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By-passing ﬂuctuation theorems with a catalyst

P. Boes

, R. Gallego

, N. H. Y. Ng

, J. Eisert

, and H. Wilming

Dahlem Center for Complex Quantum Systems, Freie Universit

at Berlin, 14195 Berlin, Germany

Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland

Fluctuation theorems impose constraints on

possible work extraction probabilities in ther-

modynamical processes. These constraints are

stronger than the usual second law, which is

concerned only with average values. Here,

we show that such constraints, expressed in

the form of the Jarzysnki equality, can be by-

passed if one allows for the use of catalysts—

additional degrees of freedom that may be-

come correlated with the system from which

work is extracted, but whose reduced state re-

mains unchanged so that they can be re-used.

This violation can be achieved both for small

systems but also for macroscopic many-body

systems, and leads to positive work extrac-

tion per particle with ﬁnite probability from

macroscopic states in equilibrium. In addition

to studying such violations for a single system,

we also discuss the scenario in which many par-

ties use the same catalyst to induce local tran-

sitions. We show that there exist catalytic pro-

cesses that lead to highly correlated work dis-

tributions, expected to have implications for

stochastic and quantum thermodynamics.

1 Introduction

Consider a physical system in thermal equilibrium with

its environment. The second law of thermodynamics dic-

tates that it is impossible to extract positive average work

from this system using reversible processes that are cyclic

in the Hamiltonian. More precisely, if the system’s ini-

tial state is represented by a canonical ensemble and we

consider many iterations of a probabilistic process during

which the Hamiltonian of the system is varied but returned

to the initial Hamiltonian at the end, then it holds that

hW i ≤ 0, (1)

where hW i is the average work extracted during the pro-

cess. We will refer to (1) as the Average Second Law (Av-

SL),

However, there exist signiﬁcantly stronger constraints

on the possible extracted work in the above type of pro-

cesses, namely those imposed by ﬂuctuation theorems

[1, 2, 3]. Indeed, using such theorems, one can show

that the probability of extracting a ﬁnite amount of pos-

itive work per particle is exponentially suppressed with

the number of particles in a system [1]. Once these dif-

ferent types of constraints are recognized, an interesting

questions arises: What are physically meaningful settings

in which the probabilistic constraints imposed by ﬂuctua-

tion theorems can be circumvented, while still respecting

the Av-SL? In particular, do ﬂuctuation theorems also hold

when an additional, cyclically evolving auxiliary system is

allowed for?

In this work, we present an answer to this question, by

introducing a class of processes that generalize the above

reversible processes, are physically well motivated, com-

patible with (1), and yet allow for the extraction of positive

work per particle with a probability that is independent of

system size. We do so via the notion of a catalytic process,

in which we allow for the reversible process to not only act

on the system as such, but additionally on an auxiliary sys-

tem that can be initially prepared in an arbitrary state, but

whose marginal state has to be left invariant by the process.

Such catalysts are well-motivated – they allow a general

description of thermodynamic processes in which the sys-

tem may be interacting with some experimental apparatus

(such as a quantum clock [4, 5]), however not extracting

energetic/information resources from such an ancilla. In

terms of our discussion of the Av-SL above, catalysts cor-

respond to the cyclically evolving auxiliary system. De-

spite being studied frequently in resource-theoretic formu-

lations of thermodynamics [6, 7, 8, 9], catalytic processes

have never been studied in the context of ﬂuctuation theo-

rems until now. Furthermore, even in previous works of

catalysis, the exact form of the catalyst is highly state-

dependent and therefore rarely studied explicitly [6, 8].

In this work, we make progress in the signiﬁcant gaps in

the knowledge of catalysis, by presenting and discussing

constructive examples of such catalytic processes in the

framework where ﬂuctuation theorems are commonly de-

rived. We show that, by sharing the same catalyst, a group

of agents can follow collective strategies to achieve highly

correlated work-distributions. This makes these processes

interesting for the ﬁeld of quantum and stochastic ther-

modynamics and potentially also for certain negentropic

processes in biology. On the overall, our work provides a

rigorous footing for the further study of thermodynamical

processes that systematically exploit the notion of cataly-

sis in order to achieve certain patterns of work ﬂuctuations

in an environment that is governed by the Av-SL. Given

the broad applicability of our results, we believe that the

study of such processes will produce many further inter-

esting results of both foundational and practical interest.

Accepted in Quantum 2020-02-06, click title to verify. Published under CC-BY 4.0. 1

arXiv:1904.01314v4 [quant-ph] 14 Feb 2020

2 Setup

2.1 Formulation of the physical situation

We formulate our arguments and results in the language of

quantum mechanics, but all of our results similarly apply

to classical, stochastic systems. We consider the setting

depicted in Fig. 1: A d-dimensional system S with Hamil-

tonian H =

i=1

ihE

| is initalized in the Gibbs

state

(H) :=

−βH

Z(β, H)

where Z(β, H) := Tr(e

−βH

). This state describes a sys-

tem initially in thermal equilibrium with its environment

at inverse temperature β := 1/(k

T ). An agent (some

experimenter) ﬁrst performs an energy measurement on

this system which produces a measurement outcome E

According to quantum mechanics, the post-measurement

state is described by the density matrix |E

ihE

|. The

agent then performs a physical operation on the system

which does not depend on the outcome of the measure-

ment. Such an operation can always be represented by a

general quantum channel C (i.e., a trace-preserving, com-

pletely positive map that takes density matrices to den-

sity matrices) applied to the post-measurement state. This

operation is then followed by a second energy measure-

ment with respect to the same Hamiltonian with outcome

. This procedure results in a channel-dependent joint

distribution P (E

, E

) = P (E

)P (E

). In general, a

given quantum channel may be realized in different ways.

Whether the change of energy E

−E

can be interpreted

as work from a thermodynamic point of view will depend

on how exactly the quantum channel C was physically re-

alized. We will assume that this is the case in the follow-

ing, but will comment on this assumption again later on.

In particular, we can then deﬁne the work distribution P

for the above process as

P (W ) :=

i,f

P (E

, E

)δ(W − (E

− E

)),

where δ is the Dirac delta distribution. We are interested in

investigating possible distributions P (W ) that arise from

different channels C. To do so, it is useful to note the rela-

tion

βW

i =

−βE

|C[I] |E

i, (2)

which is straightforwardly derived using the above deﬁni-

tions, where I denotes the identity matrix.

In the standard setting of Tasaki-type ﬂuctuation the-

orems, C is considered to be a unitary channel C[·] =

U(·)U

†

, since these are generated by changing the Hamil-

tonian over time [3]. For such channels, (2) becomes

βW

i = 1, (3)

It is possible to extend the setup and our further results

to the more general case of diﬀerent Hamiltonians for the ini-

tial and ﬁnal measurement. We present our results within this

restricted settings for conceptual and notational simplicity.



(H)

P (E

)

ihE

C(|E

ihE

P (E

)

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ihE

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Figure 1: The basic setup for all processes in this work: An

agent with access to a system S equipped with Hamiltonian

H that is assumed to be initially in thermal equilibrium with a

heat bath at inverse temperature β samples from S (by mea-

suring in the energy basis), then implements a process that

maps the post-measurement state |E

ihE

| to C( |E

ihE

|),

where C is a quantum channel. Finally, the agent repeats the

energy measurement on S with respect to the same Hamil-

tonian H.

which is the well-known Jarzynski equality (JE) for cyclic,

reversible processes [1]. Eq. (3) is strictly stronger than

(1), the latter being implied by (3) via Jensen’s inequality.

2.2 No macroscopic work

One of the reasons for the importance of the JE derives

from the fact that it gives strong bounds on the possibil-

ity of extracting work from a large system in a thermal

state [10, 11, 12]. To see this, let S be an N-particle sys-

tem and deﬁne the probability of extracting work w per

particle as

p(w) := P (wN ).

Plugging this into (3) yields that for any  > 0,

1 = he

βW

i =

βwN

P (wN) ≥ e

βN

w≥

p(w),

which implies that events which extract signiﬁcant posi-

tive work per particle from a macroscopic system at equi-

librium are exponentially unlikely in N . For later use, we

formalize this property.

