# By-passing fluctuation theorems with a catalyst

P. Boes  Dahlem Center for Complex Quantum Systems, Freie Universität Berlin,
14195 Berlin, Germany  R. Gallego  Dahlem Center for Complex Quantum Systems,
Freie Universität Berlin, 14195 Berlin, Germany  N. H. Y. Ng  Dahlem Center
for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
J. Eisert  Dahlem Center for Complex Quantum Systems, Freie Universität
Berlin, 14195 Berlin, Germany  H. Wilming  Institute for Theoretical Physics,
ETH Zurich, 8093 Zurich, Switzerland

###### Abstract

Fluctuation theorems impose constraints on possible work extraction
probabilities in thermodynamical 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 become correlated with the system from which work
is extracted, but whose reduced state remains 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 extraction 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 parties use the same catalyst to induce local
transitions. We show that there exist catalytic processes that lead to highly
correlated work distributions, 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 dictates 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 initial 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

$\langle W\rangle\leq 0,$ (1)

where $\langle W\rangle$ is the average work extracted during the process. 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 processes, 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 positive work per
particle is exponentially suppressed with the number of particles in a system
[1]. Once these different types of constraints are recognized, an interesting
questions arises: What are physically meaningful settings in which the
probabilistic constraints imposed by fluctuation 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, compatible 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 system 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 system 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 correspond to the cyclically evolving
auxiliary system. Despite being studied frequently in resource-theoretic
formulations of thermodynamics [6, 7, 8, 9], catalytic processes have never
been studied in the context of fluctuation theorems 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 derived. 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 thermodynamics_
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
_catalysis_ 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 interesting results of both foundational and practical interest.

##  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
Hamiltonian
$H=\sum_{i=1}^{d}E_{i}\left.\left|{E_{i}}\right.\right\rangle\\!\\!\left.\left\langle{E_{i}}\right.\right|$
is initalized in the Gibbs state

$\omega_{\beta}(H):=\frac{\mathrm{e}^{-\beta H}}{Z(\beta,H)},$ (2)

where $Z(\beta,H):=\mathrm{Tr}(\mathrm{e}^{-\beta H})$. This state describes a
system initially in thermal equilibrium with its environment at inverse
temperature $\beta:=1/(k_{B}T)$. An agent (some experimenter) first performs
an energy measurement on this system which produces a measurement outcome
$E_{i}$. According to quantum mechanics, the post-measurement state is
described by the density matrix
$\left.\left|{E_{i}}\right.\right\rangle\\!\\!\left.\left\langle{E_{i}}\right.\right|$.
The agent then performs a physical operation on the system which does not
depend on the outcome of the measurement. Such an operation can always be
represented by a general quantum channel $\mathcal{C}$ (i.e., a trace-
preserving, completely positive map that takes density matrices to density
matrices) applied to the post-measurement state. This operation is then
followed by a second energy measurement with respect to the same Hamiltonian
with outcome $E_{f}$ 111It is possible to extend the setup and our further
results to the more general case of different Hamiltonians for the initial and
final measurement. We present our results within this restricted settings for
conceptual and notational simplicity.. This procedure results in a channel-
dependent joint distribution ${P(E_{f},E_{i})=P(E_{f}|E_{i})P(E_{i})}$. In
general, a given quantum channel may be realized in different ways. Whether
the change of energy $E_{f}-E_{i}$ can be interpreted as work from a
thermodynamic point of view will depend on how exactly the quantum channel
$\mathcal{C}$ was physically realized. We will assume that this is the case in
the following, but will comment on this assumption again later on. In
particular, we can then define the work distribution $P$ for the above process
as

$P(W):=\sum_{i,f}P(E_{f},E_{i})\delta(W-(E_{i}-E_{j})),$ (3)

where $\delta$ is the Dirac delta distribution. We are interested in
investigating possible distributions $P(W)$ that arise from different channels
$\mathcal{C}$. To do so, it is useful to note the relation

$\langle\mathrm{e}^{\beta W}\rangle=\sum_{j}\frac{e^{-\beta
E_{j}}}{Z_{H}}\left.\left\langle{E_{j}}\right.\right|\mathcal{C}[\mathbf{I}]\left.\left|{E_{j}}\right.\right\rangle,$
(4)

which is straightforwardly derived using the above definitions, where
$\mathbf{I}$ denotes the identity matrix.

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 $\beta$ samples
from $S$ (by measuring in the energy basis), then implements a process that
maps the post-measurement state
$\left.\left|{E_{i}}\right.\right\rangle\\!\\!\left.\left\langle{E_{i}}\right.\right|$
to
$\mathcal{C}(\left.\left|{E_{i}}\right.\right\rangle\\!\\!\left.\left\langle{E_{i}}\right.\right|)$,
where $\mathcal{C}$ is a quantum channel. Finally, the agent repeats the
energy measurement on $S$ with respect to the same Hamiltonian $H$.

In the standard setting of _Tasaki-type fluctuation theorems_ , $\mathcal{C}$
is considered to be a unitary channel
$\mathcal{C}[\cdot]=U(\cdot)U^{\dagger}$, since these are generated by
changing the Hamiltonian over time [3]. For such channels, (4) becomes

$\langle\mathrm{e}^{\beta W}\rangle=1,$ (5)

which is the well-known Jarzynski equality (JE) for cyclic, reversible
processes [1]. Eq. (5) is strictly stronger than (1), the latter being implied
by (5) 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 possibility of extracting work from a large system
in a thermal state [10, 11, 12]. To see this, let $S$ be an $N$-particle
system and define the probability of extracting work $w$ per particle as

$p(w):=P(wN).$ (6)

Plugging this into (5) yields that for any $\epsilon>0$,

$1=\langle\mathrm{e}^{\beta W}\rangle=\sum_{w}\mathrm{e}^{\beta
wN}\>P(wN)\geq\mathrm{e}^{\beta\epsilon N}\sum_{w\geq\epsilon}\>p(w),$ (7)

which implies that events which extract significant positive work per particle
from a macroscopic system at equilibrium are exponentially unlikely in $N$.
For later use, we formalize this property.

######  Definition 1 (No macroscopic work).

Given a sequence of $N$-particle systems initially at thermal equilibrium with
inverse temperature $\beta$ and channels $\mathcal{C}$ (implicitly depending
on $N$), we say that the processes represented by $\mathcal{C}$ fulfill the
_no macroscopic work (NMW)_ condition if the probability of an event
extracting work per particle larger or equal than $\epsilon$ is arbitrarily
small as $N\rightarrow\infty$,

$\lim_{N\rightarrow\infty}p(w\geq\epsilon):=\lim_{N\rightarrow\infty}\sum_{w>\epsilon}\>p(w)=0.$
(8)

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 physically motivated
family of channels $\mathcal{C}$ that violates both NMW and JE, but respects
the Av-SL. To aid comparison, we first briefly discuss other generalizations
of the standard setting to non-unitary channels (see also Refs. [13, 14]).

###  3.1  Violating JE with non-unitary channels

It is easy to see from (4) that a more general class of channels that satisfy
the JE are _unital_ channels, that is, channels that satisfy
$\mathcal{C}[\mathbf{I}]=\mathbf{I}$. 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 unitality is relaxed, it becomes easy to
violate JE on a formal level. For example, consider the fully-thermalizing
channel that maps every input state to the thermal state $\omega_{\beta}(H)$,
in other words $\mathcal{C}(\cdot)=\omega_{\beta}(H)$. This channel always
violates the JE whenever $\omega_{\beta}(H)\neq\mathbf{I}/d$. It is, however,
not clear how the energy-fluctuations can be interpreted as _work_ in this
example, since thermalizing processes usually 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 formally violate JE, it
is not obvious whether it is possible to do so in a physically meaningful and
operationally useful 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 (5). This means that, even if one interprets energy
fluctuations as work, one still could not use the thermalizing channel to
extract macroscopic amounts of work from a many-body system.

###  3.2 Violations of NMW and JE via $\beta$-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 respecting the Av-SL and allowing for the interpretation of the
random variable $W$ as work extracted from $S$. Such channels, 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 result of this work is to answer this question affirmatively. To this
end, we define the notion of a $\beta$-catalytic channel.

