Fermat's Library | Coherent fluctuation relations: from the abstract to the concrete annotated/explained version.

Coherent ﬂuctuation relations: from the abstract to the concrete

Zoë Holmes

1

, Sebastian Weidt

2

, David Jennings

1,3,4

, Janet Anders

5

, and Florian Mintert.

1

Controlled Quantum Dynamics Theory Group, Imperial College London, London, SW7 2BW, United Kingdom.

2

Department of Physics and Astronomy, University of Sussex, Brighton BN1 9QH, United Kingdom.

3

Department of Physics, University of Oxford, Oxford, OX1 3PU, United Kingdom.

4

School of Physics and Astronomy, University of Leeds, Leeds, LS2 9JT, United Kingdom.

5

CEMPS, Physics and Astronomy, University of Exeter, Exeter, EX4 4QL, United Kingdom.

February 20, 2019

Recent studies using the quantum informa-

tion theoretic approach to thermodynamics

show that the presence of coherence in quan-

tum systems generates corrections to classi-

cal ﬂuctuation theorems. To explicate the

physical origins and implications of such cor-

rections, we here convert an abstract frame-

work of an autonomous quantum Crooks rela-

tion into quantum Crooks equalities for well-

known coherent, squeezed and cat states. We

further provide a proposal for a concrete ex-

perimental scenario to test these equalities.

Our scheme consists of the autonomous evo-

lution 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 sys-

tems. A central question is whether quantum me-

chanical phenomena, such as coherence and entan-

glement, 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 entangle-

ment 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], gen-

eralisations 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 [8–11]. Much of this research [5–11]

utilised the resource theory framework and in par-

ticular the concept of thermal operations [12, 13].

However, despite signiﬁcant theoretical progress, the

results have been seen as rather abstract and una-

mendable 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 be-

tween the resultant thermal ﬂuctuations [16]. Ex-

perimental tests of ﬂuctuation theorems, in partic-

ular the Jarzynski [17] and Crooks [18] equalities,

have been conducted in classical systems, including

stretched RNA molecules [19, 20], over-damped col-

loidal particles in harmonic potentials [21] and clas-

sical 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 re-

cently been performed [25].

However, the use of two point energy measure-

ments or continual weak measurements to obtain a

work probability distribution reduces a system’s co-

herence with respect to the energy eigenbasis [26–31].

This limits the extent to which these previous exper-

iments 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 sys-

tem explicitly as a quantum battery. Rather than

implicitly appealing to an additional classical system

to drive the system out of equilibrium, the quan-

tum system and battery here evolve autonomously

under a time independent Hamiltonian [33–36]. This

proposal does not require projective measurements

onto energy eigenstates and so does not destroy

coherences. As a result coherence actively con-

tributes to the Autonomous Quantum Crooks equal-

ity (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

Accepted in Quantum 2018-02-12, click title to verify 1

arXiv:1806.11256v4 [quant-ph] 19 Feb 2019

harmonic oscillator battery that is prepared in op-

tical 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. (15), 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 po-

sition dependent AC Stark shift on the internal en-

ergy 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

i

S

to H

f

S

. The classical

Crooks equality [18],

P

+

(W )

P

−

(−W )

= exp



−

∆F

k

B

T



exp



W

k

B

T



, (1)

quantiﬁes the ratio of the probability P

+

(W ) of the

work W done on a system in such a non-equilibrium

process, to the probability P

−

(−W ) to extract the

work W in the time reversed process (where the

Hamiltonian is changed back from H

f

S

to H

i

S

). This

ratio is a function of the change in the system’s free

energy,

∆F

:

= F (H

f

S

) − F (H

i

S

) with

F (H

S

)

:

= −k

B

T ln



Tr

S



exp



−

H

S

k

B

T



,

(2)

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 sys-

tem. 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

i

S

to H

f

S

is realised.

2.2 The autonomous quantum Crooks equality

The realisation of a time-dependent Hamiltonian in

the classical Crooks equality, Eq. (1), necessarily im-

plies an interaction with a classical agent. The au-

tonomous framework [32] proposed by Johan Åberg

makes this control explicit by introducing a battery.

Speciﬁcally, the system evolves together with the bat-

tery according to the time-independent Hamiltonian,

H

SB

= H

S

⊗

B

+

S

⊗ H

B

+ V

SB

, (3)

comprised of the Hamiltonians H

S

and H

B

for sys-

tem 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

i

S

⊗ Π

i

B

+ H

f

S

⊗ Π

f

B

+ V

⊥

SB

, (4)

where Π

i

B

and Π

f

B

are projectors onto two orthog-

onal subspaces, R

i

and R

f

, of the battery’s Hilbert

space, and V

⊥

SB

has support only outside those two

subspaces, i.e. (X

S

⊗Π

i

B

)V

⊥

SB

= (X

S

⊗Π

f

B

)V

⊥

SB

= 0

for any system operator X

S

. The system Hamilto-

nian H

S

can always be absorbed into V

SB

, which we

will do in the following. Assuming the battery is ini-

tialised in a state in subspace R

i

only and evolves to a

ﬁnal state in subspace R

f

only, the system Hamilto-

nian evolves from H

i

S

to H

f

S

, i.e. the system Hamilto-

nian 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 Hamilto-

nian evolves from H

i

S

to H

f

S

is necessarily provided

or absorbed by the battery.

The initially thermal system in the Crooks equal-

ity is incoherent in energy and so, to extend the rela-

tion to the quantum regime in this autonomous set-

ting, coherence

1

must begin in an additional non-

equilibrium quantum system. Here the coherence is

provided by the battery which we assume can be pre-

pared in superpositions of energy states. The evolu-

tion under H

SB

generates correlations between the

system and battery reducing the coherence that can

be attributed to the battery alone. Since the dynam-

ics 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 co-

herence under interactions with a thermal system.

1

We use the term ‘coherence’ in the sense of a ‘superposi-

tion of states belonging to diﬀerent energy eigenspaces’. Such

energetic coherences are formally quantiﬁed in the thermal op-

erations framework [12, 13] and by measures such as the l

1

norm of coherence [39], C

l

1

(ρ)

:

=

P

j6=k

| hE

j

| ρ |E

k

i |, the sum

of the absolute value of the oﬀ diagonal elements of a state in

the energy eigenbasis.

Accepted in Quantum 2018-02-12, click title to verify 2

The diﬃculties surrounding how to deﬁne work in

the quantum regime [40–45] are avoided by formu-

lating the AQC in terms of transition probabilities

between battery states. Considering a system ini-

tialised in the thermal state γ

i

and the battery ini-

tialised in the pure state |φ

i

i, the probability to ob-

serve the battery after a time-interval t in the state

|φ

f

i reads

P(φ

f

|φ

i

, γ

i

) = hφ

f

|Tr

S

[U(γ

i

⊗ |φ

i

ihφ

i

|)U

†

] |φ

f

i ,

(5)

with U = exp(−iH

SB

t/~) the propagator of the sys-

tem and battery and the trace over the system de-

noted by Tr

S

. If we consider the transition probabil-

ity between energy eigenstates, the scheme consid-

ered here reduces to the usual two-point projective

energy measurement scheme [26–28] and as such we

can recover the classical Crooks relation, Eq. (1). For

transition probabilities between more general quan-

tum 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 en-

ergy eigenbasis and the measurement can induce a

non-classical updating of the state of the system.