Deﬁnition 1 (No macroscopic work). Given a se-

quence of N-particle systems initially at thermal equi-

librium with inverse temperature β and channels C

(implicitly depending on N), we say that the pro-

cesses represented by C fulﬁll the no macroscopic work

(NMW) condition if the probability of an event ex-

tracting work per particle larger or equal than  is ar-

bitrarily small as N → ∞,

lim

N→∞

p(w ≥ ) := lim

N→∞

w>

p(w) = 0.

As is clear from the above, channels that satisfy the JE,

such as unitary channels, also satisfy NMW and Av-SL.

We now turn to investigate violations of JE and NMW for

non-unitary channels.

3 Violations of NMW and JE

The ﬁrst main result of this work is to introduce a phys-

ically motivated family of channels C that violates both

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NMW and JE, but respects the Av-SL. To aid comparison,

we ﬁrst brieﬂy discuss other generalizations of the stan-

dard setting to non-unitary channels (see also Refs. [13,

14]).

3.1 Violating JE with non-unitary channels

It is easy to see from (2) that a more general class of chan-

nels that satisfy the JE are unital channels, that is, channels

that satisfy C[I] = I. Consequently, neither JE, nor in turn

NMW or Av-SL can be violated in settings which give rise

to a unital channel. However, once this condition on uni-

tality is relaxed, it becomes easy to violate JE on a formal

level. For example, consider the fully-thermalizing chan-

nel that maps every input state to the thermal state ω

(H),

in other words C(·) = ω

(H). This channel always vi-

olates the JE whenever ω

(H) 6= I/d. It is, however,

not clear how the energy-ﬂuctuations can be interpreted

as work in this example, since thermalizing processes usu-

ally occur due to contact with a heat bath, in which case

one would naturally interpret the changes of energy on the

system being due to heat. Thus, while it is trivial to for-

mally violate JE, it is not obvious whether it is possible

to do so in a physically meaningful and operationally use-

ful manner. Nevertheless, in Appendix A, we show that

the fully-thermalizing channel, in fact any channel with

the thermal state as a ﬁxed point, cannot violate the NMW

condition for typical many-body systems, even if they may

violate (3). This means that, even if one interprets energy

ﬂuctuations as work, one still could not use the thermaliz-

ing channel to extract macroscopic amounts of work from

a many-body system.

3.2 Violations of NMW and JE via β-catalytic

channels

The above ﬁndings raise the important question whether

there exist channels for which the above procedure leads

to a violation of NMW (and hence JE), while still respect-

ing the Av-SL and allowing for the interpretation of the

random variable W as work extracted from S. Such chan-

nels, if they exist, promise to be of great interest because

they could allow for a systematic exploitation of relatively

likely events extracting work from heat baths. The ﬁrst re-

sult of this work is to answer this question afﬁrmatively.

To this end, we deﬁne the notion of a β-catalytic channel.

Deﬁnition 2 (β-catalytic channel). A completely

positive, trace-preserving map C is a β-catalytic chan-

nel on S, if there exists a quantum state σ

on a sys-

tem C with Hamiltonian H

, together with a unitary

U such that [σ

, H

] = 0 and

C(·) = Tr

(U( · ⊗ σ

†

s.t. Tr

(U(ω

(H) ⊗ σ

†

) = σ

. (4)

Before stating our ﬁrst main result, let us make some

comments about this deﬁnition. First of all, we already

assumed that the initial and ﬁnal Hamiltonian coincides.

This means that while during the process, C may couple

system and catalyst for example by introducing interac-

tion terms H

, nevertheless at the end of the process, the

channel must also turn off such interaction terms. Sec-

ondly, note that β-catalytic channels describe reversible

processes, in the sense that they do not change the entropy

of the joint-system SC and can be undone by acting on

this joint-system by a unitary process. We refer to the sys-

tem C as being the “catalyst”, understanding that it may

be some by-stander system involving additional degrees

of freedom. This terminology is motivated by the fact that,

on average, i.e., if we do not condition on the outcomes of

the energy measurements, then C is returned, at the end of

the procedure, to its original state. It can therefore be re-

used for further rounds of the protocol with new copies of

S. Note, however, that the invariance of the reduced state

on C under the channel is required not for all initial states

of S, but only for ω

(H). As such, β-catalytic channels

depend on β and H through the second condition.

While Deﬁnition 2 does not require the catalyst to be

uncorrelated with S at the end of the protocol, and in

this sense goes beyond the conventional notion of catal-

ysis discussed in the resource-theoretic literature on quan-

tum thermodynamics [6, 7], the more general notion of

catalysis that we employ here is receiving increasing in-

terest in quantum thermodynamics, where it was shown to

single out the quantum relative entropy, free energy and

von Neumann entropy [15, 8, 16], to be useful in the con-

text of algorithmic cooling [16, 17] and to show the en-

ergetic instability of passive states [18]. Finally, let us

brieﬂy comment on the interpretation of the random vari-

able W as work in the setting of β-catalytic channels and

the role of the Hamiltonian of the catalyst. Since the pro-

cess on C and S is unitary, it is meaningful to denote the

total changes of energies of the two systems as work mea-

sured by a two-point measurement scheme on each sys-

tem. This gives rise to a joint-distribution of work on the

two systems P (W

(S)

, W

(C)

). The probability distribution

of work P (W ) discussed above then simply corresponds

to the marginal distribution P (W

(S)

) on S. Importantly,

this distribution is independent of the Hamiltonian on C

(see Sec. G in the Appendix). In particular, we can as-

sume that the catalyst has trivial Hamiltonian H

= 0,

which in turn implies [σ

, H

] = 0 for any σ

. It is then

clear that no energy ﬂows from the catalyst to the system,

not even probabilistically. For the rest of the article, we

hence assume that H

= 0.

Given these constraints, it may, at ﬁrst glance, be un-

clear how such a catalyst would offer any advantage to

violating JE. For instance, one apparent way to make use

of the catalyst is to perform a controlled unitary on S, con-

ditioned on C: For some σ

|iihi|, one uses a

unitary in Eq. (4) of the form

⊗ |i ihi |

This special case of β-catalytic channels by construction

produces random unitary channels [19, 16] on S, which

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have the form C

(·) =

(·)U

†

. But random uni-

tary channels are always unital, and therefore automati-

cally satisfy JE.

In the following, we show that there exist non-unital β-

catalytic channels that allow for a meaningful violation of

both NMW and JE, while at the same time they always

respect the Av-SL. To see the latter, we note that these

channels necessarily increase the von Neumann entropy of

the input Gibbs state. This follows from the sub-additivity

of entropy and the fact that C remains locally unchanged.

Now, since ω

(H) is the state with the least energy given

a ﬁxed entropy [20, 21], then we also have that

Tr(HC(ω

(H)) ≥ Tr(Hω

(H))

which is just the Av-SL, concomitant with the ﬁndings of

Ref. [9]. We stress that despite this property, β-catalytic

channels are in general not unital. It remains to be shown

that β-catalytic channels that violate JE and NMW do

exist. We ﬁrst show that JE can be violated already with

small quantum systems, and then turn to the violation

of NMW for macroscopic many-body systems with

physically realistic Hamiltonians.

Microscopic violation of JE. As a toy-like example of

violating the JE with β-catalytic channels, we consider a

system with three states – two degenerate (but distinguish-

able) ground states and an excited state with energy E. As

catalyst, we consider a system with two states and the uni-

tary is a simple permutation between two pairs of energy

eigenvalues of the joint system (for details, see App. B). It

is straightforward to compute the probability distribution

of work for such small systems, which in this case leads to

βW

i =

Z + 5 + 2(Z − 2)(Z − 1)

Z(Z + 1)

≥ 1,

where Z = 2 + e

−βE

is the partition function of the sys-

tem and we used 2 ≤ Z ≤ 3. We hence ﬁnd he

βW

i > 1

whenever E > 0 (since then Z < 3) and we obtain a

moderate maximum violation in the limit E → ∞ given

by he

βW

i = 7/6.