######  Definition 2 ($\beta$-catalytic channel).

A completely positive, trace-preserving map $\mathcal{C}$ is a _$\beta$
-catalytic channel_ on S, if there exists a quantum state $\sigma_{C}$ on a
system $C$ with Hamiltonian $H_{C}$, together with a unitary $U$ such that
$[\sigma_{C},H_{C}]=0$ and

$\displaystyle\mathcal{C}(\cdot)=\mathrm{Tr}_{C}(U(\>\cdot\otimes\>\sigma_{C})U^{\dagger}),$
(9)
$\displaystyle\text{s.t.}\>\>\mathrm{Tr}_{S}(U(\omega_{\beta}(H)\otimes\sigma_{C})U^{\dagger})=\sigma_{C}.$
(10)

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, $\mathcal{C}$
may couple system and catalyst for example by introducing interaction terms
$H_{SC}$, nevertheless at the end of the process, the channel must also turn
off such interaction terms. Secondly, note that $\beta$-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 system $C$ 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 $\omega_{\beta}(H)$. As such, $\beta$-catalytic channels depend
on $\beta$ 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 catalysis discussed in the resource-theoretic literature on quantum
thermodynamics [6, 7], the more general notion of catalysis that we employ
here is receiving increasing interest 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 context of algorithmic cooling [16,
17] and to show the energetic instability of passive states [18]. Finally, let
us briefly comment on the interpretation of the random variable $W$ as work in
the setting of $\beta$-catalytic channels and the role of the Hamiltonian of
the catalyst. Since the process on $C$ and $S$ is unitary, it is meaningful to
denote the total changes of energies of the two systems as work measured by a
two-point measurement scheme on each system. 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 assume that the catalyst has trivial Hamiltonian $H_{C}=0$,
which in turn implies $[\sigma_{C},H_{C}]=0$ for any $\sigma_{C}$. 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
$H_{C}=0$.

Given these constraints, it may, at first glance, be unclear 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 $S$,
conditioned on $C$: For some
$\sigma_{C}=\sum_{i}p_{i}\left.\left|{i}\right.\right\rangle\\!\\!\left.\left\langle{i}\right.\right|$,
one uses a unitary in Eq. (10) of the form

$U_{\rm
SC}:=\sum_{i}U_{i}\otimes\left.\left|{i}\right.\right\rangle\\!\\!\left.\left\langle{i}\right.\right|_{\rm
C}.$ (11)

This special case of $\beta$-catalytic channels by construction produces
random unitary channels [19, 16] on $S$, which have the form $\mathcal{C}_{\rm
RU}(\cdot)=\sum_{i}p_{i}U_{i}(\cdot)U_{i}^{\dagger}$. But random unitary
channels are always unital, and therefore automatically satisfy JE.

In the following, we show that there exist non-unital $\beta$-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 $\omega_{\beta}(H)$ is the state with
the least energy given a fixed entropy [20, 21], then we also have that

$\mathrm{Tr}(H\mathcal{C}(\omega_{\beta}(H))\geq\mathrm{Tr}(H\omega_{\beta}(H))$
(12)

which is just the Av-SL, concomitant with the findings of Ref. [9]. We stress
that despite this property, $\beta$-catalytic channels are in general _not_
unital. It remains to be shown that $\beta$-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
$\beta$-catalytic channels, we consider a system with three states – two
degenerate (but distinguishable) ground states and an excited state with
energy $E$. As catalyst, we consider a system with two states and the unitary
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

$\langle\mathrm{e}^{\beta W}\rangle=\frac{Z+5+2(Z-2)(Z-1)}{Z(Z+1)}\geq 1,$
(13)

where $Z=2+\mathrm{e}^{-\beta E}$ is the partition function of the system and
we used $2\leq Z\leq 3$. We hence find $\langle\mathrm{e}^{\beta W}\rangle>1$
whenever $E>0$ (since then $Z<3$) and we obtain a moderate maximum violation
in the limit $E\rightarrow\infty$ given by $\langle\mathrm{e}^{\beta
W}\rangle=7/6$.

_Macroscopic violation of NMW condition._ We now show that one can violate the
NMW principle using catalysts.

######  Proposition 1 (Violation of no macroscopic work with catalysts).

Let $(S^{(N)})_{N}$ be a sequence of $N$-particle locally interacting lattice
systems with Hamiltonian $H^{(N)}$ that satisfy mild assumptions. Then, for
sufficiently large $N$, there exist values of $\epsilon>0$, such that

$p(w\geq\epsilon)$ (14)

can be brought arbitrarily close to $\frac{1}{2}$ with $\beta$-catalytic
channels.

We provide a proof and full statement of the assumptions 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 $\delta>0$, there exists an $N$ and a
$\beta$-catalytic channel $\mathcal{C}$ such that

$\mathcal{C}(\omega_{\beta})\approx_{\delta}\frac{1}{2}\left.\left|{E_{-}}\right.\right\rangle\\!\\!\left.\left\langle{E_{-}}\right.\right|+\frac{1}{2}\tau,$
(15)

where $\approx_{\delta}$ denotes equality of the states on LHS and RHS up to
$\delta$ in trace distance, $\left.\left|{E_{-}}\right.\right\rangle$ is some
eigenvector of $H$ with $E_{-}<e^{(N)}N$ and $\tau$ is some other “fail”-state
the details of which are irrelevant. We can interpret Eq. (15) as describing
the approximation of a work extraction protocol that results in the state
$\left.\left|{E_{-}}\right.\right\rangle$ with probability $1/2$. Now, as the
result of standard concentration bounds, for large $N$ the mass of the thermal
state $\omega_{\beta}$ will be highly concentrated around energy $e^{(N)}N$.
This implies that every 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 $\epsilon\equiv e^{(N)}-E_{-}/N$,
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. (15), 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 transition 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. (15), 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, highlighting 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 Appendix 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 $\beta$-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 JE – the violation of JE is in fact exponential in the system
size. In particular, (14) implies that there exist values $\epsilon>0$, such
that

$\langle\mathrm{e}^{\beta W}\rangle\>\geq\frac{1}{2}\mathrm{e}^{\beta
N\epsilon}\gg 1$ (16)

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 realize a certain violation.
This is clarified by the following result.

######  Proposition 2 (Bound on violation of JE).

Let $\mathcal{C}$ be any $\beta$-catalytic channel with $d_{C}=\dim(H_{C})$.
Then,

$\displaystyle\langle\mathrm{e}^{\beta W}\rangle$
$\displaystyle\leq\mathrm{min}\\{d_{C}\left\|\sigma\right\|_{\infty},d\left\|\omega_{\beta}(H)\right\|_{\infty}\\}$
(17) $\displaystyle\leq\mathrm{min}\\{d_{C},d\\},$ (18)

where $\|\cdot\|_{\infty}$ denotes the $\infty$-norm, which, for density
matrices, equals the largest absolute value of the input’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 $\sigma$ on a
growing auxiliary system and, more importantly, also have control over the
increasingly large joint system. Hence, in practice, 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 $\epsilon\geq 0$
we have

$\langle\mathrm{e}^{\beta W}\rangle\geq
P(W\geq\epsilon)\mathrm{e}^{\beta\epsilon}.$ (19)

Hence, Prop. 2 implies

$P(W\geq\epsilon)\leq
d_{C}\left\|\sigma\right\|_{\infty}\mathrm{e}^{-\beta\epsilon}.$ (20)

In particular this means that to extract a macroscopic amount of work, $W\geq
wN$, with finite probability, $d_{C}$ has to grow exponentially with $N$ (note
that $\left\|\sigma\right\|_{\infty}\leq 1$).

##  4 Multi-partite work extraction

As emphasized before, even though the state of the catalyst 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 system
allow for processes in which many agents re-use the same catalyst to obtain
highly inter-correlated work distributions.

Consider $n$ agents, each with identical systems $S_{i},i\in\\{1,\dots,n\\}$
that are initialized in the Gibbs state $\omega(\beta,H)$. For a given
$\beta$-catalytic channel $\mathcal{C}$ with state $\sigma$ on the catalyst,
consider the following protocol: Agent $1$ runs the standard process from Fig.
1 using the catalyst and hence implementing $\mathcal{C}$ 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 $\mathcal{C}$, is is clear
that, for each agent, the same marginal distribution of work is obtained.
However, the joint work distribution for all agents will be correlated, due to
individual correlations between each $S_{i}$ 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
$p(w_{1},\dots,w_{n})$ denote the global distribution over the extracted work
per particle, assuming that all $S_{i}$ are copies of the same $N$-particle
system. We have the following, proven in Appendix E.

######  Proposition 3 (Multiple agents).

Let each $\\{S_{i}\\}_{i=1}^{n}$ be a sequence of N-particle systems that
satisfy the conditions of Proposition 1. Then, for sufficiently large $N$,
there exists an $\epsilon>0$, such that

$\begin{split}p(\epsilon,-\epsilon,\epsilon,-\epsilon,\dots)&=\lambda,\\\
p(-\epsilon,\epsilon,-\epsilon,\epsilon,\dots)&=1-\lambda,\end{split}$ (21)

where $\lambda$ can be brought arbitrarily close to $1/2$ using a sequence of
$\beta$-catalytic channels on $S_{i}$ and $C$.

