−iHt
ρe
+iHt
for all t ∈
R. For any state ρ and any t ∈ R we have
Λ(β, ρ(t)) = Λ(β, ρ(0)). Moreover, we have that
Λ(β, ρ) = Λ(β, D(ρ)) where D(ρ) =
P
k
Π
k
ρΠ
k
is
the dephasing of ρ in the energy basis, with Π
k
be-
ing the projector onto the k’th energy eigenspace of
H.
2. For any state ρ, we have that
βhHi
ρ
−
β
2
8
(E
max
−E
min
)
2
≤ Λ(β, ρ) ≤ βhHi
ρ
. (155)
3. Let E
min
and E
max
be the smallest and largest
eigenvalues of H respectively in the support of ρ.
Then
βE
min
≤ Λ(β, ρ) ≤ βE
max
. (156)
4. (Variational form) For states ρ of full rank and
[ρ, H] = 0, we have that
Λ(β, ρ) = min
σ
[βhHi
σ
+ S(σ||ρ)] , (157)
where the minimization is taken over all quantum
states σ of the system S.
Accepted in Quantum 2019-10-24, click title to verify 33
5. (Additivity) For a bipartite system S
1
S
2
with to-
tal Hamiltonian H
12
= H
1
⊗
2
+
1
⊗ H
2
and
a bipartite product state ρ
12
= ρ
1
⊗ ρ
2
we have
Λ(β, ρ
1
⊗ρ
2
) = Λ
1
(β, ρ
1
) + Λ
2
(β, ρ
2
), where the ef-
fective potentials for subsystems are defined solely
with respect to the corresponding Hamiltonian on
that system.
Proof. 1. For any t ∈ R we have Λ(β, ρ(t)) =
−ln tr(e
−βH
e
−iHt
ρe
+iHt
) = Λ(β, ρ), by cyclicity of the
trace. Moreover, we can evaluate the trace in an energy
eigenbasis of H, and so denoting by Π
k
the projector
onto the k’th eigenspace of H we have that
−ln tr(e
−βH
ρ) = −ln tr(
X
k
Π
k
Π
k
(e
−βH
ρ))
= −ln tr(
X
k
Π
k
(e
−βH
ρ)Π
k
)
= −ln tr(
X
k
e
−βH
Π
k
ρΠ
k
)
= −ln tr(e
−βH
D(ρ)) = Λ(β, D(ρ)).
(158)
2. The proof of the upper bound is a direct result of
Jensen’s inequality, he
−βH
i
ρ
≥ e
−βhHi
ρ
, which implies
Λ(β, ρ) ≤ −ln e
−βhHi
ρ
= βhHi
ρ
, (159)
achieving the desired result.
The lower bound is obtained as follows. Given a
state ρ, let us define the zero mean observable H
0
=
H − hHi
ρ
1. It is readily seen that the effective poten-
tial satisfies Λ(β, ρ) = Λ
0
(β, ρ) + βhHi
ρ
, where Λ
0
(β, ρ)
is evaluated for the Hamiltonian H
0
. Let us denote the
random variable
ˆ
X
0
with sample space eigs(H
0
), dis-
tributed by ρ. Then
ˆ
X
0
has zero mean and E
min
≤
ˆ
X
0
≤ E
max
almost surely. Thus we can apply Hoeffd-
ing’s lemma [73], which states
E[e
λ
ˆ
X
0
] ≤ e
λ
2
8
(E
max
−E
min
)
2
(160)
for any λ ∈ R. Taking the logarithm and re-ordering
the inequality, Λ
0
(β, ρ) ≥ −
β
2
8
(E
max
− E
min
)
2
, which
gives the lower bound.