Macroscopic violation of NMW condition. We now

show that one can violate the NMW principle using cat-

alysts.

Proposition 1 (Violation of no macroscopic work

with catalysts). Let (S

(N)

)

be a sequence of N-

particle locally interacting lattice systems with Hamil-

tonian H

(N)

that satisfy mild assumptions. Then, for

suﬃciently large N, there exist values of  > 0, such

that

p(w ≥ ) (5)

can be brought arbitrarily close to

with β-catalytic

channels.

We provide a proof and full statement of the assump-

tions in Appendix D. Our assumptions are satisﬁed by

typical many-body Hamiltonians with energy windows in

which the density of states grows exponentially [22].

While the formal proof of Proposition 1 is given in the

Appendix, the idea behind it is simple and we sketch it

here on a higher level. For a given N , let e

(N)

denote

the mean energy per particle of an N -particle system that

satisﬁes our assumptions. In the proof, we show that for

systems that satisfy the above assumptions and any δ > 0,

there exists an N and a β-catalytic channel C such that

C(ω

) ≈

−

ihE

−

| +

τ, (6)

where ≈

denotes equality of the states on LHS and RHS

up to δ in trace distance, |E

−

i is some eigenvector of H

with E

−

< e

(N)

N and τ is some other “fail”-state the

details of which are irrelevant. We can interpret Eq. (6)

as describing the approximation of a work extraction pro-

tocol that results in the state |E

−

i with probability 1/2.

Now, as the result of standard concentration bounds, for

large N the mass of the thermal state ω

will be highly

concentrated around energy e

(N)

N. This implies that ev-

ery time the above work extraction protocol succeeds to

prepare the ground state, for sufﬁciently high values of

N the extracted work per particle is arbitrarily close to

 ≡ e

(N)

− E

−

/N, leading to the statement of Prop. 1.

We note that it is remarkable that catalytic channels,

which are guaranteed to satisfy the Av-2nd law, allow for

the preparation of states like the one described in Eq. (6),

in which a pure low-energy state carries much of the

weight, from a thermal state. Indeed, it has recently been

conjectured that with the help of catalysts any state tran-

sition between full-rank states that increases the entropy

is possible [9], a statement known as the catalytic entropy

conjecture. Prop. 1, and in particular the ability to prepare

the state in Eq. (6), further supports this conjecture, which

has not been proven so far (even though strong evidence

has been established).

Similar results as above also apply to the case in which

the initial state of the system is described by a micro-

canonical ensemble rather than the Gibbs state, highlight-

ing a similar contrast to ﬂuctuation theorem results in the

micro-canonical regime [23]. For detailed discussions and

proves of corresponding statements in this regime, see Ap-

pendix C.

One may wonder whether the creation of correlations

between system and catalyst is in fact necessary to violate

the NMW principle. This is indeed true, when one simply

forces the catalyst to remain uncorrelated in the deﬁnition

of β-catalytic channels. A proof of this statement along

with further discussion on this problem can be found in

Appendix I. Interestingly, such processes at the same time

allow for a violation of the Jarzynski equality. A particular

example is given by the fully thermalizing channel, which

can be realized using a catalyst that is simply a copy of the

Gibbs state of the system and the unitary simply swapping

the system and catalyst.

Required size of the catalyst. Proposition 1 not only

shows that there exist catalytic procedures that allow an

agent to bypass the work extraction bounds imposed by the

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JE – the violation of JE is in fact exponential in the system

size. In particular, (5) implies that there exist values  > 0,

such that

βW

i ≥

βN

 1

in the limit of large N. It is natural to wonder how far the

JE can be violated and how big the catalyst has to be to re-

alize a certain violation. This is clariﬁed by the following

result.

Proposition 2 (Bound on violation of JE). Let C be

any β-catalytic channel with d

= dim(H

). Then,

βW

i ≤ min{d

kσk

∞

, d kω

(H)k

∞

}

≤ min{d

, d},

where k· k

∞

denotes the ∞-norm, which, for density

matrices, equals the largest absolute value of the in-

put’s eigenvalues.

This proposition, the simple proof of which is given

in Appendix F, shows that in order to extract a growing

amount of work from a single run of a process, an external

agent will have to be able to prepare a state σ on a growing

auxiliary system and, more importantly, also have control

over the increasingly large joint system. Hence, in prac-

tice, the ability to violate JE will still be constrained by

operational limitations. To illustrate the implications of

Prop. 2, let us show how it immediately implies a bound

on P (W ). As noticed when deriving the NMW principle,

for any  ≥ 0 we have

βW

i ≥ P (W ≥ )e

β

Hence, Prop. 2 implies

P (W ≥ ) ≤ d

kσk

∞

−β

In particular this means that to extract a macroscopic

amount of work, W ≥ wN , with ﬁnite probability, d

has to grow exponentially with N (note that kσk

∞

≤ 1).

4 Multi-partite work extraction

As emphasized before, even though the state of the cata-

lyst remains unchanged in a catalytic process, in general

it builds up correlations with the system. We now show

that the correlations established between catalyst and sys-

tem allow for processes in which many agents re-use the

same catalyst to obtain highly inter-correlated work distri-

butions.

Consider n agents, each with identical systems S

, i ∈

{1, . . . , n} that are initialized in the Gibbs state ω(β, H).

For a given β-catalytic channel C with state σ on the cat-

alyst, consider the following protocol: Agent 1 runs the

standard process from Fig. 1 using the catalyst and hence

implementing C between the two measurements. After the

procedure, she then passes C on to agent 2 who repeats

this process, and so on, until the last agent has received C

and performed the process. From the catalytic nature of C,

is is clear that, for each agent, the same marginal distribu-

tion of work is obtained. However, the joint work distribu-

tion for all agents will be correlated, due to individual cor-

relations between each S

with C. We now show that the

agents can use these correlations to systematically achieve

certain global work distributions. Using the same notation

as before, let p(w

, . . . , w

) denote the global distribution

over the extracted work per particle, assuming that all S

are copies of the same N -particle system. We have the

following, proven in Appendix E.

Proposition 3 (Multiple agents). Let each {S

}

i=1

be a sequence of N-particle systems that satisfy the

conditions of Proposition 1. Then, for suﬃciently

large N , there exists an  > 0, such that

p(, −, , −, . . . ) = λ,

p(−, , −, , . . . ) = 1 − λ,

(7)

where λ can be brought arbitrarily close to 1/2 using

a sequence of β-catalytic channels on S

and C.

While (7) is clearly consistent with (1), this proposi-

tion shows that the agents can achieve joint work distri-

butions that are strongly correlated and in which subsets

of agents, in the above proposition one half of them, can

violate JE arbitrarily, at the cost of the other half. Such

distributions of work could, for example, be of interest in

situations where the target is to maximize the probability

that a subset of players extracts a positive amount work, at

the ready cost of the others, for instance in order to surpass

an activation energy. Importantly, the size of the catalyst

needed to realize the distribution (7) is ﬁxed, i.e., it does

not scale with the number of agents n.

Proposition 3 shows the existence of catalytic processes

that produce very interesting global work distributions.

This naturally raises the question what other global dis-

tributions can be obtained in a setting without making the

size of the catalyst depend on the number of rounds. Our

results, however, already imply that not every distribution

compatible with the Second Law can be obtained in such

a way. For instance, Proposition 2 implies that the distri-

bution

p(, , , , . . . ) = p(−, −, −, −, . . . ) ≈ 1/2

cannot be obtained via β-catalytic channels, since other-

wise there would exist a catalyst of ﬁxed size that would

allow, for any n, the total work W = n to be extracted

with probability approximately 1/2, in violation of Propo-

sition 2.

5 Summary and future work.

In this work we have studied work extraction protocols

from states at thermal equilibrium. We signiﬁcantly ex-

pand the common setting of ﬂuctuation theorems under

Accepted in Quantum 2020-02-06, click title to verify. Published under CC-BY 4.0. 5

cyclic, reversible processes by introducing a catalyst—an

additional system which, on average, remains unchanged

after the protocol and can thus be re-used. This extension

enables for distributions of work extraction that are not

attainable without a catalyst. More precisely, one can by-

pass the stringent conditions imposed by the JE, achieving

positive work per particle with high probability, even for

macroscopic systems. Furthermore, it allows for interest-

ing, correlated work distributions when many agents use

the same catalyst.

