Imperfect Thermalizations Allow for Optimal Thermody-
namic Processes
Elisa Bäumer
1
, Martí Perarnau-Llobet
2
, Philipp Kammerlander
1
, Henrik Wilming
1
, and Renato Ren-
ner
1
1
Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland
2
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, D-85748 Garching, Germany
June 21, 2019
Optimal (reversible) processes in ther-
modynamics can be modelled as step-by-
step processes, where the system is succes-
sively thermalized with respect to different
Hamiltonians by an external thermal bath.
However, in practice interactions between
system and thermal bath will take finite
time, and precise control of their inter-
action is usually out of reach. Motivated
by this observation, we consider finite-time
and uncontrolled operations between sys-
tem and bath, which result in thermaliza-
tions that are only partial in each step. We
show that optimal processes can still be
achieved for any non-trivial partial ther-
malizations at the price of increasing the
number of operations, and characterise the
corresponding tradeoff. We focus on work
extraction protocols and show our results
in two different frameworks: A collision
model and a model where the Hamilto-
nian of the working system is controlled
over time and the system can be brought
into contact with a heat bath. Our results
show that optimal processes are robust to
noise and imperfections in small quantum
systems, and can be achieved by a large set
of interactions between system and bath.
1 Introduction
Recent years have experienced a renewed in-
terest in understanding thermodynamics in the
quantum regime [1, 2]. A common assumption
in the field is that a system thermalizes to a
Gibbs state when put in contact with a ther-
mal bath. While this is perfectly reasonable, it
requires some implicit assumptions: For exam-
Elisa Bäumer: ebaeumer@itp.phys.ethz.ch
ple, that a sufficiently long time is available to
ensure full thermalization of the system [3]; or,
within approaches based on repeated interactions
between system-bath, that control is available on
the system-bath interaction [4]. The goal of this
article is to study how optimal thermodynamic
processes are modified when one does not have
access to such “perfect thermalizations". Our re-
sults thereby contribute to a recent research line
which aims at understanding the level of con-
trol required in thermodynamic protocols [5–11].
To ensure that our findings are not particular to
a specific model for thermodynamic transforma-
tions, we consider two different frameworks in this
article.
Before giving an overview of the specific frame-
works considered, let us briefly explain the main
insight of this article. We consider discrete pro-
cesses consisting of N small steps following a
smooth curve between two fixed points in param-
eter space, where the system interacts for a cer-
tain time t
th
with the thermal bath at each step.
Let us introduce a parameter α, such that for
α = 1, there is no interaction, and for α = 0 the
system thermalizes perfectly in each step. In gen-
eral, α = α(t
th
) is a function of the interaction
time with the bath in each step. For α = 0 the
second law of thermodynamics implies that for
generic initial states the extracted work satisfies
(see e.g. [12–15])
W = ∆F −
Γ
N
+ O
1
N
2
(1)
where ∆F is the difference of non-equilibrium free
energy between the initial and final state as de-
fined in (13) and Γ depends on the specific pro-
tocol and system of interest. Here we find
W = ∆F −
Γ + Λ
α
1 − α
1
N
+ O
1
N
2
, (2)
Accepted in Quantum 2019-06-10, click title to verify 1
arXiv:1712.07128v4 [quant-ph] 20 Jun 2019
where Λ is independent of α and N . This implies
that the scaling of the dissipation ∆F − W with
N remains the same even with imperfect thermal-
izations, and it shows how N can be rescaled in
order to keep the dissipation controlled for differ-
ent α’s.
The result (2) is derived in two different frame-
works. We start by considering a framework
where thermalization takes place due to repeated
interactions between the system (S) and elements
of a thermal bath (B), in the spirit of collision
models [4, 16–18]. We study work extraction from
a qubit system S when having access to B, which
consists of a collection of qubit systems in a ther-
mal state. This allows for a fully inclusive ap-
proach, where S, B and the work storage device
are described as quantum systems evolving uni-
tarily. In this case, we find that α in (2) behaves
as
α = cos
2
(gt
th
/~) (3)
where g is the coupling between S and each qubit
of B. For this model, we also compute explicitly
Γ and Λ, finding the simple relation
Γ =
Λ
2
, (4)
for a large class of protocols. This result is then
generalized to arbitrary qudit systems. Further-
more, we also study work fluctuations, showing
that for a large number of steps the final work
distribution using such uncontrolled interactions
converges to the same form as for the optimal
protocol [14, 19, 20].
In a second part of the article, we study the
same question in a different and widely used
framework, namely when B is a macroscopic bath
and work is extracted by changing the Hamilto-
nian of S over time. In this case, partial ther-
malizations naturally appear due to finite-time
contacts with B. However, we generalize from the
partial thermalizations presented above, by de-
scribing the action of B on S as essentially any
non-trivial quantum channel with the Gibbs state
as a unique fixed point. In this case, we also show
(2), where α now quantifies how much closer (in
trace norm) the state becomes to the correspond-
ing Gibbs state upon application of the channel.
Here, one may naturally expect (at least for suf-
ficiently large interaction times)
α(t
th
) ≈ e
−t
th
/τ
th
, (5)
where τ
th
is a thermalization time-scale.
Finally, we study the implications of (3) and
(5) in (2) when we fix the total time T = Nt
th
of the process and optimize t
th
to minimize the
dissipation W
dis
≡ ∆F − W (this definition en-
sures that W
dis
> 0 given our sign convention,
i.e., W > 0 when work is extracted from the sys-
tem). We find very different behaviours of W
dis
as a function of the thermalization time t
th
de-
pending on the model in consideration: while in
collision-like models short t
th
are not desirable,
the opposite holds for an exponential relaxation
as in (5), which is expected in situations where S
is constantly coupled to B.
The paper is structured as follows. In Section
2.1, after introducing the framework, we describe
the ideal protocol for work extraction and later
introduce a protocol which also works for imper-
fect operations. In Section 2.2, we show that the
noisy protocol, like the ideal protocol, allows for
maximal work extraction in the case where sys-
tem and bath consist of qubits, and calculate the
corresponding work fluctuations. These results
are later generalized in Section 2.2.3 to generic
qudit systems. In section 3, we discuss the sec-
ond framework. In Sec. 4 we discuss how W
dis
is
minimized for the different frameworks, and con-
clude in Section 5.
2 Collision Model
Collision models have been extensively used to
describe open quantum systems [4, 17, 18], in par-
ticular thermalization processes [16, 21–24] and
non-equilibrium scenarios in different thermody-
namic contexts [25–30]. The main assumption is
that when the system interacts with a bath, it
collides and thereby interacts with only one par-
ticle of the bath at a time. This means that we
can model a thermalization process as successive
interactions of the system with different parts of
the bath.
Recently isothermal processes have also been de-
scribed in the framework of collision models as
a concatenation of interactions between the sys-
tem S and elements of the bath B [14, 15, 31].
A difference between the latter and the standard
description of thermalization via collision models
[16] is that in the latter the system S interacts
with non-identical Gibbs states, in order to con-
struct an isothermal process. We follow this ap-
Accepted in Quantum 2019-06-10, click title to verify 2
proach in the first part of the paper. We do so in
an inclusive way, where all elements entering in
the process are described explicitly as quantum
systems, in the spirit of the resource theory of
thermodynamics [5, 32–34].
2.1 Framework
The model we consider for describing optimal
processes consists of a system S, which interacts
with a thermal bath B at fixed temperature T ,
and a work battery W, in which the extracted
work can be stored. Most of our results concern
the scenario in which S is a two-level system, a
qubit – later we will consider generic qudit sys-
tems. The Hamiltonian of S is hence taken to be
H
S
=
S
|1ih1|
S
. Similarly, the bath consists of a
collection of N + 1 qubits, with internal Hamil-
tonians H
(k)
B
= E
k
|1ih1|
B
(k)
, k = 0, ..., N. For
better readability, we will in the following de-
note any object X acting on B
(k)
by X
(k)
B
. For
W, we consider a weight that stores the work
as potential energy and take the Hamiltonian
H
W
= mgˆx, where ˆx is the continuous position
operator, ˆx =
R
dx x |xihx|, and for simplicity we
choose mg = 1 J/m. The total Hamiltonian on
SBW (which is a shorthand for S ⊗B ⊗W ) hence
reads
H = H
S
+
X
k
H
(k)
B
+ H
W
. (6)
The initial state is taken to be a product state,
ρ
(0)
= ρ
(0)
S
⊗
N
O
k=0
τ
(k)
B
!
⊗ ρ
(0)
W
. (7)
We take for S a diagonal state,
ρ
(0)
S
= (1 − p
0
) |0ih0|
S
+ p
0
|1ih1|
S
. (8)
The case of coherent states is discussed in [19,
35], where it is shown that the coherent content
of the free energy can be extracted by using an
external source of coherence, a process where B
does not intervene. Hence, here we only consider
diagonal states. The states of the bath τ
(k)
B
are
Gibbs states at temperature k
B
T = 1/β,
τ
(k)
B
=
e
−βH
(k)
B
Tr[e
−βH
(k)
B
]
. (9)
Since we are dealing with qubits, we can also
write
τ
(k)
B
= (1 − q
k
) |0ih0|
B
+ q
k
|1ih1|
B
, (10)
where we note the relation
E
k
= k
B
T ln ((1 − q
k
)/q
k
). Finally, the ini-
tial state for W is taken to be ρ
(0)
W
= |0ih0|
W
.
As we now argue, this choice is in fact irrelevant
for the extracted work.
Thermodynamic operations consist of unitary
operations U on SBW. The average extracted
work is then defined as the energy gained by W,
W := Tr
H
W
(ρ
0
W
− ρ
W
)
(11)
where ρ
W
= Tr
SB
[ρ] is the initial state, and ρ
0
W
=
Tr
SB
[UρU
†
] is the state after the transformation.
Importantly, we impose two conditions on U:
1. The total energy must be preserved by U,
more concretely [5],
[U, H] = 0. (12)
This condition ensures that all energy fluxes
are accounted for, as there cannot be energy
flowing out of SBW, and also justifies defini-
tion (11).
2. U must commute with the displacement op-
erator on W. If ρ
W
is diagonal, this implies
that the extracted work W is independent of
the initial state ρ
W
of W. Since we will only
consider diagonal states for ρ
W
, this condi-
tion ensures that the state of the weight can-
not be used as a source of free energy, so that
the extracted work comes solely from SB.
2.1.1 Maximal Work Extraction
The maximum extractable work of ρ
(0)
S
is given
by the change of (non-equilibrium) free energy
between initial and final state [36, 37]
∆F ≡ F (ρ
(0)
S
, H
S
) − F (τ
S
, H
S
), (13)
where F (ρ, H) = Tr(ρH) −T S(ρ), with S(ρ) the
von Neumann entropy S(ρ) = −Tr[ρ ln ρ] and
τ
S
:=
e
−βH
S
Tr[e
−βH
S
]
. (14)
Note that ∆F in (13) is the difference between
initial and final free energy (and not vice versa),
Accepted in Quantum 2019-06-10, click title to verify 3
Thermal Bath
ρ
(0)
W
ρ
( N )
W
ρ
(0)
S
ρ
(1)
S
ρ
(2)
S
ρ
( N − 1)
S
ρ
( N )
S
W
(1)
W
(2)
W
( N )
U
( N )
SBW
U
(1)
SBW
U
(2)
SBW
ρ
(1)
B
ρ
(2)
B
ρ
( N )
B
τ
( N )
B
τ
(2)
B
τ
(1)
B
τ
(0)
B
System
Work Battery
Figure 1: The setting described in this paper consists
of three subsystems: The system S initially in a non-
thermal state, a work battery W in which the work is
stored and a thermal bath B that provides thermal en-
ergy to be converted to work. They interact sequentially
by means of U
(k)
SBW
such that in each step another part of
the thermal bath is involved. The total extracted work
can then be computed as the sum of the extracted work
in each step, W
(k)
.
which is convenient for our sign convention in
which W > 0 when work is extracted from the
system.