The AQC relates the probabilities P(φ

f

|φ

i

, γ

i

) of a

forward process and P(ψ

i

|ψ

f

, γ

f

) of a reverse process

for two thermal system states

γ

x

∝ exp



−

H

x

S

k

B

T



, with x = i, f (6)

and two pairs of pure initial and ﬁnal battery states

satisfying

|ψ

f

i ∝ T exp



−

H

B

2k

B

T



|φ

f

i ,

|φ

i

i ∝ T exp



−

H

B

2k

B

T



|ψ

i

i .

(7)

The states |ψ

i

i and |ψ

f

i correspond to a time-

reversed process and are deﬁned in terms of the time-

reversal operator

2

T [47]. This relationship between

the initial and ﬁnal states of the forwards and re-

versed processes, Eq. (7), is forced by the deriva-

tion of the AQC, i.e. the structure arises natu-

rally when one reverses a general quantum process

in an autonomous setting. If the battery states

|φ

i

i and |φ

f

i are eigenstates of H

B

, the states |ψ

f

i

and |ψ

i

i of the reversed process are regular time-

reversed states. However, for any coherent super-

position of energy eigenstates the additional term

2

The time-reversal [47] operation T on a battery state |ψi

is deﬁned as complex conjugation in the energy eigenbasis of

the battery, T |ψi = |ψ

∗

i. We introduce T in more detail in

Appendix A.2.

exp(−H

B

/2k

B

T ), which is linked with the Petz re-

covery map [48, 49], is essential to capture the inﬂu-

ence of quantum coherence.

The AQC [32] reads

P(φ

f

|φ

i

, γ

i

)

P(ψ

i

|ψ

f

, γ

f

)

= exp



−

∆F

k

B

T



exp

∆

˜

E

k

B

T

!

, (8)

where ∆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

∆

˜

E(ψ

i

, φ

f

)

:

=

˜

E(ψ

i

) −

˜

E(φ

f

) (9)

is a quantum mechanical generalisation of the energy

ﬂow from the battery to the system. The function

˜

E(ψ)

:

= −k

B

T ln



hψ|exp



−

H

B

k

B

T



|ψi



(10)

is an eﬀective potential for the battery state |ψi that

speciﬁes the relevant energy value within the ﬂuctu-

ation theorem context. We provide a discussion of

its properties in Appendix A.7.

For the AQC to hold exactly we require |ψ

i

i and

|φ

f

i to have support only within subregions Π

i

B

and Π

f

B

respectively and for the battery Hamilto-

nian not to induce evolution between subregions, i.e.

(

B

−Π

i

B

)H

B

Π

i

B

= 0 and (

B

−Π

f

B

)H

B

Π

f

B

= 0. In

general, the second of these conditions does not hold

because a battery Hamiltonian that evolves states be-

tween subregions is required. However, despite this,

numerical simulations (discussed in Section 3.2) in-

dicate that any error can be made negligible as long

as |ψ

i

i and |φ

f

i are well localised in subregions Π

i

B

and Π

f

B

respectively. In Section 2.5 we discuss how

the accuracy of the equality can be quantiﬁed; how-

ever, 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

=

P

n

~ω



n +

1

2



|nihn|. In order to derive

explicit expressions for the ratio of transition prob-

abilities, 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 re-

lation between the initial and ﬁnal states of the for-

ward and reversed process, Eq. (7). While the bat-

tery Hamiltonian and states are ﬁxed in the following

sections, the system Hamiltonians, H

i

S

and H

f

S

, and

the system and battery interaction, V

⊥

SB

, can still be

chosen at will.

Accepted in Quantum 2018-02-12, click title to verify 3

2.3 Coherent state Crooks equality

The physical content of the AQC is neatly illustrated

by considering transition probabilities between co-

herent states of the battery, |αi = exp(αa

†

−α

∗

a) |0i

with a

†

and a the creation and annihilation opera-

tors of the oscillator respectively, and α a complex

number. In the forwards process, we assume the

battery is initialised in |α

i

exp(−χ)i with χ =

~ω

2k

B

T

and we are interested in the transition probability

P(α

f

|α

i

exp(−χ), γ

i

) to the coherent state |α

f

i. Fol-

lowing Eq. (7), the initial and ﬁnal states of the

reversed process are given by the coherent states

|α

∗

f

exp(−χ)i and |α

∗

i

i. The symbol

∗

denotes com-

plex conjugation in the Fock basis and arises from the

time reversal operation on a coherent state, T |αi =

|α

∗

i.

The dimensionless parameter χ is the ratio of the

magnitude of quantum ﬂuctuations,

~ω

2

, to the mag-

nitude of thermal ﬂuctuations, k

B

T , and quantiﬁes

the degree to which any regime is quantum mechan-

ical. In the limit in which thermal ﬂuctuations dom-

inate, χ tends to 0, and the prefactor exp(−χ) tends

to 1. Consequently, in this limit the reverse process

is the exact time-reversed process; however, in gen-

eral, the pairs of states considered in the forwards

and reverse process diﬀer by an amount determined

by exp(−χ).

As derived in more detail in B.1, the energy ﬂow,

Eq. (9), takes the explicit form

∆

˜

E(α

f

, α

i

) = k

B

T



|α

i

|

2

− |α

f

|

2

)(1 − exp(−2χ)



.

(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 ∆E

+

and gain ∆E

−

of the forward and reverse

processes. We deﬁne the prefactor

q

:

=

∆

˜

E

W

q

(12)

as the ratio between ∆

˜

E and W

q

= (∆E

+

−∆E

−

)/2.

As shown in B.1, we ﬁnd that for coherent states the

prefactor takes the form

q (χ) =

1

χ

tanh(χ) (13)

and can be related to the average frequency, ω

T

, of

the oscillator battery in a thermal state, γ(H

B

) ∝

exp



−

H

B

k

B

T



, at temperature T ,

~ω

T

:

= hH

B

i

γ(H

B

)

=

k

B

T

q(χ)

. (14)

0.5 1.0 1.5 2.0

χ

0.5

1.0

1.5

2.0

〈H

B

〉

γ

B

ℏω

T

1

2

ℏω

h

2

mλ

th

2

k

B

T

Figure 1: The solid, dark blue line shows the average energy,

~ω

T

, of the oscillator in a thermal state at temperature T

as a function of χ. The red lines indicate the contribution

of thermal (dotted) and quantum (dashed) contributions to

~ω

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 coherent state AQC can thus be written as

P



α

f



α

i

exp(−χ), γ

i



P



α

∗

i



α

∗

f

exp(−χ), γ

f



= exp



−

∆F

k

B

T



exp



W

q

~ω

T



.