3. From 2, we know that Λ(β, ρ) ≤ βhHi
ρ
≤ βE
max
as the expectation value cannot exceed the greatest
eigenvalue. To prove the lower bound, consider a state
ρ with diag(ρ) = p and assume E
k
= E
min
=⇒ p
k
6= 0
without loss of generality. Then
Λ(β, ρ) = −ln
e
−βE
min
[p
k
+
X
j6=k
p
j
e
−β(E
j
−E
min
)
]
(161)
= βE
min
− ln
p
k
+
X
j6=k
p
j
e
−β(E
j
−E
min
)
(162)
By assumption, E
min
is the smallest eigenvalue of H
in the support of ρ and therefore e
−β(E
n
−E
min
)
≤ 1
for all n. From the property
P
i
p
i
= 1, the argument
of the logarithm is smaller than one and therefore
the latter term is positive. It therefore follows that
βE
min
≤ Λ(β, ρ) ≤ βE
max
.
4. With an appropriate choice of operators A =
−βH and Y = ρ, Theorem B.1 implies Λ(β, ρ) =
−max
σ
{−βhHi
σ
−S(σ||ρ)} = min
σ
{βhHi
σ
+ S(σ||ρ)}
for states σ. However as ln ρ will diverge for non-full
rank states, and so the assumption that ρ must be full
rank is required.
5. Since H
12
= H
1
⊗
2
+
1
⊗ H
2
we obtain
Λ(β, ρ
1
⊗ ρ
2
) = −ln tr(e
−β(H
1
⊗1+1⊗H
2
)
ρ
1
⊗ ρ
2
)
(163)
= −ln Tr(e
−βH
1
ρ
1
)tr(e
−βH
2
ρ
2
) (164)
= Λ
1
(β, ρ
1
) + Λ
2
(β, ρ
2
). (165)
The effective potential has the following properties in
terms of the parameter β.
Lemma B.3. Given a quantum system S with Hamil-
tonian H. For any state ρ of S, the effective potential
Λ(β, ρ) obeys
1. Λ(β, ρ) is concave in β for all β ≥ 0.
2. Λ(β, ρ) is monotone increasing in the variable β,
for all states ρ, if and only if H ≥ 0.
3. (High temperature regime) lim
s→0
s
−1
Λ(sβ, ρ) =
βhHi
ρ
.
4. (Low temperature regime) lim
s→∞
s
−1
Λ(sβ, ρ) =
βE
min
, where E
min
is the smallest eigenvalue of
H in the support of ρ.
Proof. 1. Let a, b > 0 satisfy a+b = 1. For two different
inverse temperatures β
1
and β
2
and Hamiltonian H, we
have:
Λ(aβ
1
+ bβ
2
, ρ) = −ln tr(ρe
−aβ
1
H
e
−bβ
2
H
). (166)
Let us define the operators X = e
−aβ
1
H
and Y =
e
−bβ
2
H
, where X, Y ≥ 0. Then Hölder’s inequal-
ity tells us expectation values satisfy E[|XY |] ≤
(E[|X|
p
])
1/p
(E[|Y |
q
])
1/q
, where
1
p
+
1
q
= 1. Choosing
p =
1
a
and q =
1
b
, we get:
Λ(aβ
1
+ bβ
2
, ρ) ≥ aΛ(β
1
, ρ) + bΛ(β
2
, ρ). (167)
Therefore Λ is concave in the inverse temperature.
2. First suppose that H ≥ 0, then
∂Λ(β, ρ)
∂β
=
tr(He
−βH
ρ)
tr(e
−βH
ρ)
(168)
Accepted in Quantum 2019-10-24, click title to verify 34
For β ∈ R, the denominator is always strictly positive.
The nominator can be expanded using the spectral de-
composition H =
P
k
E
k
Π
k
and so
tr(He
−βH
ρ) =
X
k
E
k
e
−βE
k
tr(Π
k
ρΠ
k
). (169)
Since H ≥ 0 all terms in this expression are non-
negative for all β and thus ∂
β
(Λ(β, ρ) ≥ 0 which implies
the function is monotonely increasing in β.
Conversely, assume ∂
β
Λ(β, ρ) ≥ 0 for all states ρ.
This implies that tr(He
−βH
ρ) ≥ 0 for all states ρ.