Following Refs. [14, 31], we now briefly dis-
cuss an explicit protocol leading to (13). In or-
der to construct an optimal process, the excita-
tion probabilities {q
k
}
N
k=0
are chosen to start at
q
0
= p
0
and increase strictly monotic to q
N
=
e
−β
S
/(1 + e
−β
S
). Hence, τ
(0)
B
= ρ
(0)
S
and in the
case of maximal work extraction τ
(N)
B
= τ
S
. The
total process is described by the following unitary
operation on SBW,
U =
N
Y
k=1
U
(k)
SBW
(15)
where each U
(k)
SBW
acts on S, W and the k
th
bath
qubit, see Figure 1. Using the short hand no-
tation, |a, b, ci := |ai
S
⊗ |bi
B
⊗ |ci
W
and ω
k
:=
E
k
−
S
, U
(k)
SBW
takes the form
U
(k)
SBW
=
Z
x
dx
|0, 1, xih1, 0, x + ω
k
| + h.c.
+ |0, 0, xih0, 0, x| + |1, 1, xih1, 1, x|
(16)
Let us also write this expression
in matrix form using the basis
Figure 2: This picture shows the ideal case, where in
the k
th
step the unitary U
(k)
SBW
swaps the energy degen-
erate states |0, 1, xi
SBW
and |1, 0, x + ω
k
i
SBW
for all x,
where only the k
th
bath qubit is involved. The states
|0, 0, xi
SBW
and |1, 1, xi
SBW
are mapped to themselves.
{|0, 0, xi, |0, 1, xi, |1, 0, x + ω
k
i, |1, 1, xi}
U
(k)
SBW
=
Z
x
dx
1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
x
, (17)
where the subscript x indicates that the basis de-
pends on x. The action of U
(k)
SBW
is illustrated in
Figure 2. Note that
• U
(k)
SBW
acts upon a degenerate energy sub-
space, and hence condition 1., as given by
(12), is satisfied. It is straightforward to see
that condition 2. is satisfied as well.
• At the global level, the final state after ap-
plying U
(k)
SBW
becomes correlated. At the
level of the reduced states, it leads to a swap
between the states ρ
S
↔ τ
(k)
B
,
• As k increases, the state of S progressively
changes from ρ
(0)
S
to τ
S
, which mimics an
isothermal process.
Furthermore, the total extracted work W =
Tr
H
W
(Uρ
(0)
U
†
− ρ
(0)
)
from (15) satisfies,
W = ∆F − O(1/N), (18)
hence becoming maximal in the limit N → ∞
(see e.g. [14, 31]).
2.1.2 Imperfect Operations
Let us now describe the set of imperfect opera-
tions we consider. For clarity, we start by pre-
senting a simple model to describe lack of con-
trol on the operations U
(k)
SBW
. For that, note that
the unitary (16) can be implemented by turning
Accepted in Quantum 2019-06-10, click title to verify 4
on the interaction gσ
x
in the subspace |0, 1, xi,
|1, 0, x + ω
x
i for a time π~/2g. If the timing
is not precise, the corresponding unitary
˜
U
(k)
SBW
takes the form,
˜
U
(k)
SBW
=
Z
dx
1 0 0 0
0
√
α
k
i
√
1 − α
k
0
0 i
√
1 − α
k
√
α
k
0
0 0 0 1
,
(19)
where α
k
∈ [0, 1] quantifies the error in the k
th
step: for α
k
= 0 we apply the desired full rota-
tion and for α
k
= 1 we do nothing on the state.
After (19), the reduced states of S and B, given
by ρ
(k)
S/B
= Tr
BW/SW
˜
U
(k)
SBW
ρ
(k)
˜
U
(k)†
SBW
, take the
form
ρ
(k)
S
= α
k
ρ
(k−1)
S
+ (1 − α
k
)τ
(k)
B
ρ
(k)
B
= α
k
τ
(k)
B
+ (1 − α
k
)ρ
(k−1)
S
.
(20)
Hence, here the lack of control essentially leads
to a partial thermalization process.
In fact, there are more general processes that
lead to (20). In Appendix A, we consider ar-
bitrary thermal operations [5] on the relevant
degenerate subspaces |0, 1, xi, |1, 0, x + ω
x
i. By
this, we aim to model the situation where the rele-
vant subspace where the experimentalist is acting
upon is not thermally isolated, e.g., there is de-
coherence in the unitary operations being imple-
mented. We then show that all operations of this
form lead to the same model of noise (20), with
the α
k
’s being complex functions of the applied
specific thermal operation. Therefore, we con-
clude that (20) is generally valid when we restrict
ourselves to the operations described above which
map states from the degenerate energy subspaces
|0, 1, xi, |1, 0, x + ω
x
i to states on that same sub-
space while acting arbitrarily on an auxiliary sys-
tem. This is the model we will consider for the
rest of our work. As a remark, note that (20) nat-
urally allows for generalizations when considering
systems beyond qubits.
Before moving further, it is important to keep
in mind that the set of imperfect operations we
described is of course not the most general noisy,
or thermal, operation. The motivation behind
our choice is that, as we will now show, ther-
modynamic processes can still be constructed de-
spite the lack of control. Conversely, if we lift the
restrictions made above, it is not difficult to con-
vince oneself that other types of noise, e.g., ther-
mal noise acting on S or W only, are detrimental
for thermodynamic purposes and cannot be com-
pensated for. This is discussed in Appendix B.
2.2 Results
We now proceed to derive our main results, tak-
ing (20) as a starting point. We recall that the
α
k
’s characterize the noise level of the operation
in the k
th
step. One may think of the α
k
as being
equal to a fixed value α. For purpose of general-
ization, we treat the α
k
’s as independent random
variables with average value α. More precisely,
let the average of a function f over {α
j
}
N
j=1
be
hf({α
j
})i
{α
j
}
=
Z
...
Z
N
Y
i=1
dα
i
P
i
(α
i
)f({α
j
}),
(21)
where P
i
(α
i
) are some unknown probability dis-
tributions that satisfy
R
dα
i
P
i
(α
i
)α
i
= α for all
i. Since the α
j
’s are independent, in particular
h
k
Y
j=1
α
j
i
{α
j
}
= α
k
. (22)
It is then straightforward to obtain from (20) that
hρ
(k)
S
i
{α
j
}
k
j=1
= αhρ
(k−1)
S
i
{α
j
}
k−1
j=1
+ (1 − α)τ
(k)
B
hρ
(k)
B
i
{α
j
}
k
j=1
= ατ
(k)
B
+ (1 − α)hρ
(k−1)
S
i
{α
j
}
k−1
j=1
.
(23)
where we used that τ
(k)
B
and ρ
(k−1)
S
are indepen-
dent of α
k
. That is, when considering the aver-
age over {α
j
}
N
j=1
, we can just replace α
k
by its
average α. Given that all quantities of interest
(average work, fluctuations) are linear functions
of ρ
(k)
SB
, the same rule follows for their average. In
what follows, we hence replace the model (20) by
ρ
(k)
S
= αρ
(k−1)
S
+ (1 − α)τ
(k)
B
ρ
(k)
B
= ατ
(k)
B
+ (1 − α)ρ
(k−1)
S
,
(24)
where the corresponding results should be under-
stood as the average over uncontrolled parame-
ters {α
j
}.
Now, starting from the recursive definition of
ρ
(k)
S
in (24), we can determine its explicit form as
ρ
(k)
S
= (1 − α)
k
X
i=1
α
k−i
τ
(i)
B
+ α
k
ρ
(0)
S
. (25)
Accepted in Quantum 2019-06-10, click title to verify 5
Furthermore, all τ
(k)
B
are diagonal, and as we as-
sumed that ρ
(0)
S
is also diagonal, so is ρ
(k)
S
, and
hence the same relation holds for the excitation
probabilities p
k
, with ρ
(k)
S
= (1 − p
k
)|0ih0|
S
+
p
k
|1ih1|
S
,
p
k
= (1 − α)
k
X
i=1
α
k−i
q
i
+ α
k
p
0
. (26)
2.2.1 Average Work Extraction
We first show that it is possible to obtain W →
∆F for any α < 1 in the limit of N → ∞. For
clarity of the presentation, we find it convenient
to present here the case where ρ
S
= |0ih0|
S
and
S
= 0 and discuss the general case for arbitrary
initial states of S and higher dimensional systems
in Section 2.2.3 and Appendix D. Note that the
case ρ
S
= |0ih0|
S
and
S
= 0 corresponds to well-
known information to energy conversion, where
∆F = k
B
T ln 2 work can be extracted from one
bit of information. The opposite direction, reset-
ting one bit of information while investing work
∆F = k
B
T ln 2, the so-called erasure, works anal-
ogously since we are looking at reversible pro-
cesses.
To compute the average energy gain of the
weight W, we first notice that the work extracted
in step k is
W
(k)
= Tr
h
ρ
(k)
W
− ρ
(k−1)
W
H
W
i
= Tr
h
H
(k)
B
τ
(k)
B
−ρ
(k)
B
−H
S
ρ
(k)
S
−ρ
(k−1)
S
i
,
(27)
which follows from energy conservation. Thus,
the total average work extracted after N steps is
the sum
W =
N
X
k=1
Tr
h
H
(k)
B
τ
(k)
B
−ρ
(k)
B
−H
S
ρ
(k)
S
−ρ
(k−1)
S
i
= (1−α)
N
X
k=1
Tr
h
H
(k)
B
−H
S
τ
(k)
B
−ρ
(k−1)
S
i
.
(28)
For the qubit case, we have
W = (1 −α)
N
X
k=1
(E
k
−
S
)(q
k
− p
k−1
). (29)
Let us now use our choice of initial conditions
S
= 0 and p
0
= 0, i.e., initially ρ
S
= |0ih0|
S
.
The distribution of bath qubits is taken to be
q
k
=
k
2N
⇒ ∆q
k
:= q
k
− q
k−1
=
1
2N
∀k,
(30)
so that τ
(0)
B
= ρ
S
and τ
(N)
B
= τ
S
. Note that this
specific distribution is chosen here for simplicity
and to give an explicit upper bound – we later
discuss more general settings. Using (26), we get
an explicit form for the excitation probabilities of
ρ
(k)
S
,
p
k
= (1 − α)
k
X
i=1
α
k−i
i
2N
= q
k
−
α(1 − α
k
)
1 − α
∆q
k
,
(31)
where we can already see that they only differ by
O(1/N) compared to the excitation probabilities
q
k
we would get using error-free unitaries. That
is, for large N our state ρ
S
is at every step close
to the corresponding thermal state, and hence it
follows approximately the isothermal trajectory
of the state transformation. We now show that
for large N this also happens at an optimal work
cost, i.e., that the work extracted becomes arbi-
trary close to the one in the error-free case. Using
(31), the average extracted work simplifies to
W = (1 −α)
N
X
k=1
E
k
(q
k
− p
k−1
)
= (1 − α)
N
X
k=1
E
k
q
k
− q
k−1
+
α − α
k
1 − α
∆q
k
!