(15)

In this form, it is analogous to the classical Crooks

equality, Eq. (1), with the diﬀerence encoded in ω

T

.

The thermal frequency ω

T

can be understood in

terms of the thermal wavelength λ

th

=

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 λ

th

is small the

system is well localised and can be considered clas-

sical; 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

vac

=

1

2

~ω, it follows that

~ω

T

=

h

2

mλ

2

th

+

1

2

~ω . (16)

This equality describes the splitting of ~ω

T

into its

thermal,

h

2

mλ

2

th

, and quantum,

1

2

~ω, contributions, as

is shown in Fig. 1.

In the classical limit the thermal wavelength λ

th

is

small and the dominant contribution to ~ω

T

is ther-

mal. In this limit ~ω

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 λ

th

the thermal con-

tribution tends to zero and vacuum ﬂuctuations dom-

inate. These vacuum ﬂuctuation induce quantum

Accepted in Quantum 2018-02-12, click title to verify 4

corrections to ~ω

T

which increase monotonically with

χ. 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 dynam-

ics quantiﬁed by the AQC are unitary and so fully

reversible, the irreversibility quantiﬁed here stems

from the choice in the prepared and measured bat-

tery states [51].

2.4 Squeezed state and cat state Crooks equalities

Analogously to Eq. (15), we now discuss quantum

Crooks equalities that quantify the ratio of transi-

tion probabilities between pairs of squeezed states

of the battery and pairs of cat states of the bat-

tery. Squeezed displaced states, |r, αi

:

= exp(αa

†

−

α

∗

a) exp(

r

2

(a

2

− a

†

2

)) |0i, are the unbalanced mini-

mum uncertainty states where for a given squeezing

parameter r, the variance of the oscillator’s position

and momentum are rescaled as ∆x = exp(−r)

q

~

2mω

and ∆p = exp(+r)

q

~mω

2

. Cat states |Cati ∝ |αi +

|βi are superpositions of coherent states |αi and |βi.

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 for-

wards (|φ

i/f

i) and reverse (|ψ

i/f

i) processes.

Forwards Preparation Measurement

Coherent |α

i

e

−χ

i |α

f

i

Squeezed |s

i

, µ

i

i |r

f

, α

f

i

Cat η

|α

i

|

2

χ

|α

i

e

−χ

i + η

|β

i

|

2

χ

|β

i

e

−χ

i |α

f

i + |β

f

i

Reverse Preparation Measurement

Coherent |α

∗

f

e

−χ

i |α

∗

i

Squeezed |s

f

, µ

∗

f

i |r

i

, α

∗

i

Cat η

|α

f

|

2

χ

|α

∗

f

e

−χ

i + η

|β

f

|

2

χ

|β

∗

f

e

−χ

i |α

∗

i

i + |β

∗

i

In the above table the cat states are unnor-

malized for brevity and we have deﬁned η

χ

:

=

exp



−

1

2

(1 − exp(−2χ))



, s

:

= tanh

−1

(exp(−2χ) tanh(r))

and µ

:

=

exp(−χ)(1+tanh(r))

1+exp(−2χ) tanh(r)

. We do not show the ex-

plicit expressions of ∆

˜

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.

We compare the additional eﬀect of squeezing to

the coherent state cases in Fig. 2 by plotting q(χ),

1 2 3 4 5

χ

0.2

0.4

0.6

0.8

1.0

q

(a)

1 2 3 4 5

χ

0.2

0.4

0.6

0.8

1.0

q

(b)

Figure 2: We plot q

:

=

∆

˜

E

W

q

as a function of χ for squeezed

displaced states. As indicated by the grey Wigner plot in-

sets, in a) the prepared and measured squeezed displaced

battery state are displaced with respect to position only,

i.e. α

i,f

= <(α

i,f

) and µ

i,f

= <(µ

i,f

); whereas in b)

the state is displaced with respect to momentum only, i.e.

α

i,f

= =(α

i,f

) and µ

i,f

= =(µ

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.

the ratio of ∆

˜

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 χ tends to 0. How-

ever, q decreases monotonically for increasing χ, and

vanishes as χ 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 posi-

tion and squeezed with respect to position, the be-

haviour of q for large χ again replicates the coherent

state case; however, there is now a regime of low χ

in which q is greater than the classical value of 1.

In Fig. 2b the prepared and measured states are dis-

placed with respect to momentum and we see identi-

cal 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 prob-

ability for the reverse process is again larger than

expected classically in the limit of large χ. How-

ever, there is now an intermediary regime where the

probability for the reverse process is even more sup-

pressed 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 χ. Consequently, the analogous plot to Fig. 2 for

cat states is not a straightforward correction to the

Accepted in Quantum 2018-02-12, click title to verify 5

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 quanti-

ﬁed by error bounds. Here we establish three such

error measures, D,  and (1 − R).

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

rewritten as

Z

i

hψ

i

|exp



−

H

B

k

B

T



|ψ

i

iP (φ

f

|φ

i

, γ

i

)

− Z

f

hφ

f

|exp



−

H

B

k

B

T



|φ

f

iP (ψ

i

|ψ

f

, γ

f

) ≈ 0

(17)

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(−∆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 mag-

nitude of the left hand side of Eq (17) is approxi-

mately zero, to a statement that it is less than or

equal to some, hopefully small, error bound,

D

:

= |Z

i

hψ

i

|exp



−

H

B

k

B

T



|ψ

i

iP (φ

f

|φ

i

, γ

i

)

− Z

f

hφ

f

|exp



−

H

B

k

B

T



|φ

f

iP (ψ

f

|ψ

f

, γ

f

) | ≤ .

(18)

It is possible to determine analytically the terms

hψ

i

|exp



−

H

B

k

B

T



|ψ

i

i and hψ

f

|exp



−

H

B

k

B

T



|ψ

f

i in D

for the coherent, squeezed and cat state equalities by

arranging their approximate forms.

The total error, , is the sum of the initial and ﬁnal

factorisation errors,

 = ||

i

SB

||

1

+ ||

f

SB

||

1

(19)

where ||||

1

= Tr

h

√



†



i

denotes the trace norm of

the operator . These factorisation errors 

i,f

cap-

ture 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

i,f

S

, when acting on the battery states that are mea-

sured in subregions Π

i,f

B

(i.e. |ψ

f

i and |φ

f

i respec-

tively). More accurately, and as derived in Appendix

A, having deﬁned the map

J

K

(σ) = e

−

K

2k

B

T

σe

−

K

2k

B

T

, (20)

the factorisation errors are deﬁned as



i

SB

= J

H

SB

(

S

⊗ |ψ

i

ihψ

i

|) − J

K

i

SB

(

S

⊗ |ψ

i

ihψ

i

|)



f

SB

= J

H

SB

(

S

⊗ |φ

f

ihφ

f

|) − J

K

f

SB

(

S

⊗ |φ

f

ihφ

f

|)

(21)

where K

i

SB

= H

i

S

⊗

B

+

S

⊗ H

B

and K

f

SB

=

H

f

S

⊗

B

+

S

⊗ H

B

. As calculating  involves ex-

ponentiating H

SB

, this generally needs to be done

numerically.

