# Coherent fluctuation relations: from the abstract to the concrete

Zoë Holmes  Controlled Quantum Dynamics Theory Group, Imperial College London,
London, SW7 2BW, United Kingdom.  Sebastian Weidt  Department of Physics and
Astronomy, University of Sussex, Brighton BN1 9QH, United Kingdom.  David
Jennings  Controlled Quantum Dynamics Theory Group, Imperial College London,
London, SW7 2BW, United Kingdom.  Department of Physics, University of Oxford,
Oxford, OX1 3PU, United Kingdom.  School of Physics and Astronomy, University
of Leeds, Leeds, LS2 9JT, United Kingdom.  Janet Anders  CEMPS, Physics and
Astronomy, University of Exeter, Exeter, EX4 4QL, United Kingdom.  Florian
Mintert  Controlled Quantum Dynamics Theory Group, Imperial College London,
London, SW7 2BW, United Kingdom.

###### Abstract

Recent studies using the quantum information theoretic approach to
thermodynamics show that the presence of coherence in quantum systems
generates corrections to classical fluctuation theorems. To explicate the
physical origins and implications of such corrections, we here convert an
abstract framework of an autonomous quantum Crooks relation into quantum
Crooks equalities for well-known coherent, squeezed and cat states. We further
provide a proposal for a concrete experimental scenario to test these
equalities. Our scheme consists of the autonomous evolution of a trapped ion
and uses a position dependent AC Stark shift.

##  1 Introduction

The emergent field of quantum thermodynamics seeks to extend the laws of
thermodynamics and non-equilibrium statistical mechanics to quantum systems. A
central question is whether quantum mechanical phenomena, such as coherence
and entanglement, generate corrections to classical thermal physics.

A quantum information theoretic approach has proven a fruitful means of
incorporating genuinely quantum mechanical effects into thermal physics [1].
For example, the consequences of quantum entanglement for Landauer erasure
[2], the thermodynamic arrow of time [3] and thermalisation [4] have been
investigated. In addition, for incoherent quantum systems, general criteria
for state conversion [5], generalisations of the second laws [6], and limits
to work extraction protocols [7] have been established. More recently, these
results have been extended to coherent quantum states [8, 9, 10, 11]. Much of
this research [5, 6, 7, 8, 9, 10, 11] utilised the resource theory framework
and in particular the concept of thermal operations [12, 13]. However, despite
significant theoretical progress, the results have been seen as rather
abstract and unamendable to experimental implementation [14].

A separate line of enquiry has explored fluctuation theorems, which can be
seen as generalisations of the second law of thermodynamics to non-equilibrium
processes [15]. They consider systems that are driven out of equilibrium and
establish exact relations between the resultant thermal fluctuations [16].
Experimental tests of fluctuation theorems, in particular the Jarzynski [17]
and Crooks [18] equalities, have been conducted in classical systems,
including stretched RNA molecules [19, 20], over-damped colloidal particles in
harmonic potentials [21] and classical two-state systems [22], and quantum
systems such as trapped ions [23] and NMR systems [24]. An experiment to test
a quantum Jarzynski equality in a weakly measured system using circuit QED has
recently been performed [25].

However, the use of two point energy measurements or continual weak
measurements to obtain a work probability distribution reduces a system’s
coherence with respect to the energy eigenbasis [26, 27, 28, 29, 30, 31]. This
limits the extent to which these previous experiments probe quantum mechanical
phenomena that arise from coherences.

A recent theoretical proposal of a new quantum fluctuation relation [32] that
connects naturally with quantum information theory, models the work system
explicitly as a quantum battery. Rather than implicitly appealing to an
additional classical system to drive the system out of equilibrium, the
quantum system and battery here evolve autonomously under a time independent
Hamiltonian [33, 34, 35, 36]. This proposal does not require projective
measurements onto energy eigenstates and so does not destroy coherences. As a
result coherence actively contributes to the Autonomous Quantum Crooks
equality (AQC) [32], which will be defined in Eq. (8).

In this paper, we exploit the combined strengths of the quantum information
and fluctuation theorem approaches to quantum thermodynamics [37, 38]. We
develop quantum Crooks equalities for a quantum harmonic oscillator battery
that is prepared in optical coherent states, squeezed states or cat states and
explore the physical content of these equalities. In particular, we present
the coherent state Crooks equality, see Eq. (LABEL:eq:_coherentstaterewrite),
in which quantum corrections to the classical Crooks equality can be
attributed to the presence of quantum vacuum fluctuations.

We propose an experiment to test the AQC through which the role of coherence
in thermal physics can be probed. Using a trapped ion, one can realise a two
level system with a pair of the ion’s internal energy levels and an oscillator
battery with the ion’s axial phonon mode. The motion of the ion through an
off-resonance laser beam induces a position dependent AC Stark shift on the
internal energy levels. In this way an effectively time-dependent Hamiltonian
for the two level system can be realised.

##  2 The Classical and Quantum Crooks Equalities

###  2.1 The classical Crooks equality

An initially thermal system, at temperature $T$, can be driven from
equilibrium with a time-dependent Hamiltonian, changing from $H_{S}^{i}$ to
$H_{S}^{f}$. The classical Crooks equality [18],

$\displaystyle\frac{{\cal P}^{+}(W)}{{\cal P}^{-}(-W)}=\exp\left(-\frac{\Delta
F}{k_{B}T}\right)\exp\left(\frac{W}{k_{B}T}\right)\,,$ (1)

quantifies the ratio of the probability ${\cal P}^{+}(W)$ of the work $W$ done
on a system in such a non-equilibrium process, to the probability ${\cal
P}^{-}(-W)$ to extract the work $W$ in the time reversed process (where the
Hamiltonian is changed back from $H_{S}^{f}$ to $H_{S}^{i}$). This ratio is a
function of the change in the system’s free energy,

$\displaystyle\Delta F\mathrel{\mathop{\mathchar
58\relax}}=F(H^{f}_{S})-F(H^{i}_{S})\ \ \ \ \text{with}$ (2) $\displaystyle
F(H_{S})\mathrel{\mathop{\mathchar
58\relax}}=-k_{B}T\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ln}}\left(\operatorname{Tr}_{S}\left[\exp\left(-\frac{H_{S}}{k_{B}T}\right)\right]\right)\,,$

and hence measurement of $P(W)$ and $P(-W)$ for a non-equilibrium process
provides a means of inferring free energy changes, an equilibrium property of
a system. More generally, Eq. (1) asserts that processes that produce work are
exponentially less likely than processes that require work, irrespective of
how the change from $H_{S}^{i}$ to $H_{S}^{f}$ is realised.

###  2.2 The autonomous quantum Crooks equality

The realisation of a time-dependent Hamiltonian in the classical Crooks
equality, Eq. (1), necessarily implies an interaction with a classical agent.
The autonomous framework [32] proposed by Johan Åberg makes this control
explicit by introducing a battery. Specifically, the system evolves together
with the battery according to the time-independent Hamiltonian,

$\displaystyle H_{SB}=H_{S}\otimes\mathbbm{1}_{B}+\mathbbm{1}_{S}\otimes
H_{B}+V_{SB}\,,$ (3)

comprised of the Hamiltonians $H_{S}$ and $H_{B}$ for system and battery, and
their interaction $V_{SB}$.

For the initial and final Hamiltonians of the system to be well defined, we
need the system and battery not to interact at these times. To ensure this, we
consider an interaction of the form

$V_{SB}=H_{S}^{i}\otimes\Pi_{B}^{i}+H_{S}^{f}\otimes\Pi_{B}^{f}+V_{SB}^{\perp}\,,$
(4)

where $\Pi_{B}^{i}$ and $\Pi_{B}^{f}$ are projectors onto two orthogonal
subspaces, $R_{i}$ and $R_{f}$, of the battery’s Hilbert space, and
$V_{SB}^{\perp}$ has support only outside those two subspaces, i.e.
$(X_{S}\otimes\Pi_{B}^{i})V_{SB}^{\perp}=(X_{S}\otimes\Pi_{B}^{f})V_{SB}^{\perp}=0$
for any system operator $X_{S}$. The system Hamiltonian $H_{S}$ can always be
absorbed into $V_{SB}$, which we will do in the following. Assuming the
battery is initialised in a state in subspace $R_{i}$ only and evolves to a
final state in subspace $R_{f}$ only, the system Hamiltonian evolves from
$H_{S}^{i}$ to $H_{S}^{f}$, i.e. the system Hamiltonian is effectively time-
dependent. Since the system and battery are non-interacting in the regions
$R_{i}$ and $R_{f}$, and since energy is globally conserved, the energy
required or produced by the system as its Hamiltonian evolves from $H_{S}^{i}$
to $H_{S}^{f}$ is necessarily provided or absorbed by the battery.

The initially thermal system in the Crooks equality is incoherent in energy
and so, to extend the relation to the quantum regime in this autonomous
setting, coherence111We use the term ‘coherence’ in the sense of a
‘superposition of states belonging to different energy eigenspaces’. Such
energetic coherences are formally quantified in the thermal operations
framework [12, 13] and by measures such as the $l_{1}$ norm of coherence [39],
$C_{l_{1}}(\rho)\mathrel{\mathop{\mathchar 58\relax}}=\sum_{j\neq
k}|\bra{E_{j}}\rho\ket{E_{k}}|$, the sum of the absolute value of the off
diagonal elements of a state in the energy eigenbasis. must begin in an
additional non-equilibrium quantum system. Here the coherence is provided by
the battery which we assume can be prepared in superpositions of energy
states. The evolution under $H_{SB}$ generates correlations between the system
and battery reducing the coherence that can be attributed to the battery
alone. Since the dynamics conserves coherence globally, the coherence of the
final joint state of the system and battery is equal to that initially
provided by the battery. Thus this framework allows us to describe the
evolution of coherence under interactions with a thermal system.

The difficulties surrounding how to define work in the quantum regime [40, 41,
42, 43, 44, 45] are avoided by formulating the AQC in terms of transition
probabilities between battery states. Considering a system initialised in the
thermal state $\gamma_{i}$ and the battery initialised in the pure state
$\ket{\phi_{i}}$, the probability to observe the battery after a time-interval
$t$ in the state $\ket{\phi_{f}}$ reads

$\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})=\bra{\phi_{f}}\operatorname{Tr}_{S}[U(\gamma_{i}\otimes\ket{\phi_{i}}\bra{\phi_{i}})U^{\dagger}]\ket{\phi_{f}}\,,$
(5)

with
$U=\exp(-iH_{SB}t/{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\hbar}})$
the propagator of the system and battery and the trace over the system denoted
by $\operatorname{Tr}_{S}$. If we consider the transition probability between
energy eigenstates, the scheme considered here reduces to the usual two-point
projective energy measurement scheme [26, 27, 28] and as such we can recover
the classical Crooks relation, Eq. (1). For transition probabilities between
more general quantum states, such as between coherent states, squeezed states
and cat states [46] as we will consider, the post measurement battery state
has coherences in the energy eigenbasis and the measurement can induce a non-
classical updating of the state of the system.

The AQC relates the probabilities $\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})$
of a forward process and $\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})$ of a
reverse process for two thermal system states

$\gamma_{x}\propto\exp\left(-\frac{H_{S}^{x}}{k_{B}T}\right)\,,\mbox{ with }\
x=i,f\ $ (6)

and two pairs of pure initial and final battery states satisfying

$\begin{split}\ket{\psi_{f}}&\propto{\cal
T}\exp\left(-\frac{H_{B}}{2k_{B}T}\right)\ket{\phi_{f}}\ \,,\\\
\ket{\phi_{i}}&\propto{\cal
T}\exp\left(-\frac{H_{B}}{2k_{B}T}\right)\ket{\psi_{i}}\ .\end{split}$ (7)

The states $\ket{\psi_{i}}$ and $\ket{\psi_{f}}$ correspond to a time-reversed
process and are defined in terms of the time-reversal operator222The time-
reversal [47] operation ${\cal T}$ on a battery state $\ket{\psi}$ is defined
as complex conjugation in the energy eigenbasis of the battery, ${\cal
T}\ket{\psi}=\ket{\psi^{*}}$. We introduce ${\cal T}$ in more detail in
Appendix A.2. ${\cal T}$ [47]. This relationship between the initial and final
states of the forwards and reversed processes, Eq. (7), is forced by the
derivation of the AQC, i.e. the structure arises naturally when one reverses a
general quantum process in an autonomous setting. If the battery states
$\ket{\phi_{i}}$ and $\ket{\phi_{f}}$ are eigenstates of $H_{B}$, the states
$\ket{\psi_{f}}$ and $\ket{\psi_{i}}$ of the reversed process are regular
time-reversed states. However, for any coherent superposition of energy
eigenstates the additional term $\exp(-H_{B}/2k_{B}T)$, which is linked with
the Petz recovery map [48, 49], is essential to capture the influence of
quantum coherence.

The AQC [32] reads

$\frac{\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})}{\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})}=\exp\left(-\frac{\Delta
F}{k_{B}T}\right)\exp\left(\frac{\Delta\tilde{E}}{k_{B}T}\right)\,,$ (8)

where $\Delta F$ is the change in the system’s equilibrium free energy, Eq.
(2), and thus the AQC, similarly to the classical Crooks equality, provides a
means of inferring free energy changes. The term

$\Delta\tilde{E}(\psi_{i},\phi_{f})\mathrel{\mathop{\mathchar
58\relax}}=\tilde{E}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\psi_{i}}})-\tilde{E}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\phi_{f}}})$
(9)

is a quantum mechanical generalisation of the energy flow from the battery to
the system. The function

$\tilde{E}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\psi}})\mathrel{\mathop{\mathchar
58\relax}}=-k_{B}T\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ln}}\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\bra{\psi}}}\exp\left(-\frac{H_{B}}{k_{B}T}\right){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ket{\psi}}}\right)\
$ (10)

is an effective potential for the battery state $\ket{\psi}$ that specifies
the relevant energy value within the fluctuation theorem context. We provide a
discussion of its properties in Appendix A.7.

For the AQC to hold exactly we require $\ket{\psi_{i}}$ and $\ket{\phi_{f}}$
to have support only within subregions $\Pi^{i}_{B}$ and $\Pi^{f}_{B}$
respectively and for the battery Hamiltonian not to induce evolution between
subregions, i.e. $(\mathbbm{1}_{B}-\Pi_{B}^{i})H_{B}\Pi_{B}^{i}=0$ and
$(\mathbbm{1}_{B}-\Pi_{B}^{f})H_{B}\Pi_{B}^{f}=0$. In general, the second of
these conditions does not hold because a battery Hamiltonian that evolves
states between subregions is required. However, despite this, numerical
simulations (discussed in Section 3.2) indicate that any error can be made
negligible as long as $\ket{\psi_{i}}$ and $\ket{\phi_{f}}$ are well localised
in subregions $\Pi_{B}^{i}$ and $\Pi_{B}^{f}$ respectively. In Section 2.5 we
discuss how the accuracy of the equality can be quantified; however, the
material is relatively technical and can be skipped.

To make the predictions of the AQC, Eq. (8), more concrete we now choose the
battery to be a quantum harmonic oscillator with Hamiltonian
$H_{B}=\sum_{n}\hbar\omega\,\left(n+\frac{1}{2}\right)\,\ket{n}\bra{n}$. In
order to derive explicit expressions for the ratio of transition
probabilities, Eq. (8), we consider common states of the harmonic oscillator,
specifically coherent states, squeezed states and cat states [46]. For each of
these three states we derive a specific AQC which is based on the generalised
energy flow, Eq. (9), and the relation between the initial and final states of
the forward and reversed process, Eq. (7). While the battery Hamiltonian and
states are fixed in the following sections, the system Hamiltonians,
$H_{S}^{i}$ and $H_{S}^{f}$, and the system and battery interaction,
$V_{SB}^{\perp}\,$, can still be chosen at will.