=
N
X
k=1
E
k
∆q
k
(1 − α
k
)
= ∆F −
Γ
N
− ε − O
1
N
2
, (32)
where ∆F −
Γ
N
− O
1
N
2
with Γ = O(E
max
) is
the maximum average work that can be extracted
with N steps in the error-free protocol, i.e., with
α = 0. The loss due to the noise characterized
by α is described by ε =
α
1−α
Λ
N
+ O
1
N
2
and
corresponds to the difference between the work
extracted in the optimal protocol (α = 0) and
the one with faulty unitaries (α 6= 0). It can be
Accepted in Quantum 2019-06-10, click title to verify 6
Figure 3: Main figure: The loss ε due to noise as a
function of the total number of steps N for α = 1/2,
p
k
= k/2N , k
B
T ln 2 = 1. The green line corresponds
to the exact loss ε computed numerically with N steps,
whereas the orange (dashed) line corresponds to the up-
per bound (33). Inset figure: Errors as a function of
α for N = 1000, p
k
= k/2N, k
B
T ln 2 = 1. The green
line corresponds to ε, while the purple (dashed) one is
the total dissipation W
dis
.
upper bounded by
ε =
1
2N
N
X
k=1
E
k
α
k
=
k
B
T
2N
N
X
k=1
ln
2N − k
k
α
k
<
k
B
T
2N
N
X
k=1
ln (2N) α
k
<
α
1 − α
k
B
T ln(2N)
2N
,
(33)
which implies Λ = O(E
max
). In the limit N →
∞, we get ε → 0 and hence W → ∆F for any
α < 1. In Figure 3 we plot the numerical value
of loss ε and the upper bound (33). We observe
that the deviation is small and the bound almost
tight. We also plot ε together with the total dis-
sipation W
dis
= ∆F − W for finitely many steps
as a function of α.
2.2.2 Fluctuations
The fluctuations of work are captured by the
probability distribution P (w) = hw|ρ
W
|wi,
where ρ
W
is the final state of W, ρ
W
=
Tr
SB
Uρ
(0)
U
†
and ρ
(0)
is defined in (7). That is,
the work fluctuations are mapped to energy fluc-
tuations of the weight, which initially starts in a
well-defined energy
1
.
1
Note that, since we consider non-coherent states of S,
the same fluctuations would be obtained by using the more
standard two-energy-measurement scheme on SB together
[38]. The two approaches are equivalent here.
In Appendix C, a lengthy calculation shows
that for ρ
S
= |0ih0|
S
,
S
= 0, α
k
≤ α < 1 ∀k
and distribution (30), the variance of P (w) de-
creases as O(ln(N)/N), and hence disappears in
the limit N → ∞. This result implies that an
amount k
B
T ln 2 of work can be extracted with
certainty from one bit of information, even when
dealing with partial thermalizations as in (24).
The work probability distribution of this process
is illustrated in Figure 4.
Given this result, the case of ρ
S
= (1 −
p
0
)|0ih0|
S
+ p
0
|1ih1|
S
and
S
6= 0 can be easily
understood qualitatively. First of all, recall that
for general ρ
S
the optimal protocol is such that
H
(0)
B
satisfies τ
(0)
B
= ρ
S
and H
(N)
B
= H
S
. Sec-
ondly, since P (w) is linear in ρ
S
, we can write
P (w) = (1 − p
0
)P
0
(w) + p
0
P
1
(w) where P
0,1
(w)
is the probability distribution starting the proto-
col with |0ih0|
S
and |1ih1|
S
, respectively. From
our previous result, it follows that P
0
(w) tends
to a peaked distribution, which is now centered
at w
0
= F (ρ
S
, H
(0)
B
) − F (τ
S
, H
S
) (recall that we
consider the protocol for maximal work extrac-
tion from ρ
S
). Similarly, P
1
(w) tends also to
a peaked distribution centered at w
1
=
S
−
E
0
+ F (ρ
S
, H
(0)
B
) − F (τ
S
, H
S
). Putting every-
thing together, and noting that w
0
= k
B
T ln((1−
p
0
)/(1 − p
(eq)
S
)) and w
1
= k
B
T ln(p
0
/p
(eq)
S
), we
can write the work probability distribution in the
limit N → ∞ as
P (w)
N→∞
−→ p
0
δ
w − k
B
T ln
p
0
p
(eq)
S
!
+ (1 − p
0
)δ
w − k
B
T ln
1 − p
0
1 − p
(eq)
S
!
(34)
where δ(w − x) is the Delta function. Crucially,
this distribution is independent of α, and hence
we recover previous results where perfect ther-
malization operations (α = 0) were assumed
[14, 20].
2.2.3 Generalizations and Discussion
Although our results have been derived for qubit
systems, they can be naturally extended to qu-
dits. For that we consider the model of partial
thermalizations (24). This implies for ρ
(k)
S
in (25)
Accepted in Quantum 2019-06-10, click title to verify 7
N = 100, hW i = 0.939, σ = 0.34 N = 200, hW i = 0.967,
σ = 0.27
N = 500, hW i = 0.985, σ = 0.18 N = 1000, hW i = 0.993,
σ = 0.14
Figure 4: Histograms of the extracted work for ρ
S
=
|0ih0|
S
,
S
= 0, α = 1/2 and with k
B
T ln 2 = 1. The
fluctuations were sampled over 10000 runs to obtain
each plot. In agreement with our results, the average
work tends to k
B
T ln 2 and the fluctuations decrease
with increasing N .
that (assuming ρ
(0)
S
= τ
(0)
B
)
ρ
(k)
S
= (1 − α)
k
X
j=0
α
j
τ
(k−j)
B
+ α
k
τ
(0)
B
= τ
(k)
B
−(1−α)
k
X
j=0
α
j
(τ
(k)
B
−τ
(k−j)
B
)+O(α
k
).
(35)
Let us define the continuously differentiable fam-
ily of Hamiltonians H(s) for s ∈ [0, 1], such that
we can draw H
(k)
B
:= H(k/N), and similarly the
family of states τ(s) for s ∈ [0, 1], such that
τ
(j)
B
= τ(j/N) and ˙τ(s) ≡
d
ds
τ(s). We can then
expand τ
(k)
B
− τ
(k−j)
B
over 1/N in (35) to obtain,
ρ
(k)
S
= τ
(k)
B
− (1 − α)
k
X
j=0
j
N
α
j
˙τ(k/N) + O(1/N
2
)
≈ τ
(k)
B
−
α
N(1 −α)
˙τ(k/N) + O(1/N
2
)
(36)
Hence, we see how the factor α/(1 − α)N in (2)
naturally arises under imperfect thermalizations.
It is a consequence of the state ρ
(k)
S
lagging behind
the thermal state τ
(k)
B
.
In general, computing the average extracted
work is more involved as it depends both on
τ
(k)
B
and the Hamiltonians H
(k)
B
. This is done
in Appendix D by using the continuous expan-
sion described above. In particular, it is con-
venient to introduce the continuous limit of the
non-equilibrium free energy function,
F (s) := F (τ(s), H
S
) (37)
with s ∈ [0, 1]. In Appendix D, we show (in the
case of a full-rank initial state) that W
dis
≡ ∆F −
W satisfies at order 1/N
W
dis
= −
1
2N
1 +
2α
1 − α
Z
1
0
ds Tr(
˙
H(s) ˙τ(s))
+
α
N(1 −α)
˙
F (1) −2
˙
F (0
+
)
. (38)
The minus sign in the first term arises, since
Z
1
0
ds Tr(
˙
H(s) ˙τ(s)) ≤ 0 (39)
for any trajectory of Hamiltonians H(s), which,
for example, can be seen using the techniques of
Refs. [39, 40]. The second contribution to the dis-
sipation takes place only at the extremes of the
trajectory, when S is put in contact with B for the
first/last time. In particular, it is worth notic-
ing that if τ (1) = e
−βH
S
/Z
S
, then we have that
˙
F (1) = 0 because F (s) is minimized when the
state is thermal. By taking an appropriate trajec-
tory, we can also guarantee
˙
F (0) = 0. The more
interesting contribution is the first term, which
depends on the whole trajectory H(s). Crucially,
the dependence on α of the error is independent
of the trajectory and is always given by an overall
factor α/(1 − α). An important consequence of
(38) is that protocols that minimize dissipation
for α = 0 (see [40] for examples) will also mini-
mize dissipation for imperfect thermalizations.
In the above discussion we assumed ρ
(0)
S
= τ
(0)
B
.
This is always possible to fulfill if ρ
(0)
S
has full
rank. If this condition is not fulfilled, one ob-
tains an additional error term in the dissipated
work, which is detailled in Appendix D. Let us
here discuss the limit of optimal protocols. For
any δ > 0 it is possible to choose H(s) such that
kτ
(0)
B
−ρ
(0)
S
k
1
≤ δ 1. In this case, one finds that
the additional error term scales as O(δ log(1/δ))
and similarly
−
Z
1
0
ds Tr(
˙
H(s) ˙τ(s)) = O(log(1/δ)) . (40)
If we now choose δ = O (1/N) we thus recover
the scaling behaviour
W
dis
= O
log N
N
, (41)
Accepted in Quantum 2019-06-10, click title to verify 8
in accordance with our explicit calculation for the
qubit example.
Let us also briefly comment on our assumption
that the α
k
in (20) are random variables with
identical expectation value α, which is indepen-
dent of N. This assumption is in fact not cru-
cial for the main results –although it is very con-
venient for the computations–, the reason being
that our result holds for any α < 1. Note that
we can always decrease the individual values α
k
without compromising the work extraction pro-
tocol. As a consequence, even if the expectation
value of the α
k
would change with k, one can
guarantee convergence to an optimal thermody-
namic process as long as α
k
≤ α < 1, ∀k. Fur-
thermore, even if α = 1 −C/N for some constant
C > 0, one still obtains the optimal expected
work in the limit N → ∞ (also see section 3).
Similarly, we strongly expect that our results also
hold for typical realizations of the α
k
, provided
that each α
k
is chosen independently with expec-
tation values bounded away from 1. Indeed, we
expect that our results still hold even if a diverg-
ing number of α
k
is equal to 1 provided that these
α
k
are a vanishing fraction of all the α
k
in the
limit N → ∞.
It is worth pointing out that even when dealing
with qubits, there exist uncontrolled interactions
that are thermodynamically detrimental and can-
not be compensated for. For example, letting S
simply thermalize with B without raising the en-
ergy in W can only decrease the extractable work.
Similarly, if B and W exchange energy, W may
lose energy to the bath. In order to build an op-
timal thermodynamic process, it is needed that B
continuously thermalizes S (and not W) while W
is extracting work from it. A more detailed de-
scription of interactions that cannot be tolerated
can be found in Appendix B.