While Eq. (18) enables the AQC to be tested, D

and  are not always reliable indicators of the ac-

curacy of the AQC. For example, D decreases as

the transition probabilities 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

:

=

P(φ

f

|φ

i

, γ

i

)

P(ψ

i

|ψ

f

, γ

f

)

exp

∆

˜

E − ∆F

k

B

T

!!

−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 interac-

tion, 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’ inter-

action that replicates the classical Crooks equality

setup most closely. We then take a broader perspec-

tive 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 inter-

acting with a harmonic oscillator battery via

V

SB

= σ

z

S

⊗ E(x

B

) , (23)

where σ

z

S

cited and ground states of the two level system re-

spectively. E(x

B

) is an energetic level-shift that de-

pends on the position operator, x

B

, of the oscillator.

By choosing E(x) to be constant for x ≤ x

i

and for

x ≥ x

f

, two distinct eﬀective Hamiltonians, H

i

S

and

H

f

S

in Eq. (4), can be realised for the two level sys-

tem. The independence of the AQC on the form of

V

⊥

SB

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,

E(x) =











E

i

x ≤ x

i

E

f

−E

i

x

f

−x

i

(x − x

i

) + E

i

x

i

< x < x

f

E

f

x ≥ x

f

.

(24)

Accepted in Quantum 2018-02-12, click title to verify 6

α

i

e

-Χ

α

f

H

S

i

=

E

i

σ

S

z

H

S

f

=

E

f

σ

S

z

x

i

x

f

(a) P (α

f

|α

i

exp(−χ), γ

i

)

α

i

*

α

f

*

e

-Χ

H

S

i

=

E

i

σ

S

z

H

S

f

=

E

f

σ

S

z

x

i

x

f

(b) P(α

∗

i

|α

∗

f

exp(−χ), γ

f

)

Figure 3: Diagrammatic representation of the protocol to

test the coherent state Crooks equality. The blue lines rep-

resent the ground and excited state of the system as a func-

tion 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

.

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

|.

In order to measure the transition probability,

P(φ

f

|φ

i

, γ

i

), for the forwards process one has to be

able to implement the following three steps:

1. The battery is prepared in a state |φ

i

i that is

localised such that hφ

i

|x

B

|φ

i

i < x

i

and the sys-

tem is prepared in a thermal state with respect

to its eﬀective Hamiltonian H

i

S

= E

i

σ

z

S

.

2. The system and battery are allowed to evolve for

some time τ. The time τ 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

f

S

= E

f

σ

z

S

.

3. The battery is measured to determine whether

its evolved state is |φ

f

i.

The forwards transition probability, P(φ

f

|φ

i

, γ

i

), is

the relative frequency with which the battery is found

in the state |φ

f

i when the steps 1-3 are repeated a

large number of times. We discuss how the prepa-

ration and measurement steps might be practically

implemented in a trapped ion system in Section IV

B.

The procedure to obtain the reverse process transi-

tion probability, P(ψ

i

|ψ

f

, γ

f

), is entirely analogous.

The battery is prepared in the state |ψ

f

i in the region

beyond x

f

and the system is prepared in a thermal

state with respect to H

f

S

= E

f

σ

z

S

. A ﬁnal measure-

ment, after evolving for time τ, determines whether

the battery is in state |ψ

i

i.

A sketch of the test for the coherent state Crooks

equality is shown in Fig. 3. The procedure for the co-

herent, 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 ap-

proximate 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.

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 mea-

sures D and R, as deﬁned in Eq. (18) and Eq. (22).

We further numerically calculated the error bound ,

deﬁned in Eq. (19). As expected we ﬁnd that D for

the coherent, squeezed and cat states is less than ,

for a wide range of simulated parameters (k

B

T , ω,

α

i

, α

f

, r

i

, r

f

, β

i

, β

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 α

i

and real α

f

= −α

i

, the inset of Fig. 4a shows

that 1 −R tends to 0 as α

i

is increased. As is shown

in red and blue respectively, for a given α

i

squeezing

the position variance of the prepared and measured

states decreases 1 − R; while, squeezing the momen-

tum variance increases 1 − R. Coherent states with

also a complex displacement α corresponding to ﬁ-

nite momentum result in similar behaviour, as we

have explicitly veriﬁed. For the cat state equality,

we ﬁrstly consider ﬁnal states of the form |ψ

i

i ∝

|−αi + |−(α + 1)i and |φ

f

i ∝ |αi + |α + 1i. 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 α is increased. However, for

the cat states ∝ |αi+|−αi, we ﬁnd that 1−R = 0.59

for all α, as shown by blue line. In contrast to the

Accepted in Quantum 2018-02-12, click title to verify 7

x

f

x

i

x

f

x

i

●

4 6 8

-α

i

= α

f

0.2

0.4

0.6

1 - R

●

▲

●

▲

2 3 4 5 6 7 8

-α

i

= α

f

10

-32

10

-22

10

-2

D , ϵ

(a) Coherent State and Squeezed State Equalities

●

● ● ●

● ● ● ● ● ● ●

4 6 8

α

0.2

0.4

0.6

1 - R

▲

●

▲

●

3 4 5 6 7 8

α

10

-28

10

-23

10

-18

10

-13

10

-8

0.001

D , ϵ

(b) Cat State Equality

Figure 4: The solid and dashed lines of the main plots

show the error measures D and  respectively, and the in-

sets plot the error measure 1 − R, for evolution under the

total Hamiltonian with the position dependent level split-

ting interaction and a harmonic oscillator battery. In (a)

these are plotted as a function of −α

i

= α

f

with the coher-

ent 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,

|ψ

i

i = |φ

f

i ∝ |αi+|−αi, is shown in blue and for cat states

prepared and measured on a single side of the interaction re-

gion, |ψ

i

i ∝ |−αi+ |−(α + 1)i and |φ

f

i ∝ |αi+ |α + 1i, is

shown in red. In this simulation we have used the following

parameters: x

i

= −4, x

f

= 4, ~ω = 1, E

i

= 1, E

f

= 2 and

k

B

T = 1. The displacement parameters, α

i,f

, and the posi-

tions x

i

and x

f

, are given in units of dimensionless position

X =

p

mω

2~

x.

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 probabil-

ity to ﬁnd the battery in both regions, x < x

i

and

x > x

f

.

We conclude that decreasing the overlap of the pre-

pared and measured battery states from the interac-

tion region increases the accuracy of the AQC. More-

over, the convergence to the exact limit is reached

quickly. For example, we see in Fig. 4a that for coher-

ent states with |α

i

| = |α

f

| ≥ 6 the error measures are

very small, i.e. D, , (1 − R) < 10

−6

. Such discrep-

ancies are lower than experimental error margins and

consequently the approximate nature of these equal-

ities can eﬀectively be disregarded when the prepa-

ration and measurement states are chosen appropri-

ately.