While (21) is clearly consistent with (1), this proposition shows that the
agents can achieve joint work distributions 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 (21) 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 distributions 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
distribution

$p(\epsilon,\epsilon,\epsilon,\epsilon,\dots)=p(-\epsilon,-\epsilon,-\epsilon,-\epsilon,\dots)\approx
1/2$ (22)

cannot be obtained via $\beta$-catalytic channels, since otherwise there would
exist a catalyst of fixed size that would allow, for any $n$, the total work
$W=n\epsilon$ to be extracted with probability approximately $1/2$, in
violation of Proposition 2.

##  5 Summary and future work.

In this work we have studied work extraction protocols from states at thermal
equilibrium. We significantly expand the common setting of fluctuation
theorems under 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
bypass the stringent conditions imposed by the JE, achieving positive work per
particle with high probability, even for macroscopic systems. Furthermore, it
allows for interesting, correlated work distributions when many agents use the
same catalyst.

Our constructions illustrate in a striking way that the absence of
correlations, sometimes referred to as ‘stochastic independence’, can also be
a powerful thermodynamic resource [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 $\beta$-catalytic channels
will yield both foundational and practical insights.

We further believe that it is an interesting open problem 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 $\sqrt{N}$ if in exchange for that one can either increase the
probability for it to happen significantly or can reduce the size of the
catalyst considerably (and hence the complexity of the unitary required to be
implemented).

It would be interesting to understand the relation between 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 techniques developed in this
work provide a promising ansatz for the study of this and similar questions.

_Acknowledgements._ We thank Markus P. Müller and Alvaro M. Alhambra for
valuable discussions and anonymous referees for interesting comments. P. B.
acknowledges support from the John Templeton Foundation. H. W. acknowledges
support from the Swiss National Science Foundation through SNSF project No.
200020_165843 and through the National Centre of Competence in Research
_Quantum Science and Technology_ (QSIT). N. H. Y. N. acknowledges support from
the Alexander von Humboldt Foundation. R. G. has been supported by the DFG (GA
2184/2-1). J. E. acknowledges support by the DFG (FOR 2724), dedicated to
quantum thermodynamics, and the FQXi.

## References

  * Jarzynski [1997] C. Jarzynski, Phys. Rev. Lett. 78, 2690 (1997).
  * Crooks [1998] G. E. Crooks, J. Stat. Phys. 90, 1481 (1998).
  * Tasaki [2000] H. Tasaki, ArXiv e-prints (2000), arXiv:1303.6393 .
  * Erker  _et al._ [2017] P. Erker, M. T. Mitchison, R. Silva, M. P. Woods, N. Brunner, and M. Huber, Phys. Rev. X 7, 031022 (2017).
  * Woods  _et al._ [2019] M. P. Woods, R. Silva, and J. Oppenheim, Ann Hen. Poin. 20, 125 (2019).
  * Brandão  _et al._ [2015] F. G. S. L. Brandão, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, PNAS 112, 3275 (2015).
  * Ng  _et al._ [2015] N. H. Y. Ng, L. Mančinska, C. Cirstoiu, J. Eisert, and S. Wehner, New J. Phys. 17, 085004 (2015).
  * Müller [2018] M. P. Müller, Phys. Rev. X 8 (2018), 10.1103/physrevx.8.041051.
  * Boes  _et al._ [2019] P. Boes, J. Eisert, R. Gallego, M. P. Mueller, and H. Wilming, Phys. Rev. Lett. 122, 210402 (2019).
  * Jarzynski [2011] C. Jarzynski, Annu. Rev. Condens. Matter Phys. 2, 329 (2011).
  * Cavina  _et al._ [2016] V. Cavina, A. Mari, and V. Giovannetti, Scientific Rep. 6, 29282 (2016).
  * Maillet  _et al._ [2019] O. Maillet _et al._ , Phys. Rev. Lett. 122, 150604 (2019).
  * Rastegin [2013] A. E. Rastegin, J. Stat. Mech. 2013, P06016 (2013).
  * Rastegin and Życzkowski [2014] A. E. Rastegin and K. Życzkowski, Phys. Rev. E 89, 012127 (2014).
  * Wilming  _et al._ [2017] H. Wilming, R. Gallego, and J. Eisert, Entropy 19, 241 (2017).
  * Boes  _et al._ [2018] P. Boes, H. Wilming, R. Gallego, and J. Eisert, Phys. Rev. X 8, 041016 (2018).
  * Alhambra  _et al._ [2019] Á. M. Alhambra, M. Lostaglio, and C. Perry, Quantum 3, 188 (2019).
  * Sparaciari  _et al._ [2017] C. Sparaciari, D. Jennings, and J. Oppenheim, Nat. Commun. 8, 1895 (2017).
  * Audenaert and Scheel [2008] K. M. R. Audenaert and S. Scheel, New J. Phys. 10, 023011 (2008).
  * Lenard [1978] A. Lenard, J. Stat. Phys. 19, 575 (1978).
  * Pusz and Woronowicz [1978] W. Pusz and S. L. Woronowicz, Comm. Math. Phys. 58, 273 (1978).
  * Huang [1987] K. Huang,  _Statistical mechanics_ (Wiley, 1987).
  * Talkner  _et al._ [2008] P. Talkner, P. Hänggi, and M. Morillo, Phys. Rev. E 77, 051131 (2008).
  * Lostaglio  _et al._ [2015] M. Lostaglio, M. P. Müller, and M. Pastena, Phys. Rev. Lett. 115, 150402 (2015).
  * Sagawa and Ueda [2012a] T. Sagawa and M. Ueda, Phys. Rev. Lett. 109, 180602 (2012a).
  * Sagawa and Ueda [2013] T. Sagawa and M. Ueda, New J. Phys. 15, 125012 (2013).
  * Sagawa and Ueda [2012b] T. Sagawa and M. Ueda, Phys. Rev. Lett. 109, 180602 (2012b).
  * Toyabe  _et al._ [2010] S. Toyabe, T. Sagawa, M. Ueda, E. Muneyuki, and M. Sano, Nature Phys. 6, 988 (2010).
  * Horodecki and Oppenheim [2013] M. Horodecki and J. Oppenheim, Nature Comm. 4, 2059 (2013).
  * Perry  _et al._ [2018] C. Perry, P. Ćwikliński, J. Anders, M. Horodecki, and J. Oppenheim, Phys. Rev. X 8, 041049 (2018).
  * Faist  _et al._ [2015] P. Faist, J. Oppenheim, and R. Renner, New J. Phys. 17, 043003 (2015).
  * Gogolin  _et al._ [2011] C. Gogolin, M. P. Müller, and J. Eisert, Phys. Rev. Lett. 106, 040401 (2011).
  * Anshu [2016] A. Anshu, New J. Phys. 18, 083011 (2016).
  * Nielsen and Chuang [2000] M. A. Nielsen and I. L. Chuang,  _Quantum Computation and Quantum Information_ (Cambridge University Press, 2000).
  * Goldstein  _et al._ [2013] S. Goldstein, T. Hara, and H. Tasaki, ArXiv e-prints (2013), arXiv:1303.6393 .
  * Kliesch  _et al._ [2014] M. Kliesch, C. Gogolin, M. J. Kastoryano, A. Riera, and J. Eisert, Phys. Rev. X 4, 031019 (2014).
  * Watrous [2018] J. Watrous,  _The theory of quantum information_ (Cambridge University Press, 2018).
  * Perarnau-Llobet  _et al._ [2017] M. Perarnau-Llobet, E. Bäumer, K. V. Hovhannisyan, M. Huber, and A. Acin, Phys. Rev. Lett. 118, 070601 (2017).
  * Koski  _et al._ [2014] J. V. Koski, V. F. Maisi, T. Sagawa, and J. P. Pekola, Phys. Rev. Lett. 113, 030601 (2014).