Our constructions illustrate in a striking way that the ab-

sence of correlations, sometimes referred to as ‘stochastic

independence’, can also be a powerful thermodynamic re-

source [24]. This complements ﬁndings where the initial

presence of correlations between a system and an ancilla

are used to bypass the standard constraints imposed by

ﬂuctuation theorems [25, 26]. We discuss the connection

of our work to these ﬁndings in more detail in Appendix

H. We believe that the further study of work distributions

that can be obtained by collaborating agents by means of

β-catalytic channels will yield both foundational and prac-

tical insights.

We further believe that it is an interesting open prob-

lem to study how the size of the catalyst has to scale if

one wishes to maximize the probability to extract a certain

amount of work. For example, in the context of a many-

body system one might be content with extracting only an

amount of work of the order of

√

N if in exchange for

that one can either increase the probability for it to happen

signiﬁcantly or can reduce the size of the catalyst consid-

erably (and hence the complexity of the unitary required

to be implemented).

It would be interesting to understand the relation be-

tween our results and a more generalized type of JE in

the presence of information exchange [27], for example

in a Maxwell demon scenario. In particular, in Ref. [28]

it was also demonstrated that by using feedback control,

one may also violate JE while respecting the Av-SL. More

generally, our results also raise the question whether other

phenomena –usually described as forbidden by the second

law, or as occurring with vanishing probability– can be

made to occur with high probability using catalysts. For

example, is it possible to reverse the mixing process of

two gases or induce heat ﬂow from a cold to a hot system

with ﬁnite probability in macroscopic systems? The tech-

niques developed in this work provide a promising ansatz

for the study of this and similar questions.

Acknowledgements. We thank Markus P. M

uller and

Alvaro M. Alhambra for valuable discussions and anony-

mous referees for interesting comments. P. B. ac-

knowledges support from the John Templeton Founda-

tion. H. W. acknowledges support from the Swiss Na-

tional Science Foundation through SNSF project No.

200020 165843 and through the National Centre of Com-

petence in Research Quantum Science and Technology

(QSIT). N. H. Y. N. acknowledges support from the

Alexander von Humboldt Foundation. R. G. has been sup-

ported by the DFG (GA 2184/2-1). J. E. acknowledges

support by the DFG (FOR 2724), dedicated to quantum

thermodynamics, and the FQXi.

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A NMW for Gibbs preserving maps

Thermalizing quantum maps, in particular those studied

in the resource theoretic framework, are maps that model

the evolution of a non-equilibrium quantum state as it ex-

changes heat with its surrounding thermal bath. Several

variants of these maps exist [29, 6, 7, 30, 31], but a com-

mon feature is that they are Gibbs preserving (GP), namely

that the Gibbs canonical state is a ﬁxed point of such maps.

Thermalizing maps are often viewed as “free operations”

in a resource theoretic context, since they allow only for

heat (instead of work) exchange with an environment in

thermal equilibrium. In this section, we demonstrate two

things: First, that even such thermodynamically “cheap”

channels may violate the JE very strongly, due to non-

unitality. Secondly, that they cannot be used to violate the

NMW condition. A diagrammatic overview over the vari-

ous properties of channels with respect to JE and NMW is

given in Fig. 2.

We now turn to the ﬁrst point. Given a d-dimensional

system S with Hamiltonian H, the violation of JE can be

calculated for the thermalizing channel as

−βW

i = d

−βE

|C[I/d] |E

= d

−βE

|ω

(H) |E

i =

eﬀ

where d

eﬀ

:= 1/Tr(ω

(H)

) is known as the effective di-

mension [32] of the thermal state. One sees from the above

that JE is always violated for β > 0, since d

eﬀ

≤ d, with

equality only when ω

(H) = I/d is maximally mixed.

For N non-interacting i.i.d. systems, both d and Tr(ρ

)

scale exponentially with N, leading to an exponential vio-

lation in N for JE.

Turning to the second point, one may wonder how this

notion of thermodynamically free channels can be recon-

ciled with the fact that JE is violated. However, note that

in the standard JE setting, the work variable is traditionally

deﬁned in terms of a ﬂuctuating (measured) energy dif-

ference in the system, and does not inherently distinguish

between work and heat contributions – unlike resource-

theoretic settings where heat ﬂow is allowed for free, but

measurements incur a thermodynamic cost. Here, we con-

sider an operationally more meaningful characterization

(NMW as deﬁned in Def. 1 of the main text), and show

that NMW cannot be violated using channels that preserve

the Gibbs state in generic many-body systems. The only

assumptions that we make are that i) the system has uni-

formly bounded, local interactions on a D-dimensional

regular lattice and ii) a ﬁnite correlation length, i.e., the

temperature is non-critical.

Lemma 3 (Non-violation of NMW for Gibbs-pre-

serving maps). No channel E that preserves the Gibbs

state can violate NMW for locally interacting many-

body systems at a non-critical temperature.

Proof. We aim at showing that for any a > 0,

p(w ≥ a) = p(W ≥ aN) → 0 as N → ∞. The

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basic idea behind our proof is to make use of typ-

icality. Let e

(N)

denote the energy density of the

N-particle system and denote by Π

(N)

the projector

onto energy eigenstates with energies in the interval

N,δ

:= [(e

(N)

− δ)N, (e

(N)

+ δ)N]. Finally, denote

by p(·) the initial probability distribution of energy of

the thermal state τ

(N)

, e.g., the probability that the

initial energy measurement yields E

∈ T

N,δ

is given

p(T

N,δ

) := Tr



(N)



A theorem by Anshu [33] shows that under the

given conditions most weight of the thermal state τ

(N)

of the N-particle system is contained in a typical sub-

space. More precisely, for a many-body system de-

scribed by a D-dimensional lattice, there exist con-

stants C, K > 0 such that for any δ > 0 we have

p(T

N,δ

) ≥ 1 − Ce

−

(δ

1+D

. (8)

This is equivalent to saying that

p(T

N,δ

) ≤ Ce

−

(δ

1+D

where T

N,δ

= R \ T

N,δ

. In particular, in the case of

D = 0, i.e., N non-interacting systems, we ﬁnd the

usual scaling obtained from Hoeﬀding’s inequality. In

the following, for simplicity of notation, we write σ

(N)

and consider the normalized state σ

obtained by

restricting τ

(N)

to the subspace Π

(N)

p(T

N,δ

)

Let us further write E(σ

1(2)

) = σ

1(2)

, where σ

= σ

by assumption. Since the trace distance d(ρ

, ρ

)

Tr(|ρ

− ρ

|) fulﬁlls the data processing inequality,

d(σ

, σ

) = d(σ

, σ

) ≤ d(σ

, σ

) = p(T

N,δ

Using the operational meaning of trace distance

d(ρ

, ρ

) = max

0≤M≤I

|Tr(M(ρ

− ρ

))| [34], this means

that

|Tr(Π

(N)

) − Tr(Π

(N)

)| ≤ p(T

N,δ

) (9)

and, in turn,

Tr(Π

(N)

) ≥ p(T

N,δ

) − p(T

N,δ

) = 1 − 2p(T

N,δ

).(10)

To see this, note that (10) follows from (9) directly if

Tr(Π

(N)

) ≤ Tr(Π

(N)

), and as

Tr(Π

(N)

) > Tr(Π

(N)

) ≥ Tr(Π

(N)

) − p(T

N,δ

)

otherwise. This means that, conditioned on the fact

that the initial state was within the typical energy

Jarzynski

Unital channels

Average Second Law

- catalytic channels

No Macroscopic

- Gibbs preserving

maps

Equality

Work

Figure 2: A summary of diﬀerent criteria (Av-SL, NMW and

JE) mentioned in the main text, with examples of maps ac-

cording to this characterization.

window (E

∈ T

N,δ

), the ﬁnal energy E

is also within

this energy window except with probability 2p(T

N,δ

which is (sub-)exponentially small in N . We will use

this later.