The behaviour of (1 − R) in Fig. 4 makes phys-

ical sense. Fundamentally, the approximate nature

of the AQC arises because, by considering the au-

tonomous evolution under a time-independent inter-

acting 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

i

S

and H

f

S

. 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 prop-

erly thermalise with respect to its initial Hamilto-

nian. 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 sys-

tem 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 ≤ x

i

and x ≥ x

f

to ensure that the initial

and ﬁnal Hamiltonians are well-deﬁned and so the

AQC holds to a high degree of approximation. How-

ever, 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 de-

viation 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

:

= ∆

˜

E/W

q

, as per Eq. (12), that appears in the

Accepted in Quantum 2018-02-12, click title to verify 8

-5 5

x

-1

1

E(x)

0.5 1.0 1.5 2.0 2.5

χ

0.2

0.4

0.6

0.8

1.0

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 =

tanh(χ)

χ

(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 cho-

sen the following physically plausible parameters, x

i

= −2,

x

f

= 2, E

i

= 1MHz, E

f

= 2MHz, α

i

= −5 and α

f

= 4.

The displacement parameters α

i,f

and the positions, x

i

and

x

f

, are given in units of dimensionless position.

term exp



q

W

q

k

B

T



of the predicted ratio of transition

probabilities. The prefactor q can always be inferred

from the transition probabilities using

q =

k

B

T

W

q

ln

P(φ

f

|φ

i

, γ

i

)

P(ψ

i

|ψ

f

, γ

f

)

exp



∆F

k

B

T



!

; (25)

however, when the AQC does not hold exactly, it will

deviate from ∆

˜

E/W

q

.

We numerically simulated the evolution of a co-

herent 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 coher-

ent 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

σ

z

S

to E

f

σ

z

S

can be

approximately replicated by using a sin(x)

2

function

and choosing α

i

and α

f

to sit in the troughs and

peaks of the potential where the eﬀective Hamilto-

nian 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 abol-

ish the ﬂat ends entirely and instead enact a change

in eﬀective Hamiltonian using a solely linear poten-

tial 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 out-

line a potential experimental implementation utilis-

ing 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. Practically, we require forwards and reverse

transition probabilities of a measurable order of

magnitude, i.e.

10

−2

. P(φ

f

|φ

i

, γ

i

) . 1 − 10

−2

and

10

−2

. P(ψ

i

|ψ

f

, γ

f

) . 1 − 10

−2

.

2. To practically simulate a non-trivial thermal sys-

tem state we require non-negligible populations

of both the ground and excited states. This re-

quires the initial and ﬁnal level splittings, 2E

i

and 2E

f

respectively, to be a similar order of

magnitude to the temperature, i.e.

2E

i

≈ k

B

T ≈ 2E

f

.

3. To non-trivially test the AQC we require a mod-

erately 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.

|2E

f

− 2E

i

| ≈ ~ω .

4. To cleanly test the AQC, the decoherence and

heating rates of the battery need to be consider-

ably less than the trap frequency. If long-lived

internal levels are used to represent the two-level

system then the primary constraint is the heat-

ing rate of the phonon mode battery, ν

therm

, and

thus we need,

ω  ν

therm

.

Accepted in Quantum 2018-02-12, click title to verify 9

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.

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 re-

gion, 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 inter-

nal electronic levels and a position dependent mag-

netic ﬁeld, B(x) ∝ E(x). However, it is challenging

to obtain the required magnetic ﬁeld environment for

currently obtainable ion heights and corresponding

motional heating rates.

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 ex-

change of photons (and thus energy) between the ion

and the light ﬁeld, and therefore is not perfectly au-

tonomous. However, the energy exchange quickly de-

cays for increasing detuning and for experimentally

achievable Rabi-frequencies and detunings the sys-

tem can nonetheless be considered autonomous for

all practical purposes.

To replicate Eq. (24) as closely as possible we re-

quire 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

i

S

, while in the region x > x

f

a ﬂat laser beam pro-

ﬁle induces a constant AC Stark shift realising H

f

S

.

In the region x

i

< x < x

f

the tail of the laser beam

changes the Hamiltonian from H

i

S

to H

f

S

. In Fig. 6 we

sketch this for a smoothed top hat potential. How-

ever, 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 implementa-

tion to maintain a closer resemblance to the classi-

cal Crooks equality setup, it is also possible, as dis-

cussed in Section 3.3, to detect the quantum devi-

ation 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 mag-

netic ﬁeld that increases strength linearly with po-

sition B(x) ∝ x could be used to realise a Zeeman

shift that increases linearly with position [54–56].

4.1 Experimental Parameters

To support the plausibility of our proposal we present

a set of potential parameters that satisfy the ﬁve re-

quirements listed above and could as such enable the

AQC to be veriﬁed.

We propose using a single

171

Yb

+

ion trapped in

a linear Paul trap

3

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. Re-

quirements 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 pro-

pose using the F = 1, M

F

= 1 and M

F

= −1 levels

of the

2

S

1/2

ground state. A magnetic ﬁeld can be

applied to lift the degeneracy of the F = 1 mani-

fold [58] and set E

i

with typical splittings ranging

between approximately 0.5 MHz and 10 MHz.

3

The 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].

Accepted in Quantum 2018-02-12, click title to verify 10

We suggest using a laser ﬁeld with an intensity of

4.5 Wmm

−2

that is red detuned from the 369nm

2

S

1/2

-

2

P

1/2

transition by 0.1×10

14

Hz. Using the methods

and data from [60, 61], this is expected to induce an

AC Stark shift on the M

F

= 1 and M

F

= −1 states

of -0.2 MHz and -1.2 MHz respectively.

The position variance of

171

Yb

+

for an axial secu-

lar frequency of 0.3 MHz is 0.01 µm [46]. Therefore,

to satisfy requirement 5, a top hat potential with the

size of the ‘ﬂat’ region being greater than 0.05 µm

is required. There is some ﬂexibility in how accu-

rately the beam proﬁle needs to be shaped as our

simulations indicate that 1 − R scales linearly

4

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

i

S

to

H

f

S

, 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 signiﬁ-

cant 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 ≈ 25 ~ω regime can be probed with

x

f

− x

i

= 1µm. 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 tem-

perature regime it would be possible to probe and

the greater the χ induced quantum deviations can

be detected

5

.

4.2 Experimental Techniques

The procedure outlined in 3.1 is realisable for trapped

ions using the following currently available experi-

4

The exact constant of proportionality depends on the tem-

perature, the magnitude of the Stark shift and the battery

wavepacket parameters; however, it is generally of the order of

1 − R ≈

δI

I

where δI is the change in laser intensity over a dis-

tance of 0.1 µm 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.