###  2.3 Coherent state Crooks equality

The physical content of the AQC is neatly illustrated by considering
transition probabilities between coherent states of the battery,
$\ket{\alpha}=\exp(\alpha a^{\dagger}-\alpha^{\ast}a)\ket{0}$ with
$a^{\dagger}$ and $a$ the creation and annihilation operators of the
oscillator respectively, and $\alpha$ a complex number. In the forwards
process, we assume the battery is initialised in $\ket{\alpha_{i}\exp(-\chi)}$
with $\chi=\frac{\hbar\omega}{2k_{B}T}$ and we are interested in the
transition probability
$\mathcal{P}(\alpha_{f}|\alpha_{i}\exp(-\chi),\gamma_{i})$ to the coherent
state $\ket{\alpha_{f}}$. Following Eq. (7), the initial and final states of
the reversed process are given by the coherent states
$\ket{\alpha_{f}^{\ast}\exp(-\chi)}$ and $\ket{\alpha_{i}^{\ast}}$. The symbol
∗ denotes complex conjugation in the Fock basis and arises from the time
reversal operation on a coherent state, ${\cal
T}\ket{\alpha}=\ket{\alpha^{\ast}}$.

The dimensionless parameter $\chi$ is the ratio of the magnitude of quantum
fluctuations, $\frac{\hbar\omega}{2}$, to the magnitude of thermal
fluctuations, $k_{B}T$, and quantifies the degree to which any regime is
quantum mechanical. In the limit in which thermal fluctuations dominate,
$\chi$ tends to 0, and the prefactor $\exp(-\chi)$ tends to 1. Consequently,
in this limit the reverse process is the exact time-reversed process; however,
in general, the pairs of states considered in the forwards and reverse process
differ by an amount determined by $\exp(-\chi)$.

As derived in more detail in B.1, the energy flow, Eq. (9), takes the explicit
form

$\displaystyle\Delta\tilde{E}(\alpha_{f},\alpha_{i})=k_{B}T\left(|\alpha_{i}|^{2}-|\alpha_{f}|^{2})(1-\exp(-2\chi)\right)\
.$ (11)

In order to highlight similarities and differences to the classical situation,
it is instructive to explicitly introduce the difference between the average
energy cost $\Delta E_{+}$ and gain $\Delta E_{-}$ of the forward and reverse
processes. We define the prefactor

$q\mathrel{\mathop{\mathchar 58\relax}}=\frac{\Delta\tilde{E}}{W_{q}}$ (12)

as the ratio between $\Delta\tilde{E}$ and $W_{q}=(\Delta E_{+}-\Delta
E_{-})/2$. As shown in B.1, we find that for coherent states the prefactor
takes the form

$q\left(\chi\right)=\frac{1}{\chi}\tanh(\chi)\ $ (13)

and can be related to the average frequency, $\omega_{T}$, of the oscillator
battery in a thermal state,
$\gamma(H_{B})\propto\exp\left(-\frac{H_{B}}{k_{B}T}\right)$, at temperature
$T$,

$\hbar\omega_{T}\mathrel{\mathop{\mathchar 58\relax}}=\langle
H_{B}\rangle_{{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\gamma(H_{B})}}}=\frac{k_{B}T}{q(\chi)}\
.$ (14)

The coherent state AQC can thus be written as

$\displaystyle\frac{\mathcal{P}\left(\alpha_{f}\big{\rvert}\alpha_{i}\exp(-\chi),\gamma_{i}\right)}{\mathcal{P}\left(\alpha_{i}^{*}\big{\rvert}\alpha_{f}^{*}\exp(-\chi),\gamma_{f}\right)}=\exp\left(-\frac{\Delta
F}{k_{B}T}\right)\exp\left(\frac{W_{q}}{\hbar\omega_{T}}\right)\,.$ (15)

In this form, it is analogous to the classical Crooks equality, Eq. (1), with
the difference encoded in $\omega_{T}$.

Figure 1: The solid, dark blue line shows the average energy,
$\hbar\omega_{T}$, of the oscillator in a thermal state at temperature $T$ as
a function of $\chi$. The red lines indicate the contribution of thermal
(dotted) and quantum (dashed) contributions to $\hbar\omega_{T}$ as determined
by Eq. (16). The grey line is the classical limit in which the energy of the
harmonic oscillator equals $k_{B}T$. Energies are given in units of $k_{B}T$.

The thermal frequency $\omega_{T}$ can be understood in terms of the thermal
wavelength $\lambda_{th}=\frac{h}{p}$ [50] given in terms of the momentum $p$
of a particle. This is a spatial scale that quantifies the regime in which a
thermal system should be treated as classical or quantum. Broadly speaking,
when $\lambda_{th}$ is small the system is well localised and can be
considered classical; however, quantum effects dominate when it is larger.

The average energy of the oscillator is composed of kinetic, potential, and
vacuum fluctuation terms. For an oscillator in a thermal state the kinetic and
potential energies are equal, and with the vacuum energy $E_{\mbox{\tiny
vac}}=\frac{1}{2}\hbar\omega$, it follows that

$\hbar\omega_{T}=\frac{h^{2}}{m\lambda_{th}^{2}}+\frac{1}{2}\hbar\omega\ .$
(16)

This equality describes the splitting of $\hbar\omega_{T}$ into its thermal,
$\frac{h^{2}}{m\lambda_{th}^{2}}$, and quantum, $\frac{1}{2}\hbar\omega$,
contributions, as is shown in Fig. 1.

In the classical limit the thermal wavelength $\lambda_{th}$ is small and the
dominant contribution to $\hbar\omega_{T}$ is thermal. In this limit
$\hbar\omega_{T}$ tends to $k_{B}T$ as predicted by the equipartition theorem
and in agreement with the classical Crooks equality.

In the quantum limit of large $\lambda_{th}$ the thermal contribution tends to
zero and vacuum fluctuations dominate. These vacuum fluctuation induce quantum
corrections to $\hbar\omega_{T}$ which increase monotonically with $\chi$.
Consequently, the exponential dependence of the ratio of transition
probabilities on the difference in energy, $W_{q}$, is suppressed and the
probability for the reverse process is larger than expected classically. We
conclude that the presence of quantum coherence in a sense makes
irreversibility milder. As the dynamics quantified by the AQC are unitary and
so fully reversible, the irreversibility quantified here stems from the choice
in the prepared and measured battery states [51].

###  2.4 Squeezed state and cat state Crooks equalities

Analogously to Eq. (LABEL:eq:_coherentstaterewrite), we now discuss quantum
Crooks equalities that quantify the ratio of transition probabilities between
pairs of squeezed states of the battery and pairs of cat states of the
battery. Squeezed displaced states, $\ket{r,\alpha}\mathrel{\mathop{\mathchar
58\relax}}=\exp(\alpha
a^{\dagger}-\alpha^{\ast}a)\exp(\frac{r}{2}(a^{2}-{a^{\dagger}}^{2}))\ket{0}$,
are the unbalanced minimum uncertainty states where for a given squeezing
parameter $r$, the variance of the oscillator’s position and momentum are
rescaled as $\Delta x=\exp(-r)\sqrt{\frac{\hbar}{2m\omega}}$ and $\Delta
p=\exp(+r)\sqrt{\frac{\hbar m\omega}{2}}$. Cat states $\ket{\mbox{\small
Cat}}\propto\ket{\alpha}+\ket{\beta}$ are superpositions of coherent states
$\ket{\alpha}$ and $\ket{\beta}$.

The coherent, squeezed, and cat state equalities differ in the choice of
preparation and measurement states of the battery. The following tables list
the form of these battery states respectively for the forwards
($\ket{\phi_{i/f}}$) and reverse ($\ket{\psi_{i/f}}$) processes.

Forwards | Preparation | Measurement
---|---|---
Coherent | $\ket{\alpha_{i}e^{-\chi}}$ | $\ket{\alpha_{f}}$
Squeezed | $\ket{s_{i},\mu_{i}}$ | $\ket{r_{f},\alpha_{f}}$
Cat | $\eta_{\chi}^{|\alpha_{i}|^{2}}\ket{\alpha_{i}e^{-\chi}}+\eta_{\chi}^{|\beta_{i}|^{2}}\ket{\beta_{i}e^{-\chi}}$ | $\ket{\alpha_{f}}+\ket{\beta_{f}}$

Reverse | Preparation | Measurement
---|---|---
Coherent | $\ket{\alpha_{f}^{*}e^{-\chi}}$ | $\ket{\alpha_{i}^{*}}$
Squeezed | $\ket{s_{f},\mu_{f}^{*}}$ | $\ket{r_{i},\alpha_{i}^{*}}$
Cat | $\eta_{\chi}^{|\alpha_{f}|^{2}}\ket{\alpha_{f}^{*}e^{-\chi}}+\eta_{\chi}^{|\beta_{f}|^{2}}\ket{\beta_{f}^{*}e^{-\chi}}$ | $\ket{\alpha_{i}^{*}}+\ket{\beta_{i}^{*}}$

In the above table the cat states are unnormalized for brevity and we have
defined $\eta_{\chi}\mathrel{\mathop{\mathchar
58\relax}}=\exp\left(-\frac{1}{2}\left(1-\exp(-2\chi)\right)\right)$,
$s\mathrel{\mathop{\mathchar
58\relax}}=\tanh^{-1}\left(\exp(-2\chi)\tanh(r)\right)$ and
$\mu\mathrel{\mathop{\mathchar
58\relax}}=\frac{\exp(-\chi)\left(1+\tanh(r)\right)}{1+\exp(-2\chi)\tanh(r)}$.
We do not show the explicit expressions of $\Delta\tilde{E}$ for the cat and
squeezed state equalities here as they are reasonably long and uninstructive.
However, Appendix B contains full derivations and statements of the two
equalities.

(a)  (b)

Figure 2: We plot $q\mathrel{\mathop{\mathchar
58\relax}}=\frac{\Delta\tilde{E}}{W_{q}}$ as a function of $\chi$ for squeezed
displaced states. As indicated by the grey Wigner plot insets, in a) the
prepared and measured squeezed displaced battery state are displaced with
respect to position only, i.e. $\alpha_{i,f}=\Re(\alpha_{i,f})$ and
$\mu_{i,f}=\Re(\mu_{i,f})$; whereas in b) the state is displaced with respect
to momentum only, i.e. $\alpha_{i,f}=\Im(\alpha_{i,f})$ and
$\mu_{i,f}=\Im(\mu_{i,f})$. In a) the red, blue and turquoise lines plot
$r=1$, $r=0$ and $r=-1$ respectively. In b) the red, blue and turquoise lines
plot $r=-1$, $r=0$ and $r=1$ respectively. In both figures the grey line
indicates the classical limit in which $q=1$.

We compare the additional effect of squeezing to the coherent state cases in
Fig. 2 by plotting $q(\chi)$, the ratio of $\Delta\tilde{E}$ and $W_{q}$ as
defined in Eq. (12). As is shown in dark blue, for coherent states $q$ reduces
to 1 in the classical limit where $\chi$ tends to 0. However, $q$ decreases
monotonically for increasing $\chi$, and vanishes as $\chi$ tends to infinity.

When the battery is prepared and measured in states that are displaced with
respect to position and squeezed with respect to momentum, we see in Fig. 2a
that $q$ behaves in a similar way to the coherent state case but with a
quicker initial rate of decrease. When the battery is displaced with respect
to position and squeezed with respect to position, the behaviour of $q$ for
large $\chi$ again replicates the coherent state case; however, there is now a
regime of low $\chi$ in which $q$ is greater than the classical value of 1. In
Fig. 2b the prepared and measured states are displaced with respect to
momentum and we see identical behaviour to Fig. 2a except the roles of
squeezing with respect to position and momentum are reversed.

Thus we see that for a squeezed battery the probability for the reverse
process is again larger than expected classically in the limit of large
$\chi$. However, there is now an intermediary regime where the probability for
the reverse process is even more suppressed than in the classical limit. In
the presence of squeezing, irreversibility can thus be stronger than expected
classically.

Cat states are inherently quantum mechanical and do not reduce to a semi-
classical state in the limit of low $\chi$. Consequently, the analogous plot
to Fig. 2 for cat states is not a straightforward correction to the classical
limit. The plot is thus uninstructive and so we do not include it here.

###  2.5 Quantifying accuracy of the AQC

The approximate nature of the AQC can be quantified by error bounds. Here we
establish three such error measures, $D$, $\epsilon$ and $(1-R)$.

It is instructive to first note that Eq. (8) can be rewritten as

$\displaystyle
Z_{i}\bra{\psi_{i}}\exp\left(-\frac{H_{B}}{k_{B}T}\right)\ket{\psi_{i}}\mathcal{P}\left(\phi_{f}|\phi_{i},\gamma_{i}\right)$
(17) $\displaystyle-
Z_{f}\bra{\phi_{f}}\exp\left(-\frac{H_{B}}{k_{B}T}\right)\ket{\phi_{f}}\mathcal{P}\left(\psi_{i}|\psi_{f},\gamma_{f}\right)\approx
0\,$

using the definition of the generalised energy flow, Eq. (9), and the
definition of the system’s free energy which can be rephrased as $\exp(-\Delta
F/k_{B}T)=Z_{f}/Z_{i}$ in terms of the initial and final partition functions
$Z_{i}$ and $Z_{f}$. The degree of approximation is then made precise by
switching from a statement that the magnitude of the left hand side of Eq (17)
is approximately zero, to a statement that it is less than or equal to some,
hopefully small, error bound,

$\displaystyle D\mathrel{\mathop{\mathchar
58\relax}}=|Z_{i}\bra{\psi_{i}}\exp\left(-\frac{H_{B}}{k_{B}T}\right)\ket{\psi_{i}}\mathcal{P}\left(\phi_{f}|\phi_{i},\gamma_{i}\right)$
(18) $\displaystyle-
Z_{f}\bra{\phi_{f}}\exp\left(-\frac{H_{B}}{k_{B}T}\right)\ket{\phi_{f}}\mathcal{P}\left(\psi_{f}|\psi_{f},\gamma_{f}\right)|\leq\epsilon.$

It is possible to determine analytically the terms
$\bra{\psi_{i}}\exp\left(-\frac{H_{B}}{k_{B}T}\right)\ket{\psi_{i}}$ and
$\bra{\psi_{f}}\exp\left(-\frac{H_{B}}{k_{B}T}\right)\ket{\psi_{f}}$ in $D$
for the coherent, squeezed and cat state equalities by arranging their
approximate forms.