Finally, it is important to realize that, although
we have focused our attention on optimal work
extraction, our results also allow for optimal state
transformations in the other direction, i.e., they
can be used for converting a thermal state to a
given final state at optimal work cost. More gen-
erally, they apply to the interconversion of any
two states at minimal work cost. Furthermore,
we note that although our results have been de-
rived for a particular model within the resource
theory of thermodynamics, we expect that simi-
lar results can be obtained in other frameworks,
such as [13, 15, 20, 41, 42]. One such example is
illustrated in the next section.
3 Time-dependent Hamiltonians and
Thermalization Maps
The collision model presented above has the ad-
vantage of being very explicit, allowing us to de-
rive precise results both about average work ex-
traction and fluctuations. However, it has the
drawback of requiring a fine-tuned bath. We
therefore now show similar results in a different
framework, in which work is extracted by chang-
ing the Hamiltonian of S over time and contacts
with the bath are described by a thermalizing
channel, which can essentially be any channel
that brings the system closer to the thermal state.
The purpose of this section is two-fold (i) this
model is experimentally more relevant and realis-
tic, as no control over B is assumed (in the spirit
of open quantum systems), and (ii) in this model
we are able to generalize our results to arbitrary
interactions with the bath that bring the system
closer to the thermal state, with partial thermal-
izations as in the previous section being a specific
instance.
3.1 Framework
In this framework, a protocol consists of a choice
of a trajectory of Hamiltonians H(t) and a choice
of N times {t
i
}
N
i=1
at which the system S is
brought in contact with the bath B (the contin-
uum limit, where (t
i
− t
i+1
) → 0 so that S is
essentially always in contact with B, will be dis-
cussed below). For simplicity, in the following we
only consider cyclic protocols, for which the ini-
tial Hamiltonian coincides with the final Hamilto-
nian and denote by t
0
the time at which the proto-
col starts. We further assume that while the sys-
tem is in contact with the bath, the Hamiltonian
is not changed and the system thus (partially)
thermalizes to the Hamiltonian H
(i)
= H(t
i
).
The contact to the bath at time t
i
is effectively
described by a quantum channel G
(i)
and results
in a state
σ
(i)
= G
(i)
(ρ
(i)
), (42)
where ρ
(i)
is the state of S right before the thermal
contact. Inbetween the contacts to the bath, the
system evolves according to the time-dependent
Accepted in Quantum 2019-06-10, click title to verify 9
Hamiltonian H(t) from time t
i
to time t
i+1
, repre-
sented by the unitary U
t
i+1
,t
i
. The states ρ
(i)
and
σ
(i−1)
are therefore related by the time-dependent
evolution under H(t) as
ρ
(i)
= U
t
i
,t
i−1
σ
(i−1)
U
†
t
i
,t
i−1
. (43)
This time-evolution is associated with a work-
gain
W
(i)
= Tr
H
(i−1)
σ
(i−1)
− H
(i)
ρ
(i)
, (44)
with σ
(0)
= ρ
(0)
being the initial state. Given
an initial state ρ
(0)
and trajectory of Hamiltoni-
ans H(t), we denote the total work in an N-step
protocol by
W (ρ
(0)
, H(t)) =
N
X
i=1
W
(i)
. (45)
Let us finally discuss the assumption that we
make on the thermalizing maps G
(i)
. Clearly, the
thermalizing maps should have the thermal state
τ
(i)
of the Hamiltonian H
(i)
as a fixed-point. It
then follows in general from the monotonicity of
the trace-distance under quantum channels that
kG
(i)
(ρ) − τ
(i)
k
1
= kG
(i)
(ρ) − G
(i)
(τ
(i)
)k
1
≤ kρ − τ
(i)
k
1
. (46)
In the following we want to model that the ther-
malizing maps have some non-trivial effect on any
non-thermal initial state. We will therefore as-
sume that there exists a fixed number α < 1 such
that
kG
(i)
(ρ) − τ
(i)
k
1
≤ αkρ − τ
(i)
k
1
. (47)
This expresses formally the physical requirement
that the interaction with the bath brings the sys-
tem closer to the thermal state by at least some,
even arbitrarily small, amount. In the following
we call α the thermalization parameter of the pro-
tocol.
Before stating our results, let us emphasize
that even though we model each thermalization
process using a single map G
(i)
, this does not
mean that the physical time t
(i)
th
the thermaliza-
tion step takes is zero. It merely means that we
assume that during the contact with the heat
bath the Hamiltonian is constant and the over-
all effect of the thermalization is effectively mod-
elled by the quantum channel G
(i)
. In partic-
ular, our set-up includes the commonly studied
case where the changes of the Hamiltonians be-
tween the thermalization steps is instantaneous,
so-called quenches, and the total time of the pro-
tocol is simply given by T =
P
i
t
(i)
th
.
3.2 Results
Having introduced the framework, we can now
easily state our result. Let us mention before,
though, that the maximum amount of work that
can be extracted using a cyclic protocol of the
above form is bounded as [6, 20]
W (ρ
(0)
, H(t)) ≤ F (ρ
(0)
, H
(0)
) − F (τ
(0)
, H
(0)
)
= ∆F. (48)
Furthermore, it has been known that for perfectly
thermalizing maps (α = 0), this bound can be
achieved to arbitrary accuracy [13, 20, 42, 43].
The optimal protocol consists of a first rapid
change of the Hamiltonian to a Hamiltonian
˜
H
(0)
such that (to arbitrary accuracy)
ρ
(0)
=
e
−β
˜
H
(0)
Z
, (49)
followed by an isothermal process back to the
initial Hamiltonian, meaning a process where
t
i
− t
i−1
= 1/N and N → ∞, so that the system
stays in the thermal state of the current Hamil-
tonian throughout the protocol. The first quench
yields a work-gain
W
quench
= Tr
ρ
(0)
H
(0)
−
˜
H
(0)
= F (ρ
(0)
, H
(0)
) − F (ρ
(0)
,
˜
H
(0)
). (50)
The optimality of the above protocol follows from
the fact that an optimal isothermal process be-
tween two Hamiltonians H(0) and H(1) with
thermal states τ (0) and τ (1) has work-gain
∆F
iso
:= F (τ (0), H(0)) − F (τ (1), H(1)). (51)
Therefore, the total work-gain in the optimal pro-
tocol is given by (using H(0) =
˜
H
(0)
, H(1) =
H
(0)
):
W
optimal
= W
quench
+ ∆F
iso
= ∆F. (52)
3.2.1 Main Result
Our following result shows that the protocol dis-
cussed above remains optimal for any thermaliz-
ing parameter α < 1, since isothermal processes
Accepted in Quantum 2019-06-10, click title to verify 10
remain optimal in the limit N → ∞: In the above
formulation, consider a protocol with a smooth
curve of Hamiltonians H(t), thermalization pa-
rameter α < 1 and initial state ρ
(0)
. Choose the
time-steps as t
i
= i/N. Then the dissipated work
can be expressed (to leading order in 1/N) as
W
dis
= ∆F
iso
− W
iso
(ρ
(0)
, H(t))
= −
1
2N
Z
1
0
ds Tr
˙τ(s)
˙
H(s)
+
α
1 − α
Λ
N
+
K
N
.
(53)
The dependence on the initial state ρ
(0)
only ap-
pears in the constant K. The (relatively involved)
proof of (53) is given in Appendix E. Importantly,
we have K = 0 in the case where we assume in-
stantaneous quenches instead of continuous uni-
tary time-evolution between the thermalization
steps. Furthermore, a calculation similar to that
provided in Appendix D, shows that for the case
of instantenous quenches and partial thermaliza-
tions as in (20), one recovers the same result as
(38) with
˙
F (1) = 0 and
˙
F (0
+
) = 0, making evi-
dent the connection between both frameworks.
3.2.2 Fluctuations
So far, we have only discussed average work
extraction. This has a good reason: In the
current framework it is in general not possible
to associate a classical random variable that
measures the fluctuating work to the process
due to the coherences that can be created by the
unitaries U
t
i
,t
i−1
and the channels G
(i)
[44]. Let
us emphasize, however, that in the limit N → ∞
in which the extracted work becomes optimal,
these coherences become arbitrarily small. It
then follows that we can associate a distribution
of fluctuating work to the limiting process, which
coincides with the limiting process when perfect
thermalization and quenches of the Hamilto-
nian are assumed. It follows that this limiting
distribution is universal, i.e., independent of
the trajectory H(t) and coincides with the one
obtained from the collision model, see e.g. [20].
4 Optimal Thermalization Times to
Minimize Dissipation
In this section, we study how the interaction time
with the bath in each step affects the total dissi-
pated work when we fix the total time T of the
protocol. We focus on (38) with
˙
F (1) =
˙
F (0
+
) =
0, i.e. the case where the dissipation satisfies,
W
dis
= 2 Γ
1
2
+
α(t
th
)
1 − α(t
th
)
t
th
T
, (54)
provided that N = T /t
th
is sufficiently large, and
with Γ = −
1
2
R
1
0
ds Tr
h
˙τ(s)
˙
H(s)
i
. Since the dissi-
pation is proportional to Γ and T
−1
, we can focus
on the function,
G(t
th
) =
1
2
+
α(t
th
)
1 − α(t
th
)
t
th
. (55)
Minimizing G(t
th
) will provide the optimal ther-
malization time t
th
that minimizes dissipation. In
fact, our goal is to study how G(t
th
) behaves de-
pending on the form of equilibration between sys-
tem and bath, which is encapsulated in α(t
th
).
For the collision model described in Sec. 2, we
consider (3), i.e.,
α(t
th
) = cos
2
(gt
th
/~), (56)
which arises due to the unitary evolution between
S and each qubit of B for each step of the proto-
col. Focusing on (56), the optimal t
th
can be
numerically found to satisfy gt
th
/~ ≈ 1.4, for
which α ≈ 0.03, very close to the value of per-
fect thermalization. Note that the optimal α is
independent of g/~. In Fig. 5, we plot G(t
th
) as
a function of t
th
for the model (56) with g~ = 1.
Clearly, we note that G(t
th
) leads to low dissi-
pation when taking t
th
∈ (1, 1.5), whereas it di-
verges in the limit t
th
→ 0. To understand this
behaviour, note that for short times,
α(t
th
) ≈ 1 − O(t
2
th
), (57)
and hence W
dis
∝ t
−1
th
. This behaviour is ex-
pected in circumstances where S interacts with
an independent element of B at each step, as it
naturally happens in collision models. Hence we
conclude that in such models it is desirable to
avoid too short thermalization times to ensure
low dissipation in the process.
We now move to a thermalization model of the
form
α(t
th
) ≈ e
−t
th
/τ
th
, (58)
where τ
th
is the thermalization time-scale. This
behaviour is expected for many physical systems
Accepted in Quantum 2019-06-10, click title to verify 11
Figure 5: G(t
th
) as a function of t
th
for the relaxation
model (56). The purple point indicates the optimal ther-
malization time t
th,opt
.
Figure 6: G(t
th
) as a function of t
th
for the relaxation
model (58).
(e.g., in open systems), at least for sufficiently
large times t
th
. For this model of thermalization,
we plot G(t
th
) in Fig. 6, which shows that G(t
th
)
increases monotonically with t
th
. To minimize
dissipation, it is hence desirable to take small
t
th
, as long as (58) remains valid. Keeping that
in mind, taking small but finite t
th
in (54), we
find that at leading order W
dis
∝ T
−1
. That
is, the dissipation becomes inversely propor-
tional to the total time in the continuous limit
t
th
→ 0, a behaviour that is often assumed
on phenomenological grounds [45], or obtained
by means of Markovian master equations [46, 47].