##  Appendix 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 exchanges heat with its surrounding thermal bath. Several
variants of these maps exist [29, 6, 7, 30, 31], but a common 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 various 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

$\displaystyle\langle e^{-\beta W}\rangle$
$\displaystyle=d\sum_{j}\frac{e^{-\beta
E_{j}}}{Z_{H}}\left.\left\langle{E_{j}}\right.\right|\mathcal{C}\left[\mathbf{I}/d\right]\left.\left|{E_{j}}\right.\right\rangle,$
(23) $\displaystyle=d\sum_{j}\frac{e^{-\beta
E_{j}}}{Z_{H}}\left.\left\langle{E_{j}}\right.\right|\omega_{\beta}(H)\left.\left|{E_{j}}\right.\right\rangle=\frac{d}{d_{\rm
eff}},$ (24)

where $d_{\rm eff}:={1}/{\mathrm{Tr}(\omega_{\beta}(H)^{2})}$ is known as the
effective dimension [32] of the thermal state. One sees from the above that JE
is always violated for $\beta>0$, since $d_{\rm eff}\leq d$, with equality
only when $\omega_{\beta}(H)=\mathbf{I}/d$ is maximally mixed. For $N$ non-
interacting i.i.d. systems, both $d$ and $\mathrm{Tr}(\rho^{2})$ scale
exponentially with $N$, leading to an exponential violation in $N$ for JE.

Turning to the second point, one may wonder how this notion of
thermodynamically free channels can be reconciled 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 difference
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 consider 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 uniformly 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-preserving maps).

No channel $\mathcal{E}$ 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 $a>0$, $p(w\geq a)=p(W\geq aN)\rightarrow 0$ as
$N\rightarrow\infty$. The basic idea behind our proof is to make use of
typicality. Let $e^{(N)}$ denote the energy density of the $N$-particle system
and denote by $\Pi_{\delta}^{(N)}$ the projector onto energy eigenstates with
energies in the interval
$T_{N,\delta}:=[(e^{(N)}-\delta)N,(e^{(N)}+\delta)N]$. Finally, denote by
$p(\cdot)$ the initial probability distribution of energy of the thermal state
$\tau_{S}^{(N)}$, e.g., the probability that the initial energy measurement
yields $E_{i}\in T_{N,\delta}$ is given by

$\displaystyle
p(T_{N,\delta}):=\mathrm{Tr}\left(\tau_{S}^{(N)}\Pi_{\delta}^{(N)}\right).$
(25)

A theorem by Anshu [33] shows that under the given conditions most weight of
the thermal state $\tau_{S}^{(N)}$ of the $N$-particle system is contained in
a typical subspace. More precisely, for a many-body system described by a
$D$-dimensional lattice, there exist constants $C,K>0$ such that for any
$\delta>0$ we have

$\displaystyle p(T_{N,\delta})\geq
1-C\mathrm{e}^{-\frac{(\delta^{2}N)^{\frac{1}{1+D}}}{K}}.$ (26)

This is equivalent to saying that

$p(T_{N,\delta}^{c})\leq
C\mathrm{e}^{-\frac{(\delta^{2}N)^{\frac{1}{1+D}}}{K}},$ (27)

where $T_{N,\delta}^{c}=\mathbb{R}\setminus T_{N,\delta}$. 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 $\sigma_{1}=\tau_{S}^{(N)}$ and consider the normalized
state $\sigma_{2}$ obtained by restricting $\tau_{S}^{(N)}$ to the subspace
$\Pi_{\delta}^{(N)}$ as

$\displaystyle\sigma_{2}:=\frac{\Pi_{\delta}^{(N)}\tau_{S}^{(N)}}{p(T_{N,\delta})}.$
(28)

Let us further write $\mathcal{E}(\sigma_{1(2)})=\sigma_{1(2)}^{\prime}$,
where $\sigma_{1}^{\prime}=\sigma_{1}$ by assumption. Since the trace distance
$d(\rho_{1},\rho_{2})\coloneqq\frac{1}{2}\mathrm{Tr}(|\rho_{1}-\rho_{2}|)$
fulfills the data processing inequality,

$d(\sigma_{1},\sigma_{2}^{\prime})=d(\sigma_{1}^{\prime},\sigma_{2}^{\prime})\leq
d(\sigma_{1},\sigma_{2})=p(T_{N,\delta}^{c}).$ (29)

Using the operational meaning of trace distance
$d(\rho_{1},\rho_{2})=\displaystyle\max_{0\leq M\leq
I}|\mathrm{Tr}(M(\rho_{1}-\rho_{2}))|$ [34], this means that

$|\mathrm{Tr}(\Pi_{\delta}^{(N)}\sigma_{1})-\mathrm{Tr}(\Pi_{\delta}^{(N)}\sigma_{2}^{\prime})|\leq
p(T_{N,\delta}^{c})$ (30)

and, in turn,

$\displaystyle\mathrm{Tr}(\Pi_{\delta}^{(N)}\sigma_{2}^{\prime})\geq
p(T_{N,\delta})-p(T_{N,\delta}^{c})=1-2p(T_{N,\delta}^{c}).$ (31)

To see this, note that (31) follows from (30) directly if
$\mathrm{Tr}(\Pi_{\delta}^{(N)}\sigma_{2}^{\prime})\leq\mathrm{Tr}(\Pi_{\delta}^{(N)}\sigma_{1})$,
and as

$\mathrm{Tr}(\Pi_{\delta}^{(N)}\sigma_{2}^{\prime})>\mathrm{Tr}(\Pi_{\delta}^{(N)}\sigma_{1})\geq\mathrm{Tr}(\Pi_{\delta}^{(N)}\sigma_{1})-p(T_{N,\delta}^{c})$
(32)

otherwise. This means that, conditioned on the fact that the initial state was
within the typical energy window ($E_{i}\in T_{N,\delta}$), the final energy
$E_{f}$ is also within this energy window except with probability
$2p(T_{N,\delta}^{c})$, which is (sub-)exponentially small in $N$. We will use
this later.

We are now ready to evaluate the probability of obtaining macroscopic work.

$\displaystyle p(w\geq a)$ $\displaystyle=p(T_{N,\delta})\cdot p(w\geq
a|E_{i}\in T_{N,\delta})$ $\displaystyle\quad\quad+p(T_{N,\delta}^{c})\cdot
p(w\geq a|E_{i}\in T_{N,\delta}^{c})$ $\displaystyle\leq p(w\geq a|E_{i}\in
T_{N,\delta})+p(T_{N,\delta}^{c}).$

We can estimate the first term as

$\displaystyle p(w\geq a|E_{i}\in T_{N,\delta})$ $\displaystyle\leq
p(E_{f}\leq(e^{(N)}+\delta-a)N|E_{i}\in T_{N,\delta}).$

We now choose $\delta=a/2$ and get

$\displaystyle p(w\geq a|E_{i}\in T_{N,\delta})$ $\displaystyle\leq
p(E_{f}\leq(e^{(N)}-a/2)N|E_{i}\in T_{N,\delta})$
$\displaystyle\leq\mathrm{Tr}\left[\sigma_{2}^{\prime}\left(\mathbf{I}-\Pi_{a/2}^{(N)}\right)\right]$
$\displaystyle\leq 2p(T_{N,a/2}^{c}),$

where we have used (31) in the last step. Altogether, we thus find

$\displaystyle p(w\geq a)$ $\displaystyle\leq 3p(T_{N,a/2}^{c}),$

which decays to zero (sub-)exponentially by (26). 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.

Figure 2: A summary of different criteria (Av-SL, NMW and JE) mentioned in the
main text, with examples of maps according to this characterization.

##  Appendix 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 constructing 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 permutations of those
values. We consider a 3-level system with energy levels
$E_{1}=0,E_{2}=0,E_{3}=\Delta$ in the thermal state

$\displaystyle w=\left(\frac{1}{Z},\frac{1}{Z},\frac{Z-2}{Z}\right),$ (33)

where $Z=2+\exp(-\beta\Delta)$ is the partition function and we express the
state as a probability vector, such that $w_{i}$ denotes the $i$th eigenvalue
of the thermal state. For later, we observe that $2\leq Z\leq 3$.

We are going to construct a simple non-unital catalytic channel that involves
a 2-dimensional catalyst. Let $e_{i}$ and $f_{j}$ denote the basis states for
the vector spaces $\mathcal{V}_{S}$ and $\mathcal{V}_{C}$ describing the
system and catalyst respectively. We define the permutation $\pi$ acting on
the joint vector space $\mathcal{V}_{S}\otimes\mathcal{V}_{C}$ as that
permutation which exchanges the respective levels $e_{1}\otimes
f_{1}\Leftrightarrow e_{2}\otimes f_{2}$ and $e_{2}\otimes
f_{1}\Leftrightarrow e_{3}\otimes f_{2}$ 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

$\displaystyle q=\left(\frac{Z-1}{Z+1},\frac{2}{Z+1}\right).$ (34)

Now, the catalytic channel $\mathcal{C}$ induced by this catalyst and
permutation on the system has the general effect

$\displaystyle\mathcal{C}(p_{1},p_{2},p_{3})=(q_{1}p_{2}+q_{1}p_{3},q_{1}p_{1}+q_{2}p_{3},q_{2}p_{1}+q_{2}p_{2}),$
(35)

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

$\displaystyle\mathcal{C}(\mathbf{I}/3)=\frac{2}{3}\left(\frac{Z-1}{Z+1},\frac{1}{2},\frac{2}{Z+1}\right),$
(36)

which is different from the maximally mixed vector for any $\Delta>0$.