The total error, $\epsilon$, is the sum of the initial and final factorisation
errors,

$\epsilon=||\epsilon_{SB}^{i}||{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{{}_{1}}}+||\epsilon_{SB}^{f}||{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{{}_{1}}}\
$ (19)

where
$||\epsilon||_{1}=\operatorname{Tr}\left[\sqrt{\epsilon^{\dagger}\epsilon}\right]$
denotes the trace norm of the operator $\epsilon$. These factorisation errors
$\epsilon_{i,f}$ capture the extent to which the exponential of the total
Hamiltonian, $H_{SB}$, factorises into the exponential of effective battery
and system Hamiltonians, $H_{B}$ and $H_{S}^{i,f}$, when acting on the battery
states that are measured in subregions $\Pi_{B}^{i,f}$ (i.e. $\ket{\psi_{f}}$
and $\ket{\phi_{f}}$ respectively). More accurately, and as derived in
Appendix A, having defined the map

$J_{K}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\sigma}})=e^{-\frac{K}{2k_{B}T}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\sigma}}e^{-\frac{K}{2k_{B}T}}\,,$
(20)

the factorisation errors are defined as

$\displaystyle\epsilon_{SB}^{i}=J_{H_{SB}}(\mathbbm{1}_{S}\otimes\ket{\psi_{i}}\bra{\psi_{i}})-J_{K_{SB}^{i}}(\mathbbm{1}_{S}\otimes\ket{\psi_{i}}\bra{\psi_{i}})$
(21)
$\displaystyle\epsilon_{SB}^{f}=J_{H_{SB}}(\mathbbm{1}_{S}\otimes\ket{\phi_{f}}\bra{\phi_{f}})-J_{K_{SB}^{f}}(\mathbbm{1}_{S}\otimes\ket{\phi_{f}}\bra{\phi_{f}})$

where $K_{SB}^{i}=H_{S}^{i}\otimes\mathbbm{1}_{B}+\mathbbm{1}_{S}\otimes
H_{B}$ and $K_{SB}^{f}=H_{S}^{f}\otimes\mathbbm{1}_{B}+\mathbbm{1}_{S}\otimes
H_{B}$. As calculating $\epsilon$ involves exponentiating $H_{SB}$, this
generally needs to be done numerically.

While Eq. (18) enables the AQC to be tested, $D$ and $\epsilon$ are not always
reliable indicators of the accuracy of the AQC. For example, $D$ decreases as
the transition probabilities ${\cal P}$ decrease but this need not correspond
to the AQC holding more accurately. Therefore, we also introduce $1-R$ as an
alternative means of quantifying its accuracy, with $R$ the ratio of the two
sides of the AQC,

$R\mathrel{\mathop{\mathchar
58\relax}}=\frac{\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})}{\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})}\left(\exp\left(\frac{\Delta\tilde{E}-\Delta
F}{k_{B}T}\right)\right)^{-1}\ .$ (22)

The quantity $1-R$ equals 0 when the AQC holds exactly.

##  3 Proposed Physical Implementation

To make concrete the quantum and thermal physics governed by the AQC we
specify a particular interaction, Eq. (4), between system and battery for
which we expect the equality to hold and the procedure for testing it. We
start by considering an ‘ideal’ interaction that replicates the classical
Crooks equality setup most closely. We then take a broader perspective and
suggest alternate potentials that could be used to verify the physics
quantified by the AQC.

###  3.1 Position dependent level shift

Following [32], we consider a two level system interacting with a harmonic
oscillator battery via

$V_{SB}=\sigma^{z}_{S}\otimes E(x_{B})\,,$ (23)

where $\sigma^{z}_{S}=\ket{e}\bra{e}-\ket{g}\bra{g}$ with $\ket{e}$ and
$\ket{g}$ the excited and ground states of the two level system respectively.
$E(x_{B})$ is an energetic level-shift that depends on the position operator,
$x_{B}$, of the oscillator. By choosing $E(x)$ to be constant for $x\leq
x_{i}$ and for $x\geq x_{f}$, two distinct effective Hamiltonians, $H_{S}^{i}$
and $H_{S}^{f}$ in Eq. (4), can be realised for the two level system. The
independence of the AQC on the form of $V_{SB}^{\perp}$ in Eq. (4) means that
the choice of the form of $E(x)$ in the region $x_{i}<x<x_{f}$ is arbitrary
and so for simplicity we assume a linear increase,

$\displaystyle E(x)=\left\\{\begin{array}[]{ll}E_{i}&\ \ \ x\leq x_{i}\\\
\frac{E_{f}-E_{i}}{x_{f}-x_{i}}(x-x_{i})+E_{i}&\ \ \ x_{i}<x<x_{f}\\\ E_{f}&\
\ \ x\geq x_{f}\ \ \ \ .\end{array}\right.$ (24)

This energy level splitting profile is shown in Fig. 3. Without loss of
generality, we chose to center the interaction region around $x=0$ such that
$|x_{i}|=|x_{f}|$.

(a) $\mathcal{P}\left(\alpha_{f}|\alpha_{i}\exp(-\chi),\gamma_{i}\right)$

(b) $\mathcal{P}(\alpha_{i}^{*}|\alpha_{f}^{*}\exp(-\chi),\gamma_{f})$

Figure 3: Diagrammatic representation of the protocol to test the coherent
state Crooks equality. The blue lines represent the ground and excited state
of the system as a function of $x$. The solid Gaussians represent the coherent
states that are prepared at the start of the protocols (red) and the
measurements (grey) that are performed at the end of the protocols. The dashed
lines represent the evolved states. The harmonic trap that drives the
evolution is centered at the midpoint of $x_{i}$ and $x_{f}$.

In order to measure the transition probability,
$\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})$, for the forwards process one has
to be able to implement the following three steps:

  1. 1.

The battery is prepared in a state $\ket{\phi_{i}}$ that is localised such
that $\bra{\phi_{i}}x_{B}\ket{\phi_{i}}<x_{i}$ and the system is prepared in a
thermal state with respect to its effective Hamiltonian
$H_{S}^{i}=E_{i}\,\sigma_{S}^{z}$.

  2. 2.

The system and battery are allowed to evolve for some time $\tau$. The time
$\tau$ is chosen such that the final battery wavepacket is localised in the
region beyond $x_{f}$ in order to ensure that the effective system Hamiltonian
has changed to $H_{S}^{f}=E_{f}\sigma_{S}^{z}$.

  3. 3.

The battery is measured to determine whether its evolved state is
$\ket{\phi_{f}}$.

The forwards transition probability,
$\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})$, is the relative frequency with
which the battery is found in the state $\ket{\phi_{f}}$ when the steps 1-3
are repeated a large number of times. We discuss how the preparation and
measurement steps might be practically implemented in a trapped ion system in
Section IV B.

The procedure to obtain the reverse process transition probability,
$\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})$, is entirely analogous. The
battery is prepared in the state $\ket{\psi_{f}}$ in the region beyond $x_{f}$
and the system is prepared in a thermal state with respect to
$H_{S}^{f}=E_{f}\sigma_{S}^{z}$. A final measurement, after evolving for time
$\tau$, determines whether the battery is in state $\ket{\psi_{i}}$.

A sketch of the test for the coherent state Crooks equality is shown in Fig.
3. The procedure for the coherent, squeezed, and cat state equalities differ
only in the choice of preparation and measurement states, as specified by the
table in 2.4.

###  3.2 Numerical calculations of the error bounds

Here we numerically determine the error in the approximate equalities. While
coherent, squeezed and cat states have support outside of the regions
$x<x_{i}$ and $x>x_{f}$, and so the AQC is not obeyed perfectly, we find that
there are regimes where the errors fall below the expected experimental error
margins. In such regimes, which we characterise below, the AQC is exact for
all practical purposes.

(a) Coherent State and Squeezed State Equalities

(b) Cat State Equality

Figure 4: The solid and dashed lines of the main plots show the error measures
$D$ and $\epsilon$ respectively, and the insets plot the error measure $1-R$,
for evolution under the total Hamiltonian with the position dependent level
splitting interaction and a harmonic oscillator battery. In (a) these are
plotted as a function of $-\alpha_{i}=\alpha_{f}$ with the coherent state
Crooks equality data shown in grey and squeezed state Crooks equality data
shown in blue ($r=-1$) and red ($r=1$) (as defined in the table in 2.4). In
(b) the data for cat states prepared and measured straddling the interaction,
$\ket{\psi_{i}}=\ket{\phi_{f}}\propto\ket{\alpha}+\ket{-\alpha}$, is shown in
blue and for cat states prepared and measured on a single side of the
interaction region, $\ket{\psi_{i}}\propto\ket{-\alpha}+\ket{-(\alpha+1)}$ and
$\ket{\phi_{f}}\propto\ket{\alpha}+\ket{\alpha+1}$, is shown in red. In this
simulation we have used the following parameters: $x_{i}=-4$, $x_{f}=4$,
$\hbar\omega=1$, $E_{i}=1$, $E_{f}=2$ and $k_{B}T=1$. The displacement
parameters, $\alpha_{i,f}$, and the positions $x_{i}$ and $x_{f}$, are given
in units of dimensionless position $X=\sqrt{\frac{m\omega}{2\hbar}}x$.

We simulated the full quantum state evolution of the system and harmonic
oscillator battery under the total Hamiltonian with the position dependent
level splitting, $E(x)$, and used this to find the error measures $D$ and $R$,
as defined in Eq. (18) and Eq. (22). We further numerically calculated the
error bound $\epsilon$, defined in Eq. (19). As expected we find that $D$ for
the coherent, squeezed and cat states is less than $\epsilon$, for a wide
range of simulated parameters ($k_{B}T$, $\omega$, $\alpha_{i}$, $\alpha_{f}$,
$r_{i}$, $r_{f}$, $\beta_{i}$, $\beta_{f}$, $E_{i}$, $E_{f}$, $x_{i}$,
$x_{f}$). Fig. 4 presents some of these results.

For the coherent states and squeezed states, with real $\alpha_{i}$ and real
$\alpha_{f}=-\alpha_{i}$, the inset of Fig. 4a shows that $1-R$ tends to 0 as
$\alpha_{i}$ is increased. As is shown in red and blue respectively, for a
given $\alpha_{i}$ squeezing the position variance of the prepared and
measured states decreases $1-R$; while, squeezing the momentum variance
increases $1-R$. Coherent states with also a complex displacement $\alpha$
corresponding to finite momentum result in similar behaviour, as we have
explicitly verified. For the cat state equality, we firstly consider final
states of the form $\ket{\psi_{i}}\propto\ket{-\alpha}+\ket{-(\alpha+1)}$ and
$\ket{\phi_{f}}\propto\ket{\alpha}+\ket{\alpha+1}$. On doing so, as is shown
by the red line in the inset of Fig. 4b, we find that, as with the coherent
state case, $1-R$ tends to 0 as $\alpha$ is increased. However, for the cat
states $\propto\ket{\alpha}+\ket{-\alpha}$, we find that $1-R=0.59$ for all
$\alpha$, as shown by blue line. In contrast to the other states we simulated
that are initially displaced from the origin, these cat states are
symmetrically delocalised such that the average battery position is 0. In
particular, there may be a significant probability to find the battery in both
regions, $x<x_{i}$ and $x>x_{f}$.

We conclude that decreasing the overlap of the prepared and measured battery
states from the interaction region increases the accuracy of the AQC.
Moreover, the convergence to the exact limit is reached quickly. For example,
we see in Fig. 4a that for coherent states with $|\alpha_{i}|=|\alpha_{f}|\geq
6$ the error measures are very small, i.e. $D,\,\epsilon,\,(1-R)<10^{-6}$.
Such discrepancies are lower than experimental error margins and consequently
the approximate nature of these equalities can effectively be disregarded when
the preparation and measurement states are chosen appropriately.

The behaviour of $(1-R)$ in Fig. 4 makes physical sense. Fundamentally, the
approximate nature of the AQC arises because, by considering the autonomous
evolution under a time-independent interacting Hamiltonian, the system never
strictly has a well-defined local Hamiltonian. The crux of the issue is the
battery Hamiltonian induces evolution between subregions with different
effective Hamiltonians, $H_{S}^{i}$ and $H_{S}^{f}$. As a result, the thermal
state of the system at the start of the forward and reverse protocols is not
well defined. However, when the battery is initially prepared with support in
a single region far from the interaction region the system can nonetheless
properly thermalise with respect to its initial Hamiltonian. In this limit the
AQC is exact.

###  3.3 Alternative position dependent level splittings

Thus far we have specified one particular choice in $E(x)$, Eq. (24), to
induce an effective change in system Hamiltonian. Here we outline how
alternative choices in $E(x)$ can also be used to probe the physics quantified
by the AQC.

In Eq. (24) we took $E(x)$ to be constant in the regions $x\leq x_{i}$ and
$x\geq x_{f}$ to ensure that the initial and final Hamiltonians are well-
defined and so the AQC holds to a high degree of approximation. However, there
is more flexibility as to the choice of $E(x)$ if, rather than aiming to
directly verify the AQC, we instead focus simply on detecting the quantum
deviation from the classical Crooks equality that the AQC predicts.

The deviation of the AQC from the classical Crooks equality is encapsulated by
the prefactor $q\mathrel{\mathop{\mathchar 58\relax}}=\Delta\tilde{E}/W_{q}$,
as per Eq. (12), that appears in the term
$\exp\left(q\frac{W_{q}}{k_{B}T}\right)$ of the predicted ratio of transition
probabilities. The prefactor $q$ can always be inferred from the transition
probabilities using

$\displaystyle
q=\frac{k_{B}T}{W_{q}}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ln}}\left(\frac{\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})}{\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})}\exp\left(\frac{\Delta
F}{k_{B}T}\right)\right)\ ;$ (25)

however, when the AQC does not hold exactly, it will deviate from
$\Delta\tilde{E}/W_{q}$.

Figure 5: The predicted quantum deviation $q$ deduced from the transition
probabilities between coherent states, using Eq.(25), for different position
dependent level splittings, $E(x)$. The turquoise circles show the inferred
$q$ for the usual ‘flat ends’ potential, the purple squares show $q$ for a
$\sin(x)^{2}$ potential, and the red triangles show $q$ for a purely linear
potential. The blue line is the predicted analytical form of
$q=\frac{\tanh(\chi)}{\chi}$ (also sketched in blue in Fig. 2). The grey line
indicates the classical limit in which $q=1$. The inset plots the level
splittings simulated. In this simulation we have chosen the following
physically plausible parameters, $x_{i}=-2$, $x_{f}=2$, $E_{i}=1$MHz,
$E_{f}=2$MHz, $\alpha_{i}=-5$ and $\alpha_{f}=4$. The displacement parameters
$\alpha_{i,f}$ and the positions, $x_{i}$ and $x_{f}$, are given in units of
dimensionless position.

We numerically simulated the evolution of a coherent state for different
choices of $E(x)$ and used the obtained transition probabilities to infer $q$,
which we plot in Fig. 5. The analytic form of $q$ for coherent states, Eq.
(13), is indicated by the dark blue line. As shown by the turquoise circles,
we find for $E(x)$ with the ‘flat ends’, Eq. (24), that $q$ closely replicates
the predicted analytic form. The change in effective Hamiltonian from
$E_{i}\sigma_{S}^{z}$ to $E_{f}\sigma_{S}^{z}$ can be approximately replicated
by using a $\sin(x)^{2}$ function and choosing $\alpha_{i}$ and $\alpha_{f}$
to sit in the troughs and peaks of the potential where the effective
Hamiltonian is approximately constant. As shown in purple, for this choice in
$E(x)$ the behaviour of $q$ reasonably closely replicates the analytic
expression. If we abolish the flat ends entirely and instead enact a change in
effective Hamiltonian using a solely linear potential then, as shown by the
red triangles, we find that $q$ deviates further from the predicted analytic
form but is still of a similar functional form.

As such, we conclude that it should be possible to detect quantum deviations
using sinusoidal and linear potentials as well as plausibly other potentials
that have yet to be explored. This flexibility as to the choice of $E(x)$
opens up many possible realisations to verify the physics quantified by the
AQC.

##  4 Experimental Outline: Trapped Ion and AC Stark Shift

To make the proposal more concrete we will now outline a potential
experimental implementation utilising an ion confined to a linear Paul trap. A
pair of internal electronic levels and the elongated phonon mode represent the
system and battery respectively.

For an experimental verification of the AQC to be convincing and practical we
have the following six conditions.