Summarizing, we have observed markedly dif-
ferent behaviours of W
dis
as a function of the
thermalization time t
th
in each thermalization
step. While in collision-like models short t
th
are
not desirable, the opposite holds for an expo-
nential relaxation as in (58), which is expected
in situations where S is constantly coupled to
B. These considerations make a connection be-
tween isothermal processes in discrete processes,
where perfect thermalization is taken for granted
[6, 13, 20, 39, 42, 48], and in continuous ones
[46, 47].
5 Conclusion
In this work we have considered thermodynamic
processes in which contacts between S and B
correspond to imperfect thermalizations, which
capture large classes of possible error models or
finite-time equilibration processes. The degree of
thermalization is quantified by a parameter α,
such that for α = 0 we recover the standard case
of full thermalization and for α = 1 there is no in-
teraction between S and B (and W in the collision
model). We have shown that optimal processes
can be constructed for any α < 1 with protocols
consisting of sufficiently many steps, and char-
acterized the corresponding tradeoff between the
number of extra steps and α. This result has
been obtained in two frameworks: one based on
collision models and one where the energy lev-
els of S and interactions with B are described
through quantum channels that provide an effec-
tive model for thermalization. Finally, we have
discussed how optimizing the thermalisation time
t
th
in each thermalization step leads to notably
different results for each of the considered frame-
works.
6 Acknowledgments
We thank Guillem Perarnau, Matteo Lostaglio,
Kamil Korzekwa, and Chris Perry for inter-
esting discussions and useful comments on the
manuscript. We acknowledge contributions
from the Swiss National Science Foundation
via the NCCR QSIT as well as project No.
200020_165843. M.P.-L. acknowledges support
from the Alexander von Humboldt Foundation.
All authors are grateful for support from the EU
COST Action MP1209 on Thermodynamics in
the Quantum Regime.
Accepted in Quantum 2019-06-10, click title to verify 12
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Accepted in Quantum 2019-06-10, click title to verify 16
Appendices
A Models of Noise
In the k
th
step of the protocol, the joint state of S, W and the k
th
bath qubit can be written as,
ρ
(k)
0
=
Z
dx p
(k)
W
(x)
φ
(deg)
X
+ φ
(non-deg)
X
,
with φ
(deg)
X
= r
k
|0, 1, xih0, 1, x| + s
k
|1, 0, x + ω
k
ih1, 0, x + ω
k
|,
φ
(non-deg)
X
=
r
k
s
k
q
k
p
k−1
|0, 0, xih0, 0, x| +
r
k
s
k
(1 − q
k
)(1 − p
k−1
)
|1, 1, x + ω
k
ih1, 1, x + ω
k
|,
(59)
where r
k
:= q
k
(1−p
k−1
), s
k
:= (1−q
k
)p
k−1
, and note that r
k
> s
k
. p
(k)
W
(x) is a probability distribution
that describes the population of the different work storage levels |xihx|
W
after the k
th
step. The non-
normalized states φ
(deg)
X
represent the subspaces of relevant degenerate states where we act upon for a
given x, and φ
(non-deg)
X
are the rest of states. Now we consider thermal operations acting only on each
φ
(deg)
X
,
ψ
(deg)
X
= Tr
A
V
k
φ
(deg)
X
⊗ τ
A
V
†
k
(60)
where τ
A
= e
−βH
A
/ Tr(e
−βH
A
) is an arbitrary thermal state. From the second law of thermodynamics,
we know that the free energy of φ
(deg)
X
can only decrease which here implies that the entropy of each
φ
(deg)
X
must increase –recall that φ
deg
X
has a trivial Hamiltonian as it lives in an energy degenerate
subspace. Hence we can write without loss of generality
ψ
(deg)
X
=
(1 − α
k
(x))s
k
+ α
k
(x)r
k
z
k
(x)
z
∗
k
(x) (1 − α
k
(x))r
k
+ α
k
(x)s
k
!
, (61)
where z
k
(x) and α
k
(x) are in general functions of the specific τ
A
and V
k
, and α
k
(x) ∈ [0, 1]. On the
other hand, the total state after the thermal operation reads
ρ
(k)
=
Z
dx p
(k)
W
(x)
ψ
(deg)
X
+ φ
(non-deg)
X
. (62)
For future convenience, let us also define
α
k
:=
Z
dxp
(k)
W
(x)α
k
(x). (63)
Note that, for a specific choice of V , where z
k
(x) =
p
α
k
(x)(1 − α
k
(x))(r
k
+ s
k
), we would get exactly
the same result as after the application of (19). Another "extreme" case would be for z
k
(x) = 0, where
the resulting state on SBW would be diagonal, corresponding to the resulting state of the map
ρ −→ (1 − α
k
)U
k
ρU
†
k
+ α
k
ρ. (64)
That is: with a given probability (1 − α
k
), we perform the desired operation, and with an error rate
α
k
we do nothing.
At the level of S, i.e., after tracing out also the bath qubit B and W, the coherence term z
k
(x)
Accepted in Quantum 2019-06-10, click title to verify 17
becomes irrelevant, and analogous to (20) the final state of S is given by
ρ
(k)
S
= Tr
BW
h
ρ
(k)
i
=
Z
dx p
(k)
W
(x)
(1 − α
k
(x))s
k
+ α
k
(x)r
k
+ (1 − p
k−1
)(1 − q
k
)
|0ih0|
S
+
(1 − α
k
(x))r
k
+ α
k
(x)s
k
+ p
k−1
q
k
|1ih1|
S
=
(1 − α
k
)(1 − q
k
) + α
k
(1 − p
k−1
)
|0ih0|
S
+
(1 − α
k
)q
k
+ α
k
p
k−1
|1ih1|
S
= (1 − α
k
)τ
(k)
B
+ α
k
ρ
(k−1)
S
. (65)
Hence, for any unitary V
k
this more general model gives the same description of the resulting system
state as the specific model we considered before (19), where V
k
only determines the value of α
k
. In
any non-trivial unitary operation V
k
, i.e., for any α < 1, S and B get mixed, meaning that it brings
the state of S closer to the thermal state of B. Analogously, when tracing out the system S and work
battery W, we get the same evolution for the k
th
bath qubit,
ρ
(k)
S
= α
k
ρ
(k−1)
S
+ (1 − α
k
)τ
(k)
B
ρ
(k)
B
= α
k
τ
(k)
B
+ (1 − α
k
)ρ
(k−1)
S
.
(66)
Here, ρ
(k)
S
and ρ
(k)
B
represent the final state of S and B after applying the k
th
unitary operation.
Finally, we stress that the unitary V does not encompass the most general energy preserving unitary
on SBWA. In particular, we note that in our previous considerations S interacts successively with each
bath qubit and always on all three systems S, B and W together. This sequence is kept as we are
interested here in quasistatic isothermal processes.
B Discussion of Noise Types that the Protocol Cannot Tolerate
Here we can briefly discuss other imperfect operations which will lead to non-ideality of the process.
The error models discussed previously are not the most general energy conserving unitaries. Indeed,
since we assumed that W has a continuous Hamiltonian, there are many other degenerate subspaces.
Therefore, there are in principle other energy-preserving operations, namely
• First of all, note that we assume that in each step the correct bath qubit is chosen, i.e., the system
interacts successively with each of the N bath qubits, such that we a have quasistatic isothermal
process. Changing considerably the order would lead to energy-preserving dissipative processes.
• There can be energy-preserving interactions between S and W only, which would on average
excite the system and therefore decrease the extracted work. The same would happen for random
interactions between B and W, where in addition the correlations between S and W would be
destroyed leading to uncontrolled fluctuations.
• When considering all three systems S, B and W together, there is not only the degeneracy be-
tween |0i
S
|1i
B
|xi
W
↔ |1i
S
|0i
B
|x + E
k
−
S
i
W
, but there would also be a degeneracy between
|0i
S
|0i
B
|xi
W
↔ |1i
S
|1i
B
|x − E
k
−
S
i
W
due to the continuous Hamiltonian H
W
. Since starting
in |0i
S
|0i
B
|xi
W
is much more likely than starting in |1i
S
|1i
B
|x − E
k
−
S
i, we would also loose
work on average during this transition.
We can see that errors on all other kind of degeneracies prevent us from extracting maximal work.
The only degeneracy on which errors can be compensated for is essentially the one on which the ideal
protocol is interacting. While these considerations provide a qualitative analysis on the kind of errors
that can/cannot be tolerated, we leave as an interesting task for the future to characterize a scenario
where both kind of errors are present simultaneously.
Accepted in Quantum 2019-06-10, click title to verify 18
C Fluctuations
In order to calculate the fluctuations, we need to list each possible path with its corresponding prob-
ability and energy. We use vectors
~
p
[0]
k
and
~
p
[1]
k
in which the probabilities of all paths that end in
state 0, or 1, respectively, are listed as well as vectors
~
W
[0]
k
and
~
W
[1]
k
that list the corresponding
energies. With each step, the number of possible paths doubles, such that after k steps, we will have
2
k
-dimensional vectors.
As we have shown before, any of our considered noise models can be seen as "apply a swap" with
probability (1 − α) and “apply identity" with probability α. Using (24), we can find a recursive way
to describe the vectors:
~
p
[0]
k
=
~
p
[0]
k−1
((1 − q
k
) + αq
k
)
~
p
[1]
k−1
(1 − α)(1 − q
k
)
!
,
~
W
[0]
k
=
~
W
[0]
k−1
~
W
[1]
k−1
− E
k
I
!
,
~
p
[1]
k
=
~
p
[1]
k−1
(q
k
+ α(1 −q
k
))
~
p
[0]
k−1
(1 − α)q
k
!
,
~
W
[1]
k
=
~
W
[1]
k−1
~
W
[0]
k−1
+ E
k
I
!
,
with
~
p
[0]
0
=
1
,
~
p
[1]
0
= (),
~
W
[0]
0
=
0
,
~
W
[1]
0
= (),
(67)
where q
k
corresponds to the excitation probability of the k
th
bath qubit and E
k
= ln
1−q
k
q
k
to the
corresponding energy gap.