${\color[rgb]{1,0,0}q_{2}p_{3}}$ | $q_{1}p_{3}$ | $p_{3}$
---|---|---
${\color[rgb]{1,0,0}q_{2}p_{2}}$ | ${\color[rgb]{0,0,1}q_{1}p_{2}}$ | $p_{2}$
$q_{2}p_{1}$ | ${\color[rgb]{0,0,1}q_{1}p_{1}}$ | $p_{1}$
$q_{2}$ | $q_{1}$ |

$\>\rightarrow\>$ ${\color[rgb]{0,0,1}q_{1}p_{2}}$ $q_{1}p_{3}$
$q_{1}p_{2}+q_{1}p_{3}$ ${\color[rgb]{0,0,1}q_{1}p_{1}}$
${\color[rgb]{1,0,0}q_{2}p_{3}}$ $q_{1}p_{1}+q_{2}p_{3}$ $q_{2}p_{1}$
${\color[rgb]{1,0,0}q_{2}p_{2}}$ $q_{2}p_{1}+q_{2}p_{2}$ $q_{2}$ $q_{1}$

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 $i$th
eigenvalue of the system and $j$th eigenvalue of the catalyst. _Right:_ After
applying the permutation 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 remain invariant. For a two-dimensional
catalyst, specifying the permutation and initial system state fixes the
catalyst state.

What is more, we can also directly calculate the work-distribution $p(w)$,
yielding

$\displaystyle p(0)$
$\displaystyle=\frac{1}{Z(Z+1)}\left[Z+3+2(Z-2)(Z-1)\right],$ (37)
$\displaystyle p(\Delta)$ $\displaystyle=\frac{2(Z-2)}{Z(Z+1)},$ (38)
$\displaystyle p(-\Delta)$ $\displaystyle=\frac{Z-1}{Z(Z+1)}.$ (39)

We now want to compute $\langle\mathrm{e}^{\beta W}\rangle$. To do so, it is
useful to note that $e^{-\beta\Delta}=Z-2$ and hence
$\mathrm{e}^{\beta\Delta}=1/(Z-2)$. We find

$\displaystyle\langle\mathrm{e}^{\beta
W}\rangle=\frac{Z+5+2(Z-2)(Z-1)}{Z(Z+1)}\geq 1.$ (40)

In fact, this quantity is larger than $1$ whenever $Z<3$, corresponding to
$\Delta>0$. Its maximum is given as $7/6$ for $Z=3$, which corresponds to
$\Delta\rightarrow\infty$. Thus, the Jarzynski inequality is violated. At the
same time the second law is fulfilled as expected, since $p(-\Delta)\geq
p(\Delta)$.

##  Appendix 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 microcanonical
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 conceptually 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 $I\subset\mathbb{R}$ an energy window, by
$g(I)$ the number of energy eigenstates in this window,

$\displaystyle g(I)=\sum_{E_{i}\in I}1,$ (41)

and the corresponding micro-canonical state by

$\displaystyle\Omega_{\rm S}(I)=\frac{1}{g(I)}\sum_{E_{i}\in
I}\left.\left|{E_{i}}\right.\right\rangle\\!\\!\left.\left\langle{E_{i}}\right.\right|.$
(42)

A micro-canonical energy window around energy density $e$ is any energy window
$I(e)$ of the form $[e-O(\sqrt{N}),e]$, 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
$\omega_{\beta}(H)$. Instead, it is given by the micro-canonical ensemble. In
other words, given a micro-canonical energy window $I$, we consider channels
$\mathcal{C}$ of the form

$\displaystyle\mathcal{C}(\cdot)={\rm Tr}_{\rm C}(U(\cdot\otimes\sigma_{\rm
C})U^{\dagger})$ (43) $\displaystyle{\rm s.t.}~{}{\rm Tr}_{\rm
S}(U(\Omega_{\rm S}(I)\otimes\sigma_{\rm C})U^{\dagger})=\sigma_{\rm C}.$ (44)

We carry over notation from the main text, so that $p(w\geq\epsilon)$ denotes
the probability of measuring the system’s energy per particle decrease by at
least an amount $\epsilon$, and so on. Furthermore, we take the catalyst
Hamiltonian in our construction to be $H_{C}=\mathbf{I}$.

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
circumvented 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 before. Thus, the following proof is essentially a
reproduction for the convenience of the reader. We consider a sequence of
many-body Hamiltonians $H_{S}^{(N)}$ on $N$ particles with the generic
property of having an exponential density of states:

$\displaystyle g((-\infty,E])\coloneqq\sum_{E_{i}\leq
E}1=\mathrm{e}^{N\mu(E/N)-o(N)},$ (45)

where $\mu$ is a strictly monotonic and differentiable function independent of
$N$ and $o(N)$ denotes terms small compared to $N$,
$\lim_{N\to\infty}o(N)/N=0$.

######  Proposition 4 (NMW for micro-canonical states).

Consider a sequence of $N$-particle Hamiltonians fulfilling (45) and a
sequence of micro-canonical energy-windows $I^{(N)}=[eN,eN+\delta\sqrt{N}]$
around energy density $e$ (with $\delta>0$ fixed). Then for any unital channel
acting on the $N$-particle system, the probability of extracting work $w$ per
particle is bounded as

$\displaystyle p(w>\epsilon)\leq C\mathrm{e}^{-\mu^{\prime}(e)\epsilon
N+o(N)},$ (46)

where $C>0$ is a constant and $\mu^{\prime}$ denotes the derivative of $\mu$.

###### Proof.

Let $I_{\leq}:=(-\infty,(e-\epsilon)N+\delta\sqrt{N}]$, denote by $P_{\rm
S}(I_{\leq})$ the projector onto energy-eigenstates with energies below
$(e-\epsilon)N+\delta\sqrt{N}$ and let $\mathcal{U}$ denote a unital channel.
In the following, we write $I$ instead of $I^{(N)}$ to simplify notation. Then

$\displaystyle p(w>\epsilon)$ $\displaystyle\leq\mathrm{Tr}\left(P_{\rm
S}(I_{\leq})\mathcal{U}[\Omega_{\rm S}(I)]\right)$ (47)
$\displaystyle=\sum_{E_{i}\in I}\frac{1}{g(I)}\mathrm{Tr}\left(P_{\rm
S}(I_{\leq})\mathcal{U}\left[\left.\left|{E_{i}}\right.\right\rangle\\!\\!\left.\left\langle{E_{i}}\right.\right|\right]\right)$
(48) $\displaystyle\leq\frac{1}{g(I)}\mathrm{Tr}\left(P_{\rm
S}(I_{\leq})\mathcal{U}\left[\mathbf{I}\right]\right)=\frac{g(I_{\leq})}{g(I)}.$
(49)

Writing $\tilde{e}:=e+\delta N^{-1/2}$, we have

$\displaystyle g(I)$
$\displaystyle=\mathrm{e}^{N\mu(\tilde{e})-o(N)}-\mathrm{e}^{N\mu(e)-o(N)}$
(50)
$\displaystyle=\mathrm{e}^{N\mu(\tilde{e})-o(N)}\left(1-\mathrm{e}^{-N(\mu(\tilde{e})-\mu(e))+o(N)}\right)$
(51) $\displaystyle\approx\mathrm{e}^{N\mu(\tilde{e})-o(N)},$ (52)

where in the last estimation we use that $\mu$ is strictly monotonic. In
particular, we can estimate the exponential in the parenthesis as

$\displaystyle\mathrm{e}^{-N(\mu(\tilde{e})-\mu(e))-o(N)}=O\left(\mathrm{e}^{-\delta\mu^{\prime}(e)N^{1/2}}\right),$
(53)

where $\mu^{\prime}$ denotes the derivative of $\mu$. Using
$g(I_{\leq})=\mathrm{e}^{N(\mu(\tilde{e}-\epsilon)-o(N)}$ we then find

$\displaystyle p(w>\epsilon)$
$\displaystyle\leq\frac{\mathrm{e}^{-N\left(\mu(\tilde{e})-\mu(\tilde{e}-\epsilon)\right)+o(N)}}{1-O(\mathrm{e}^{-\delta\mu^{\prime}(e)\sqrt{N}})}$
(54) $\displaystyle\leq C\mathrm{e}^{-\mu^{\prime}(e)\epsilon N}.$ (55)

∎

We have here used that $\mu$ is differentiable to prove this result. Similar
results would follow for weaker notions of regularity of $\mu$, such as
Lipschitz-continuity. Having proven the NMW principle for generic many-body
systems, let us now show how to circumvent it using catalysts.