Again, using the spectral decomposition of H and the
fact that we have that
X
k
E
k
e
−βE
k
p
k
≥ 0 (170)
for all distributions p
k
over the eigenvalues E
k
of H. For
p = (p
k
) sharp on an eigenvalue E
m
the above equation
implies that E
m
≥ 0, and since this holds for all m this
in turn implies that H ≥ 0 as required.
3. To prove this, we Taylor expand in orders of s. We
find:
lim
s→0
s
−1
Λ(sβ, ρ) = lim
s→0
(−s
−1
) lnhe
−sβH
i
ρ
(171)
= lim
s→0
(−s
−1
) ln[1 − sβhHi
ρ
+ O(s
2
)]
(172)
= lim
s→0
(−s
−1
)[−sβhHi
ρ
+ O(s
2
)]
(173)
= βhHi
ρ
(174)
as claimed.
4. Let E
min
= E
k
be the smallest energy eigenvalue
in the support of ρ, and let ρ
i
:= tr(Π
i
ρ) for any i.
Then:
lim
s→∞
s
−1
Λ(sβ, ρ) = lim
s→∞
(−s
−1
) ln
X
i
e
−sβE
i
ρ
i
!
(175)
= lim
s→∞
(−s
−1
) ln
e
−sβE
min
[ρ
k
+
X
i6=k
e
−sβ(E
i
−E
min
)
ρ
i
]
= lim
s→∞
(−s
−1
)[−sβE
min
+ ln(ρ
k
+
X
i6=k
e
−sβ(E
i
−E
min
)
ρ
i
)]
= βE
min
+ lim
s→∞
(−s
−1
) ln(ρ
k
+
X
i6=k
e
−sβ(E
i
−E
min
)
ρ
i
).
(176)
Now since E
i
−E
k
> 0 we have that e
−sβ(E
i
−E
min
)
< 1,
for all i 6= k and for all s > 0, β > 0 and this term
decreases to zero as s → ∞. Moreover ρ
k
6= 0 by as-
sumption, and therefore
lim
s→∞
(−s
−1
) ln(ρ
k
+
X
i6=k
e
−sβ(E
i
−E
min
)
ρ
i
)
= lim
s→∞
(−s
−1
) ln(ρ
k
) = 0. (177)
which implies that lim
s→∞
s
−1
Λ(sβ, ρ) = βE
min
as re-
quired.
Accepted in Quantum 2019-10-24, click title to verify 35
B.2 Mean coherence representation
As in the main text, we have that χ
m
, the mean coher-
ence at inverse temperature β, is given by the equation
Λ(β, ρ) = βhHi
ρ
−
1
8π
β
2
χ
m
(˜ρ), (178)
where χ
m
is evaluated at the inverse temperature β
m
≤
β determined from the Mean Value Theorem.
Theorem B.4. Given a quantum system S with Hamil-
tonian H
S
with spectral decomposition H
S
=
P
k
E
k
Π
k
,
the mean coherence at inverse temperature β of a quan-
tum state |ψi is given by
β
2
8π
χ
m
(
˜
ψ) = β(hH
S
i
p
− hH
S
i
˜
p
) − S(
˜
p||p). (179)
where p is the distribution p
k
:= tr[Π
k
ψ] over energy of
ψ and
˜
p is given by ˜p
k
:= tr[Π
k
˜
ψ], the distribution over
energy for |
˜
ψih
˜
ψ| = Γ(|ψihψ|).
Proof. From Lemma B.2 we have that Λ(β, ρ) =
Λ(β, D(ρ)) where D(ρ) is the dephased state across the
energy eigenspaces, and so because of this it will suffice
to restrict to this dephased state, as we now show.