We are now ready to evaluate the probability of

obtaining macroscopic work.

p(w ≥ a) = p(T

N,δ

) · p(w ≥ a|E

∈ T

N,δ

)

+ p(T

N,δ

) · p(w ≥ a|E

∈ T

N,δ

)

≤ p(w ≥ a|E

∈ T

N,δ

) + p(T

N,δ

We can estimate the ﬁrst term as

p(w ≥ a|E

∈ T

N,δ

) ≤ p(E

≤ (e

(N)

+ δ − a)N |E

∈ T

N,δ

We now choose δ = a/2 and get

p(w ≥ a|E

∈ T

N,δ

) ≤ p(E

≤ (e

(N)

− a/2)N|E

∈ T

N,δ

)

≤ Tr



I − Π

(N)

a/2

i

≤ 2p(T

N,a/2

where we have used (10) in the last step. Altogether,

we thus ﬁnd

p(w ≥ a) ≤ 3p(T

N,a/2

which decays to zero (sub-)exponentially by (8). This

concludes the proof.

As a side-remark, we note that if the Gibbs-preserving

channels that appear here are interpreted as modelling the

interaction with a heat bath, then the above result can be

interpreted as a ”no macroscopic heat” statement: If a

macroscopic system is brought in thermal contact with a

heat bath at the same temperature, then the probability of

an exchange of a macroscopic amount of heat is arbitrarily

small in the system size.

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B Microscopic toy example

In this section, we show that already for small systems and

using catalysts, the JE can be violated. We do so by con-

structing non-unital catalytic channels. Indeed, such maps

can be realized “quasi-classically”, in the sense that in the

construction it is sufﬁcient to consider the energy spectra

of the involved states and that all unitaries are simple per-

mutations of those values. We consider a 3-level system

with energy levels E

= 0, E

= ∆ in the ther-

mal state

w =



Z − 2



where Z = 2 + exp(−β∆) is the partition function and

we express the state as a probability vector, such that w

denotes the ith eigenvalue of the thermal state. For later,

we observe that 2 ≤ Z ≤ 3.

We are going to construct a simple non-unital catalytic

channel that involves a 2-dimensional catalyst. Let e

and

denote the basis states for the vector spaces V

and V

describing the system and catalyst respectively. We deﬁne

the permutation π acting on the joint vector space V

⊗V

as that permutation which exchanges the respective levels

⊗ f

⇔ e

⊗ f

and e

⊗ f

⇔ e

⊗ f

and leaves

all other entries unchanged (see Fig. 3). For the catalyst to

remain unchanged for this permutation and initial system

state, it is easy to check that the catalyst has to be given by

the vector

q =



Z − 1

Z + 1



Now, the catalytic channel C induced by this catalyst and

permutation on the system has the general effect

C(p

, p

) = (q

+ q

, q

+ q

, q

+ q

so that, in particular, the maximally mixed input state is

mapped to

C(I/3) =



Z − 1

Z + 1



which is different from the maximally mixed vector for

any ∆ > 0.

What is more, we can also directly calculate the work-

distribution p(w), yielding

p(0) =

Z(Z + 1)

[Z + 3 + 2(Z − 2)(Z − 1)] ,

p(∆) =

2(Z − 2)

Z(Z + 1)

p(−∆) =

Z − 1

Z(Z + 1)

We now want to compute he

βW

i. To do so, it is useful to

note that e

−β∆

= Z −2 and hence e

β∆

= 1/(Z −2). We

ﬁnd

βW

i =

Z + 5 + 2(Z − 2)(Z − 1)

Z(Z + 1)

≥ 1.

→

+ q

Figure 3: We represent the joint state of system and catalyst

by means of a table. Left: At the beginning the joint system

starts out in a product state, so that the entry (i, j) is given

by the product of the ith eigenvalue of the system and jth

eigenvalue of the catalyst. Right: After applying the permu-

tation highlighted in red, the marginal state of the system,

given by the rows sums, has changed, while the marginal

state of the catalyst (given by the column sums), has to re-

main invariant. For a two-dimensional catalyst, specifying

the permutation and initial system state ﬁxes the catalyst

state.

In fact, this quantity is larger than 1 whenever Z < 3, cor-

responding to ∆ > 0. Its maximum is given as 7/6 for

Z = 3, which corresponds to ∆ → ∞. Thus, the Jarzyn-

ski inequality is violated. At the same time the second law

is fulﬁlled as expected, since p(−∆) ≥ p(∆).

C Work extraction for initial micro-

canonical ensembles

In this appendix, we show that a statement similar to

Proposition 1 of the main text holds in the slightly different

setting of a micro-canonical initial state. This serves two

purposes: i) in statistical mechanics, one often assumes

that closed, macroscopic systems are described by micro-

canonical ensembles due to the postulate of equal a priori

probabilities of microstates corresponding to a macrostate.

ii) The proof for the microcanonical initial state is con-

ceptually simpler, but also provides the blueprint for the

slightly more involved proof in the case of a canonical

state, which is provided in Sec. D.

In the following, we denote by I ⊂ R an energy win-

dow, by g(I) the number of energy eigenstates in this win-

dow,

g(I) =

∈I

and the corresponding micro-canonical state by

Ω

(I) =

g(I)

∈I

ihE

A micro-canonical energy window around energy density

e is any energy window I(e) of the form [e − O(

√

N), e],

where N is the number of particles.

The only difference to the standard setting described in

the main text (as depicted in Fig. 1) is that the initial state

differs from the thermal state ω

(H). Instead, it is given

by the micro-canonical ensemble. In other words, given a

micro-canonical energy window I, we consider channels

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C of the form

C(·) = Tr

(U(· ⊗ σ

†

)

s.t. Tr

(U(Ω

(I) ⊗ σ

†

) = σ

We carry over notation from the main text, so that p(w ≥

) denotes the probability of measuring the system’s en-

ergy per particle decrease by at least an amount , and so

on. Furthermore, we take the catalyst Hamiltonian in our

construction to be H

= I.

We will now ﬁrst show that the NMW principle also

holds for micro-canonical states of generic many-body

systems. After that we will show that it can be circum-

vented using catalysts. To show the validity of the NMW

principle we will use the same reasoning as presented in

Ref. [35], where the NMW principle has been studied be-

fore. Thus, the following proof is essentially a reproduc-

tion for the convenience of the reader. We consider a se-

quence of many-body Hamiltonians H

(N)

on N particles

with the generic property of having an exponential density

of states:

g((−∞, E])

≤E

1 = e

Nµ(E/N )−o(N )

, (11)

where µ is a strictly monotonic and differentiable function

independent of N and o(N) denotes terms small compared

to N, lim

N→∞

o(N)/N = 0.

Proposition 4 (NMW for micro-canonical states).

Consider a sequence of N-particle Hamiltonians ful-

ﬁlling (11) and a sequence of micro-canonical energy-

windows I

(N)

= [eN, eN + δ

√

N] around energy den-

sity e (with δ > 0 ﬁxed). Then for any unital channel

acting on the N-particle system, the probability of ex-

tracting work w per particle is bounded as

p(w > ) ≤ Ce

−µ

(e)N+o(N)

where C > 0 is a constant and µ

denotes the deriva-

tive of µ.

Proof. Let I

≤

:= (−∞, (e − )N + δ

√

N], denote by

≤

) the projector onto energy-eigenstates with en-

ergies below (e−)N +δ

√

N and let U denote a unital

channel. In the following, we write I instead of I

(N)

to simplify notation. Then

p(w > ) ≤ Tr (P

≤

)U[Ω

(I)])

∈I

g(I)

Tr (P

≤

)U [ |E

ihE

|])

≤

g(I)

Tr (P

≤

)U [I]) =

g(I

≤

)

g(I)

Writing ˜e := e + δN

−1/2

, we have

g(I) = e

Nµ(˜e)−o(N)

− e

Nµ(e)−o(N)

= e

Nµ(˜e)−o(N)



1 − e

−N(µ(˜e)−µ(e))+o(N)



≈ e

Nµ(˜e)−o(N)

where in the last estimation we use that µ is strictly

monotonic. In particular, we can estimate the expo-

nential in the parenthesis as

−N(µ(˜e)−µ(e))−o(N)

= O



−δµ

(e)N

1/2



where µ

denotes the derivative of µ. Using g(I

≤

) =

N(µ(˜e−)−o(N)

we then ﬁnd

p(w > ) ≤

−N(µ(˜e)−µ(˜e−))+o(N)

1 − O(e

−δµ

(e)

√

)

≤ Ce

−µ

(e)N

We have here used that µ is differentiable to prove this

result. Similar results would follow for weaker notions

of regularity of µ, such as Lipschitz-continuity. Having

proven the NMW principle for generic many-body sys-

tems, let us now show how to circumvent it using catalysts.

