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
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, axi
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
n0
|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
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
=
a
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 , 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
nZ
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
j
q
j
|ji
S
(13)
|ωi
A
=
X
k
e
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
nZ
n|ni
A
hn|, if and
only the distribution (p
n
) over Z can be written as
p
n
=
X
jZ
r
j
(∆
j
q)
n
.
=
X
jZ
r
j
q
nj
(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
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
(φψ) = 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 : ((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 : ((x), x). Classical mechanical work occurs when
this distribution is sharp, µ(x) = δ(x w). However one may