We can calculate the average work after the m
th
step,
hW
m
i =
~
p
[0]
m
·
~
W
[0]
m
+
~
p
[1]
m
·
~
W
[1]
m
=
~
p
[0]
m−1
·
~
W
[0]
m−1
+
~
p
[1]
m−1
·
~
W
[1]
m−1
+ (1 − α)E
m
q
m
k
~
p
[0]
m−1
k
1
− (1 − q
m
)k
~
p
[1]
m−1
k
1
= hW
m−1
i + (1 −α)E
m
q
m
− k
~
p
[1]
m−1
k
1
= (1 − α)
m
X
i=1
E
i
q
i
− k
~
p
[1]
i−1
k
1
, (68)
as well as the square of it, since this is needed for calculating the variance,
hW
m
i
2
= hW
m−1
i
2
+ (1 − α)
2
E
2
m
q
m
− k
~
p
[1]
m−1
k
1
2
+ 2(1 − α)hW
m−1
iE
m
q
m
− k
~
p
[1]
m−1
k
1
=
m
X
i=1
(1 − α)
2
E
2
i
q
i
− k
~
p
[1]
i−1
k
1
2
+ 2(1 − α)hW
i−1
iE
i
q
i
− k
~
p
[1]
i−1
k
1
. (69)
Next, we calculate the sum of all elements of
~
p
[1]
k
, which corresponds to the excitation probability p
k
of the system qubit after the k
th
step:
k
~
p
[1]
k
k
1
= k
~
p
[1]
k−1
k
1
(q
k
+ α(1 −q
k
)) + k
~
p
[0]
k−1
k
1
(1 − α)q
k
= (1 − α)q
k
k
~
p
[1]
k−1
k
1
+ k
~
p
[0]
k−1
k
1
+ αk
~
p
[1]
k−1
k
1
= (1 − α)q
k
+ αk
~
p
[1]
k−1
k
1
(70)
Using this recursive form, we can determine the explicit form:
p
k
= k
~
p
[1]
k
k
1
= (1 − α)
k
X
i=1
α
k−i
q
i
+ α
k
q
0
, (71)
Accepted in Quantum 2019-06-10, click title to verify 19
which gives us the same result as (26). In addition, we need the average value of the squared work after
m steps, where in the following
~
W
2 [1]
m
denotes the entry-wise square product of each vector element
hW
2
m
i =
~
p
[0]
m
·
~
W
2 [0]
m
+
~
p
[1]
m
·
~
W
2 [1]
m
=
~
p
[0]
m−1
·
~
W
2 [0]
m−1
+
~
p
[1]
m−1
·
~
W
2 [1]
m−1
+
E
2
m
I − 2E
m
~
W
[1]
m−1
(1 − α)(1 − q
m
)
~
p
[1]
m−1
+
E
2
m
I + 2E
m
~
W
[0]
m−1
(1 − α)q
m
~
p
[0]
m−1
= hW
2
m−1
i+2E
m
(1−α)
q
m
hW
m−1
i−
~
p
[1]
m−1
~
W
[1]
m−1
+E
2
m
(1 − α)
q
m
+k
~
p
[1]
m−1
k
1
−2q
m
k
~
p
[1]
m−1
k
1
=
m
X
i=1
2E
i
(1 − α)
q
i
hW
i−1
i −
~
p
[1]
i−1
~
W
[1]
i−1
+ E
2
i
(1 − α)
q
i
+ k
~
p
[1]
i−1
k
1
− 2q
i
k
~
p
[1]
i−1
k
1
. (72)
Now we can calculate the variance after the total N steps as
Var(W
N
) = hW
2
N
i − hW
N
i
2
=
N
X
m=1
E
2
m
(1 − α)
q
m
(1 − q
m
) + k
~
p
[1]
m−1
k
1
(1 − k
~
p
[1]
m−1
k
1
) + α(q
m
−
~
p
[1]
m−1
)
2
+ 2E
m
(1 − α)
k
~
p
[1]
m−1
k
1
hW
m−1
i −
~
p
[1]
m−1
~
W
[1]
m−1
(73)
Writing the last term explicitly, which determines the average work after m steps when ending in the
excited state |1i
S
, we get
~
p
[1]
m
~
W
[1]
m
=
~
p
[1]
m−1
~
W
[1]
m−1
(q
m
+ α(1 −q
m
)) +
~
p
[0]
m−1
~
W
[0]
m−1
+ E
m
I
(1 − α)q
m
= q
m
(1 − α)
hW
m−1
i + (1 −k
~
p
[1]
m−1
k
1
)E
m
+ α
~
p
[1]
m−1
~
W
[1]
m−1
= (1 − α)
m
X
k=1
α
m−k
q
k
hW
k−1
i + (1 −k
~
p
[1]
k−1
k
1
)E
k
. (74)
To simplify these expressions and show that the variance indeed goes to zero for N → ∞, we need to
choose a probability distribution q
k
for the excitation probabilities of the bath qubits. For this proof
we will use q
k
=
k
2N
and the corresponding energy levels are then given by E
k
= k
B
T ln
1−q
k
q
k
=
k
B
T ln
2N−k
k
, but looking at simulations, it should also work for all the other distributions allowed
in this paper (i.e., those mentioned in Appendix D for the case of two-level systems).
First, we will simplify some expressions that will help us later on:
• k
~
p
[1]
k
k
1
= (1 − α)
k
X
i=1
α
k−i
i
2N
=
k
2N
−
α
1 − α
1 − α
k
2N
(75)
• q
k
− k
~
p
[1]
k−1
k
1
=
1
2N
1 − α
k
1 − α
(76)
• hW
N
i = (1 − α)
N
X
i=1
E
k
q
k
− k
~
p
[1]
k−1
k
1
=
1
2N
N
X
k=1
E
k
(1 − α
k
). (77)
•
1
N
2
N
X
m=1
E
2
m
=
1
N
2
N
X
m=1
O
ln
2
(N)
= O
ln
2
N
N
!
(78)
• E
m
−E
m+1
= k
B
T ln
(2N −m)(m+1)
m(2N −m−1)
=k
B
T ln
1+
1
2N −m−1
+k
B
T ln
1+
1
m
= O
1
m
(79)
• E
2
m
−E
2
m+1
=k
B
T ln
2
2N −m
m
−k
B
T
ln
2N −m
m
+ln
2N −m−1
2N −m
| {z }
=O
(
1
N
)
+ln
m
m + 1
| {z }
=O
(
1
m
)
2
=O
ln N
m
,
(80)
Accepted in Quantum 2019-06-10, click title to verify 20
and estimate the sum:
m
X
i=k
E
i
= k
B
T ln
m
Y
i=k
2N − i
i
!
= k
B
T ln
(2N − k)!
(2N − m − 1)!
(k − 1)!
m!
≤ k
B
T
(2N − k) ln(2N − k) + (k − 1) ln(k − 1) − (2N − m −1) ln(2N − m − 1) − m ln(m)
+
1
2
ln
(2N − k)(k − 1)
(2N − m − 1)m
+ O
1
k
(Stirling approx.)
= k
B
T
(2N − k + 1) ln
2N − k
k
− (2N − m) ln
2N − m
m
− 2N ln
m
k
−(k − 1) ln
k
k−1
| {z }
=−1+O
(
1
k
)
+ (2N −m−1) ln
2N − m
2N −m−1
| {z }
=1+O
(
1
N
)
+
1
2
ln
2N −m−1
2N − k
k − 1
m
| {z }
≤0
+ O
1
k
≤ (2N − k + 1)E
k
− (2N − m)E
m
− 2Nk
B
T ln
m
k
+ O
1
k
. (81)
Using (76) and (78) we can already see that the last term in the first line of (73) goes as O
ln
2
N
N
and
thus disappears for N → ∞. Now we rewrite the other terms,
Var(W
N
) = (1 − α)
N−1
X
m=1
E
2
m
q
m
(1 − q
m
) + E
2
m+1
k
~
p
[1]
m
k
1
(1 − k
~
p
[1]
m
k
1
)
+ 2E
m+1
k
~
p
[1]
m
k
1
hW
m
i −
~
p
[1]
m
~
W
[1]
m
+ O
ln
2
N
N
!
,
(82)
where the change of indices does not matter since E
N
= p
0
= 0. Writing the last two terms in the
brackets explicitly, we get
k
~
p
[1]
m
k
1
hW
m
i−
~
p
[1]
m
~
W
[1]
m
(74)
= (1 − α)
m
X
k=1
α
m−k
q
k
hW
m
i − hW
k−1
i − (1 −k
~
p
[1]
k−1
k
1
)E
k
(75),(77)
= (1−α)
m
X
k=1
α
m−k
q
k
1
2N
m
X
i=k
E
i
(1−α
i
)
| {z }
≤1
−
1−
k−1
2N
+
α
1−α
1−α
k−1
2N
!
E
k
(81)
≤ −(1 − α)
m
X
k=1
α
m−k
q
k
(1 − q
m
)E
m
+ k
B
T ln
m
k
+
α
1 − α
1
2N
E
k
+ O
1
kN
+ O
α
k
ln N
N
(75)
= −k
~
p
[1]
m
k
1
(1 − q
m
)E
m
−(1−α)
m
X
k=1
α
m−k
q
k
k
B
T ln
m
k
+
α
1 − α
1
2N
E
k
+ O
1
N
2
+ O
m
2
α
m
ln N
N
2
. (83)
Accepted in Quantum 2019-06-10, click title to verify 21
Next, we separately calculate the sum, using −ln(1 + x) ≤ −
x −
x
2
2
,
− (1 − α)k
B
T
m
X
k=1
α
m−k
q
k
ln
m
k
+
α
1 − α
1
2N
ln
2N − k
k
= − (1 −α)k
B
T
m−1
X
d=0
α
d
m − d
2N
ln
m
m − d
+
α
1 − α
1
2N
ln
2N − m + d
m − d
=−(1−α)k
B
T
m−1
X
d=0
α
d
m − d
2N
ln
1+
d
m−d
+
α
1−α
1
2N
ln
2N −m
m
+ln
2N −m+d
2N − m
+ln
m
m−d
=−(1−α)k
B
T
m−1
X
d=0
α
d
m−d
2N
ln
1+
d
m−d
+
α
1−α
1
2N
ln
2N −m
m
+ln
1+
d
2N −m
+ln
1+
d
m−d
≤−(1−α)k
B
T
m−1
X
d=0
α
d
m−d
2N
d
m−d
−
d
2
2(m−d)
2
+
α
1−α
1
2N
ln
2N −m
m
+O
d
N
+
d
m−d
−
d
2
2(m−d)
2
!!
≤ −
α
(1 − α)
k
B
T
2N
+ O
mα
m
N
+ (1 − α)k
B
T
m/2
X
d=0
α
d
d
2
2Nm
| {z }
=O
(
1
mN
)
+O
mα
m/2
N
+ (1 − α)k
B
T
m−1
X
d=m/2
α
m/2
d
2
4N
| {z }
=O
m
3
α
m/2
N
− k
B
T ln
2N − m
m
1
4N
2
mα
(1 − α)
+ O
ln N
N
2
+ O
m
N
3
+ O
1
N
2
+ O
m
2
a
m
N
2
!
= −
α
(1 − α)
k
B
T
2N
− E
m
m
4N
2
α
(1 − α)
+ O
1
mN
+ O
m
3
α
m/2
N
!
+ O
ln N
N
2
, (84)
which we can insert into (83),
k
~
p
[1]
m
k
1
hW
m
i −
~
p
[1]
m
~
W
[1]
m
≤ −k
~
p
[1]
m
k
1
(1 − q
m
)E
m
−
α
(1 − α)
k
B
T
2N
− E
m
m
4N
2
α
(1 − α)
+ O
1
mN
+ O
m
3
α
m/2
N
!
+ O
ln N
N
2
.
(85)
Inserting this into the expression for the variance (82) and rewriting the first line, we get
Var(W
N
) = (1−α)
N−1
X
m=1
(E
2
m
+E
2
m+1
)k
~
p
[1]
m
k
1
(1−q
m
)+
E
2
m
(1 − q
m
) + E
2
m+1
k
~
p
[1]
m
k
1
(q
m
− k
~
p
[1]
m
k
1
)
− 2E
m+1
k
~
p
[1]
m
k
1
(1 − q
m
)E
m
+
α
(1 − α)
k
B
T
2N
+ E
m
m
4N
2
α
(1 − α)
+ O
ln
2
N
N
!