######  Proposition 5 (Overcoming NMW using catalysts).

Consider a Hamiltonian $H_{S}$ and a microcanonical state $\Omega_{\rm S}(I)$,
with $I$ a micro-canonical energy window around energy density $e$. Suppose
there exists an energy window $I_{+}$ with $g(I_{+})=g(I)^{2}$. Then, for any
$0\leq e_{-}<e$, there exists a catalytic channel such that

$\displaystyle p(w\geq e-e_{-})=\frac{1}{2}.$ (56)

Before giving the proof of the proposition, let us emphasize again that the
required conditions on the Hamiltonian are very weak. In particular, the
conditions are (approximately) fulfilled if the density of states is well
approximated 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.

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
$\left.\left|{E_{-}}\right.\right\rangle$: 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 $d_{\rm
C}-1$ 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. (64) 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.

###### 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 $g:=g(I)$, $g_{+}:=g(I_{+})=g^{2}$ and let
$P_{\rm S}(I)$ and $P_{\rm S}(I_{+})$ be the projectors onto the corresponding
energy subspaces. Let $\left.\left|{E_{-}}\right.\right\rangle$ be _any_
eigenstate of the Hamiltonian such that $0\leq E_{-}/N=e_{-}\leq e$. Following
this notation, the initial state of the system is

$\displaystyle\Omega_{\rm S}(I)=\frac{1}{g}P_{\rm S}(I).$ (57)

The aim is to bring the system to a state that is an equal mixture of
$\left.\left|{E_{-}}\right.\right\rangle\\!\\!\left.\left\langle{E_{-}}\right.\right|$
and $\Omega(I_{+})$. To do this, we employ a catalyst of dimension $d_{\rm
C}=g+1$. Let $\\{\left.\left|{i}\right.\right\rangle_{\rm C}\\}_{i=1}^{d_{\rm
C}}$ be an arbitrary orthonormal basis on the Hilbert-space of the catalyst
and let $P_{\rm
C}=\sum_{i=1}^{g}\left.\left|{i}\right.\right\rangle\\!\\!\left.\left\langle{i}\right.\right|$.
The initial state on the catalyst is given by

$\displaystyle\sigma=\frac{1}{2g}P_{\rm C}+\frac{1}{2}\left.\left|{d_{\rm
C}}\right.\right\rangle\\!\\!\left.\left\langle{d_{\rm C}}\right.\right|_{\rm
C}.$ (58)

We define the unitary $U$ by the conditions

$\displaystyle U[P_{\rm S}(I)\otimes\left.\left|{d_{\rm
C}}\right.\right\rangle\\!\\!\left.\left\langle{d_{\rm C}}\right.\right|_{\rm
C}]U^{\dagger}$
$\displaystyle=\left.\left|{E_{-}}\right.\right\rangle\\!\\!\left.\left\langle{E_{-}}\right.\right|\otimes
P_{\rm C}$ (59) $\displaystyle U[P_{\rm S}(I)\otimes P_{\rm C}]U^{\dagger}$
$\displaystyle=P_{\rm S}(I_{+})\otimes\left.\left|{d_{\rm
C}}\right.\right\rangle\\!\\!\left.\left\langle{d_{\rm C}}\right.\right|_{\rm
C}.$ (60)

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 unitaries achieving this, and any of them is fine
for our purposes.

Applying $U$ to the state $\Omega_{\rm S}(I)\otimes\sigma_{\rm C}$ one obtains

$\displaystyle U\left(\Omega_{\rm S}(I)\otimes\sigma_{\rm
C}\right)U^{\dagger}$ $\displaystyle=\frac{1}{2g^{2}}U\left(P_{\rm
S}(I)\otimes P_{\rm C}\right)U^{\dagger}+\frac{1}{2g}U\left(P_{\rm
S}(I)\otimes\frac{1}{2}\left.\left|{d_{\rm
C}}\right.\right\rangle\\!\\!\left.\left\langle{d_{\rm C}}\right.\right|_{\rm
C}\right)U^{\dagger}$ (61) $\displaystyle=\frac{1}{2}\Omega_{\rm
S}(I_{+})\otimes\left.\left|{d_{\rm
C}}\right.\right\rangle\\!\\!\left.\left\langle{d_{\rm C}}\right.\right|_{\rm
C}+\frac{1}{2g}\left.\left|{E_{-}}\right.\right\rangle\\!\\!\left.\left\langle{E_{-}}\right.\right|\otimes
P_{\rm C}.$ (62)

It is clear from (62) that

$\displaystyle\mathrm{Tr}_{\rm S}(U(\Omega_{\rm S}(I)\otimes\sigma_{\rm
C})U^{\dagger})=\sigma_{\rm C},$ (63)

as required for a catalytic channel. Moreover, the quantity of interest
$P(w\geq e-e_{-})$ given by this channel $\mathcal{C}$ (defined by $U$ and
$\sigma_{\rm C}$) can be derived by noting that

$\displaystyle\mathcal{C}(\Omega_{\rm
S}(I))=\frac{1}{2}\Omega(I_{+})+\frac{1}{2}\left.\left|{E_{-}}\right.\right\rangle\\!\\!\left.\left\langle{E_{-}}\right.\right|,$
(64)

so that $p(W\geq e-E^{\prime}/n)=\frac{1}{2}$. ∎

##  Appendix 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 previous 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 catalytic channels as introduced in the
main text. In the following, we denote by $P_{\rm S}(I)$ the projector onto a
specific energy-window $I$. Then $g(I)$ is equal to the rank of $P_{\rm
S}(I)$. We consider Hamiltonians $H_{S}^{(N)}$ on a regular lattice
$\Lambda^{(N)}$ of $N$ sites and assume that the $H_{S}^{(N)}$ (for different
values of $N$) constitute a sequence of _local_ , _uniformly bounded_
Hamiltonians:

$\displaystyle H_{S}^{(N)}=\sum_{x\in\Lambda^{(N)}}h_{x},$ (65)

where each term $h_{x}$ acts on sites at most a distance $l$ away from $x$ and
the norm of each term is bounded as $\left\|h_{x}\right\|\leq h$ independent
of the system size for some constant $h$.

######  Proposition 6 (Lower bound to the probability of work extraction).

Fix an inverse temperature $\beta>0$ and consider a sequence of local,
uniformly bounded $N$-particle Hamiltonians $H_{S}^{(N)}$ on a regular,
$D$-dimensional lattice. Assume that the states $\omega_{\beta}(H_{S}^{(N)})$
have a finite correlation length bounded by a constant and denote by $e^{(N)}$
the energy density corresponding to $\beta$. Let $\delta>0$ be fixed and
consider $I^{(N)}:=[e^{(N)}N-\delta\sqrt{N},e^{(N)}N]$. Further assume that
there exist micro-canonical energy windows $I^{(N)}_{+}$ with
$g(I^{(N)}_{+})=g(I^{(N)})^{2}$. Then, for sufficiently large $N$, there
exists, for any $0<e_{-}<e^{(N)}$, a corresponding sequence of catalytic
channels such that

$\displaystyle p(w\geq e^{(N)}-e_{-})\geq
1/2-C\mathrm{e}^{-\frac{\left(\delta^{2}N\right)^{\frac{1}{1+D}}}{K}},$ (66)

where $C,K>0$ are constants.