Proposition 5 (Overcoming NMW using catalysts).

Consider a Hamiltonian H

and a microcanonical

state Ω

(I), with I a micro-canonical energy window

around energy density e. Suppose there exists an en-

ergy window I

with g(I

) = g(I)

. Then, for any

0 ≤ e

−

< e, there exists a catalytic channel such that

p(w ≥ e − e

−

) =

Before giving the proof of the proposition, let us em-

phasize again that the required conditions on the Hamilto-

nian are very weak. In particular, the conditions are (ap-

proximately) fulﬁlled if the density of states is well ap-

proximated by an exponential in the range of energies that

we are working in, a condition that is typically fulﬁlled in

many-body systems and, as we have seen above, leads to

an NMW principle if we do not allow for catalysts.

Proof. A sketch of the proof is given in Fig. 4. The

proof is constructive in the sense that we provide an

explicit catalyst and unitary. We ﬁrst introduce some

useful notation. Deﬁne g := g(I), g

:= g(I

) = g

and let P

(I) and P

) be the projectors onto the

corresponding energy subspaces. Let |E

−

i be any

eigenstate of the Hamiltonian such that 0 ≤ E

−

/N =

−

≤ e. Following this notation, the initial state of

the system is

Ω

(I) =

(I).

The aim is to bring the system to a state that

is an equal mixture of |E

−

ihE

−

| and Ω(I

). To

do this, we employ a catalyst of dimension d

g + 1. Let {|i i

}

i=1

be an arbitrary orthonormal

basis on the Hilbert-space of the catalyst and let

i=1

|iihi|. The initial state on the catalyst

is given by

σ =

ihd

Accepted in Quantum 2020-02-06, click title to verify. Published under CC-BY 4.0. 10

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

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Final

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...

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Initial

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...

{



Figure 4: Proof sketch for Proposition 5: Top: We represent the initial product state of system and catalyst by means of

a table, using the fact that both are initially diagonal in the energy eigenbasis: Ordering the spectra of both states non-

increasingly, the entry (i, j) of the table corresponds to the product of the i-th eigenvalue of the system (corresponding to the

a particular energy eigenstate) and the j-th energy eigenvalue of the catalyst. We focus on three regions in the table—denoted

top (T), middle (M), bottom (B)—corresponding to two degeneracy bands I and I

and (the projector onto) a single energy

eigenvector |E

−

i: Since the system is initially in the micro-canonical ensemble with energy window I, the support of the

joint state is initially contained in the coloured middle band. The catalyst is constructed as carrying half of its mass uniformly

distributed over d

− 1 of its entries and the other half in a single entry. This means that the middle band is divided into two

subregions, middle left (ML) and middle right (MR), where the total probability mass coloured in blue equals the mass coloured

in red. Furthermore, each of these subregions has its mass uniformly distributed over its entries. Bottom: By construction,

both the subregions BL and MR as well as ML and TR have the same number of entries. Hence, we can swap BL and MR by

means of a permutation, and similarly for ML and TR. This permutation results in a reduced state on S of the form Eq. (13)

and hence produces the claimed work extraction probability. Moreover, it leaves the marginal state of the catalyst unchanged,

so that the permutation induces a valid catalytic channel.

Accepted in Quantum 2020-02-06, click title to verify. Published under CC-BY 4.0. 11

We deﬁne the unitary U by the conditions

U[P

(I) ⊗ |d

ihd

†

= |E

−

ihE

−

| ⊗ P

U[P

(I) ⊗ P

†

= P

) ⊗ |d

ihd

This is possible since i) the corresponding subspaces

have the same dimension, ii) the subspaces in the

two equations are orthogonal and iii) subspaces of the

same dimension can always be mapped into each other

by a unitary. In fact there will be many diﬀerent uni-

taries achieving this, and any of them is ﬁne for our

purposes.

Applying U to the state Ω

(I) ⊗ σ

one obtains

U (Ω

(I) ⊗ σ

) U

†

U (P

(I) ⊗ P

) U

†



(I) ⊗

ihd



†

Ω

) ⊗ |d

ihd

−

ihE

−

| ⊗ P

. (12)

It is clear from (12) that

(U(Ω

(I) ⊗ σ

†

) = σ

as required for a catalytic channel. Moreover, the

quantity of interest P (w ≥ e−e

−

) given by this chan-

nel C (deﬁned by U and σ

) can be derived by noting

that

C(Ω

(I)) =

Ω(I

) +

−

ihE

−

|, (13)

so that p(W ≥ e − E

/n) =

D Proof of Proposition 1 in the main

text

In this section, we provide the proof and full statement of

Proposition 1 in the main text. This proof is very similar

to that of the micro-canonical case presented in the previ-

ous section, we will hence only describe the adjustments

that have to be made. Also, unlike in Appendix C, we now

again consider the standard setting and deﬁnition of cat-

alytic channels as introduced in the main text. In the fol-

lowing, we denote by P

(I) the projector onto a speciﬁc

energy-window I. Then g(I) is equal to the rank of P

(I).

We consider Hamiltonians H

(N)

on a regular lattice Λ

(N)

of N sites and assume that the H

(N)

(for different values

of N) constitute a sequence of local, uniformly bounded

Hamiltonians:

(N)

x∈Λ

(N)

where each term h

acts on sites at most a distance l away

from x and the norm of each term is bounded as kh

k ≤ h

independent of the system size for some constant h.

Proposition 6 (Lower bound to the probability of

work extraction). Fix an inverse temperature β > 0

and consider a sequence of local, uniformly bounded

N-particle Hamiltonians H

(N)

on a regular, D-

dimensional lattice. Assume that the states ω

(N)

)

have a ﬁnite correlation length bounded by a con-

stant and denote by e

(N)

the energy density corre-

sponding to β. Let δ > 0 be ﬁxed and consider

(N)

:= [e

(N)

N − δ

√

N, e

(N)

N]. Further assume that

there exist micro-canonical energy windows I

(N)

with

g(I

(N)

) = g(I

(N)

)

. Then, for suﬃciently large N,

there exists, for any 0 < e

−

< e

(N)

, a corresponding

sequence of catalytic channels such that

p(w ≥ e

(N)

− e

−

) ≥ 1/2 − Ce

−

(

)

1+D

where C, K > 0 are constants.

Before giving the proof, we again emphasize the weak-

ness of the assumptions in the statement, which, in the

limit of large N, can be satisﬁed to arbitrary precision if

the density of states grows exponentially within I

(N)

, as is

typically the case. Furthermore, let us emphasize that the

energy densities e

(N)

ﬂuctuate arbitrarily little (for sufﬁ-

ciently large N ) from a constant e due to the locality of

temperature [36].

Proof. The proof follows the proof for the micro-

canonical case in Appendix C. In particular, the uni-

tary that we use is exactly the same as that con-

structed in the proof for the micro-canonical case.