Let σ = D(ρ) for compactness. Noting that [σ, H] =
0 we now expand S(˜σ) as
S(˜σ) = −tr[˜σ ln ˜σ]
= −tr
˜σ ln
e
−βH/2
σe
−βH/2
1
tr(e
−βH
σ)
= −tr
h
˜σ
n
ln(e
−βH/2
σe
−βH/2
) + Λ(β, σ)
oi
= −tr
h
˜σ ln(e
−βH/2
σe
−βH/2
)
i
− Λ(β, σ)
= −tr [˜σ(−βH + ln σ)] − Λ(β, σ)
= βhHi
˜σ
− tr[˜σ ln σ] − Λ(β, σ)
= βhHi
˜σ
+ S(˜σ) + S(˜σ||σ) − Λ(β, σ), (180)
which implies that
Λ(β, σ) = βhHi
˜σ
+ S(˜σ||σ). (181)
Now since σ = D(ρ) we have that tr[Π
k
σ] = tr[Π
k
ρ]
and likewise it can be checked that tr[Π
k
˜σ] = tr[Π
k
˜ρ],
where we use that D(Γ(ρ)) = Γ(D(ρ)) for any ρ. Thus
the preceding equation can be written
Λ(β, D(ρ)) = Λ(β, ρ) = βhHi
˜ρ
+ S(˜σ||σ). (182)
Restricting to ρ = ψ, a pure state, and using p
k
=
tr[Π
k
ψ] and ˜p
k
= tr[Π
k
˜
ψ] we have that
Λ(β, ψ) = βhHi
˜
p
+ S(
˜
p||p), (183)
where we use that ˜σ commutes with σ and so S(˜σ||σ)
can be replaced with the classical divergence between
the respective probability distributions over energy
S(
˜
p||p). Finally, using equation (178) we have that
1
8π
β
2
χ
m
(
˜
ψ) = β(hH
S
i
p
− hH
S
i
˜
p
) − S(
˜
p||p), (184)
as required.
From this the fluctuation relation in Theorem 3.1 can
be stated in the following way.
Theorem B.5. Given the assumptions to Theorem 3.1,
let p
k
and
˜
p
k
denote the probability distribution for an
energy measurement of the state |ψ
k
i and |
˜
ψ
k
i respec-
tively. Then
P [ψ
1
|
˜
ψ
0
]
P [ψ
∗
0
|
˜
ψ
∗
1
]
= e
−β∆F −β
˜
W
S
e
S(
˜
p
0
||p
0
)−S(
˜
p
1
||p
1
)
, (185)
where
˜
W
S
:= hH
S
i
˜
ψ
1
− hH
S
i
˜
ψ
0
and S(p||q) is the
Kullback-Leibler divergence.
B.3 Variational expression for the effective po-
tential
The following is the same as Lemma 3.3 in the main
text.
Lemma B.6. Given a quantum system S with Hamil-
tonian H
S
and {Π
k
} the projectors onto the energy
eigenspaces of H
S
. Let us denote the de-phased form of
state in the energy basis by D(ρ) =
P
k
Π
k
ρΠ
k
. Then,
for any full rank quantum state ρ, the effective potential
Λ(β, ρ) is given by
Λ(β, ρ) = min
τ
{βhH
S
i
τ
+ S(τ||D(ρ))}, (186)
where the minimization is over all quantum states τ of
S, and S(τ ||ρ) = tr[τ log τ − τ log ρ] is the relative en-
tropy function. Moreover, the minimization is attained
for the state τ = Γ(D(ρ)).
Proof. From Lemma B.2, we see that for any full rank
ρ we have
Λ(β, ρ) = Λ(β, D(ρ)) (187)
= min
τ
[βhH
S
i
τ
+ S(τ||D(ρ))].
However equation (181) says
Λ(β, ρ) = Λ(β, σ) = βhH
S
i
˜σ
+ S(˜σ||σ), (188)
where σ = D(ρ). Comparing the two equations we see
that the minimization is realised for τ = ˜σ = Γ(σ) =
Γ(D(ρ)) as claimed.
Accepted in Quantum 2019-10-24, click title to verify 36
B.4 Coherent work and the quantum fluctuation
relation.
We now prove the coherent work form of the coherent
fluctuation theorem and provide an account of how it
naturally connects with the resource theory of asym-
metry, inducing a natural measure for coherent work.