Before giving the proof, we again emphasize the weakness 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 sufficiently 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 unitary that we use is exactly the same as that constructed 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 ensures 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 distribution 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

$\displaystyle{r(I)}=\mathrm{Tr}(P_{\rm S}(I)\omega_{\beta}(H_{\rm S}))$ (67)

and by $r_{-}=\left.\left\langle{E_{-}}\right.\right|\omega_{\beta}(H_{\rm
S})\left.\left|{E_{-}}\right.\right\rangle$ the initial weight on the low-
energy eigenstate $\left.\left|{E_{-}}\right.\right\rangle$. 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 the energy-windows $I$ and $I_{+}$are disjoint. Denote by
$\\{q_{i}\\}_{i=1}^{d_{C}}$ 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

$\displaystyle q_{d_{C}}\left(r(I)+r(I_{+})\right)=(1-q_{d_{C}})(r(I)+r_{-}).$
(68)

This can be seen, for example, from Fig. 4, where the above represents the
condition of catalyticity for the right-most column. Solving in (68) for
$q_{d_{C}}$, we find that

$\displaystyle q_{d_{C}}=\frac{r(I)+r_{-}}{2r(I)+r_{-}+r(I_{+})}.$ (69)

We now invoke the result from Ref. [33] (as previously in the proof of Lemma
3) which implies that

$\displaystyle{r(I)}\geq 1-\epsilon_{N},$ (70)

where there exist constants $C,K$ such that

$\displaystyle\epsilon_{N}\leq
Ce{-\frac{\left(\delta^{2}N\right)^{\frac{1}{1+D}}}{K}}.$ (71)

For large enough $N$, the energy windows $I$ and $I_{+}$ are disjoint. Hence
$0\leq{r_{-}+r(I_{+})\leq 1-r(I)}$ and we find

$\displaystyle q_{d_{C}}$
$\displaystyle\geq\frac{r(I)}{2r(I)+1-r(I)}=\frac{r(I)}{1+r(I)}$ (72)
$\displaystyle\geq\frac{r(I)}{2}\geq\frac{1}{2}\left(1-\epsilon_{N}\right).$
(73)

Finally, we find

$\displaystyle p(w\geq e-e_{-})$ $\displaystyle\geq P({E_{f}=E_{-}}|E_{i}\in
I)w(I)=q_{d_{c}}\cdot{r(I)}$
$\displaystyle\geq\frac{1}{2}(1-\epsilon_{N})^{2}\geq\frac{1}{2}-\epsilon_{N}.$
(74)

∎

######  Lemma 4 (Existence of catalysts).

Let $\rho_{\rm S}$ be a quantum state on a finite-dimensional Hilbert-space
$\mathcal{H}_{\rm S}$ and $U$ be a unitary on the Hilbert-space
$\mathcal{H}_{\rm S}\otimes\mathcal{H}_{\rm C}$, where $\mathcal{H}_{\rm C}$
is an arbitrary finite-dimensional Hilbert-space. Then there exists a density
matrix $\sigma_{\rm C}$ such that

$\displaystyle\mathrm{Tr}_{\rm S}\left(U(\rho_{\rm S}\otimes\sigma_{\rm
C})U^{\dagger}\right)=\sigma_{\rm C}.$ (75)

###### Proof.

The map $\sigma_{\rm C}\mapsto\mathrm{Tr}_{\rm S}\left(U(\rho_{\rm
S}\otimes\sigma_{\rm C})U^{\dagger}\right)$ 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). ∎

##  Appendix E Proof of Proposition 3 in the main text

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 $\tilde{\mathcal{C}}$ acting
on $C$. We find a $\tilde{\mathcal{C}}$ and two states $\sigma_{-},\sigma_{+}$
with the following properties: (a) If the initial state of the catalyst is
$\sigma_{+}$, the result of running the standard protocol is to extract
positive work $\epsilon$ from the system, while the state of the catalyst is
changed to $\tilde{\mathcal{C}}(\sigma_{+})=\sigma_{-}$. (b) The _same_
unitary, however, for initial state $\sigma_{-}$, extracts negative work
$-\epsilon$ and changes the catalyst state to
$\tilde{\mathcal{C}}(\sigma_{-})=\sigma_{+}$. (c) Hence, if we initialize the
catalyst in the state $\sigma=\frac{1}{2}(\sigma_{+}+\sigma_{-})$, then there
are two “branches” of work extraction distributions, each occurring with
probability $1/2$, while the resulting channel on $S_{i}$ is catalytic for
every $i$. Note that, if the agent _knew_ whether her input state was
$\sigma_{+}$ or $\sigma_{-}$, 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._

Proposition 3 in the main text follows straightforwardly once we realize that
we can tune the process used in the 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 particular, in terms of notation of the
previous section, let $e_{+}^{(N)}$ denote the energy density around which the
window $I_{+}^{(N)}$ is centered. Then we choose $E_{-}$ in such a way that
$e-E_{-}/N=e_{+}-e$ to ensure that the extracted and invested 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

$\displaystyle p(w_{2},w_{3},w_{4},\dots|w_{1})$ (76)

denote the work probability distribution for agents $2$ to $n$ conditional on
the player $1$ extracting work $w_{1}$. The key recognition then is that, for
any $n$, by construction of the catalytic channel,

$p(w_{2},w_{3},\dots|w_{1})=1$ (77)

whenever $w_{i}=-w_{i-1}$ for all $i\in\\{2,\dots,n\\}$, while

$p(w_{2},w_{3},\dots|w_{1})=0$ (78)

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
$\sigma^{\prime}=\left.\left|{d}\right.\right\rangle\\!\\!\left.\left\langle{d}\right.\right|$
with probability 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 $\sigma^{\prime}$ and her copy of $\omega_{\beta}(H)$ will
result in a decrease of the system’s energy (and hence positive work
extraction) and a final catalyst state $\sigma^{\prime\prime}$ with support on
the subspace
$\sum_{i}^{g}\left.\left|{i}\right.\right\rangle\\!\\!\left.\left\langle{i}\right.\right|$,
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 (77). The claimed work distributions then follow from

$\displaystyle
p(w_{1},w_{2},\dots,w_{n})=p(w_{2},w_{3},w_{4},\dots|w_{1})p(w_{1}),$ (79)

together with Proposition 1 in the main text. We also note that a similar
conclusion holds in the case of the microscopic toy-example presented in
Section B, where this behaviour can be checked easily by explicit calculation.

##  Appendix F Proof of Proposition 2 in the main text

Given a catalytic channel $\mathcal{C}$, let $\mathcal{U}$ denote the unitary
channel applied to the joint system $SC$ when dilating the channel. The key
observation is that, if $\mathcal{U}$ is unitary, then $\mathcal{U}^{*}$ is
trace-preserving and hence maps quantum states to quantum states (in fact,
this property holds for all unital channels). Here, ∗ denotes the Hilbert-
Schmidt adjoint. We then write

$\displaystyle\langle\mathrm{e}^{\beta W}\rangle$
$\displaystyle=\mathrm{Tr}\left(\omega\mathcal{C}(\mathbf{1})\right)$ (80)
$\displaystyle=\mathrm{Tr}\left(\omega\otimes\mathbf{I}\mathcal{U}(\mathbf{1}\otimes\sigma)\right)$
(81)
$\displaystyle=d_{C}\mathrm{Tr}\left(\mathcal{U}^{*}\left(\omega\otimes\frac{\mathbf{I}}{d_{C}}\right)\mathbf{1}\otimes\sigma\right)$
(82) $\displaystyle\leq
d_{C}\left\|\mathbf{1}\otimes\sigma\right\|_{\infty}\left\|\frac{\mathbf{1}}{d_{C}}\otimes\sigma\right\|_{1}$
(83) $\displaystyle=d_{C}\left\|\sigma\right\|_{\infty}.$ (84)

Here, the first equality is simply Eqn. 2 in the main text and we write
$\omega$ instead of $\omega_{\beta}(H)$. Similarly, we get

$\displaystyle\langle\mathrm{e}^{\beta W}\rangle$
$\displaystyle=d\mathrm{Tr}\left((\omega\otimes\mathbf{I})\mathcal{U}\left(\frac{\mathbf{1}}{d}\otimes\sigma\right)\right)$
(85) $\displaystyle\leq
d\left\|\omega\otimes\mathbf{I}\right\|_{\infty}=d\left\|\omega\right\|_{\infty}.$
(86)

##  Appendix G Non-trivial Hamiltonian on the catalyst

In this section we show that the probability distribution of work done on the
system is independent of the Hamiltonian on the catalyst. To do this, let us
first assume we had a catalytic process that uses a catalyst with a non-
trivial Hamiltonian $H_{C}$ and a quasi-classical state $\sigma_{C}$, i.e.,
$[H_{C},\sigma_{C}]=0$. We assume that $\sigma_{C}$ is quasi-classical, since
it is well known that it is impossible to associate a meaningful 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

$W^{(S)}=E_{f}^{(S)}-E_{i}^{(S)}$ (87)

denotes the work done on the system and

$W^{(C)}=E_{f}^{(C)}-E_{i}^{(C)}$ (88)

the work done on the catalyst. The work distribution on the system is simply
given by the marginal

$\displaystyle P\left(W^{(S)}\right)=\int
P\left(W^{(S)},W^{(C)}\right)\,\mathrm{d}W^{(C)}.$ (89)