5

It is possible to realise standing waves and linear mag-

netic ﬁeld gradients on the scale of 10

−2

µm [52, 56]. Thus

the small spatial scales of these alternative approaches could

enable larger values of χ to be more easily probed.

mental 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 mod-

eled by running the protocol on a single ion N

g

and

N

e

times (where N

e

/N

g

= exp(−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 se-

quence 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 gen-

erated 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 us-

ing 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 ap-

paratus it would be possible to generate states sepa-

rated by µm as required here.

Measurement of oscillator states. A measure-

ment to determine the overlap of the ﬁnal state of the

phonon modes, |ψ

ﬁnal

i, with some test state, |ψ

test

i,

can be achieved by performing the inverse of the rel-

evant preparation process, U

test†

B

, and then measur-

ing to determine whether the battery is in the mo-

tional ground state |0i, i.e. the overlap hψ

test

|ψ

ﬁnal

i =

h0|U

test†

B

|ψ

ﬁnal

i. For example, to determine the over-

lap of some ﬁnal state with the coherent state |α

test

i

we would have U

test†

B

= D(α

test

)

†

= D(−α

test

) (this is

sketched in Fig. 7).

6

There is some ﬂexibility on how fast the measurement

needs to be performed. The inverse process U

†

test

needs to be

performed at time τ or when the ion returns to the same place

after another complete rotation, i.e. (2n + 1)τ ∀n ∈ Z

+

. The

measurement to determine whether the ion is in the motional

ground state need only be performed before there is a substan-

Accepted in Quantum 2018-02-12, click title to verify 11

(a) Preparation

(b) Measurement

Figure 7: Sketch of preparation and measurement stages of

the forwards protocol of the coherent state trapped ion im-

plementation. (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 displace-

ment 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 pre-

cisely 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 pro-

jective measurement on a single ion that answers

the binary question ‘is the phonon in the state n =

m

test

?’ [30, 64]. Choosing m

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.

tial probability for the state to be disturbed by decoherence

or heating.

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 sub-

systems 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. (15). The de-

pendence 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 equal-

ities can be viewed as setting out quantum correc-

tions to the classical Crooks equality. In particu-

lar, the coherent state Crooks equality is its low-

est order extension to quantum states. The coher-

ent state equality eﬀectively reduces to the classi-

cal Crooks equality in the classical limit but by in-

creasing the ratio of quantum ﬂuctuations to ther-

mal ﬂuctuations it is possible to smoothly interpo-

late from the classical to the quantum regime. In

the general quantum case the probability for the re-

verse 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 irreversibil-

ity is apparently strengthened. It would be interest-

ing 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 ex-

perimentally test resource theories for quantum ther-

modynamics. 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]. Conse-

quently, the proposed experiment allows us to probe

coherent features of the resource framework, which is

noteworthy as these frameworks [8, 12, 13] have, with

the exception of a few recent developments [65, 66],

Accepted in Quantum 2018-02-12, click title to verify 12

remained abstract.

More broadly, we have taken a highly mathemati-

cal result from within the quantum information theo-

retic approach to thermodynamics and both explored

its physical content and an experimental implemen-

tation. We hope our research encourages more such

attempts to physically ground recent quantum ther-

modynamics 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 Re-

search Council Centre for Doctoral Training in Con-

trolled Quantum Dynamics. SW is supported by the

U.K. Quantum Technology hub for Networked Quan-

tum Information Technologies, EP/M013243/1. DJ

is supported by the Royal Society. JA acknowledges

support from Engineering and Physical Sciences Re-

search Council, Grant EP/M009165/1, and the Royal

Society. FM acknowledges support from Engineer-

ing and Physical Sciences Research Council, Grant

EP/P024890/1.

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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 ρ is,

G

H

(ρ)

:

=

exp



−

H

2k

B

T



ρ exp



−

H

2k

B

T



˜

Z

H

(ρ)

, with

˜

Z

H

(ρ)

:

= Tr



exp



−

H

k

B

T



ρ



.

(26)

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

Accepted in Quantum 2018-02-12, click title to verify 16

of physically natural properties. When the energy of the input state is completely certain, as in an energy

eigenstate |E

k

i, the Gibbs map has no eﬀect and

G

H

(|E

k

ihE

k

|) = |E

k

ihE

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, ρ

:

=

1

N

, to the thermal

state,

G

H

(ρ ) = γ(H)

:

=

1

Z

H

exp



−

H

k

B

T



, with

Z

H

:

= Tr



exp



−

H

k

B

T



,

(28)

and an equal superposition of energy states, |ψ

+

i

:

=

1

√

N

P

N

k=1

|E

k

i, to the coherent thermal state (a pure

state with the same energy populations as the thermal state),

G

H

(|ψ

+

ihψ

+

|) = |ψ

γ

ihψ

γ

|, with

|ψ

γ

i =

1

√

Z

H

N

X

k

exp



−

E

k

2k

B

T



|E

k

i .

(29)

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 Hamilto-

nians, the normalisation constant

˜

Z

H

(ρ) 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

˜

Z

H

(ρ)

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. T(x) = x and T(p) = −p where T(X) to denotes the time-reversal

of any quantum operator X. In virtue of the time reversal being a symmetry operation, T is additionally

requirements entail that T is an anti-unitary operator [47].

The simplest choice in anti-unitary operation to represent T is complex conjugation. This is the ‘text-

book’ [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 oper-

ation is equivalent to complex conjugation. As the AQC quantiﬁes transition probabilities between quantum

states, we are free to use this equivalence and represent T via the transpose.

Deﬁnition 2. Quantum time reversal.

The quantum time reversal operation T on any quantum operator X is deﬁned as

T(X)

:

= X

T

,

(30)

where

T

is the transpose operation. It follows that the time reversal operation T on any pure state |ψi is

given by

T |ψi = |ψ

∗

i , (31)

where

∗

denotes complex conjugation.

Accepted in Quantum 2018-02-12, click title to verify 17

As the transpose operation is basis dependent, so are T and T . The physical situation dictates the appro-

priate choice in basis. Speciﬁcally, the basis is chosen to ensure that the time reversed battery states drive

the change in Hamiltonian from H

f

S

back to H

i

S

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, {|nihn|}, and the time reversal of the system with respect to the Pauli

z basis, {|eihe|, |gihg|}. 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 T. The ﬁrst of these is the fact that as probabilities

are scalars they are invariant under T. The second is that T reverses the order of products of operators,

T(AB) = T(B)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/~), H] = 0 and the time reversal invariance of the

evolution operator follows from the time reversal invariance of the Hamiltonian, i.e. T(H) = H entails

T (exp(−iHt/~)) = exp(−iHt/~).

Theorem 1. Global Invariance.