  1. 1.

Practically, we require forwards and reverse transition probabilities of a
measurable order of magnitude, i.e.

$\begin{array}[]{rcccll}10^{-2}&\lesssim&\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})&\lesssim&1-10^{-2}&\mbox{and}\\\
10^{-2}&\lesssim&\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})&\lesssim&1-10^{-2}\
.\end{array}$

  2. 2.

To practically simulate a non-trivial thermal system state we require non-
negligible populations of both the ground and excited states. This requires
the initial and final level splittings, $2E_{i}$ and $2E_{f}$ respectively, to
be a similar order of magnitude to the temperature, i.e.

${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{2E_{i}}}\approx
k_{B}T\approx{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{2E_{f}}}\
.$

  3. 3.

To non-trivially test the AQC we require a moderately strong system-battery
interaction. This is ensured as long as the change in the splitting of the
system energy levels is of a similar order of magnitude as the trap frequency,
i.e.

$|{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{2E_{f}-2E_{i}}}|\approx\hbar\omega\
.$

  4. 4.

To cleanly test the AQC, the decoherence and heating rates of the battery need
to be considerably less than the trap frequency. If long-lived internal levels
are used to represent the two-level system then the primary constraint is the
heating rate of the phonon mode battery, $\nu_{\mbox{\tiny therm}}$, and thus
we need,

$\omega\gg\nu_{\mbox{\tiny therm}}\ .$

  5. 5.

The interaction needs to be designed in such a way that the ion can be
prepared in a state comfortably in the regions $x<x_{i}$ and $x>x_{f}$ with
minimal overlap with the interaction region, $x_{i}<x<x_{f}$.

The simplest means of inducing our proposed Hamiltonian with the level
splitting specified by Eq. (24) would be to use a pair of Zeeman split
internal electronic levels and a position dependent magnetic field,
$B(x)\propto E(x)$. However, it is challenging to obtain the required magnetic
field environment for currently obtainable ion heights and corresponding
motional heating rates.

Figure 6: Sketch of trapped ion proposed implementation. The ion (red
wavepacket) is at the start of the evolution stage of the forwards protocol.
It is displaced in an elongated trap (grey line) that drives its evolution. An
off-resonance laser beam propagates perpendicularly across one side of the
trap (blue oval). The laser has an intensity profile that is sloped on the
edges and flat through the center (blue line in oval). The trapped ion
experiences a position dependent AC Stark shift (pair of blue lines) as it
travels autonomously through the laser beam. The red dashed line indicates the
spread of the ion and the black dashed line the length of the trap. The
preparation and measurement stages are sketched in Fig. 7.

We instead advocate using an AC Stark shift, generated by an off resonant
laser field, to induce a position dependent level splitting. While the
magnetic field realisation is perfectly autonomous, strictly speaking this
implementation allows the exchange of photons (and thus energy) between the
ion and the light field, and therefore is not perfectly autonomous. However,
the energy exchange quickly decays for increasing detuning and for
experimentally achievable Rabi-frequencies and detunings the system can
nonetheless be considered autonomous for all practical purposes.

To replicate Eq. (24) as closely as possible we require the laser beam to have
an intensity profile with a flat region corresponding to $x>x_{f}$. In the
region $x<x_{i}$ there is no laser beam and the Hamiltonian is $H_{S}^{i}$,
while in the region $x>x_{f}$ a flat laser beam profile induces a constant AC
Stark shift realising $H_{S}^{f}$. In the region $x_{i}<x<x_{f}$ the tail of
the laser beam changes the Hamiltonian from $H_{S}^{i}$ to $H_{S}^{f}$. In
Fig. 6 we sketch this for a smoothed top hat potential. However, in general
the tails of the intensity profile need not be smoothed as there are no
constraints on the interaction in the region $x_{i}$ and $x_{f}$.

The AC Stark shift typically changes the average energy of the system as well
as the splitting between the energy levels. This change in average energy and
the smoothed change in gradient at $x_{i}$ and $x_{f}$ (as shown in Fig. 6)
requires a minor change to $V_{SB}$ as specified in Eq. (23) and (24) but
makes no pertinent difference to the underlying thermal physics being
investigated.

While we focus on the tapered beam implementation to maintain a closer
resemblance to the classical Crooks equality setup, it is also possible, as
discussed in Section 3.3, to detect the quantum deviation predicted by the AQC
using a level splitting that varies sinusoidally or linearly with position. An
AC Stark shift that varies sinusoidally with position could be realised using
a standing wave generated by two counter propagating lasers [52, 53]. A
magnetic field that increases strength linearly with position $B(x)\propto x$
could be used to realise a Zeeman shift that increases linearly with position
[54, 55, 56].

###  4.1 Experimental Parameters

To support the plausibility of our proposal we present a set of potential
parameters that satisfy the five requirements listed above and could as such
enable the AQC to be verified.

We propose using a single 171Yb+ ion trapped in a linear Paul trap333The
radial trapping strength must be large enough to freeze out the radial force
on the ion that is induced by the magnetic component of the laser [57]. with
an axial secular frequency of 0.3 MHz [58]. Such secular frequencies typically
experience heating rates of the order of 40 phonons per second [59], thereby
satisfying requirement 4. Requirements 2 and 3 limit us to a pair of internal
levels with a separation of the order of MHz and thus we are constrained to
using hyperfine levels. We propose using the F = 1, M${}_{\mbox{\tiny F}}=1$
and M${}_{\mbox{\tiny F}}=-1$ levels of the 2S1/2 ground state. A magnetic
field can be applied to lift the degeneracy of the F = 1 manifold [58] and set
$E_{i}$ with typical splittings ranging between approximately 0.5 MHz and 10
MHz.

We suggest using a laser field with an intensity of $4.5$ Wmm-2 that is red
detuned from the 369nm 2S1/2 \- 2P1/2 transition by $0.1\times 10^{14}$Hz.
Using the methods and data from [60, 61], this is expected to induce an AC
Stark shift on the M${}_{\mbox{\tiny F}}=1$ and M${}_{\mbox{\tiny F}}=-1$
states of -0.2 MHz and -1.2 MHz respectively.

The position variance of 171Yb+ for an axial secular frequency of 0.3 MHz is
$0.01\ \mu m$ [46]. Therefore, to satisfy requirement 5, a top hat potential
with the size of the ‘flat’ region being greater than $0.05\ \mu$m is
required. There is some flexibility in how accurately the beam profile needs
to be shaped as our simulations indicate that $1-R$ scales linearly444The
exact constant of proportionality depends on the temperature, the magnitude of
the Stark shift and the battery wavepacket parameters; however, it is
generally of the order of $1-R\approx\frac{\delta I}{I}$ where $\delta I$ is
the change in laser intensity over a distance of $0.1\,\mu$m and $I$ is the
maximum laser intensity in that region. The sensitivity to an intensity
gradient is lower for higher temperatures, smaller Stark shifts and battery
states that are prepared comfortably outside the interaction region. with a
gradient across the ideally ‘flat’ region of the beam.

Requirement 1 gives rise to a trade off between the spatial scale on which the
beam can be engineered and the temperature regime that can be probed. As the
distance $x_{f}-x_{i}$ is the minimum the ion must travel to realise the
change in Hamiltonian $H_{S}^{i}$ to $H_{S}^{f}$, this sets the minimum
possible displacement of the initial and final states. When the ion oscillates
over large distances, its position variance is relatively small. In this
limit, it is more challenging to choose the initial and final ion states such
that there is a significant overlap between the measured states and the
evolved states in both the forwards and reverse processes. It is easier to
ensure there is a significant overlap in the high temperature regime because
the pair of states prepared in the reverse process are approximately the time
reverse of the pair of states in the forwards process. Our simulations
indicate that the $k_{B}T\approx 25\,\hbar\omega$ regime can be probed with
$x_{f}-x_{i}=1\mu$m. The experiment would be feasible with a larger
$x_{f}-x_{i}$; however, the shorter the spatial scale on which the ion
oscillates, the lower the temperature regime it would be possible to probe and
the greater the $\chi$ induced quantum deviations can be detected555It is
possible to realise standing waves and linear magnetic field gradients on the
scale of $10^{-2}\mu m$ [56, 52]. Thus the small spatial scales of these
alternative approaches could enable larger values of $\chi$ to be more easily
probed..

###  4.2 Experimental Techniques

The procedure outlined in 3.1 is realisable for trapped ions using the
following currently available experimental techniques.

##### Preparation of two-level system.

A thermal state of the internal electronic energy levels can be modeled using
‘pre’-processing [62]. Specifically, a thermal ‘ensemble’ of $N=N_{g}+N_{e}$
ions can be modeled by running the protocol on a single ion $N_{g}$ and
$N_{e}$ times (where
$N_{e}/N_{g}=\exp(-{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{2E_{i}}}/k_{B}T)$)
with the ion initialised in the ground state and excited states respectively.
The ion can be prepared in the ground state or excited state by using an
appropriate sequence of laser and microwave pulses. Simulating the thermal
state in this way makes it straightforward to ensure that we are in the low
temperature limit with respect to the trapping frequency.

##### Preparation of oscillator states.

The first step to generate coherent, squeezed and cat states is to prepare the
ion in the motional ground state using sideband cooling. A coherent state can
then be generated by shifting the trap center non-adiabatically resulting in
the effective displacement of the ion with respect to the new trapping
potential [46] (this is sketched in Fig. 7). A squeezed displaced state can
similarly be generated from the motional ground state by first generating a
squeezed vacuum state using a non-adiabatic drop in the trap frequency and
then displacing it using the non-adiabatic shift of the trap center [46].

Cat states can be generated using laser pulses that entangle the internal
electronic states and motional states of the ion [46]. Recently a cat state
separated by 259nm has been achieved [63] and with this apparatus it would be
possible to generate states separated by $\mu m$ as required here.

##### Measurement of oscillator states.

A measurement to determine the overlap of the final state of the phonon modes,
$\ket{\psi_{\mbox{\tiny final}}}$, with some test state,
$\ket{\psi_{\mbox{\tiny test}}}$, can be achieved by performing the inverse of
the relevant preparation process, $U_{B}^{\mbox{\tiny test}\dagger}$, and then
measuring to determine whether the battery is in the motional ground state
$\ket{0}$, i.e. the overlap $\braket{\psi_{\mbox{\tiny
test}}}{\psi_{\mbox{\tiny
final}}}=\bra{0}{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{U_{B}^{\mbox{\tiny
test}\dagger}}}\ket{\psi_{\mbox{\tiny final}}}$. For example, to determine the
overlap of some final state with the coherent state $\ket{\alpha_{\mbox{\tiny
test}}}$ we would have
${\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{U_{B}^{\mbox{\tiny
test}\dagger}}}=D(\alpha_{\mbox{\tiny test}})^{\dagger}=D(-\alpha_{\mbox{\tiny
test}})$ (this is sketched in Fig. 7).666There is some flexibility on how fast
the measurement needs to be performed. The inverse process $U_{\mbox{\tiny
test}}^{\dagger}$ needs to be performed at time $\tau$ or when the ion returns
to the same place after another complete rotation, i.e. $(2n+1)\tau\ \ \forall
n\,\in\mathbb{Z}^{+}$. The measurement to determine whether the ion is in the
motional ground state need only be performed before there is a substantial
probability for the state to be disturbed by decoherence or heating.

(a) Preparation

(b) Measurement

Figure 7: Sketch of preparation and measurement stages of the forwards
protocol of the coherent state trapped ion implementation. (a) To prepare a
coherent state the ion is first prepared in the motional ground state of a
harmonic trapping potential, shown here in blue. The potential center is then
shifted non-adiabatically resulting in the effective displacement of the ion
with respect to the new trapping potential, shown here in dark grey. (b) To
find the overlap between the evolved state of the ion (the dashed red
wavepacket) and some test coherent state (the blue Gaussian), the trap
potential is first shifted such that were the evolved state precisely in the
test state, then the evolved state would be in the ground state with respect
to the new trapping potential. This shifted trapping potential is shown in
blue. A projective measurement is then performed to find the overlap between
the evolved state and the ground state of the new trapping potential. In both
figures the blue oval in the background is the laser beam that induces the AC
Stark shift.

A filtering scheme can be used to perform a projective measurement on a single
ion that answers the binary question ‘is the phonon in the state
$n=m_{\mbox{\tiny test}}$?’ [30, 64]. Choosing $m_{\mbox{\tiny test}}=0$, this
method can be used to determine whether the ion is in the motional ground
state. The filtering scheme uses the dependence of the Rabi frequency of the
red and blue sidebands on phonon number.

##  5 Conclusions and Outlook

The AQC is derived from a simple set of physical principles and in virtue of
this the coherent, squeezed and cat state Crooks equalities are both natural
and general. Specifically, the derivation of the AQC only assumes that the
system and battery evolve under a microscopically energy conserving and time
reversal invariant unitary and that the system and battery obey a
‘factorisability’ condition. (This condition, defined in Appendix A,
characterises the extent to which the system and battery are independent
subsystems at the start and end of the protocol.) Given that these are a
natural set of assumptions, shared with other derivations of fluctuation
theorems, it is perhaps intriguing that for coherent states, which are often
viewed as the most classical of the motional quantum states, we find the AQC
can be written in the satisfyingly compact form of Eq.
(LABEL:eq:_coherentstaterewrite). The dependence of the coherent state Crooks
equality on the thermal de Broglie wavelength is a non-classical feature,
arising from coherent superpositions between energy eigenstates.

The coherent, squeezed and cat state Crooks equalities can be viewed as
setting out quantum corrections to the classical Crooks equality. In
particular, the coherent state Crooks equality is its lowest order extension
to quantum states. The coherent state equality effectively reduces to the
classical Crooks equality in the classical limit but by increasing the ratio
of quantum fluctuations to thermal fluctuations it is possible to smoothly
interpolate from the classical to the quantum regime. In the general quantum
case the probability for the reverse process is greater than expected
classically and correspondingly irreversibility is apparently softened. The
squeezed state and cat state Crooks equalities are higher order extensions to
the classical Crooks equality. The equalities are correspondingly more complex
and there are regimes in which irreversibility is apparently strengthened. It
would be interesting to investigate whether the squeezed and cat state
equalities can be written in a more compact form by relating them to
appropriate thermal wavelengths.

Our proposal can be seen to provide a means to experimentally test resource
theories for quantum thermodynamics. As in the protocol the system starts in a
thermal state and energy is globally conserved, the operation on the battery
is a thermal operation for a system with a changing Hamiltonian [38].
Consequently, the proposed experiment allows us to probe coherent features of
the resource framework, which is noteworthy as these frameworks [12, 13, 8]
have, with the exception of a few recent developments [65, 66], remained
abstract.

More broadly, we have taken a highly mathematical result from within the
quantum information theoretic approach to thermodynamics and both explored its
physical content and an experimental implementation. We hope our research
encourages more such attempts to physically ground recent quantum
thermodynamics theory results.

## Acknowledgements

The authors thank Johan Åberg for comments on a draft, Tom Hebdige, Erick
Hinds Mingo and Chris Ho for numerous helpful discussions and Stephen Barnett
for highlighting the radial force induced on the ion by an off resonant laser.
This research was supported in part by the COST network MP1209 “Thermodynamics
in the quantum regime.” ZH is supported by Engineering and Physical Sciences
Research Council Centre for Doctoral Training in Controlled Quantum Dynamics.
SW is supported by the U.K. Quantum Technology hub for Networked Quantum
Information Technologies, EP/M013243/1. DJ is supported by the Royal Society.
JA acknowledges support from Engineering and Physical Sciences Research
Council, Grant EP/M009165/1, and the Royal Society. FM acknowledges support
from Engineering and Physical Sciences Research Council, Grant EP/P024890/1.

