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
sults thereby contribute to a recent research line
which aims at understanding the level of con-
trol required in thermodynamic protocols [511].
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
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. [1215])
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, 1618]. We study work extraction from
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
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, 2124] and
non-equilibrium scenarios in diﬀerent thermody-
namic contexts [2530]. 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, 3234].
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
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,
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 )
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
(k1)
S
+ (1 α
k
)τ
(k)
B
ρ
(k)
B
= α
k
τ
(k)
B
+ (1 α
k
)ρ
(k1)
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
i
P
i
(α
i
)f({α
j
}),
(21)
where P
i
(α
i
) are some unknown probability dis-
tributions that satisfy
R
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
(k1)
S
i
{α
j
}
k1
j=1
+ (1 α)τ
(k)
B
hρ
(k)
B
i
{α
j
}
k
j=1
= ατ
(k)
B
+ (1 α)hρ
(k1)
S
i
{α
j
}
k1
j=1
.
(23)
where we used that τ
(k)
B
and ρ
(k1)
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
(k1)
S
+ (1 α)τ
(k)
B
ρ
(k)
B
= ατ
(k)
B
+ (1 α)ρ
(k1)
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
ki
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
ki
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
(k1)
W
H
W
i
= Tr
h
H
(k)
B
τ
(k)
B
ρ
(k)
B
H
S
ρ
(k)
S
ρ
(k1)
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
(k1)
S
i
= (1α)
N
X
k=1
Tr
h
H
(k)
B
H
S
τ
(k)
B
ρ
(k1)
S
i
.
(28)
For the qubit case, we have
W = (1 α)
N
X
k=1
(E
k
S
)(q
k
p
k1
). (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
k1
=
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
ki
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
k1
)
= (1 α)
N
X
k=1
E
k
q
k
q
k1
+
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
1α
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