Consider a Hamiltonian H and evolution U such that: T(H) = H, 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

(ρ

i

), evolves under U and a binary POVM measurement is performed

with POVM elements {ρ

f

, − ρ

f

}. The evolved state, UG

H

(ρ

i

)U

†

, collapses onto ρ

f

with the probability

P (ρ

f

|G

H

(ρ

i

))

:

= Tr[ρ

f

UG

H

(ρ

i

)U

†

] . (32)

In the reverse process the state G

H

(T(ρ

f

)) evolves under U and on measurement collapses onto T(ρ

i

) with

the probability

P (T(ρ

i

)|G

H

(T(ρ

f

)))

:

= Tr[T(ρ

i

)UG

H

(T(ρ

f

))U

†

] . (33)

The ratio of these transition probabilities is,

P (ρ

f

|G

H

(ρ

i

))

P (T(ρ

i

)|G

H

(T(ρ

f

)))

=

˜

Z

H

(T(ρ

f

))

˜

Z

H

(ρ

i

)

. (34)

The proof of global invariance starts with the deﬁnition of the forwards transition probability, and makes

use of the deﬁnition of T in combination with the fact that T(H) = H and T(U) = U ,

P (ρ

f

|G

H

(ρ

i

))

:

= Tr[ρ

f

UG

H

(ρ

i

)U

†

] = Tr[T



ρ

f

UG

H

(ρ

i

)U

†



] = Tr[U

†

G

H

(T(ρ

i

))UT(ρ

f

)] . (35)

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

P (ρ

f

|G

H

(ρ

i

)) =

1

˜

Z

H

(ρ

i

)

Tr



U

†

exp



−

H

2k

B

T



T(ρ

i

) exp



−

H

2k

B

T



UT(ρ

f

)



,

(36)

Accepted in Quantum 2018-02-12, click title to verify 18

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

P (ρ

f

|G

H

(ρ

i

)) =

1

˜

Z

H

(ρ

i

)

Tr



T(ρ

i

) exp



−

H

2k

B

T



UT(ρ

f

)U

†

exp



−

H

2k

B

T



=

1

˜

Z

H

(ρ

i

)

Tr



T(ρ

i

)U exp



−

H

2k

B

T



T(ρ

f

) exp



−

H

2k

B

T



U

†



.

(37)

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

probability,

P (ρ

f

|G

H

(ρ

i

)) =

˜

Z

H

(T(ρ

f

))

˜

Z

H

(ρ

i

)

Tr

h

T(ρ

i

)UG

H

(T(ρ

f

))U

†

i

=

˜

Z

H

(T(ρ

f

))

˜

Z

H

(ρ

i

)

P (T(ρ

i

)|G

H

(T(ρ

f

))) .

(38)

The AQC is derived by applying global invariance to a bipartite setup. By evaluating the forwards and

reverse transition probabilities, P (ρ

f

|G

H

(ρ

i

)) and P (T(ρ

i

)|G

H

(T(ρ

f

))), with respect to a carefully chosen

Hamiltonian H and states ρ

i

and ρ

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

(σ)

:

= exp



−

H

2k

B

T



σ exp



−

H

2k

B

T



(39)

for any operator σ. This map is, on any quantum state ρ, the unnormalised version of the Gibbs map, i.e.

J

H

(ρ) =

˜

Z

H

(ρ) G

H

(ρ).

Deﬁnition 3. Factorisability.

A composite system with a Hamiltonian H

AB

in the state ρ

A

⊗ ρ

B

is factorisable if

J

H

AB

(ρ

A

⊗ ρ

B

) = J

H

A

(ρ

A

) ⊗ J

H

B

(ρ

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 ρ

A

⊗ ρ

B

is factorisable then

G

H

AB

(ρ

A

⊗ ρ

B

) = G

H

A

(ρ

A

) ⊗ G

H

B

(ρ

B

) and

˜

Z

H

AB

(ρ

A

⊗ ρ

B

) =

˜

Z

H

A

(ρ

A

) ⊗

˜

Z

H

B

(ρ

B

) .

(41)

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

i

S

or H

f

S

, 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, |Ψ

i

i and |Ψ

f

i, such that the system and battery Hamiltonian H

SB

factorises as

J

H

SB

(ρ ⊗ |Ψ

i

ihΨ

i

|) = J

H

i

S

(ρ ) ⊗ J

H

B

(|Ψ

i

ihΨ

i

|) and

J

H

SB

(ρ ⊗ |Ψ

f

ihΨ

f

|) = J

H

f

S

(ρ ) ⊗ J

H

B

(|Ψ

f

ihΨ

f

|)

(42)

where H

i

S

and H

f

S

are Hamiltonians of the system and H

B

is a Hamiltonian of the battery.

Accepted in Quantum 2018-02-12, click title to verify 19

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 |Ψ

i

ihΨ

i

| = T(|ψ

i

ihψ

i

|)

and |Ψ

f

ihΨ

f

| = T(|φ

f

ihφ

f

|) and for convenience deﬁne

σ

i

= ρ ⊗ T (|ψ

i

ihψ

i

|) and

T(σ

f

) = ρ ⊗ T(|φ

f

ihφ

f

|) .

(43)

The factorisability condition, Eq. (42), entails that the Gibbs map acting on σ

i

and T(σ

f

), Eq. (43),

decomposes as

G

H

SB

(σ

i

) = γ

i

⊗ |φ

i

ihφ

i

| and

G

H

SB

(T(σ

f

)) = γ

f

⊗ |ψ

f

ihψ

f

|

(44)

where we have used the shorthand γ

k

≡ γ(H

k

S

) for k = i, f and with

|φ

i

i ∝ T exp



−

H

B

2k

B

T



|ψ

i

|ψ

f

i ∝ T exp



−

H

B

2k

B

T



|φ

f

i

(45)

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 (44), to convert a transition probability of the system

and battery to a transition probability of the battery alone. The forwards transition probability P (σ

f

|G

H

(σ

i

)),

Eq (32), of the system and battery reduces to P(φ

f

|φ

i

, γ

i

), the transition probability of the battery from |φ

i

to |φ

f

i for a system initialised in the thermal state γ

i

,

P (σ

f

|G

H

SB

(σ

i

)) = Tr

h

(ρ ⊗ |φ

f

ihφ

f

|) U G

H

SB

(ρ ⊗ T(|ψ

i

ihψ

i

|)) U

†

i

=

1

N

hφ

f

|Tr

S

h

U(γ

i

⊗ |φ

i

ihφ

i

|)U

†

i

|φ

f

i

:

=

1

N

P(φ

f

|φ

i

, γ

i

) .

(46)

(The factor

1

N

is a normalisation term from the maximally mixed system state ρ and cancels out in the ﬁnal

ﬂuctuation theorem.) Similarly, the reverse global transition probability P (T(σ

i

) |G

H

SB

(T(σ

f

)) reduces to

the transition probability of the battery for the reverse process P(ψ

i

|ψ

f

, γ

f

), in which the system is initialised

in the thermal state γ

f

,

P (T(σ

i

) |G

H

SB

(T(σ

f

)) = Tr

h

(ρ ⊗ |ψ

i

ihψ

i

|)U G

H

SB

(ρ ⊗ T(|φ

f

ihφ

f

|)) U

†

i

=

1

N

hψ

i

|Tr

S

h

U(γ

f

⊗ |ψ

f

ihψ

f

|)U

†

i

|ψ

i

:

=

1

N

P(ψ

i

|ψ

f

, γ

f

) .