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##  Appendix A Derivation of the AQC

The AQC is derived in [32] by combining two properties that we will call
‘global invariance’ and ‘factorisability’. Global invariance is a property of
any unitary evolution that is energy conserving and time reversal invariant.
Factorisability is a property that quantifies when two parts of a global
system can be considered well defined subsystems. The battery states
considered in the AQC are parameterized by a map known as the Gibbs map. To
derive the AQC we will first introduce these four concepts: the Gibbs map, the
quantum time reversal operation, global invariance and factorisability.

###  A.1 Gibbs Map

The Gibbs map is a mathematical generalisation of the thermal state that turns
out to be a convenient means of parameterising the battery states.

######  Definition 1.

Gibbs map.
Given a system with Hamiltonian $H$ at temperature $T$, the action of the
Gibbs map, $G_{H}$, on a state $\rho$ is,

$\displaystyle G_{H}(\rho)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\exp\left(-\frac{H}{2k_{B}T}\right)\rho\exp\left(-\frac{H}{2k_{B}T}\right)}{\tilde{Z}_{H}(\rho)}\,,\
\ \mbox{with}$ (26)
$\displaystyle\tilde{Z}_{H}(\rho)\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}\left[\exp\left(-\frac{H}{k_{B}T}\right)\rho\right].$

This map arises naturally as a quantum-mechanical version of the Crooks
reversal of a Markov process, and is intimately linked with the Petz recovery
map for general quantum states [48, 49]. It also has a range of physically
natural properties. When the energy of the input state is completely certain,
as in an energy eigenstate $\ket{E_{k}}$, the Gibbs map has no effect and

$\displaystyle G_{H}\left(\ket{E_{k}}\bra{E_{k}}\right)$
$\displaystyle=\ket{E_{k}}\bra{E_{k}}.$ (27)

However, when the energy of the input state is completely uncertain, as in a
maximally mixed state or in an equal superposition, the state output by the
Gibbs map has thermally distributed populations. Additionally, the map is non-
dephasing and affects the energy populations in the same way irrespective of
the coherent properties the state. In this way, the Gibbs map takes a
maximally mixed state, $\rho_{\mathbbm{1}}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{N}\mathbbm{1}$, to the thermal state,

$\displaystyle
G_{H}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\rho_{\mathbbm{1}}}})=\gamma(H)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{Z_{H}}\exp\left(-\frac{H}{k_{B}T}\right)\,,\ \mbox{with}$
(28) $\displaystyle Z_{H}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}\left[\exp\left(-\frac{H}{k_{B}T}\right)\right]\,,$

and an equal superposition of energy states,
$\ket{\psi_{+}}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\sqrt{N}}\sum_{k=1}^{N}\ket{E_{k}}$, to the coherent
thermal state (a pure state with the same energy populations as the thermal
state),

$\displaystyle
G_{H}\left(\ket{\psi_{+}}\bra{\psi_{+}}\right)={\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ket{\psi^{\gamma}}\bra{\psi^{\gamma}}}}\,,\
\mbox{with}$ (29)
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ket{\psi^{\gamma}}}}=\frac{1}{\sqrt{Z_{H}}}\sum_{k}^{N}\exp\left(-\frac{E_{k}}{2k_{B}T}\right)\ket{E_{k}}\
.$

As such we see that the Gibbs maps makes a state crudely ‘as thermal as
possible’ subject to the constraints imposed by the input state. However, this
loose claim is not intended to be taken literally but rather as a signpost
towards the map’s deeper physical significance.

It is worth noting that while the Gibbs map is invertible for bounded
Hamiltonians; for unbounded Hamiltonians, the normalisation constant
$\tilde{Z}_{H}(\rho)$ is not guaranteed to be finite and so the map is not
strictly invertible. However, this does not cause any difficulties in our work
because for coherent, squeezed and cat states $\tilde{Z}_{H}(\rho)$ is finite.
A full analysis of the invertibility of the Gibbs map is outside the scope of
this paper.

###  A.2 Time Reversal

In broad terms the time reversal operation enacts a motion reversal. This can
be visualised as taking a video of the dynamics of a process and running it in
reverse. While there are numerous subtleties involved with making this
intuitive picture precise [67, 68]; in general terms the classical time
reversal operation amounts to the replacement of $t$ by $-t$ in all pertinent
physical variables describing the dynamics of a system.

As time is a parameter rather than observable in quantum mechanics, the
quantum time reversal operation acts indirectly on $t$ via its action on the
position and momentum operators. To maintain correspondence with classical
time reversals, the position operator $x$ is required to remain invariant
under a reversal while the momentum operator $p$ changes sign, i.e.
$\mathfrak{T}(x)=x$ and $\mathfrak{T}(p)=-p$ where $\mathfrak{T}(X)$ to
denotes the time-reversal of any quantum operator $X$. In virtue of the time
reversal being a symmetry operation, $\mathfrak{T}$ is additionally required
to be length preserving in the sense that
$\operatorname{Tr}[\ket{\psi}\bra{\psi}]=\operatorname{Tr}[\mathfrak{T}(\ket{\psi}\bra{\psi})]$
for any state $\ket{\psi}$. These two requirements entail that $\mathfrak{T}$
is an anti-unitary operator [47].

The simplest choice in anti-unitary operation to represent $\mathfrak{T}$ is
complex conjugation. This is the ‘textbook’ [47] quantum time reversal
operation; however, the anti-linearity of conjugation can make it arduous to
work with. On Hermitian operators, and therefore on any quantum state or
observable, the transpose operation is equivalent to complex conjugation. As
the AQC quantifies transition probabilities between quantum states, we are
free to use this equivalence and represent $\mathfrak{T}$ via the transpose.

######  Definition 2.

Quantum time reversal.
The quantum time reversal operation $\mathfrak{T}$ on any quantum operator $X$
is defined as

$\displaystyle\mathfrak{T}(X)\mathrel{\mathop{\mathchar 58\relax}}=X^{T}\,,$
(30)

where T is the transpose operation. It follows that the time reversal
operation ${\cal T}$ on any pure state $\ket{\psi}$ is given by

${\cal T}\ket{\psi}=\ket{\psi^{*}}\,,$ (31)

where ∗ denotes complex conjugation.

As the transpose operation is basis dependent, so are $\mathfrak{T}$ and
${\cal T}$. The physical situation dictates the appropriate choice in basis.
Specifically, the basis is chosen to ensure that the time reversed battery
states drive the change in Hamiltonian from $H_{S}^{f}$ back to $H_{S}^{i}$
and to ensure that the system and battery Hamiltonian, and therefore the
system and battery evolution operator, are time reversal invariant. For the
osccilator battery and two level system implementation discussed in the main
text we take the time reversal operation of the battery with respect to the
Fock basis, $\\{\ket{n}\bra{n}\\}$, and the time reversal of the system with
respect to the Pauli $z$ basis, $\\{\ket{e}\bra{e},\ket{g}\bra{g}\\}$. This
flips the sign of the momentum of any battery oscillator state while leaving
its position unchanged and ensures that the thermal states of the system are
time reversal invariant.

The derivation of the AQC makes use of two properties of $\mathfrak{T}$. The
first of these is the fact that as probabilities are scalars they are
invariant under $\mathfrak{T}$. The second is that $\mathfrak{T}$ reverses the
order of products of operators,
$\mathfrak{T}(AB)=\mathfrak{T}(B)\mathfrak{T}(A)$.

###  A.3 Global Invariance

Global invariance is a property of transition probabilities for a single
system with a time reversal invariant Hamiltonian and an energy conserving and
time reversal invariant unitary evolution. As such, it applies to a large
class of systems because most pertinent physical situations are time reversal
invariant and all are ultimately unitary and energy conserving when the total
setup considered is made sufficiently large. Global invariance is the starting
point to derive a large family of quantum fluctuation theorems and plays an
analogous role to detailed balance for classical fluctuation theorems.

An energy conserving evolution is here defined to be an evolution that
commutes with the Hamiltonian, i.e. $[U,H]=0$. This is stronger than requiring
the average energy of the setup to remain constant as a result of the
evolution. In particular, it rules out the creation of non-degenerate
superpositions of energy states from energy eigenstates [11].

For the autonomous evolution of a system under a time independent Hamiltonian
$H$, the strict energy conservation condition is trivially satisfied as
$[\exp(-iHt/\hbar),H]=0$ and the time reversal invariance of the evolution
operator follows from the time reversal invariance of the Hamiltonian, i.e.
$\mathfrak{T}(H)=H$ entails
$\mathfrak{T}\left(\exp(-iHt/\hbar)\right)=\exp(-iHt/\hbar)$.

######  Theorem 1.

Global Invariance.
Consider a Hamiltonian $H$ and evolution $U$ such that: $\mathfrak{T}(H)=H$,
$\mathfrak{T}(U)=U$ and $[U,H]=0$. Global invariance relates a forwards and
reverse transition probability for such a system. In the forwards process, the
system is prepared in a state $G_{H}(\rho_{i})$, evolves under $U$ and a
binary POVM measurement is performed with POVM elements
$\\{\rho_{f},\mathbbm{1}-\rho_{f}\\}$. The evolved state,
$UG_{H}(\rho_{i})U^{\dagger}$, collapses onto $\rho_{f}$ with the probability

$P(\rho_{f}|G_{H}(\rho_{i}))\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}[\rho_{f}UG_{H}(\rho_{i})U^{\dagger}]\ .$ (32)

In the reverse process the state $G_{H}(\mathfrak{T}(\rho_{f}))$ evolves under
$U$ and on measurement collapses onto $\mathfrak{T}(\rho_{i})$ with the
probability

$P(\mathfrak{T}(\rho_{i})|G_{H}(\mathfrak{T}(\rho_{f})))\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}[\mathfrak{T}(\rho_{i})UG_{H}(\mathfrak{T}(\rho_{f}))U^{\dagger}]\
.$ (33)

The ratio of these transition probabilities is,

$\frac{P(\rho_{f}|G_{H}(\rho_{i}))}{P(\mathfrak{T}(\rho_{i})|G_{H}(\mathfrak{T}(\rho_{f})))}=\frac{\tilde{Z}_{H}(\mathfrak{T}(\rho_{f}))}{\tilde{Z}_{H}(\rho_{i})}\
.$ (34)

The proof of global invariance starts with the definition of the forwards
transition probability, and makes use of the definition of $\mathfrak{T}$ in
combination with the fact that $\mathfrak{T}(H)=H$ and $\mathfrak{T}(U)=U$,

$\displaystyle P(\rho_{f}|G_{H}(\rho_{i}))$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}[\rho_{f}UG_{H}(\rho_{i})U^{\dagger}]=\operatorname{Tr}[\mathfrak{T}\left(\rho_{f}UG_{H}(\rho_{i})U^{\dagger}\right)]=\operatorname{Tr}[U^{\dagger}G_{H}(\mathfrak{T}(\rho_{i}))U\mathfrak{T}(\rho_{f})]\
.$ (35)

If we now substitute in the definition of the Gibbs map, Eq. (26), we have

$\displaystyle P(\rho_{f}|G_{H}(\rho_{i}))$
$\displaystyle=\frac{1}{\tilde{Z}_{H}(\rho_{i})}\operatorname{Tr}\left[U^{\dagger}\exp\left(-\frac{H}{2k_{B}T}\right)\mathfrak{T}(\rho_{i})\exp\left(-\frac{H}{2k_{B}T}\right)U\mathfrak{T}(\rho_{f})\right]\,,$
(36)

which can be reordered using the cyclic nature of the trace operation and the
fact that $[U,H]=0$,

$\displaystyle P(\rho_{f}|G_{H}(\rho_{i}))$
$\displaystyle=\frac{1}{\tilde{Z}_{H}(\rho_{i})}\operatorname{Tr}\left[\mathfrak{T}(\rho_{i})\exp\left(-\frac{H}{2k_{B}T}\right)U\mathfrak{T}(\rho_{f})U^{\dagger}\exp\left(-\frac{H}{2k_{B}T}\right)\right]$
(37)
$\displaystyle=\frac{1}{\tilde{Z}_{H}(\rho_{i})}\operatorname{Tr}\left[\mathfrak{T}(\rho_{i})U\exp\left(-\frac{H}{2k_{B}T}\right)\mathfrak{T}(\rho_{f})\exp\left(-\frac{H}{2k_{B}T}\right)U^{\dagger}\right]\
.$

The proof is completed by reusing the definition of the Gibbs map and the
definition of the reverse transition probability,

$\displaystyle P(\rho_{f}|G_{H}(\rho_{i}))$
$\displaystyle=\frac{\tilde{Z}_{H}(\mathfrak{T}(\rho_{f}))}{\tilde{Z}_{H}(\rho_{i})}\operatorname{Tr}\left[\mathfrak{T}(\rho_{i})UG_{H}(\mathfrak{T}(\rho_{f}))U^{\dagger}\right]$
(38)
$\displaystyle=\frac{\tilde{Z}_{H}(\mathfrak{T}(\rho_{f}))}{\tilde{Z}_{H}(\rho_{i})}P(\mathfrak{T}(\rho_{i})|G_{H}(\mathfrak{T}(\rho_{f})))\,.$

The AQC is derived by applying global invariance to a bipartite setup. By
evaluating the forwards and reverse transition probabilities,
$P(\rho_{f}|G_{H}(\rho_{i}))$ and
$P(\mathfrak{T}(\rho_{i})|G_{H}(\mathfrak{T}(\rho_{f})))$, with respect to a
carefully chosen Hamiltonian $H$ and states $\rho_{i}$ and $\rho_{f}$, they
can be use to quantify a Crooks-like scenario where an initially thermal
system is driven from equilibrium with a time dependent Hamiltonian with the
energy supplied by a quantum battery. However, for the initial thermal states
of the system to be well defined, the system and battery need to be
effectively non-interacting at the start and end of the process. This
requirement is captured by an additional assumption that we call the
‘factorisability’ of the system and battery.

###  A.4 Factorisability

Factorisability characterises the extent to which multiple, potentially
interacting, systems can be considered independent subsystems. It is defined
in terms of the map

$J_{H}(\sigma)\mathrel{\mathop{\mathchar
58\relax}}=\exp\left(-\frac{H}{2k_{B}T}\right)\sigma\exp\left(-\frac{H}{2k_{B}T}\right)$
(39)

for any operator $\sigma$. This map is, on any quantum state $\rho$, the
unnormalised version of the Gibbs map, i.e.
$J_{H}(\rho)=\tilde{Z}_{H}(\rho)\,G_{H}(\rho)$.

######  Definition 3.

Factorisability.
A composite system with a Hamiltonian $H_{AB}$ in the state
$\rho_{A}\otimes\rho_{B}$ is factorisable if

$\displaystyle J_{H_{AB}}(\rho_{A}\otimes\rho_{B})=J_{H_{A}}(\rho_{A})\otimes
J_{H_{B}}(\rho_{B})$ (40)

for a pair of Hamiltonians $H_{A}$ and $H_{B}$ of subsystems $A$ and $B$
respectively.