However, first we require the conditions in which the
moments of a probability distribution uniquely deter-
mine the distribution.
Theorem B.7 ([74]). Let µ be a probability measure on
the line having finite moments m
k
=
R
∞
−∞
x
k
µ(dx) of
all orders. If the power series
P
∞
k=1
r
k
k!
m
k
has positive
radius of convergence, then µ is the unique probability
measure with the moments m
1
, m
2
, . . . .
When the full set of moments uniquely define the
probability distribution, then so too do the cumulants.
This follows since in such cases the moments can be ex-
pressed in terms of the cumulants. Furthermore, finite
moments correspond to finite cumulants since once can
obtain the cumulants from a recursive formula that is
polynomial in the moments. Thus the full set of cu-
mulants uniquely determines a distribution when the
moment generating function of the distribution has a
positive radius of convergence. When the moment gen-
erating function is finite and non-zero, then the cumu-
lant generating function is finite and non-zero as it is
obtained by taking the logarithm.
In the result that follows, we assume the moment
generating function and by extension the effective po-
tential, have infinite radius of convergence and therefore
always exist and uniquely specify the distribution. This
choice is for simplicity in the result, though one could
generalise it to hold for any positive radius of conver-
gence. Requiring an infinite radius of convergence is
a restriction on the states permitted. For example, the
negative binomial distribution NB(r, p) has finite radius
of convergence and therefore any state with such a dis-
tribution would not be included in the statement. This
can be seen from the moment generating function M (t)
for a random variable
ˆ
X ∼ NB(r, p), with [13]
M(t) =
p
1 − (1 − p)e
t
r
, (189)
which is valid for t < −ln(1 − p).
Theorem B.8. Let S be a quantum system with Hamil-
tonian H
S
and let |ψ
0
i
S
and |ψ
1
i
S
be two pure states
of S that have energy statistics with finite cumulants of
all orders. Then the following are equivalent:
1. The pure states |ψ
0
i
S
and |ψ
1
i
S
are coherently con-
nected with coherent work output/input ω on an
auxiliary system A.
2. Within the fluctuation relation context with a ther-
mal system B and quantum system S we have that
P [ψ
1
|
˜
ψ
0
]
P [ψ
∗
0
|
˜
ψ
∗
1
]
= e
−β(∆F ±kT Λ(β,ω))
, (190)
for all inverse temperatures β ≥ 0, for some state
|ωi
A
with finite cumulants on an auxiliary system
A and some choice of sign before Λ(β, ω).
Proof. First assume that |ψ
0
i
S
and |ψ
1
i
S
are coherently
connected states. Suppose there is a coherent work pro-
cess transforming |ψ
0
i
S
ω
−→ |ψ
1
i
S
. Without loss of
generality we can choose the initial state |0i
A
of the
auxiliary system A to have zero energy.
By unitary invariance of the effective potential we
have that
Λ(β, |ψ
0
i
S
⊗ |0i
A
) = Λ(β, V (|ψ
0
i
A
⊗ |0i
A
))
= Λ(β, |ψ
1
i
A
⊗ |ωi
A
), (191)
where V is a unitary that realises the pro-
cess. This conserves energy and so commutes with
e
−β(H
S
⊗1
A
+1
S
⊗H
A
)
. Since the effective potential is ad-
ditive for independent Hamiltonians and product states,
we have that
Λ(β, ψ
0
) = Λ(β, ψ
1
) + Λ(β, ω). (192)
Therefore the effective potential change in the fluctua-
tion theorem can be expressed in terms of the coherent
work output |ωi as
e
Λ(β,ψ
0
)−Λ(β,ψ
1
)
= e
Λ(β,ω)
. (193)
and so the fluctuation theorem can be expressed as
P [ψ
1
|
˜
ψ
0
]
P [ψ
∗
0
|
˜
ψ
∗
1
]
= e
Λ(β,ω)−β∆F
=
e
−β∆F
hω|e
−βH
A
|ωi
. (194)
If instead |ψ
1
i
S
ω
−→ |ψ
0
i
S
, then the same argument
applies with the only difference being the sign in front
of Λ(β, ω).