= (1−α)
N−1
X
m=1
(E
m
−E
m+1
)
2
k
~
p
[1]
m
k
1
(1−q
m
)
| {z }
=O
(
1
m
)
2
·O
(
m
N
)
=O
(
1
mN
)
+E
2
m
(q
m
− k
~
p
[1]
m
k
1
)
| {z }
=
α
1−α
1−α
m
2N
− E
2
m
(q
m
− k
~
p
[1]
m
k
1
)
2
| {z }
=O(ln N)
2
O
(
1
N
)
2
=O
ln
2
N
N
2
+(E
2
m+1
−E
2
m
)k
~
p
[1]
m
k
1
(q
m
−k
~
p
[1]
m
k
1
)
| {z }
=O
(
ln N
m
)
O
(
m
N
)
O
(
1
N
)
=O
ln N
N
2
−2E
m+1
α
(1−α)
k
B
T
2N
+E
m
m
4N
2
α
(1−α)
+O
ln
2
N
N
!
=
N−1
X
m=1
O
1
mN
+ E
2
m
α
2N
+ O
α
m
ln
2
N
N
!
− 2E
m
αk
B
T
2N
+ E
m
mα
4N
2
+ O
ln
2
N
N
!
=
α
2N
N−1
X
m=1
E
2
m
1 −
m
2N
− k
B
T E
m
+ O
ln
2
N
N
!
. (86)
Accepted in Quantum 2019-06-10, click title to verify 22
In the last step we need to calculate these two sums,
−
αk
B
T
N
N−1
X
m=1
E
m
= −
α(k
B
T )
2
N
Z
N
1
ln
2N − m
m
dm + O
ln N
N
=
α(k
B
T )
2
N
(2N − m) ln(2N − m) + m ln m
N
1
+ O
ln N
N
= −2α(k
B
T )
2
ln 2 + O
ln N
N
(87)
and
α
2N
N−1
X
m=1
E
2
m
1 −
m
N
=
α(k
B
T )
2
2N
Z
N
1
ln
2
2N − m
m
1 −
m
N
dm + O
ln
2
N
N
!
=
α(k
B
T )
2
2N
−4N ln(2N −m)+2m ln
2N −m
m
+(1−
m
2N
)m ln
2
2N −m
m
N
1
+ O
ln
2
N
N
!
= 2α(k
B
T )
2
ln 2 + O
ln
2
N
N
!
. (88)
Thus, we get for the variance,
Var(W
N
) = O
ln
2
N
N
!
N→∞
−→ 0, (89)
where we can see that for infinitely many steps N the fluctuations vanish. This is in accordance with
the Jarzynski equality e
β∆F
= he
−βW
i [49], as for N → ∞ we are looking at a quasistatic process with
∆F = −hW i = F (τ
(N)
B
, H
S
) − F (τ
(0)
B
, H
S
) and having no fluctuations implies e
−βhW i
= he
−βW
i.
D Generalization to Arbitrary Systems
We now extend our previous considerations to generic qudit systems and more general choices for the
Hamiltonians of the bath systems. We consider a continuous family of Hamiltonians H(s), s ∈ [0, 1],
as well as the corresponding continuous family of thermal states τ(s) =
e
−βH(s)
Tr[e
−βH(s)
]
, such that for
a fixed N we can draw the Hamiltonians H
(k)
B
= H(k/N) and the corresponding thermal states
τ
(k)
B
= τ(k/N). The functions needs to be continuously differentiable and for optimal processes satisfy
kτ(0) − ρ
(0)
S
k
1
≤ δ 1, where δ = 0 is always possible if the initial state ρ
(0)
S
is a full-rank state. We
also define the continuous function,
F (λ) := F (τ(λ), H
S
), (90)
with λ ∈ (0, 1). We take as a starting point the model of noise (24). The derivation up to the second
line of (28) still follows naturally. For simplicity, we will in the following first assume ρ
(0)
S
= τ
(0)
B
and
hence δ = 0. In the case δ 6= 0 we obtain an additional term, which we discuss below. To move on, we
notice the identity,
Tr ((H
S
− H
B
)(τ − ρ)) = F (τ, H
S
) − F (ρ, H
S
) − F (τ, H
B
) + F (ρ, H
B
)
= F (τ, H
S
) − F (ρ, H
S
) + T S(ρ k τ ). (91)
Accepted in Quantum 2019-06-10, click title to verify 23
Using (25) in the case ρ
(0)
S
= τ
(0)
B
, we can rewrite the second line of (28) as,
W = −(1 −α)
N−1
X
k=0
Tr
H
S
− H
(k+1)
B
τ
(k+1)
B
−
k−1
X
j=0
(1 − α)α
j
τ
(k−j)
B
− α
k
τ
(0)
B
(92)
= −(1 − α)
N−1
X
k=0
Tr
H
S
−H
(k+1)
B
(1−α)
1−α
k
1−α
τ
(k+1)
B
−
k−1
X
j=0
α
j
τ
(k−j)
B
+α
k
(τ
(k+1)
B
−τ
(0)
B
)
= −(1 − α)
2
N−1
X
k=0
k−1
X
j=0
α
j
Tr
h
H
S
− H
(k+1)
B
τ
(k+1)
B
− τ
(k−j)
B
i
− (1 − α)
N−1
X
k=0
α
k
Tr
h
H
S
− H
(k+1)
B
τ
(k+1)
B
− τ
(0)
B
i
= −(1 − α)
2
N−1
X
k=0
k−1
X
j=0
α
j
F (τ
(k+1)
B
, H
S
) − F (τ
(k−j)
B
, H
S
) + T S(τ
(k−j)
B
k τ
(k+1)
B
)
− (1 − α)
N−1
X
k=0
α
k
F (τ
(k+1)
B
, H
S
) − F (τ
(0)
B
, H
S
) + T S(τ
(0)
B
k τ
(k+1)
B
)
.
(93)
In the case δ 6= 0, would have an additional term proportional α
k
(ρ
(0)
B
− τ
(0)
B
) in (92) obtained by
replacing ρ
(0)
S
by τ
(0)
B
+ ρ
(0)
S
− τ
(0)
B
. This leads to additional work
W
0
= (1 − α)
N−1
X
k=0
α
k
Tr
h
H
(k)
B
− H
S
τ
(0)
B
− ρ
(0)
B
i
≤ δ max
s
||H(s) − H
S
||
∞
= O
δ ln
1
δ
, (94)
where the last estimate follows from the fact that kH(0)k
∞
= O(ln(1/δ)) is required to fulfill kτ(0) −
ρ
(0)
S
k
1
= δ 1 and we assumed that max
s
kH(s)k
∞
= kH(0)k
∞
. This fact will similarly lead to a
divergence of the quantity
R
f(s)ds below of the order log(1/δ). Since we are interestead to leading
order corrections, by choosing δ = 1/N we can obtain a scaling as log(N )/N instead of 1/N in what
follows. For simplicity, however, we concentrate on the case δ = 0.
Since we are interested in leading order corrections up to order 1/N, we have that
F (τ
(k)
B
, H
S
) − F (τ
(k−j)
B
, H
S
) =
j
N
˙
F (k/N) +
j
2
2N
2
¨
F (k/N) + O
j
3
N
3
!
, (95)
which will be useful to simplify (93). Similarly, we can expand S(· k ·) as
S (τ (λ + x) k τ (λ)) =
x
2
2
d
2
S(τ(λ + x) k τ (λ))
dx
2
x=0
+ O(x
3
), (96)
which follows because S(ρ k τ) has a minimum at ρ = τ and S(ρ k ρ) = 0, and define
f(λ) = T
d
2
S(τ(λ + x) k τ (λ))
dx
2
x=0
. (97)
Although here we will not be concerned with the form of f(λ), it is worth pointing out that f(λ) can
be written as
f(λ) = −Tr( ˙τ(λ)
˙
H(λ)). (98)
Accepted in Quantum 2019-06-10, click title to verify 24
Let us now consider,
W
∗
=−(1−α)
2
N−1
X
k=0
k−1
X
j=0
α
j
F (τ
(k+1)
B
, H
S
)−F (τ
(k−j)
B
, H
S
)
−(1−α)
N−1
X
k=0
α
k
F (τ
(k+1)
B
, H
S
)−F (τ
(0)
B
, H
S
)
E = −W + W
∗
(99)
We now use (95) to simplify W
∗
and work out the sum in j. Neglecting terms of order O
α
N
, we
obtain
W
∗
=−
N−1
X
k=0
(1−α)
2
k−1
X
j=0
α
j
j + 1
N
˙
F (k/N)+
(j + 1)
2
2N
2
¨
F (k/N)
!
+ (1−α)α
k
k + 1
N
˙
F (k/N)
+O(1/N
2
)
= −
N−1
X
k=0
(1 − α
k+1
)
˙
F (k/N)
N
+ (1 − α)
2
k−1
X
j=0
α
j
(j + 1)
2
2N
2
¨
F (k/N)
+ O(1/N
2
)
=−
N−1
X
k=0
˙
F (k/N)
N
+
¨
F (k/N)
2N
2
!
−α
k+1
˙
F (k/N)
N
+
¨
F (k/N)
2N
2
−1+(1−α)
2
k−1
X
j=0
α
j
(j+1)
2
+O(1/N
2
)
= ∆F −
N−1
X
k=0
−α
k+1
˙
F (k/N)
N
+
¨
F (k/N)
2N
2
−1 + (1 −α)
2
k−1
X
j=0
α
j
(j + 1)
2
+ O(1/N
2
). (100)
Using that
˙
F (k/N) =
˙
F (0
+
) + O(k/N), and again neglecting elements with O
α
N
, we simplify the
expression to
W
∗
= ∆F +
˙
F (0
+
)
α
N(1−α)
−
N−1
X
k=0
¨
F (k/N)
2N
2
2α+
2k
2
+2k−1
α
k+1
−k
2
α
k+2
−(k + 1)
2
α
k
1 − α
+ O(1/N
2
)
= ∆F −
α
N(1 −α)
−
˙
F (0
+
) +
1
N
N−1
X
k=0
¨
F (k/N)
!
+ O(1/N
2
)
= ∆F −
α
N(1 −α)
−
˙
F (0
+
) +
Z
1
0
ds
¨
F (s)
+ O(1/N
2
)
= ∆F −
α
N(1 −α)
˙
F (1) −2
˙
F (0
+
)
+ O(1/N
2
) (101)
On the other hand, we have for E
E = (1 − α)
2
N−1
X
k=0
k−1
X
j=0
α
j
T S(τ
(k−j)
B
k τ
(k+1)
B
) + (1 −α)
N−1
X
k=0
α
k
T S(τ
(0)
B
k τ
(k+1)
B
)
=
1
2N
2
N−1
X
k=0
f(k/N)(1 − α)
2
k−1
X
j=0
α
j
(j + 1)
2
+ O(1/N
2
)
=
1
2N
2
1 +
2α
1 − α
N−1
X
k=0
f(k/N) + O(1/N
2
)
=
1
2N
1 +
2α
1 − α
Z
1
0
dsf(s) + O(1/N
2
)
=:
Γ
N
1 +
2α
1 − α
+ O(1/N
2
). (102)
E Isothermal Processes and Thermal Maps
In this section, we provide the proof of the results presented in 3.2. Before starting the formal proof,
let us discuss the main ideas that go into the proof. For a fixed path of Hamiltonians, we are dealing
Accepted in Quantum 2019-06-10, click title to verify 25
with a process of a large number N of steps. For a sufficiently large number of steps, we can expect
that i) the unitary time-evolution between two thermalization steps converges to the time-evolution
generated by a constant Hamiltonian and ii) that the state of the system remains relatively close to a
thermal state during a protocol, even though each individual thermalizing map is imperfect and the
system therefore "lags behind" the actual thermal state. To simplify interpretation, we therefore split
the total dissipated work into three parts:
∆F
iso
− W
iso
= γ + ε + κ. (103)
Here, γ is the error that would arise in the case of perfect thermalizations and we would replace the
unitary time-evolution inbetween the thermalization steps by instantaneous quenches. It is imply due
to the fact that we consider a finite protocol. The term ε vanishes in the case of perfect thermalizations
and the term κ vanishes when we consider instantaneous quenches instead of unitary time-evolution.