However, here we do not construct the state of the

catalyst explicitly, but allude to Lemma 4, which en-

sures there is always some catalyst given the unitary

that we consider. What remains to be done is to show

that for every such catalyst the probability distribu-

tion of work is as claimed. To do this, we denote by

r the initial probability of an energy-window I in the

initial thermal state given by

r(I) = Tr(P

(I)ω

))

and by r

−

= hE

−

|ω

) |E

−

i the initial weight

on the low-energy eigenstate |E

−

i. Here and in the

following, we drop the explicit dependence on the

system-size for simplicity of notation. The following

arguments work as long as N is large enough such that

Accepted in Quantum 2020-02-06, click title to verify. Published under CC-BY 4.0. 12

the energy-windows I and I

are disjoint. Denote by

}

i=1

the spectrum of the catalyst. By considering

the action of the used unitary, it is easy to see that a

necessary condition for the transition being catalytic

under the given unitary is that

(r(I) + r(I

)) = (1 − q

)(r(I) + r

−

). (14)

This can be seen, for example, from Fig. 4, where the

above represents the condition of catalyticity for the

right-most column. Solving in (14) for q

, we ﬁnd

that

r(I) + r

−

2r(I) + r

−

+ r(I

)

We now invoke the result from Ref. [33] (as previously

in the proof of Lemma 3) which implies that

r(I) ≥ 1 −

where there exist constants C, K such that



≤ Ce−





1+D

For large enough N, the energy windows I and I

are disjoint. Hence 0 ≤ r

−

+ r(I

) ≤ 1 − r(I) and we

ﬁnd

≥

r(I)

2r(I) + 1 − r(I)

r(I)

1 + r(I)

≥

r(I)

≥

(1 − 

) .

Finally, we ﬁnd

p(w ≥ e − e

−

) ≥ P (E

= E

−

∈ I)w(I) = q

· r(I)

≥

(1 − 

)

≥

− 

Lemma 4 (Existence of catalysts). Let ρ

be a quan-

tum state on a ﬁnite-dimensional Hilbert-space H

and U be a unitary on the Hilbert-space H

⊗ H

where H

is an arbitrary ﬁnite-dimensional Hilbert-

space. Then there exists a density matrix σ

such

that



U(ρ

⊗ σ

†



= σ

Proof. The map σ

7→ Tr



U(ρ

⊗ σ

†



speciﬁes

a quantum-channel. Since every quantum channel is

a continuous map on the compact and convex set of

states, it has a ﬁxed point by Brouwer’s ﬁxed point

theorem ([37], Section 4.2.2).

E Proof of Proposition 3 in the main

text

Proposition 3 in the main text follows straightforwardly

once we realize that we can tune the process used in the

(a)

(b)

(c)





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Figure 5: The idea behind the proof of Proposition 3 in the

main text: For any choice of unitary, we can understand

the second condition in the Def. 2 of the main text as the

deﬁnition of a quantum channel

C acting on C. We ﬁnd

C and two states σ

−

, σ

with the following properties:

(a) If the initial state of the catalyst is σ

, the result of

running the standard protocol is to extract positive work 

from the system, while the state of the catalyst is changed to

C(σ

) = σ

−

. (b) The same unitary, however, for initial state

−

, extracts negative work − and changes the catalyst state

C(σ

−

) = σ

. (c) Hence, if we initialize the catalyst in the

state σ =

(σ

+σ

−

), then there are two “branches” of work

extraction distributions, each occurring with probability 1/2,

while the resulting channel on S

is catalytic for every i. Note

that, if the agent knew whether her input state was σ

or σ

−

then she could condition her unitary U on this knowledge and

achieve the claimed work distribution easily. Hence, the key

achievement of the proof is to show that agents can achieve

correlated work distributions without knowing the initial state

of the catalyst.

construction of the proof for Proposition 1 in the main

text in such a way that its repeated application implies

the claimed work distribution. This follows because we

have great freedom in choosing the state E

−

. In par-

ticular, in terms of notation of the previous section, let

(N)

denote the energy density around which the window

(N)

is centered. Then we choose E

−

in such a way that

e − E

−

/N = e

− e to ensure that the extracted and in-

vested amount of work in every iteration are exactly the

same. The above choice of E

−

is always possible for the

Hamiltonians with exponentially growing density of states

that we consider (for which e

will not be much greater

than e.)

Fig. 5 provides a sketch of the proof. For the many-

player process described in the main text, let

p(w

, w

, . . . |w

)

denote the work probability distribution for agents 2 to n

conditional on the player 1 extracting work w

. The key

recognition then is that, for any n, by construction of the

catalytic channel,

p(w

, w

, . . . |w

) = 1 (15)

whenever w

= −w

i−1

for all i ∈ {2, . . . , n}, while

p(w

, w

, . . . |w

) = 0

Accepted in Quantum 2020-02-06, click title to verify. Published under CC-BY 4.0. 13

in all other cases. This is because, if the extracted work in

the ﬁrst round was negative, corresponding to an increase

in the system’s energy, then by construction of the unitary,

the ﬁnal state of the catalyst is σ

= |d ihd | with proba-

bility one, since all transitions that lead to an increase in

energy on the system result in this ﬁnal state. This, in turn,

is sufﬁcient to determine that, for the second player, the

application of the unitary to this catalyst state σ

and her

copy of ω

(H) will result in a decrease of the system’s en-

ergy (and hence positive work extraction) and a ﬁnal cata-

lyst state σ

with support on the subspace

|iihi|, etc.

This reasoning can be applied to an arbitrary number of

agents and also to the case in which the extracted work in

the ﬁrst round was positive, and hence implies (15). The

claimed work distributions then follow from

p(w

, w

, . . . , w

) = p(w

, w

, . . . |w

)p(w

together with Proposition 1 in the main text. We also note

that a similar conclusion holds in the case of the micro-

scopic toy-example presented in Section B, where this be-

haviour can be checked easily by explicit calculation.

F Proof of Proposition 2 in the main

text

Given a catalytic channel C, let U denote the unitary chan-

nel applied to the joint system SC when dilating the chan-

nel. The key observation is that, if U is unitary, then U

∗

trace-preserving and hence maps quantum states to quan-

tum states (in fact, this property holds for all unital chan-

nels). Here,

∗

denotes the Hilbert-Schmidt adjoint. We

then write

βW

i = Tr (ωC(1))

= Tr (ω ⊗IU(1 ⊗ σ))

= d



∗



ω ⊗



1 ⊗ σ



≤ d

k1 ⊗ σk

∞



⊗ σ



= d

kσk

∞

Here, the ﬁrst equality is simply Eqn. 2 in the main text

and we write ω instead of ω

(H). Similarly, we get

βW

i = dTr



(ω ⊗ I)U



⊗ σ



≤ d kω ⊗ Ik

∞

= d kωk

∞

G Non-trivial Hamiltonian on the cat-

alyst

In this section we show that the probability distribution of

work done on the system is independent of the Hamilto-

nian on the catalyst. To do this, let us ﬁrst assume we

had a catalytic process that uses a catalyst with a non-

trivial Hamiltonian H

and a quasi-classical state σ

, i.e.,

, σ

] = 0. We assume that σ

is quasi-classical, since

it is well known that it is impossible to associate a mean-

ingful random variable of work in the case coherent initial

states [38]. Using the two-time measurement process on

the system and catalyst together, we can then associate a

bi-partite work-distribution P (W

(S)

, W

(C)

), where

(S)

= E

(S)

− E

(S)

denotes the work done on the system and

(C)

= E

(C)

− E

(C)

the work done on the catalyst. The work distribution on

the system is simply given by the marginal



(S)





(S)

, W

(C)



(C)

Let us write σ

(C)

ihE

(C)

| and ω

(H) =

(S)

ihE

(S)

|. We then get



(S)



(S)

−E

(S)

(C)



(S)

, E

(C)

(S)

, E

(C)



P (E

(S)

)P (E

(C)

)

(S)

−E

(S)

(C)

(S)

| ⊗ hE

(C)



(S)

ihE

(S)

| ⊗ |E

(C)

ihE

(C)



†



(S)

i ⊗ |E

(C)

(S)

−E

(S)

|Tr



(S)

ihE

(S)

| ⊗ σ



†



(S)

−E

(S)



(S)

ihE

(S)



(S)

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It is hence identical with the one obtained on the system

alone when we think of the catalyst as a system with a

trivial Hamiltonian, that is, with the distribution as deﬁned

above Eqn. 2 in the main text. This shows that we can al-

ways assume that the catalyst has a trivial Hamiltonian, in

which case it is clear that no energy ﬂows from the catalyst

to the system, even probabilistically. Therefore, such an

energy ﬂow is not necessary to implement catalytic transi-

tions.