If a composite system is factorisable then the Gibbs map and the Gibbs map
normalisation term, Eq. (26), decompose into terms acting on the separate
subsystems, i.e. if a composite system with a Hamiltonian $H_{AB}$ in the
state $\rho_{A}\otimes\rho_{B}$ is factorisable then

$\displaystyle G_{H_{AB}}(\rho_{A}\otimes\rho_{B})=G_{H_{A}}(\rho_{A})\otimes
G_{H_{B}}(\rho_{B})\ \ \ \ \text{and}$ (41)
$\displaystyle\tilde{Z}_{H_{AB}}(\rho_{A}\otimes\rho_{B})=\tilde{Z}_{H_{A}}(\rho_{A})\otimes\tilde{Z}_{H_{B}}(\rho_{B})\
.$

To ensure that system and battery are effectively non-interacting at the start
and end of non-equilibrium process, we require that given an appropriate
choice in battery states their Hamiltonian, $H_{SB}$, factorises into an
initial or final system Hamiltonian, $H_{S}^{i}$ or $H_{S}^{f}$, and battery
Hamiltonian, $H_{B}$. Specifically we assume that the following
factorisability condition holds for the system and battery.

######  Assumption 1.

Factorisability of the system and battery.
There exist a pair of pure battery states, $\ket{\Psi_{i}}$ and
$\ket{\Psi_{f}}$, such that the system and battery Hamiltonian $H_{SB}$
factorises as

$\displaystyle
J_{H_{SB}}\left(\rho_{\mathbbm{1}}\otimes\ket{\Psi_{i}}\bra{\Psi_{i}}\right)=J_{H_{S}^{i}}(\rho_{\mathbbm{1}})\otimes
J_{H_{B}}(\ket{\Psi_{i}}\bra{\Psi_{i}})\ \ \ \ \text{and}$ (42) $\displaystyle
J_{H_{SB}}\left(\rho_{\mathbbm{1}}\otimes\ket{\Psi_{f}}\bra{\Psi_{f}}\right)=J_{H_{S}^{f}}\left(\rho_{\mathbbm{1}})\otimes
J_{H_{B}}(\ket{\Psi_{f}}\bra{\Psi_{f}}\right)\ $

where $H_{S}^{i}$ and $H_{S}^{f}$ are Hamiltonians of the system and $H_{B}$
is a Hamiltonian of the battery.

This condition essentially encodes that the system is a genuine system with
concrete initial and final Hamiltonians. The AQC follows directly from global
invariance, Eq. (34), when the factorisability condition, Eq. (42), is obeyed.

###  A.5 Main Derivation

To see how the AQC follows from the factorisability of the system and battery,
we set
$\ket{\Psi_{i}}\bra{\Psi_{i}}=\mathfrak{T}(\ket{\psi_{i}}\bra{\psi_{i}})$ and
$\ket{\Psi_{f}}\bra{\Psi_{f}}=\mathfrak{T}(\ket{\phi_{f}}\bra{\phi_{f}})$ and
for convenience define

$\displaystyle\sigma_{i}=\rho_{\mathbbm{1}}\otimes\mathfrak{T}\left(\ket{\psi_{i}}\bra{\psi_{i}}\right)\
\ \ \text{and}$ (43)
$\displaystyle\mathfrak{T}(\sigma_{f})=\rho_{\mathbbm{1}}\otimes\mathfrak{T}(\ket{\phi_{f}}\bra{\phi_{f}})\
.$

The factorisability condition, Eq. (42), entails that the Gibbs map acting on
$\sigma_{i}$ and $\mathfrak{T}(\sigma_{f})$, Eq. (43), decomposes as

$\displaystyle
G_{H_{SB}}\left(\sigma_{i}\right)=\gamma_{i}\otimes\ket{\phi_{i}}\bra{\phi_{i}}\
\ \text{and}$ (44) $\displaystyle
G_{H_{SB}}\left(\mathfrak{T}(\sigma_{f})\right)=\gamma_{f}\otimes\ket{\psi_{f}}\bra{\psi_{f}}$

where we have used the shorthand $\gamma_{k}\equiv\gamma(H_{S}^{k})$ for
$k=i,f$ and with

$\displaystyle\ket{\phi_{i}}\propto{\cal
T}\exp\left(-\frac{H_{B}}{2k_{B}T}\right)\ket{\psi_{i}}$ (45)
$\displaystyle\ket{\psi_{f}}\propto{\cal
T}\exp\left(-\frac{H_{B}}{2k_{B}T}\right)\ket{\phi_{f}}$

as in the main text. In this way the factorised Gibbs map generates both the
thermal states of the system, Eq. (6), and the temperature dependent operation
that parameterises the battery states, Eq. (7), that, as we will see, appear
in the AQC.

We can use the decomposition of the Gibbs map, Eq
(LABEL:eq:_factoriseexample), to convert a transition probability of the
system and battery to a transition probability of the battery alone. The
forwards transition probability $P(\sigma_{f}|G_{H}(\sigma_{i}))$, Eq (32), of
the system and battery reduces to $\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})$,
the transition probability of the battery from $\ket{\phi_{i}}$ to
$\ket{\phi_{f}}$ for a system initialised in the thermal state $\gamma_{i}$,

$\displaystyle P\left(\sigma_{f}\,|\,G_{H_{SB}}\left(\sigma_{i}\right)\right)$
$\displaystyle=\operatorname{Tr}\left[\left(\rho_{\mathbbm{1}}\otimes\ket{\phi_{f}}\bra{\phi_{f}}\right)U\,G_{H_{SB}}\left(\rho_{\mathbbm{1}}\otimes\mathfrak{T}(\ket{\psi_{i}}\bra{\psi_{i}})\right)U^{\dagger}\right]$
(46)
$\displaystyle=\frac{1}{N}\bra{\phi_{f}}\operatorname{Tr}_{S}\left[U(\gamma_{i}\otimes\ket{\phi_{i}}\bra{\phi_{i}})U^{\dagger}\right]\ket{\phi_{f}}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{N}\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})\ .$

(The factor $\frac{1}{N}$ is a normalisation term from the maximally mixed
system state $\rho_{\mathbbm{1}}$ and cancels out in the final fluctuation
theorem.) Similarly, the reverse global transition probability
$P\left(\mathfrak{T}(\sigma_{i})\,|\,G_{H_{SB}}(\mathfrak{T}(\sigma_{f})\right)$
reduces to the transition probability of the battery for the reverse process
$\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})$, in which the system is
initialised in the thermal state $\gamma_{f}$,

$\displaystyle
P\left(\mathfrak{T}(\sigma_{i})\,|\,G_{H_{SB}}(\mathfrak{T}(\sigma_{f})\right)$
$\displaystyle=\operatorname{Tr}\left[(\rho_{\mathbbm{1}}\otimes\ket{\psi_{i}}\bra{\psi_{i}})U\,G_{H_{SB}}\left(\rho_{\mathbbm{1}}\otimes\mathfrak{T}(\ket{\phi_{f}}\bra{\phi_{f}})\right)U^{\dagger}\right]$
(47)
$\displaystyle=\frac{1}{N}\bra{\psi_{i}}\operatorname{Tr}_{S}\left[U(\gamma_{f}\otimes\ket{\psi_{f}}\bra{\psi_{f}})U^{\dagger}\right]\ket{\psi_{i}}$
$\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{N}\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})\ .$

It also follows from the factorisation of the system and battery that the
Gibbs map normalisation terms decompose as

$\displaystyle\tilde{Z}_{H_{SB}}\left(\sigma_{f}\right)=\frac{1}{N}Z_{f}\times\tilde{Z}_{\phi_{f}}\
\ \ \ \text{and}$ (48)
$\displaystyle\tilde{Z}_{H_{SB}}\left(\sigma_{i}\right)=\frac{1}{N}Z_{i}\times\tilde{Z}_{\psi_{i}}\,,$

where we have used the shorthand $Z_{k}\equiv Z_{H_{S}^{k}}$ for $k=i,f$ for
the partition functions of the system and
$\tilde{Z}_{\phi_{f}}\equiv\tilde{Z}_{H_{B}}(\ket{\phi_{f}}\bra{\phi_{f}})$
and
$\tilde{Z}_{\psi_{i}}\equiv\tilde{Z}_{H_{B}}(\mathfrak{T}(\ket{\psi_{i}}\bra{\psi_{i}}))=\tilde{Z}_{H_{B}}(\ket{\psi_{i}}\bra{\psi_{i}})$
as shorthand for the Gibbs map normalisation terms of the battery.

The AQC now follows directly from substituting the factorised global
transition probabilities, Eq. (46) and Eq. (47), and the factorised Gibbs map
normalisation terms, Eq. (LABEL:eq:_factoriseZ), into global invariance, Eq.
(34),

$\displaystyle\frac{\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})}{\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})}$
$\displaystyle=\frac{Z_{f}}{Z_{i}}\frac{\tilde{Z}_{\phi_{f}}}{\tilde{Z}_{\psi_{i}}}\
.$ (49)

The derivation is completed by rewriting the ratio of the partition functions
in terms of the change in free energy of the system,

$\displaystyle\Delta$ $\displaystyle F\mathrel{\mathop{\mathchar
58\relax}}=F(H^{f}_{S})-F(H^{i}_{S})\ \ \text{with}$ (50) $\displaystyle
F(H)\mathrel{\mathop{\mathchar
58\relax}}=-k_{B}T\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ln}}\left(Z_{H}\right)\,,$

and by analogously defining and the generalised energy flow in terms of the
Gibbs map normalisation terms,

$\displaystyle\Delta\tilde{E}\mathrel{\mathop{\mathchar
58\relax}}=\tilde{E}(\psi_{i})-\tilde{E}(\phi_{f})\ \ \text{with}$ (51)
$\displaystyle\tilde{E}(\psi)\mathrel{\mathop{\mathchar
58\relax}}=-k_{B}T\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ln}}\left(\tilde{Z}_{\psi}\right)\,=-k_{B}T\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ln}}\left(\bra{\psi}\exp\left(-\frac{H_{B}}{k_{B}T}\right)\ket{\psi}\right).$

The AQC can thus be stated as

$\displaystyle\frac{\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})}{\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})}=\exp\left(-\frac{\Delta
F}{k_{B}T}\right)\exp\left(\frac{\Delta\tilde{E}}{k_{B}T}\right)\ .$ (52)

##### Relationship to Ref. [32]

The above equality, Eq. (52), is Eq.(38) of Ref. [32] but written in the limit
where factorisability holds exactly and for an initially thermal system and a
battery that is prepared in a pure state. Our battery plays the role of both
the energy reservoir and control system in Ref. [32].

###  A.6 Autonomy and approximate factorisability

To realise an autonomous change in system Hamiltonian we consider a system and
a battery with a Hamiltonian of the form

$H_{SB}=H_{S}^{i}\otimes\Pi_{B}^{i}+H_{B}^{f}\otimes\Pi_{B}^{f}+V_{SB}^{\perp}+\mathbbm{1}_{S}\otimes
H_{B}\ \,,$ (53)

where $\Pi_{B}^{i}$ and $\Pi_{B}^{f}$ are projectors onto two orthogonal
subspaces, $R_{i}$ and $R_{f}$, of the battery’s Hilbert space, and
$V_{SB}^{\perp}$ has support only outside those two subspaces, i.e.
$(X_{S}\otimes\Pi_{B}^{i})V_{SB}^{\perp}=(X_{S}\otimes\Pi_{B}^{f})V_{SB}^{\perp}=0$
for any system operator $X_{S}$. When the battery is initialised in a state in
subspace $R_{i}$ only and evolves to a final state in subspace $R_{f}$ only,
the system Hamiltonian evolves from $H_{S}^{i}$ to $H_{S}^{f}$. The reverse
change in system Hamiltonian $H_{S}^{f}$ to $H_{S}^{i}$ is similarly realised
by a battery initialised in $R_{i}$ that evolves to subspace $R_{f}$.

This Hamiltonian, Eq. (53), is exactly factorisable on the states $\sigma_{i}$
and $\mathfrak{\sigma_{f}}$, Eq. (43), when: (i) the battery state ${\cal
T}\ket{\psi_{i}}$ has support solely in $R_{i}$ and the battery state ${\cal
T}\ket{\phi_{f}}$ has support solely in $R_{f}$, and (ii) the battery
Hamiltonian does not induce evolution between subregions, i.e.
$(\mathbbm{1}_{B}-\Pi_{B}^{i})H_{B}\Pi_{B}^{i}=0$ and
$(\mathbbm{1}_{B}-\Pi_{B}^{f})H_{B}\Pi_{B}^{f}=0$. Under these two
restrictions it is straightforward to verify that

$\displaystyle J_{H_{SB}}(\sigma_{i})$
$\displaystyle=\exp\left(-\frac{H_{S}^{i}\otimes\Pi_{B}^{i}+\mathbbm{1}_{S}\otimes
H_{B}}{2k_{B}T}\right)(\rho_{\mathbbm{1}}\otimes\mathfrak{T}(\ket{\phi_{f}}\bra{\phi_{f}}))\exp\left(-\frac{H_{S}^{i}\otimes\Pi_{B}^{i}+\mathbbm{1}_{S}\otimes
H_{B}}{2k_{B}T}\right)$ (54)
$\displaystyle=J_{H_{S}^{i}}(\rho_{\mathbbm{1}})\otimes
J_{H_{B}}(\mathfrak{T}(\ket{\phi_{f}}\bra{\phi_{f}}))\ \ \ \ \ \ $

and similarly for $J_{H_{SB}}(\mathfrak{T}(\sigma_{f}))$. Thus in this limit
the AQC holds exactly.

However, to realise an autonomous change in system Hamiltonian, the evolution
of the battery between subspaces $R_{i}$ and $R_{f}$ must be generated by the
unitary evolution of the system and battery under $H_{SB}$, i.e. by the
propagator $U=\exp(-iH_{SB}t/\hbar)$. Consequently, in practise $H_{B}$ must
induce evolution between the subregions $R_{i}$ and $R_{f}$ and so
$(\mathbbm{1}_{B}-\Pi_{B}^{i})H_{B}\Pi_{B}^{i}\neq 0$ and
$(\mathbbm{1}_{B}-\Pi_{B}^{f})H_{B}\Pi_{B}^{f}\neq 0$. As a result, the system
and battery factorisability condition, Eq. (42), does not hold exactly for
autonomous dynamics and consequently nor does the AQC.

The AQC is thus strictly only ever approximate. Its approximate nature stems
from the fact that when considering the autonomous dynamics of a system and
battery under a time-independent interacting Hamiltonian, the system never
strictly has a well-defined local Hamiltonian. This tension is implicit to all
fluctuation theorems and arises from the impossibility of perfectly
delineating the individual energies of interacting systems. Nonetheless, the
tension is rarely a practical problem because it is generally possible to
identify approximately well defined subsystems with their own concrete
Hamiltonians.

The AQC holds to a high degree of approximation whenever it is possible to
identify the system and battery as approximately well defined subsystems with
approximately well defined Hamiltonians at the start and end of the forwards
and reverse protocols. This idea is captured by the claim that with an
appropriate choice in the battery states $\ket{\psi_{i}}$ and $\ket{\phi_{f}}$
it is possible to ensure that factorisability holds to a high degree of
approximation, i.e.