Let us write $\sigma_{C}=\sum_{j}\sigma_{j}|E^{(C)}_{j}\rangle\langle
E^{(C)}_{j}|$ and $\omega_{\beta}(H)=\sum_{k}w_{k}|E^{(S)}_{k}\rangle\langle
E^{(S)}_{k}|$. We then get

$\displaystyle P\left(W^{(S)}\right)$
$\displaystyle=\sum_{E^{(S)}_{f}-E^{(S)}_{i}=W^{(S)}}\sum_{E^{(C)}_{f^{\prime}}}\sum_{E^{(C)}_{i^{\prime}}}P\left(E^{(S)}_{f},E^{(C)}_{f^{\prime}}|E^{(S)}_{i},E^{(C)}_{i^{\prime}}\right)P(E^{(S)}_{i})P(E^{(C)}_{i^{\prime}})$
$\displaystyle=\sum_{E^{(S)}_{f}-E^{(S)}_{i}=W^{(S)}}\sum_{E^{(C)}_{f^{\prime}}}\sum_{E^{(C)}_{i^{\prime}}}\langle
E^{(S)}_{f}|\otimes\langle
E^{(C)}_{f^{\prime}}|\left(U\left(w_{i}\sigma_{i^{\prime}}|E^{(S)}_{i}\rangle\langle
E^{(S)}_{i}|\otimes|E^{(C)}_{i^{\prime}}\rangle\langle
E^{(C)}_{i^{\prime}}|\right)U^{\dagger}\right)|E^{(S)}_{f}\rangle\otimes|E^{(C)}_{f^{\prime}}\rangle$
$\displaystyle=\sum_{E^{(S)}_{f}-E^{(S)}_{i}=W^{(S)}}\langle
E^{(S)}_{f}|\mathrm{Tr}_{C}\left(U\left(w_{i}|E^{(S)}_{i}\rangle\langle
E^{(S)}_{i}|\otimes\sigma\right)U^{\dagger}\right)|E^{(S)}_{f}\rangle$ (90)
$\displaystyle=\sum_{E^{(S)}_{f}-E^{(S)}_{i}=W^{(S)}}\langle
E^{(S)}_{f}|\mathcal{C}\left(w_{i}|E^{(S)}_{i}\rangle\langle
E^{(S)}_{i}|\right)|E^{(S)}_{f}\rangle.$ (91)

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
always 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 transitions.

##  Appendix H Comparison with literature on generalized Jarzynski equalities
in the presence of correlations

In recent years, the role of correlations, specifically quantified 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
demonstrations to test these theoretical results [28, 39]. One may ask, how
the results of this manuscript fit in the context 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 advantage, 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 performed on
the system that is given by

$\langle W\rangle\geq\langle\Delta F\rangle+\langle\Delta E_{\rm
int}\rangle+\beta^{-1}\langle\Delta I\rangle,$ (92)

where $\langle\Delta F\rangle$ is the difference between final and initial
equilibrium free energy on the system, $\langle\Delta E_{\rm int}\rangle$ for
the energy difference coming from the interaction Hamiltonian between system
and catalyst, and finally $\langle\Delta I\rangle$ is the change in mutual
information between system and catalyst. For our setup, both $\langle\Delta
F\rangle$ and $\langle\Delta E_{\rm int}\rangle$ are zero. Given that the
extracted work $W_{\rm ext}=-W$, the above bound reduces to

$\langle W_{\rm ext}\rangle\leq-\beta^{-1}\langle\Delta I\rangle,$ (93)

which says that if one allows the consumption of mutual information (leading
to $\Delta I<0$), then it is possible to violate the average second law,
namely extract some positive amount of $W_{\rm ext}$ 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 always have $\langle\Delta I\rangle\geq 0$.
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
allow for violations of the original JE and Second law _only_ if
$\langle\Delta I\rangle<0$. 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. Indeed, 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 $\beta$-catalytic channel allows for the evolution
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 evolutions to be implemented on the system and hence explains 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.

##  Appendix I Is it necessary to establish correlations with the catalyst?

In our definition of $\beta$-catalytic channels, we allow the catalyst to
become correlated with the system. These correlations 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, consider the set of _$\beta$ -trumping
channels_, where a quantum channel $\mathcal{T}$ is in this set iff it has the
form

$\mathcal{T}(\rho)=\mathrm{Tr}_{2}(\mathcal{N}(\rho\otimes\sigma)),$ (94)

where $\mathcal{N}(\mathbb{I})=\mathbb{I}$ is unital and
$\mathcal{N}(\omega_{\beta}(H)\otimes\sigma)=\rho^{\prime}\otimes\sigma$. Note
that in the case of $\beta$-catalytic channels, we restricted the
corresponding channel $\mathcal{N}$ to be unitary. Here, we allow instead for
the more general class of unital channels. We will prove that in the unitary
case, NMW cannot be violated by $\beta$-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 thermalizing channel is
exactly a $\beta$-trumping channel where $\sigma=\omega_{\beta}(H)$, and
$\mathcal{N}$ is a unitary swap between the system and catalyst. Thus, even in
the case of a unitary channel $\mathcal{N}$, such $\beta$-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 means of $\beta$-trumping channels. However, we
do not believe 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 $S_{\infty}$
(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]). However, in
our constructions, the final min-entropy is essentially given by
$-\log(p(w\geq\epsilon)\approx\log(2)$, 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 $\beta$-trumping
channels cannot be used to violate NMW in general: We now present an argument
that rules out violations of NMW in the case of a microcanonical initial state
$\Omega$ with energy density $e$, but we expect that similar statements hold
true for the canonical case due to equivalence of ensembles-type of arguments.
Because of the highly peaked probability distribution of the energy density
for a macroscopic, non-critical many-body system, it is easy to see that the
probability $p(w\geq\epsilon)$ to extract work per particle at least
$\epsilon$ is (up to arbitrarily small corrections for large $N$) given by the
total probability of measuring an energy below $(e-\epsilon)N$ in the final
state $\mathcal{T}(\Omega)$. Let us denote the projector onto these energies
by $P$. We then have

$\displaystyle p(w\geq\epsilon)\approx\mathrm{Tr}[P\mathcal{T}(\Omega)],$ (95)

where the approximation is arbitrarily good as $N\rightarrow\infty$. 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 $(e-\epsilon)N$ is given by $\exp(s(e-\epsilon)N)$, where
$s(e-\epsilon)$ is the microcanonical entropy density at energy density
$e-\epsilon$. Since the total weight in this subspace is $p(w\geq\epsilon)$,
the final min-entropy is upper bounded by

$\displaystyle
S_{\min}^{\mathrm{(final)}}\leq-\log(p(w\geq\epsilon))+s(e-\epsilon)N.$ (96)

However, since trumping requires $S_{\min}^{\mathrm{(final)}}\geq
S_{\min}^{\mathrm{(initial)}}=s(e)N$, we then find

$\displaystyle p(w\geq\epsilon)\leq\exp(-(s(e)-s(e-\epsilon))N)\rightarrow 0,$
(97)

as $N\rightarrow\infty$ for any $\epsilon>0$. This shows that NMW holds for
$\beta$-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
catalysts provide a strict advantage in this set-up.

iii) Finally, let us also show that if we assume that $\mathcal{N}$ is
_unitary_ , as we do in the case of $\beta$-catalytic channels, then NMW
cannot be violated if the catalyst remains uncorrelated. The reason is the
following: Since the global transformation on system and catalyst is unitary,
it leaves the spectrum invariant. Since the catalyst remains invariant 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 $\mathcal{T}[\omega_{\beta}(H)]=V\omega_{\beta}(H)V^{\dagger}$.
As argued in case ii), we then have

$\displaystyle p(w\geq\epsilon)$
$\displaystyle\approx\mathrm{Tr}[P\mathcal{T}[\omega_{\beta}(H)]]=\mathrm{Tr}[PV\omega_{\beta}(H)V^{\dagger}]$
(98) $\displaystyle\leq\mathrm{Tr}[P\omega_{\beta}(H)],$ (99)

where the last inequality follows because Gibbs states are passive states and
the first approximation holds to arbitrary accuracy as $N\rightarrow\infty$.
However, by the same concentration inequalities we used to prove of our main
results, we have

$\displaystyle\mathrm{Tr}[P\omega_{\beta}(H)]\leq
K\exp(-k(\epsilon^{2}N)^{1/(1+D)}),$ (100)

for a non-critical many-body system in $D$ spatial dimensions (with constants
$k,K>0$). Thus, NMW holds true in this case as well.