H Comparison with literature on gen-

eralized Jarzynski equalities in the pres-

ence of correlations

In recent years, the role of correlations, speciﬁcally quan-

tiﬁed by the mutual information, has been well studied,

in particular with respect to its inﬂuence on the Jarzynski

equality [25, 26], even leading up to experimental demon-

strations to test these theoretical results [28, 39]. One

may ask, how the results of this manuscript ﬁt in the con-

text of that line of research. This section provides a brief

overview of the main differences.

In Ref. [25] the core observation is that the presence

of initial correlations between a system S and an ancillary

(catalyst) C can be used to create a thermodynamic advan-

tage, in the sense that such processes obey a generalized

JE and Second law and hence can be used to by-pass the

constraints imposed by the original JE and Second law.

Speciﬁcally, [25] derives (according to their generalized

version of Jarzynski equality) a bound on the work per-

formed on the system that is given by

hW i ≥ h∆F i + h∆E

int

i + β

−1

h∆Ii,

where h∆F i is the difference between ﬁnal and initial

equilibrium free energy on the system, h∆E

int

i for the

energy difference coming from the interaction Hamilto-

nian between system and catalyst, and ﬁnally h∆Ii is the

change in mutual information between system and cata-

lyst. For our setup, both h∆F i and h∆E

int

i are zero.

Given that the extracted work W

ext

= −W , the above

bound reduces to

ext

i ≤ −β

−1

h∆Ii,

which says that if one allows the consumption of mutual

information (leading to ∆I < 0), then it is possible to

violate the average second law, namely extract some pos-

itive amount of W

ext

from a Gibbs state, for instance by

reducing the entropy of the system in the process. This

particular viewpoint of correlations (information) being a

thermodynamic resource is a mature and well-studied one.

In our setting, however, the initial state of system and

catalyst are always uncorrelated, which means that we al-

ways have h∆Ii ≥ 0. Hence it is clear that the type of

catalytic operation studied in Ref. [25] cannot correspond

to our setting, since the generalized JE and Second law al-

low for violations of the original JE and Second law only if

h∆Ii < 0. The difference to our setting, however, is easily

understood. It lies in the fact that here we allow for more

general joint evolutions of the system and the catalyst. In-

deed, it is easy to see that under the requirement that the

initial state between catalyst and system be uncorrelated,

the channels that can be implemented on the system via

the operations allowed in Ref. [25] are unital channels, for

which we show above that they cannot be used to by-pass

the JE (see Fig. 2). This is because in the above works, the

catalyst is required to not evolve over time. In contrast,

the notion of a β-catalytic channel allows for the evolu-

tion of the catalyst to be non-trivial, as long as the ﬁnal

density matrix describing the catalyst is unchanged. Since

this constraint only requires the statistical invariance of

the catalyst, this allows for a much broader class of evo-

lutions to be implemented on the system and hence ex-

plains how we can by-pass the JE and NMW in a setting

where the marginal entropy of the system has to increase.

In summary, the key differences to the line of work rooted

in Refs. [25, 26] are that we study processes that by-pass

the JE by means of the creation of correlations paired with

catalysts that evolve non-trivially over time, while in the

above work processes are studied that by-pass the JE by

means of the absorption of initial correlations paired with

catalysts that do not evolve over time.

I Is it necessary to establish correla-

tions with the catalyst?

In our deﬁnition of β-catalytic channels, we allow the cat-

alyst to become correlated with the system. These cor-

relations are certainly necessary for the correlated multi-

player strategies discussed in the main text, but one might

wonder whether they are also necessary to violate NMW

on a single system. To make this question concrete, con-

sider the set of β-trumping channels, where a quantum

channel T is in this set iff it has the form

T (ρ) = Tr

(N(ρ ⊗ σ)),

where N(I) = I is unital and N(ω

(H) ⊗ σ) = ρ

⊗ σ.

Note that in the case of β-catalytic channels, we restricted

the corresponding channel N to be unitary. Here, we al-

low instead for the more general class of unital channels.

We will prove that in the unitary case, NMW cannot be

violated by β-trumping channels even though Jarzynski’s

equality may be violated. We will also present arguments

that suggest that the same is true in the unital case.

It is worth noting, for starters, that the fully thermaliz-

ing channel is exactly a β-trumping channel where σ =

(H), and N is a unitary swap between the system and

catalyst. Thus, even in the case of a unitary channel N,

such β-trumping channels can violate Jarzynski’s equality.

On the other hand, in the main text we have demonstrated

that the thermalizing channel cannot violate NMW since

it is Gibbs preserving. Hence, the above leaves open the

question whether the NMW condition can be violated by

Accepted in Quantum 2020-02-06, click title to verify. Published under CC-BY 4.0. 15

means of β-trumping channels. However, we do not be-

lieve that this is the case, for the following reasons:

i) Our constructions of violating NMW can not work in

the trumping case. This is because in the trumping setting

the so-called min-entropy S

∞

(minus log of the largest

eigenvalue) of the ﬁnal state has to be at least as large as

that of the initial state (see for example Ref. [6]). How-

ever, in our constructions, the ﬁnal min-entropy is essen-

tially given by −log(p(w ≥ ) ≈ log(2), whereas the

initial min-entropy is extensive in N. It thus decreases by

a macroscopic amount.

ii) The previous point also suggests a route for arguing

that β-trumping channels cannot be used to violate NMW

in general: We now present an argument that rules out vi-

olations of NMW in the case of a microcanonical initial

state Ω with energy density e, but we expect that similar

statements hold true for the canonical case due to equiv-

alence of ensembles-type of arguments. Because of the

highly peaked probability distribution of the energy den-

sity for a macroscopic, non-critical many-body system, it

is easy to see that the probability p(w ≥ ) to extract work

per particle at least  is (up to arbitrarily small corrections

for large N) given by the total probability of measuring

an energy below (e − )N in the ﬁnal state T (Ω). Let us

denote the projector onto these energies by P . We then

have

p(w ≥ ) ≈ Tr[P T (Ω)],

where the approximation is arbitrarily good as N → ∞.

This insight also was an essential ingredient to the proof

that Gibbs-preserving maps cannot violate NMW. Now, to

leading order, the total number of states with energy below

(e − )N is given by exp(s(e − )N), where s(e − ) is

the microcanonical entropy density at energy density e−.

Since the total weight in this subspace is p(w ≥ ), the

ﬁnal min-entropy is upper bounded by

(ﬁnal)

min

≤ −log(p(w ≥ )) + s(e − )N.

However, since trumping requires S

(ﬁnal)

min

≥ S

(initial)

min

s(e)N, we then ﬁnd

p(w ≥ ) ≤ exp(−(s(e) − s(e −))N ) → 0,

as N → ∞ for any  > 0. This shows that NMW holds

for β-trumping channels in the micro-canonical case. Note

that when we allow the catalyst to become correlated,

NMW can be violated for microcanonical initial states, as

shown above. This already makes clear that correlated cat-

alysts provide a strict advantage in this set-up.

iii) Finally, let us also show that if we assume that N

is unitary, as we do in the case of β-catalytic channels,

then NMW cannot be violated if the catalyst remains un-

correlated. The reason is the following: Since the global

transformation on system and catalyst is unitary, it leaves

the spectrum invariant. Since the catalyst remains invari-

ant and uncorrelated, this implies that already the spectrum

of the initial density matrix on the system has to remain

invariant. Therefore there exists a unitary V , such that

T [ω

(H)] = V ω

(H)V

†

. As argued in case ii), we then

have

p(w ≥ ) ≈ Tr[P T [ω

(H)]] = Tr[P V ω

(H)V

†

]

≤ Tr[P ω

(H)],

where the last inequality follows because Gibbs states are

passive states and the ﬁrst approximation holds to arbitrary

accuracy as N → ∞. However, by the same concentration

inequalities we used to prove of our main results, we have

Tr[P ω

(H)] ≤ K exp(−k(

1/(1+D)

for a non-critical many-body system in D spatial dimen-

sions (with constants k, K > 0). Thus, NMW holds true

in this case as well.

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