We will show that
γ =
Γ
N
+ O
1
N
2
, ε =
α
1 − α
Λ
N
+ O
1
N
2
, κ =
K
N
+ O
1
N
2
. (104)
In the following we will use the simplified notation
U
i
:
= U
t
i
,t
i−1
(105)
for the unitary that is generated by H(t) from time t
i−1
to t
i
. We remind the reader that we assume
δt = t
i
− t
i−1
= 1/N. (106)
Recall that the total work that can be extracted during the protocol is given by
W
iso
=
N
X
i=1
Tr
H
(i−1)
σ
(i−1)
− H
(i)
ρ
(i)
=
N
X
i=1
Tr
H
(i−1)
σ
(i−1)
− H
(i)
U
i
σ
(i−1)
U
†
i
. (107)
Adding and substracting zeroes allows us to re-write the work as
W
iso
=
N
X
i=1
Tr
∆H
(i)
τ
(i−1)
+
N
X
i=1
Tr
∆H
(i)
σ
(i−1)
− τ
(i−1)
+
N
X
i=1
Tr
H
(i)
σ
(i−1)
− U
i
σ
(i−1)
U
†
i
, (108)
where ∆H
(i)
= H
(i−1)
− H
(i)
. Each of the terms corresponds to one of the terms for the dissipated
work above. In the following we will bound each term separately. We will assume that both the norm
of the Hamiltonian as well as the norm of the derivative of the Hamiltonian along the Hamiltonian
trajectory are bounded.
E.1 First Term
The first term has previously been shown to yield (see, e.g., [13, 20, 42, 43])
N
X
i=1
Tr
∆H
(i)
τ
(i−1)
= ∆F
iso
−
Γ
N
− O
1
N
2
, (109)
where Γ is given by [40] (also see Appendix D)
Γ = −
1
2
Z
1
0
ds Tr
˙τ(s)
˙
H(s)
. (110)
Accepted in Quantum 2019-06-10, click title to verify 26
E.2 Second Term
The second term vanishes in the case of perfect thermalization, it hence corresponds to the dissipation
ε. To obtain the bound,we first emphasize that ∆H
(i)
is of order 1/N. We therefore essentially have
to show that σ
(i−1)
− τ
(i−1)
is also of order 1/N. For notational reasons, it is useful to introduce the
quantities
∂H
(i)
∞
:
= max
t∈[t
i−1
,t
i
]
kH(t) − H
(i)
k
∞
= O(1/N), (111)
k∂Hk
∞
:
= max
i
∂H
(i)
∞
= O(1/N). (112)
Note that here we use the Schatten norms, i.e. kAk
p
=
P
n≥1
s
p
n
(A)
1/p
, where s
p
n
(A) denotes the
n-th singular values of A. In particular this implies that kAk
∞
corresponds to the operator norm and
therefore assigns the largest singular value of A, while kAk
1
= Tr |A| corresponds to the trace norm.
We can then bound the second term using Hölder’s inequality as
N
X
i=1
Tr
∆H
(i)
σ
(i−1)
− τ
(i−1)
≥ −
N
X
i=1
k∂Hk
∞
kσ
(i−1)
− τ
(i−1)
k
1
. (113)
We can now make use of the fact that the thermalizing maps have a thermalization parameter α to
write
kσ
(i)
− τ
(i)
k
1
≤ αkU
i
σ
(i−1)
U
†
i
− τ
(i)
k
1
≤ αkσ
(i−1)
− U
†
i
τ
(i)
U
i
k
1
≤ α
kσ
(i−1)
− τ
(i)
k
1
+ kU
†
i
τ
(i)
U
i
− τ
(i)
k
1
. (114)
At this point we have to make use of the fact that the unitary U
i
is close to that generated by the
Hamiltonian H
(i)
. To increase clarity of presentation, we delegate the proof of this fact to Section E.4.
Using the notation
˜
U
i
:
= e
−iH
(i)
δt
, δU
i
:
= U
i
−
˜
U
i
, (115)
we show in Section E.4 that
kδU
i
k
∞
= O
1/N
2
. (116)
Since [τ
(i)
,
˜
U
i
] = 0, we can use this result to estimate the last term in (114) as
kU
†
i
τ
(i)
U
i
− τ
(i)
k
1
= kU
†
i
τ
(i)
U
i
−
˜
U
†
i
τ
(i)
˜
U
i
k
1
≤ 2kδU
i
k
∞
= O(1/N
2
), (117)
where we made use of Hölder’s inequality. Adding and subtracting zero and using the triangle inequality
again, we then arrive at
kσ
(i)
− τ
(i)
k
1
≤ α
kσ
(i−1)
− τ
(i−1)
k
1
+ kτ
(i−1)
− τ
(i)
k
1
+ O(1/N
2
)
(118)
We now bound the second term in this expression. To do this, we first use Pinsker’s inequality
kρ − σk
1
≤
p
2S(ρkσ), where S(·k·) is again the quantum relative entropy. Using the results around
(97), we then find
kτ
(i)
− τ
(i−1)
k
1
≤
q
2S(τ
(i)
kτ
(i−1)
) =
s
δt
2
f
i
N
+ O(δt
3
)
= δt
s
f
i
N
+ O
δt
2
≤
1
N
p
f
max
+ O
1
N
2
, (119)
Accepted in Quantum 2019-06-10, click title to verify 27
where we defined f(λ) as in (97) and f
max
= max
λ
f(λ). This yields
kσ
(i)
− τ
(i)
k
1
≤ α
kσ
(i−1)
− τ
(i−1)
k
1
+
√
f
max
N
+ O(1/N
2
)
!
≤
i
X
k=1
α
k
√
f
max
N
+ O(1/N
2
)
!
+ α
i
kσ
(0)
− τ
(0)
k
1
=
α(1 − α
i
)
1 − α
√
f
max
N
+ O(1/N
2
)
!
+ α
i
kσ
(0)
− τ
(0)
k
1
. (120)
Combining all the above bounds, we finally obtain
N
X
i=1
Tr
∆H
(i)
σ
(i−1)
− τ
(i−1)
≥ −
N
X
i=1
k∂Hk
∞
kσ
(i−1)
− τ
(i−1)
k
1
≥
α
1 − α
k∂Hk
∞
p
f
max
+ O(1/N
1/2
) + kσ
(0)
− τ
(0)
k
1
≥
α
1 − α
Λ
N
+ O
1/N
2
, (121)
for a constant Λ.
E.3 Third Term
The third term vanishes in the case where unitary time-evolution is replaced by instantaneous quenches,
so that U
i
= . It hence corresponds to the dissipation κ. To bound it, we again make use of the
approximation of the unitaries U
i
by the time-evolution under H
(i)
, given by
˜
U
i
. Since [
˜
U
i
, H
(i)
] = 0,
we can make use of U
i
=
˜
U
i
+ δU
i
and bound this term using Hölder’s inequality as
N
X
i=1
Tr
H
(i)
− U
†
i
H
(i)
U
i
σ
(i−1)
≥ −
N
X
i=1
kH
(i)
− U
†
i
H
(i)
U
i
k
∞
≥ −kHk
∞
N
X
i=1
2kδU
i
k
∞
≥ −
K
N
+ O(1/N
2
), (122)
where K is a constant determined by the trajectory of Hamiltonians and where we introduced
kHk
∞
:
= max
t
kH(t)k
∞
. (123)
The second inequality follows from
kH
(i)
− U
†
i
H
(i)
U
i
k
∞
= k
˜
U
†
i
H
(i)
(U
i
− δU
i
) − (
˜
U
†
i
+ δU
†
i
)H
(i)
U
i
)k
∞
≤ k
˜
U
†
i
H
(i)
δU
i
k
∞
+ kδU
†
i
H
(i)
U
i
k
∞
≤ 2kH
(i)
k
∞
kδU
i
k
∞
, (124)
where in the last step we used kABk
∞
≤ kAk
∞
kBk
∞
, kA
†
k
∞
= kAk
∞
and kUk
∞
= 1 for any unitary
U.
E.4 Approximation of unitaries
In this section we show that the unitaries U
i
only differ from
˜
U
i
:= e
−iH
(i)
δt
by an amount of order
O(δt
2
).
To do that, we first write out the time-ordered exponential that defines U
i
:
U
i
= T e
−i
R
t
i
t
i−1
H(t)dt
=
∞
X
k=0
(−i)
k
k!
Z
t
i
t
i−1
dt
i
1
···
Z
t
i
t
i−1
dt
i
k
T {H(t
i
1
) ···H(t
i
k
)}. (125)
Accepted in Quantum 2019-06-10, click title to verify 28
Bringing
˜
U
i
into a similar form we get
˜
U
i
=
∞
X
k=0
(−i)
k
k!
(H
(i)
δt)
k
=
∞
X
k=0
(−i)
k
k!
H
(i)
k
(t
i
− t
i−1
)
k
=
∞
X
k=0
(−i)
k
k!
H
(i)
k
Z
t
i
t
i−1
dt
i
1
···
Z
t
i
t
i−1
dt
i
k
. (126)
We can then upper bound their difference as
kδU
i
k
∞
= kU
i
−
˜
U
i
k
∞
≤
∞
X
k=1
1
k!
Z
t
i
t
i−1
dt
i
1
···
Z
t
i
t
i−1
dt
i
k
T {H(t
i
1
) ···H(t
i
k
)} − H
(i)
k
∞
≤
∞
X
k=1
1
k!
δt
k
kkHk
k−1
∞
k∂H
(i)
k
∞
≤ e
kHk
∞
δt
δt k∂Hk
∞
= O(1/N
2
). (127)
Here, the second inequality follows from a telescope-like sum by iterating the bound
H(t
i
1
) ···H(t
i
k
) − H
(i)
k
∞
(128)
≤
H(t
i
1
) ···H(t
i
k
) − H(t
i
1
) ···H(t
i
k−1
)H
(i)
∞
+
H(t
i
1
) ···H(t
i
k−1
)H
(i)
− H
(i)
k
∞
≤ kH(t
i
1
)k
∞
···
H(t
i
k−1
)
∞
H(t
i
k
) − H
(i)
∞
+
H
(i)
∞
H(t
i
1
) ···H(t
i
k−1
) − H
(i)
k−1
∞
≤ kHk
k−1
∞
∂H
(i)
∞
+ kHk
∞
H(t
i
1
) ···H(t
i
k−1
) − H
(i)
k−1
∞
. (129)
This finishes the proof.
Accepted in Quantum 2019-06-10, click title to verify 29