Conversely, suppose the above statement 2 holds for
all inverse temperatures β ≥ 0, for some pure state |ωi
A
on a system A, and choice of sign in the exponent. The
exponent in Theorem 3.1 implies that we have
Λ(β, ψ
0
) = Λ(β, ψ
1
) ∓ Λ(β, ω), (195)
for all inverse temperatures β ∈ [0, ∞). In the event of
the minus sign, we write this as
Λ(β, ψ
0
) + Λ(β, ω) = Λ(β, |ψ
0
i
A
⊗ |ωi
A
)
= Λ(β, |ψ
1
i
S
⊗ |0i
A
), (196)
Accepted in Quantum 2019-10-24, click title to verify 37
where we can assume that A has a zero energy eigen-
state |0i
A
. However, since all the above cumulant gen-
erating functions have infinite radius of convergence and
are finite, the energy distributions of measuring |ψ
0
i
S
,
|ψ
1
i
S
and |ωi
A
are uniquely determined. Moreover the
distributions of a joint energy measurement of compos-
ite system SA in the state |ψ
0
i
S
⊗|ωi
A
and |ψ
1
i
S
⊗|0i
A
are identical. This implies that
ˆ
S
0
+
ˆ
A =
ˆ
S
1
, (197)
where
ˆ
S
0
,
ˆ
S
1
,
ˆ
A are the associated random variables for
measuring |ψ
0
i
S
, |ψ
1
i
S
, |ωi
A
respectively in their energy
eigenbases. By Theorem 2.5 this implies there exists a
non-trivial coherent work process
|ψ
1
i
ω
−→ |ψ
0
i, (198)
and so the states are coherently connected. In the event
of the plus sign in the exponent we instead write
Λ(β, ψ
0
) = Λ(β, ψ
1
) + Λ(β, ω), (199)
and then express the argument of the left-hand side us-
ing |ψ
0
i
S
⊗ |0i
A
as before, using the additivity of the
effective potential for the left-hand side. The same argu-
ment as before then implies the existence of a coherent
work process
|ψ
0
i
ω
−→ |ψ
1
i, (200)
and again the states are coherently connected, which
completes the proof.
B.5 Proof of semi-classical fluctuation theorem
Theorem B.9 (Semi-classical relation [25]). Let the
assumptions of Theorem 3.1 hold, and let S be a har-
monic oscillator with Hamiltonian H
S
= hν(a
†
a +
1
2
).
Then for two coherent states of the system |α
0
i and
|α
1
i, the following holds
P [α
1
|˜α
0
]
P [α
∗
0
|˜α
∗
1
]
= exp
−
∆F
kT
+
¯
W
B
hν
th
, (201)
where hν
th
= hH
S
i
γ
is the average energy of a Gibbs
state γ of a quantum harmonic oscillator, related to the
thermal de Broglie wavelength λ
dB
(T ) via
hν
th
=
h
2
mλ
dB
(T )
2
+
1
2
hν, (202)
and
¯
W
B
:= −
1
2
(W
S
+
˜
W
S
), W
S
= hH
S
i
α
1
− hH
S
i
α
0
,
˜
W
S
= hH
S
i
˜α
1
− hH
S
i
˜α
0
.
Proof. Consider a system with Hamiltonian H
S
=
hν(a
†
a +
1
2
). For simplicity, we take an arbitrary state
|α, ki ∈ C and assume k = 0, as the extension is simple
but provides little benefit. The effective potential for a
coherent state |αi:
Λ(β, α) =
βhν
2
+ |α|
2
(1 − e
−βhν
). (203)
Now consider the average energy of a thermally dis-
tributed quantum harmonic oscillator
hH
S
i
γ
=
P
n≥0
hν(n +
1
2
)e
−βhν(n+
1
2
)
P
m≥0
e
−βhν(m+
1
2
)
(204)
=
hν
2
1 + e
−βhν
1 − e
−βhν
. (205)
Let us define a thermal frequency hν
th
:= hH
S
i
γ
. Then
the effective potential change can be expressed as
Λ(β, α
1
) − Λ(β, α
0
) = (|α
1
|
2
− |α
0
|
2
)(1 − e
−βhν
)
(206)
=
ν
2ν
th
(|α
1
|
2
− |α
0
|
2
)(1 + e
−βhν
).