$\displaystyle J_{H_{SB}}(\sigma_{i})$ $\displaystyle\simeq
J_{H_{S}^{i}}(\rho_{\mathbbm{1}})\otimes
J_{H_{B}}(\mathfrak{T}(\ket{\psi_{i}}\bra{\psi_{i}}))\ \ \ \ \text{and}$ (55)
$\displaystyle J_{H_{SB}}(\mathfrak{T}(\sigma_{f}))$ $\displaystyle\simeq
J_{H_{S}^{f}}(\rho_{\mathbbm{1}})\otimes
J_{H_{B}}(\mathfrak{T}(\ket{\phi_{f}}\bra{\phi_{f}}))\ .$

From which it follows that with an appropriate choice in the battery states
$\ket{\psi_{i}}$ and $\ket{\phi_{f}}$ the AQC also holds to a high degree of
approximation

$\displaystyle\frac{\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})}{\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})}\simeq\exp\left(-\frac{\Delta
F}{k_{B}T}\right)\exp\left(\frac{\Delta\tilde{E}}{k_{B}T}\right)\ .$ (56)

The error-bounded AQC, Eq. (18), is an inequality that is obeyed when even
when the factorisability condition does not hold exactly. The extent to which
factorisability holds can be quantified by the difference between left and
right hand side of the system and battery factorisability condition, Eq. (55).
We thus define the initial and final factorisation errors, $\epsilon_{SB}^{i}$
and $\epsilon_{SB}^{f}$, as

$\displaystyle\epsilon_{SB}^{i}\mathrel{\mathop{\mathchar
58\relax}}=J_{H_{SB}}(\mathbbm{1}_{S}\otimes\ket{\psi_{i}}\bra{\psi_{i}})-J_{H_{S}^{i}}(\mathbbm{1}_{S})\otimes
J_{H_{B}}(\ket{\psi_{i}}\bra{\psi_{i}})\ \ \ \text{and}$ (57)
$\displaystyle\epsilon_{SB}^{f}\mathrel{\mathop{\mathchar
58\relax}}=J_{H_{SB}}(\mathbbm{1}_{S}\otimes\ket{\phi_{f}}\bra{\phi_{f}})-J_{H_{S}^{f}}(\mathbbm{1}_{S})\otimes
J_{H_{B}}(\ket{\phi_{f}}\bra{\phi_{f}}).$

The proof of the error-bounded AQC proceeds in the same manner as the proof of
the AQC except that we keep track of the error generated from factorisability
holding imperfectly. We start with the global invariance condition for the
states $\sigma_{i}$ and $\sigma_{f}$, Eq. (43),

$\displaystyle\tilde{Z}_{H_{SB}}(\rho_{\mathbbm{1}}\otimes\ket{\psi_{i}}\bra{\psi_{i}})\,P(\rho_{\mathbbm{1}}\otimes\ket{\phi_{f}}\bra{\phi_{f}}|G_{H_{SB}}(\rho_{\mathbbm{1}}\otimes\mathfrak{T}(\ket{\psi_{i}}\bra{\psi_{i}})))$
(58)
$\displaystyle=\tilde{Z}_{H_{SB}}(\rho_{\mathbbm{1}}\otimes\ket{\phi_{f}}\bra{\phi_{f}})\,P(\rho_{\mathbbm{1}}\otimes\ket{\psi_{i}}\bra{\psi_{i}}|G_{H_{SB}}(\rho_{\mathbbm{1}}\otimes\mathfrak{T}(\ket{\phi_{f}}\bra{\phi_{f}})))\,.$

and substitute in the definition of the factorisability errors
$\epsilon_{SB}^{i}$ and $\epsilon_{SB}^{f}$, Eq. 57,

$\displaystyle
Z_{i}\,\tilde{Z}_{\psi_{i}}\,\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})+\operatorname{Tr}[(\mathbbm{1}_{S}\otimes\ket{\phi_{f}}\bra{\phi_{f}})U\mathfrak{T}(\epsilon_{SB}^{i})U^{\dagger}]$
(59)
$\displaystyle=Z_{f}\,\tilde{Z}_{\phi_{f}}\,\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})+\operatorname{Tr}[(\mathbbm{1}_{S}\otimes\ket{\psi_{i}}\bra{\psi_{i}})U\mathfrak{T}(\epsilon_{SB}^{f})U^{\dagger}]\,,$

which on rearranging and using the triangle inequality gives

$\displaystyle|Z_{i}\,\tilde{Z}_{\psi_{i}}\,\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})-Z_{f}\,\tilde{Z}_{\phi_{f}}\,\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})|$
(60)
$\displaystyle\leq|\operatorname{Tr}[(\mathbbm{1}_{S}\otimes\ket{\phi_{f}}\bra{\phi_{f}})U\mathfrak{T}(\epsilon_{SB}^{i})U^{\dagger}]|+|\operatorname{Tr}[(\mathbbm{1}_{S}\otimes\ket{\psi_{i}}\bra{\psi_{i}})U\mathfrak{T}(\epsilon_{SB}^{f})U^{\dagger}]|\
.$

It follows from the Cauchy Schwartz identity that

$\displaystyle\operatorname{Tr}[(U^{\dagger}(\mathbbm{1}_{S}\otimes\ket{\phi_{f}}\bra{\phi_{f}})U)\,\epsilon_{SB}^{i}]\leq||\mathfrak{T}(\epsilon_{SB}^{i})||_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{1}}=||\epsilon_{SB}^{i}||_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{1}}\
\ \ \text{and}$ (61)
$\displaystyle\operatorname{Tr}[(U^{\dagger}(\mathbbm{1}_{S}\otimes\ket{\psi_{i}}\bra{\psi_{i}})U)\,\epsilon_{SB}^{f}]\leq||\mathfrak{T}(\epsilon_{SB}^{f})||_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{1}}=||\epsilon_{SB}^{f}||_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{1}}\,.$

Thus we are left with the error-bounded AQC,

$\displaystyle|Z_{i}\,\tilde{Z}_{\psi_{i}}\,\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})-Z_{f}\,\tilde{Z}_{\phi_{f}}\,\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})|\leq||\epsilon_{SB}^{f}||_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{1}}+||\epsilon_{SB}^{i}||_{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{1}}\mathrel{\mathop{\mathchar
58\relax}}=\epsilon\ .$ (62)

##### Non-autonomous variant.

It is possible to obtain an equality of the same general form as the AQC, Eq.
(52), for a non-autonomous setup where the evolution is induced by an applied
unitary $U$ rather than by evolution under $H_{SB}$, i.e.
$U\neq\exp(-iH_{SB}t/\hbar)$. This is in fact the central result of Ref. [32].
In the non-autonomous variant an additional control system $C$ is introduced
and the total Hamiltonian (up to technicalities concerning time reversals) is
of the form,

$H_{CSB}=\ket{C_{i}}\bra{C_{i}}\otimes
H_{S}^{i}\otimes\mathbbm{1}_{B}+\ket{C_{f}}\bra{C_{f}}\otimes
H_{S}^{f}\otimes\mathbbm{1}_{B}+V_{CSB}^{\perp}\
+\mathbbm{1}_{C}\otimes\mathbbm{1}_{S}\otimes H_{B}\,$ (63)

where these control states are orthogonal, $\braket{C_{f}}{C_{i}}=0$, and
$V_{CSB}^{\perp}$ has orthogonal support to both $\ket{C_{i}}$ and
$\ket{C_{f}}$, i.e. $V_{CSB}^{\perp}\ (\ket{C_{i}}\bra{C_{i}}\otimes
X_{S}\otimes X_{B})=V_{CSB}^{\perp}\ (\ket{C_{f}}\bra{C_{f}}\otimes
X_{S}\otimes X_{B})=0$ for any system and battery operators $X_{S}$ and
$X_{B}$. A change in system Hamiltonian $H_{S}^{i}$ to $H_{S}^{f}$ is induced
by applying a unitary that evolves the control from the state $\ket{C_{i}}$ to
$\ket{C_{f}}$. By requiring the applied unitary to be energy conserving and
time reversal invariant, global invariance holds as before. Furthermore, as
the above Hamiltonian, Eq. (63), is effectively non-interacting as long as the
control is prepared in the state $\ket{C_{i}}$, or the state $\ket{C_{f}}$,
the factorisability condition holds exactly. Given that these two conditions
are satisfied the rest of the derivation runs identically to that of the AQC.
The resulting quantum Crooks equality holds exactly and differs from the AQC
only by the particular definition of the transition probabilities. It follows
that the coherent, squeezed and cat state equalities can also be used for this
non-autonomous setup.

##### The role of the thermal bath.

In this paper we have treated the thermal bath as implicit: a bath is a means
of preparing the system in the thermal state but we do not model it.
Effectively this amounts to assuming that the system and bath are sufficiently
weakly interacting at the start of the protocols that they can be considered
well defined independent systems at these times.

More rigorously, we can think of the system discussed in this paper as an
enlarged system consisting of some small system of interest (s) that is driven
by a change in Hamiltonian and a thermal bath (b) with a constant Hamiltonian
$H_{b}$, i.e. $H_{S}^{i}=H_{s}^{i}\otimes H_{b}$ and
$H_{S}^{f}=H_{s}^{f}\otimes H_{b}\,$. The AQC can be re-derived explicitly
including the thermal bath if an additional factorisability condition is
assumed to hold between the system and bath. The factorisability condition
will hold if their shared Hamiltonian is non-interacting in the regions
$R_{i}$ and $R_{f}$, i.e.

$H_{sbB}=H_{s}^{i}\otimes H_{b}\otimes\Pi_{B}^{i}+H_{s}^{f}\otimes
H_{b}\otimes\Pi_{B}^{f}+V_{sbB}^{\perp}\ +\mathbbm{1}_{sb}\otimes H_{B}\ .$
(64)

Under these circumstances, the influence of the bath ‘factorises out’. The
resulting equality explicitly quantifies the transition probabilities of the
battery, when a system is driven with a change in Hamiltonian, in the presence
of a thermal bath. This quantum Crooks equality takes exactly the same form as
the usual AQC, Eq. (8), but with the relevant transition probabilities
replaced by

$\displaystyle\mathcal{P}(\phi_{f}|\phi_{i},\gamma_{i})\mathrel{\mathop{\mathchar
58\relax}}=\bra{\phi_{f}}\operatorname{Tr}_{sb}\left[U_{sbB}(\gamma(H_{s}^{f})\otimes\gamma(H_{b})\otimes\ket{\phi_{i}}\bra{\phi_{i}})U_{sbB}^{\dagger}\right]\ket{\phi_{f}}\
\ \ \text{and}$ (65)
$\displaystyle\mathcal{P}(\psi_{i}|\psi_{f},\gamma_{f})\mathrel{\mathop{\mathchar
58\relax}}=\bra{\psi_{i}}\operatorname{Tr}_{sb}\left[U_{sbB}(\gamma(H_{s}^{f})\otimes\gamma(H_{b})\otimes\ket{\psi_{f}}\bra{\psi_{f}})U_{sbB}^{\dagger}\right]\ket{\psi_{i}}\
.$

A worthwhile extension to our research would be to explicitly simulate the
bath in the experimental proposal and numerical analysis. One means of doing
so would be to investigate the master equation fluctuation theorem that is
proposed in the final version of [32]. Alternatively one could try and
incorporate the bath using the quantum jump approach explored in [36].

###  A.7 Properties of generalised energy flow

Fundamentally, the term $\tilde{E}$ appears in the AQC because the energy of a
general quantum state is not well defined and instead some statistical
estimate of the states’ energy is required. The precise form of $\tilde{E}$ is
forced by the fluctuation theorem approach of comparing a forwards and reverse
process, i.e. by the structure of the derivation. In a contrast to Eq. (10) in
the main text, we here treat the Hamiltonian $H$ and temperature $T$ as
variables,

$\tilde{E}({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\psi}},H,T)\mathrel{\mathop{\mathchar
58\relax}}=-k_{B}T\,{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ln}}\left({\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\bra{\psi}}}\exp\left(-\frac{H}{k_{B}T}\right){\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}{\ket{\psi}}}\right)\,,$
(66)

to enable us to discuss the properties of the function $\tilde{E}$ more
generally. An analysis of $\tilde{E}$ provides some physical support for our
interpretation of $\Delta\tilde{E}$ as a quantum generalisation of the energy
flow between the system and the battery. In particular, the following
properties hold:

  1. 1.

$\tilde{E}(\psi,H+\delta,T)=\tilde{E}(\psi,H,T)+\delta$.

  2. 2.

$\tilde{E}(\psi,\lambda H,T)=\lambda\tilde{E}(\psi,H,\lambda T)$.

  3. 3.

For an energy eigenstate, $\ket{E_{k}}$, $\tilde{E}$ is simply the associated
eigenstate energy, $\tilde{E}(E_{k},H,T)=E_{k}$.

  4. 4.

In the high temperature limit $\tilde{E}$ tends to the average energy of the
state, $\lim_{T\to\infty}\tilde{E}(\psi,H,T)=\bra{\psi}H\ket{\psi}$.

  5. 5.

In the general case $\tilde{E}$ is less than the average energy:
$\tilde{E}(\psi,H,T)\leq\bra{\psi}H\ket{\psi}$.

  6. 6.

$\tilde{E}$ is independent of the the phase of the state
$\tilde{E}(\psi,H,T)=\tilde{E}\left(\psi e^{i\phi},H,T\right)\ \forall\
\phi\in\Re$.

Properties 1 and 2 verify that $\tilde{E}$ scales as one would expect an
energy measure to scale. Property 3 is intuitive because when the energy of a
state is well-defined there is no need to statistically estimate it. Moreover,
property 3 is required to regain the classical limit. Property 6 ensures that
in the absence of interactions, $\tilde{E}$ is constant in time which is a
desirable condition for a statistical estimate of the energy of a quantum
state.

##  Appendix B Derivation of the coherent, squeezed and cat state AQCs

###  B.1 Coherent state AQC

The coherent state AQC is derived by considering a harmonic oscillator
battery, i.e.
$H_{B}=\hbar\omega\left(a^{\dagger}a+\frac{1}{2}\right)=\sum_{n}\hbar\omega\left(n+\frac{1}{2}\right)\ket{n}\bra{n}$,
and transition probabilities between coherent states. The measured states
$\ket{\phi_{f}}$ and $\ket{\psi_{i}}$ are set to the coherent states
$\ket{\alpha_{f}}$ and $\ket{\alpha_{i}^{*}}$ respectively,

$\displaystyle\ket{\phi_{f}}=\ket{\alpha_{f}}\ \ \text{and}$ (67)
$\displaystyle\ket{\psi_{i}}=\ket{\alpha_{i}^{*}}\,,$

where $\ket{\alpha}\mathrel{\mathop{\mathchar
58\relax}}=\exp(-\frac{}{}|\alpha|^{2}/2)\sum_{n=0}^{\infty}\frac{\alpha^{n}}{n!}\,\ket{n}$.
We calculate $\ket{\phi_{i}}$ and $\ket{\psi_{f}}$ using the relation between
states $\ket{\phi_{i,f}}$ and $\ket{\psi_{i,f}}$ as specified in Eq. (7). This
requires calculating the effect of applying the temperature dependent
operation
$\exp\left(-\frac{H_{B}}{2k_{B}T}\right)=\exp\left(-\chi\left(a^{\dagger}a+\frac{1}{2}\right)\right)$
where $\chi=\frac{\hbar\omega}{2k_{B}T}$, to a coherent state. Using operator
algebra we find that

$\displaystyle\ket{\phi_{i}}=\frac{1}{\sqrt{\tilde{Z}_{\alpha_{i}}}}\exp(-|\alpha_{i}|^{2}/2)\sum_{n}\exp\left(-\chi\left(a^{\dagger}a+\frac{1}{2}\right)\right)\frac{\alpha_{i}^{n}}{n!}\ket{n}=\ket{\alpha_{i}\exp\left(-\chi\right)}\
\ \text{and}$ (68) $\displaystyle\ket{\psi_{f}}={\cal
T}\left(\frac{1}{\sqrt{\tilde{Z}_{\alpha_{f}}}}\exp(-|\alpha_{f}|^{2}/2)\sum_{n}\exp\left(-\chi\left(a^{\dagger}a+\frac{1}{2}\right)\right)\frac{\alpha_{f}^{n}}{n!}\ket{n}\right)=\ket{\alpha_{f}^{*}\exp\left(-\chi\right)}\
.$

Additionally, given the definition of $\Delta\tilde{E}$ in Eq. (9), it can be
shown using operator algebra that

$\displaystyle\exp\left(\frac{\Delta\tilde{E}}{k_{B}T}\right)=\frac{\tilde{Z}_{\alpha_{f}}}{\tilde{Z}_{\alpha_{i}}}=\frac{\left(\frac{\exp\left(-|\alpha_{f}|^{2}/2\right)}{\exp(-|\alpha_{f}\exp(-\chi)|^{2}/2)}\right)^{2}}{\left(\frac{\exp(-|\alpha_{i}|^{2}/2)}{\exp\left(-|\alpha_{i}\exp\left(-\chi\right)|^{2}/2\right)}\right)^{2}}=\exp\left((|\alpha_{f}|^{2}-|\alpha_{i}|^{2})(\exp(-2\chi)-1)\right)\
.$ (69)

The error-bounded coherent state AQC follows from Eq. (67), Eq. (68) and Eq.
(69) and the error bounded AQC, Eq. (18).