(207)
Since the Gibbs rescaled coherent state |˜αi =
|αe
−βhν/2
i, the change in effective potential can be ex-
pressed in terms of the unscaled and rescaled coherent
state energy transfers,
W
S
= hH
S
i
α
1
− hH
S
i
α
0
(208)
˜
W
S
= hH
S
i
˜α
1
− hH
S
i
˜α
0
, (209)
as
Λ(β, α
1
) − Λ(β, α
0
) =
1
hν
th
W
S
+
˜
W
S
2
. (210)
which gives the claimed result upon insertion into Theo-
rem 3.1. For the breakdown of hν
th
in terms of λ
dB
(T ),
we refer to [25].
For completeness, we can also show the form of the
fluctuation theorem using the coherent work represen-
tation. Consider an auxillary weight system A with
Hamiltonian H
A
= hνa
†
a. Once again, this system ex-
ists as a hypothetical process in which |ψ
0
i
ω
−→ |ψ
1
i
under a coherent work process. We note that
e
Λ(β,ω)
=
hα
1
|e
−βH
S
|α
1
i
hα
0
|e
−βH
S
|α
0
i
. (211)
From the semi-classical result Theorem 2.4, we know
that |α
0
i
S
ω
−→ |α
1
i
S
under a coherent work process if
and only if |α
1
| ≤ |α
0
|. Assuming this condition to be
true, then we can simply write down:
P [α
1
|˜α
0
]
P [α
∗
0
|˜α
∗
1
]
= e
|α|
2
(1−e
−βhν
)−β∆F
(212)
Accepted in Quantum 2019-10-24, click title to verify 38
where |α|
2
= |α
0
|
2
−|α
1
|
2
and ||α|i is the coherent work
output of the hypothetical process. Owing to the sim-
ple form of the effective potential for coherent states,
this is fairly trivial to write down. However generally,
the effective potential fails to obtain such a simplified
form, and simplifying the change in effective potential
Λ(β, ψ
1
) −Λ(β, ψ
0
) may be harder than finding the an-
alytic form for Λ(β, ω).
B.6 Macrosopic limit of the quantum fluctuation
relation
For simplicity, we choose to make the random variable
dimensionless. We have that hψ
i
|H
i
|ψ
i
i = µ, which is
independent of i. We define the operator X
i
:=
1
µ
H
i
− ,
which has mean zero in the state |ψ
i
i. Note that the
variance of X
i
is also dimensionless and given by
hX
2
i
i − hX
i
i
2
= hX
2
i
i (213)
= h
1
µ
2
H
2
i
−
2
µ
H
i
+ i (214)
=
1
µ
2
hH
2
i
i − 2 + 1 (215)
=
1
µ
2
(hH
2
i
i − µ
2
) (216)
Var(X
i
) =
σ
2
µ
2
≡ ˜σ
2
. (217)
The cumulant generating function of the total state
is
e
−Λ(β,ψ
n
)
= tr[e
−β
P
i
H
i
ψ
⊗n
] (218)
= tr[e
−βµ
P
i
(X
i
+ )
ψ
⊗n
] (219)
= e
−βnµ
tr[e
−βµ
P
i
X
i
ψ
⊗n
] (220)
= e
−βnµ
tr[e
−β
√
nµ
P
i
1
√
n
X
i
ψ
⊗n
] (221)
= e
−βnµ
e
−
˜
Λ(β
√
nµ,ψ
n
)
. (222)
Here
˜
Λ is the CGF corresponding to the dimensionless
mean zero observable
1
√
n
(X
1
+···+X
n
). Thus we have
that
Λ(β, ψ
n
) = βnµ +
˜
Λ(β
√
nµ, ψ
n
), (223)
for all β, n.