To compare the coherent state Crooks equality to the classical Crooks
equality, Eq. 1, we rewrite $\Delta\tilde{E}$ in terms of the difference
between the average energy change of the battery in the forwards and reverse
processes, $W_{q}\mathrel{\mathop{\mathchar 58\relax}}=(\Delta E_{+}-\Delta
E_{-})/2$, with

$\displaystyle\Delta E_{+}\mathrel{\mathop{\mathchar
58\relax}}=\bra{\phi_{i}}H_{B}\ket{\phi_{i}}-\bra{\phi_{f}}H_{B}\ket{\phi_{f}}\
\ \ \text{and}$ (70) $\displaystyle\Delta E_{-}\mathrel{\mathop{\mathchar
58\relax}}=\bra{\psi_{f}}H_{B}\ket{\psi_{f}}-\bra{\psi_{i}}H_{B}\ket{\psi_{i}}\,.$

Since energy is globally conserved, $-W_{q}$ is also the difference between
the average energy changes of the system in the forwards and reverse
processes. When the battery is prepared and measured in energy eigenstates,
$W_{q}$ reduces from an average to a well defined energy change. Specifically,
if $\ket{\psi_{i,f}}=\ket{E_{i,f}}$ then it follows from Eq.
(LABEL:eq:_relationbetweenstates) that $\ket{\phi_{i,f}}=\ket{E_{i,f}}$ and
therefore

$W_{q}=\frac{1}{2}\left((E_{i}-E_{f})--(E_{i}-E_{f})\right)\equiv\frac{1}{2}\left(W-(-W)\right)=W$
(71)

where $(E_{i}-E_{f})\equiv W$ is the energy gained (lost) by the system
(battery). Beyond this classical limit, $W_{q}$ is a generalisation of $W$
that accounts for coherence by considering the battery’s average energy and
allows for the fact that the AQC quantifies different average energy
gains/losses in the forwards and reverse processes (i.e. that in general
$\Delta E_{+}\neq-\Delta E_{-}$).

By calculating $\Delta E_{+}$ and $\Delta E_{-}$ for coherent states,

$\displaystyle\Delta E_{+}\mathrel{\mathop{\mathchar
58\relax}}=(\exp(-2\chi)|\alpha_{i}|^{2}-|\alpha_{f}|^{2})\hbar\omega$ (72)
$\displaystyle\Delta E_{-}\mathrel{\mathop{\mathchar
58\relax}}=(\exp(-2\chi)|\alpha_{f}|^{2}-|\alpha_{i}|^{2})\hbar\omega\,,$ (73)

we can rewrite $\Delta\tilde{E}$ for coherent states, Eq. 69, in terms of
$W_{q}$,

$\displaystyle\Delta\tilde{E}=k_{B}T(|\alpha_{i}|^{2}-|\alpha_{f}|^{2})(1-\exp(-2\chi))=\frac{1}{\chi}\frac{1-\exp(-2\chi)}{1+\exp(-2\chi)}\frac{1}{2}(\Delta
E_{+}-\Delta E_{-})=\frac{1}{\chi}\frac{1-\exp(-2\chi)}{1+\exp(-2\chi)}W_{q}\
.$ (74)

Thus the coherent state AQC can be written as

$\displaystyle\frac{\mathcal{P}\left(\alpha_{f}|\alpha_{i}\exp(-\chi),\gamma_{i}\right)}{\mathcal{P}\left(\alpha_{i}^{*}|\alpha_{f}^{*}\exp(-\chi),\gamma_{f}\right)}=\exp\left(-\frac{\Delta
F}{k_{B}T}\right)\,\exp\left(q\left(\chi\right)\frac{W_{q}}{k_{B}T}\right)\,,$
(75)

where we have defined the quantum prefactor,

$q\left(\chi\right)\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\chi}\frac{1-\exp(-2\chi)}{1+\exp(-2\chi)}=\frac{1}{\chi}\tanh(\chi)\
.$ (76)

The quantum prefactor, Eq. (76), can be related to the average energy of a
thermal battery $\langle
H_{B}\rangle_{\gamma(H_{B})}\mathrel{\mathop{\mathchar
58\relax}}=\operatorname{Tr}[H_{B}\gamma(H_{B})]$. We calculate the explicit
algebraic form of $\langle H_{B}\rangle_{\gamma(H_{B})}$ and find that

$\displaystyle\langle H_{B}\rangle_{\gamma({H_{B}})}$
$\displaystyle=\sum_{n}\hbar\omega(n+1/2)\frac{\exp\left(-\frac{\hbar\omega(n+\frac{1}{2})}{k_{B}T}\right)}{Z_{H_{B}}}$
(77)
$\displaystyle=\frac{1}{2}\hbar\omega\left(\frac{1+\exp(-2\chi)}{1-\exp(-2\chi)}\right)$
$\displaystyle=\frac{k_{B}T}{q(\chi)}\ .$

Rearranging we are left with

$q\left(\chi\right)=\frac{k_{B}T}{\langle
H\rangle_{\gamma(H_{B})}}\mathrel{\mathop{\mathchar
58\relax}}=\frac{k_{B}T}{\hbar\omega_{T}}\ .$ (78)

where we have implicitly defined the ‘thermal frequency’ $\omega_{T}$ as the
average frequency of a harmonic oscillator in a thermal state. The rewritten
coherent state Crooks equality in the main text, Eq.
(LABEL:eq:_coherentstaterewrite), follows from Eq.
(LABEL:eq:_coherentstaterewriteqfactor) and Eq. (78).

##### Comment on classical limit.

For the coherent state Crooks equality, the classical limit corresponds to
$\chi=\frac{\hbar\omega}{2k_{B}T}\rightarrow 0$. In this limit, the core
physics of the coherent state Crooks equality is equivalent to that of the
classical Crooks equality; however, there are technical distinctions resulting
from their different formulations. In particular, the coherent state Crooks
equality quantifies state transitions of the battery rather than energy
changes of the system and these are related many-to-one. However, in the
classical limit the coherent state equality quantifies transitions between
battery states with sharp energies and as energy is globally conserved this
corresponds to the sharp energy changes of the system quantified by the
classical Crooks equality. As such, battery states can be viewed as pointer
states to determine the energy flow in or out of the system during the
protocols.

###  B.2 Cat state Crooks equality

We take a cat state to be an equal superposition of two arbitrary coherent
states, $\ket{\mbox{\small Cat}}\propto(\ket{\alpha}+\ket{\beta})$. This is a
more general definition than that sometimes seen in the literature where a cat
state is taken to be only a superposition of a pair of coherent states
$\ket{\alpha}$ and $\ket{-\alpha}$. To derive a cat state AQC the measured
states $\ket{\phi_{f}}$ and $\ket{\psi_{i}}$ are set to

$\displaystyle\ket{\phi_{f}}$ $\displaystyle=\ket{\mbox{\small
Cat}_{f}}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\sqrt{2+2\Re(\exp(-\frac{1}{2}(|\alpha_{f}|^{2}+|\beta_{f}|^{2}-2\beta_{f}^{*}\alpha_{f})))}}(\ket{\alpha_{f}}+\ket{\beta_{f}})\
\ \text{and}$ (79) $\displaystyle\ket{\psi_{i}}$
$\displaystyle=\ket{\mbox{\small Cat}_{i}^{*}}\mathrel{\mathop{\mathchar
58\relax}}=\frac{1}{\sqrt{2+2\Re(\exp(-\frac{1}{2}(|\alpha_{i}|^{2}+|\beta_{i}|^{2}-2\beta_{i}^{*}\alpha_{i})))}}(\ket{\alpha_{i}^{*}}+\ket{\beta_{i}^{*}})\,,$

where the normalisation term is calculated using the fact that the overlap
between two coherent states $\ket{\alpha}$ and $\ket{\beta}$ is given by
$\braket{\beta}{\alpha}=\exp(-\frac{1}{2}(|\alpha|^{2}+|\beta|^{2}-2\beta^{*}\alpha))$.
The prepared states $\ket{\phi_{i}}$ and $\ket{\psi_{f}}$ are calculated using
the relation between states $\ket{\phi_{i,f}}$ and $\ket{\psi_{i,f}}$ as
specified in Eq. (7). Using operator algebra we find that

$\displaystyle\ket{\phi_{i}}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\exp\left(-\chi\left(a^{\dagger}a+\frac{1}{2}\right)\right)\ket{\mbox{\small
Cat}_{i}}\propto\left(\exp\left(-\chi
a^{\dagger}a\right)\ket{\alpha_{i}}+\exp\left(-\chi
a^{\dagger}a\right)\ket{\beta_{i}}\right)$ (80)
$\displaystyle=\frac{1}{\sqrt{\tilde{Z}_{\mbox{\tiny
Cat}_{i}}}}\left(\eta_{\chi}^{|\alpha_{i}|^{2}}\ket{\exp(-\chi)\alpha_{i}}+\eta_{\chi}^{|\beta_{i}|^{2}}\ket{\exp(-\chi)\beta_{i}}\right)\
\ \text{and}$ $\displaystyle\ket{\psi_{f}}$
$\displaystyle=\frac{1}{\sqrt{\tilde{Z}_{\mbox{\tiny
Cat}_{f}}}}\left(\eta_{\chi}^{|\alpha_{f}|^{2}}\ket{\exp(-\chi)\alpha_{f}^{*}}+\eta_{\chi}^{|\beta_{f}|^{2}}\ket{\exp(-\chi)\beta_{f}^{*}}\right)$

where we have defined

$\displaystyle\eta_{\chi}$ $\displaystyle\mathrel{\mathop{\mathchar
58\relax}}=\exp\left(-\frac{1}{2}\left(1-\exp(-2\chi)\right)\right)\ $ (81)

and the normalisation term $\tilde{Z}_{\mbox{\tiny Cat}_{i}}$ (and
equivalently for $\tilde{Z}_{\mbox{\tiny Cat}_{f}}$) takes the form

$\displaystyle\tilde{Z}_{\mbox{\tiny
Cat}_{i}}=\frac{\eta_{\chi}^{2|\alpha_{i}|^{2}}+\eta_{\chi}^{2|\beta_{i}|^{2}}+\eta_{\chi}^{|\alpha_{i}|^{2}+|\beta_{i}|^{2}}2\Re\left(\exp\left(-\frac{1}{2}\exp(-2\chi)(|\beta_{i}|^{2}+|\alpha_{i}|^{2}-2\beta_{i}^{*}\alpha_{i})\right)\right)}{2+2\Re(\exp(-\frac{1}{2}(|\alpha_{i}|^{2}+|\beta_{i}|^{2}-2\beta_{i}^{*}\alpha_{i})))}\
.$ (82)

The normalisation terms $\tilde{Z}_{\mbox{\tiny Cat}_{i}}$ and
$\tilde{Z}_{\mbox{\tiny Cat}_{f}}$ also provide the
$\exp\left(\frac{\Delta\tilde{E}}{k_{B}T}\right)$ term. It follows from the
definition of $\Delta\tilde{E}$ in Eq. (9) that

$\displaystyle\exp\left(\frac{\Delta\tilde{E}}{k_{B}T}\right)=\frac{\bra{\phi_{f}}\exp\left(-2\chi\right)\ket{\phi_{f}}}{\bra{\psi_{i}}\exp\left(-2\chi\right)\ket{\psi_{i}}}=\frac{\tilde{Z}_{\mbox{\tiny
Cat}_{f}}}{\tilde{Z}_{\mbox{\tiny Cat}_{i}}}\ .$ (83)

As such, the cat state AQC follows from Eq.(B12 -B16). (We do not state the
explicit form of $\exp\left(\frac{\Delta\tilde{E}}{k_{B}T}\right)$ in terms of
$\alpha_{i,f}$, $\beta_{i,f}$, $\chi$ and $\eta_{\chi}$ as the expression does
not simplify.) The error-bounded cat state Crooks equality follows from the
error bounded autonomous Crooks equality, Eq. (62), with $\tilde{Z}_{i,f}$
defined in Eq. (82).

###  B.3 Squeezed state Crooks equality

The squeezed state AQC quantifies transition probabilities between battery
oscillator states that are not only displaced but also squeezed. As such, the
measured states $\ket{\phi_{f}}$ and $\ket{\psi_{i}}$ are set to the squeezed
displaced states $\ket{\alpha_{f},r_{f}}$ and $\ket{\alpha_{i}^{*},r_{i}}$
respectively, i.e.

$\displaystyle\ket{\phi_{f}}=\ket{\alpha_{f},r_{f}}=D(\alpha_{f})S(r_{f})\ket{0}\
\ \text{and}$ (84)
$\displaystyle\ket{\psi_{i}}=\ket{\alpha_{i}^{*},r_{i}}=D(\alpha_{i}^{*})S(r_{i})\ket{0}\,,$

where we have introduced the displacement operator $D(\alpha)=\exp(\alpha
a^{\dagger}-\alpha^{*}a)$, the squeeze operator
$S(r)=\exp(\frac{r}{2}(a^{2}-{a^{\dagger}}^{2}))$, and for simplicity we
assume from the outset that the squeezing parameters $r_{i}$ and $r_{f}$ are
real. We calculate $\ket{\phi_{i}}$ and $\ket{\psi_{f}}$ using the relation
between states $\ket{\phi_{i,f}}$ and $\ket{\psi_{i,f}}$ as specified in Eq.
(7). This amounts to calculating the effect of applying the operator
$\exp\left(-\chi a^{\dagger}a\right)$ to a squeezed displaced state, i.e.

$\displaystyle\ket{\phi_{i}}=\frac{1}{\sqrt{\tilde{Z}_{\alpha_{i},r_{i}}}}\exp\left(-\chi\left(a^{\dagger}a+\frac{1}{2}\right)\right)D(\alpha_{i})S(r_{i})\ket{0}\equiv
D(\mu_{i})S(t_{i})\ket{0}=\ket{\mu_{i},t_{i}}\ \ \text{and}$ (85)
$\displaystyle\ket{{\psi_{f}}^{*}}=\frac{1}{\sqrt{\tilde{Z}_{\alpha_{f},r_{f}}}}\exp\left(-\chi\left(a^{\dagger}a+\frac{1}{2}\right)\right)D(\alpha_{f}^{*})S(r_{f})\ket{0}\equiv
D(\mu_{f}^{*})S(t_{f})\ket{0}=\ket{\mu_{f}^{*},t_{f}}\ .$

We introduce $\mu_{i,f}$ and $t_{i,f}$ to denote the displacement and