A subtlety comes in here – if one wants to take a limit
and obtain a Central Limit Theorem (CLT) statement
we need to scale β also in order to get a non-trivial
statement. This is a mathematical re-formulation so as
to maintain the correct scaling to obtain a non-trivial
statement the mean value of the sequence (in n) of ran-
dom variables.
Since β has units of reciprocal energy we can set
β =
β
0
√
nλ
, for some arbitrary energy scale λ, where β
0
is dimensionless and we count in energy units of
√
nλ
when we have n systems. It might look like the inverse
temperature is decreasing as we increase the number of
systems, but this is not what is intended by this – in
the description of the sequence of systems one scales
the units of energy as we increase n when we specify
the temperature. Note also that the first and second
cumulants scale differently with energy, and it is this
that makes the scaling trick needed to obtain a state-
ment containing both in the large n limit.
Expressed in these units we have
Λ(
β
0
√
nλ
, ψ
n
) = β
0
µ
0
√
n +
˜
Λ(β
0
µ
0
, ψ
n
), (224)
where µ
0
:=
µ
λ
is dimensionless, and β
0
does not vary
in n.
This holds for all n, and in the n → ∞ limit we have
that
lim
n→∞
Λ(
β
0
√
nλ
, ψ
n
) − β
0
µ
0
(ψ)
√
n
= −
1
2
˜σ
2
ψ
β
2
0
µ
0
(ψ)
2
,
(225)
and also
lim
n→∞
Λ(
β
0
√
nλ
, φ
n
) − β
0
µ
0
(φ)
√
n
= −
1
2
˜σ
2
φ
β
2
0
µ
0
(φ)
2
,
(226)
where we have included the state variables to distin-
guish the energies of the two states.
This implies that for any > 0 there is an integer
M such that for all n > M we have the following two
conditions hold
Λ(
β
0
√
nλ
, ψ
n
) − [β
0
µ
0
(ψ)
√
n −
1
2
σ
2
ψ
β
2
0
]
≤ (227)
Λ(
β
0
√
nλ
, φ
n
) − [β
0
µ
0
(φ)
√
n −
1
2
σ
2
φ
β
2
0
]
≤ , (228)
where we also used the definition of ˜σ
2
. We therefore
have that for any > 0 there is an M > 0 such that for
all n > M we have
|Λ(
β
0
√
nλ
, ψ
n
)−Λ(
β
0
√
nλ
, φ
n
)−[β
0
√
n∆µ
0
−
1
2
β
2
0
∆σ
2
0
]| ≤ 2.
(229)
Thus, for any > 0 there is an M such that for all
n > M we have
P [ψ
n
|φ
n
]
P [φ
∗
n
|ψ
∗
n
]
− e
−
√
n(β
0
∆f
0
+β
0
∆µ
0
)+
1
2
β
2
0
∆σ
2
0
≤ (230)
where ∆f
0
:=
∆F
nλ
is a dimensionless number. We could
put back in β, but only with the proviso that it is as-
sumed to scale with n in the way stated. This gives
P [ψ
n
|
˜
φ
n
]
P [φ
∗
n
|
˜
ψ
∗
n
]
− e
−n(β∆f+β∆µ−
1
2
β
2
∆σ
2
)
≤ (231)
Accepted in Quantum 2019-10-24, click title to verify 39
with the proviso that the scaling β =
β
0
√
nλ
is adopted.
This captures the informal statement that for the IID
case the states “become more like Gaussians” and thus
the fluctuation theorem behaves as
P [ψ
n
|
˜
φ
n
]
P [φ
∗
n
|
˜
ψ
∗
n
]
n→∞
∼ e
−n(β∆f+β∆µ−
1
2
β
2
∆σ
2
)
. (232)
Accepted in Quantum 2019-10-24, click title to verify 40