(47)

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

decompose as

˜

Z

H

SB

(σ

f

) =

1

N

Z

f

×

˜

Z

φ

f

and

˜

Z

H

SB

(σ

i

) =

1

N

Z

i

×

˜

Z

ψ

i

,

(48)

Accepted in Quantum 2018-02-12, click title to verify 20

where we have used the shorthand Z

k

≡ Z

H

k

S

for k = i, f for the partition functions of the system and

˜

Z

φ

f

≡

˜

Z

H

B

(|φ

f

ihφ

f

|) and

˜

Z

ψ

i

≡

˜

Z

H

B

(T(|ψ

i

ihψ

i

|)) =

˜

Z

H

B

(|ψ

i

ihψ

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. (48), into global invariance, Eq. (34),

P(φ

f

|φ

i

, γ

i

)

P(ψ

i

|ψ

f

, γ

f

)

=

Z

f

Z

i

˜

Z

φ

f

˜

Z

ψ

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,

∆F

:

= F (H

f

S

) − F (H

i

S

) with

F (H)

:

= −k

B

T ln (Z

H

) ,

(50)

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

∆

˜

E

:

=

˜

E(ψ

i

) −

˜

E(φ

f

) with

˜

E(ψ)

:

= −k

B

T ln



˜

Z

ψ



= −k

B

T ln



hψ|exp



−

H

B

k

B

T



|ψi



.

(51)

The AQC can thus be stated as

P(φ

f

|φ

i

, γ

i

)

P(ψ

i

|ψ

f

, γ

f

)

= exp



−

∆F

k

B

T



exp

∆

˜

E

k

B

T

!

. (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

i

S

⊗ Π

i

B

+ H

f

B

⊗ Π

f

B

+ V

⊥

SB

+

S

⊗ H

B

, (53)

where Π

i

B

and Π

f

B

are projectors onto two orthogonal subspaces, R

i

and R

f

, of the battery’s Hilbert space,

and V

⊥

SB

has support only outside those two subspaces, i.e. (X

S

⊗ Π

i

B

)V

⊥

SB

= (X

S

⊗ Π

f

B

)V

⊥

SB

= 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

i

S

to H

f

S

. The reverse change in system

Hamiltonian H

f

S

to H

i

S

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 σ

i

and σ

f

, Eq. (43), when: (i) the battery

state T |ψ

i

i has support solely in R

i

and the battery state T |φ

f

i has support solely in R

f

, and (ii) the battery

Hamiltonian does not induce evolution between subregions, i.e. (

B

−Π

i

B

)H

B

Π

i

B

= 0 and (

B

−Π

f

B

)H

B

Π

f

B

=

0. Under these two restrictions it is straightforward to verify that

J

H

SB

(σ

i

) = exp

−

H

i

S

⊗ Π

i

B

+

S

⊗ H

B

2k

B

T

!

(ρ ⊗ T(|φ

f

ihφ

f

|)) exp

−

H

i

S

⊗ Π

i

B

+

S

⊗ H

B

2k

B

T

!

= J

H

i

S

(ρ ) ⊗ J

H

B

(T(|φ

f

ihφ

f

|))

(54)

and similarly for J

H

SB

(T(σ

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/~). Consequently, in practise H

B

must induce evolution between the

Accepted in Quantum 2018-02-12, click title to verify 21

subregions R

i

and R

f

and so (

B

− Π

i

B

)H

B

Π

i

B

6= 0 and (

B

− Π

f

B

)H

B

Π

f

B

6= 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 con-

sidering 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 the-

orems 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 |ψ

i

i and |φ

f

i it is possible to ensure that factorisability holds to a high degree of approximation,

i.e.

J

H

SB

(σ

i

) ' J

H

i

S

(ρ ) ⊗ J

H

B

(T(|ψ

i

ihψ

i

|)) and

J

H

SB

(T(σ

f

)) ' J

H

f

S

(ρ ) ⊗ J

H

B

(T(|φ

f

ihφ

f

|)) .

(55)

From which it follows that with an appropriate choice in the battery states |ψ

i

i and |φ

f

i the AQC also holds

to a high degree of approximation

P(φ

f

|φ

i

, γ

i

)

P(ψ

i

|ψ

f

, γ

f

)

' exp



−

∆F

k

B

T



exp

∆

˜

E

k

B

T

!

. (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, 

i

SB

and 

f

SB

, as



i

SB

:

= J

H

SB

(

S

⊗ |ψ

i

ihψ

i

|) − J

H

i

S

(

S

) ⊗ J

H

B

(|ψ

i

ihψ

i

|) and



f

SB

:

= J

H

SB

(

S

⊗ |φ

f

ihφ

f

|) − J

H

f

S

(

S

) ⊗ J

H

B

(|φ

f

ihφ

f

|).

(57)

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 σ

i

and σ

f

, Eq. (43),

˜

Z

H

SB

(ρ ⊗ |ψ

i

ihψ

i

|) P (ρ ⊗ |φ

f

ihφ

f

||G

H

SB

(ρ ⊗ T(|ψ

i

ihψ

i

|)))

=

˜

Z

H

SB

(ρ ⊗ |φ

f

ihφ

f

|) P (ρ ⊗ |ψ

i

ihψ

i

||G

H

SB

(ρ ⊗ T(|φ

f

ihφ

f

|))) .

(58)

and substitute in the deﬁnition of the factorisability errors 

i

SB

and 

f

SB

, Eq. 57,

Z

i

˜

Z

ψ

i

P(φ

f

|φ

i

, γ

i

) + Tr[(

S

⊗ |φ

f

ihφ

f

|)UT(

i

SB

)U

†

]

= Z

f

˜

Z

φ

f

P(ψ

i

|ψ

f

, γ

f

) + Tr[(

S

⊗ |ψ

i

ihψ

i

|)UT(

f

SB

)U

†

] ,

(59)

which on rearranging and using the triangle inequality gives

|Z

i

˜

Z

ψ

i

P(φ

f

|φ

i

, γ

i

) − Z

f

˜

Z

φ

f

P(ψ

i

|ψ

f

, γ

f

)|

≤ |Tr[(

S

⊗ |φ

f

ihφ

f

|)UT(

i

SB

)U

†

]| + |Tr[(

S

⊗ |ψ

i

ihψ

i

|)UT(

f

SB

)U

†

]| .

(60)

It follows from the Cauchy Schwartz identity that

Tr[(U

†

(

S

⊗ |φ

f

ihφ

f

|)U) 

i

SB

] ≤ ||T(

i

SB

)||

1

= ||

i

SB

||

1

and

Tr[(U

†

(

S

⊗ |ψ

i

ihψ

i

|)U) 

f

SB

] ≤ ||T(

f

SB

)||

1

= ||

f

SB

||

1

.

(61)

Thus we are left with the error-bounded AQC,

|Z

i

˜

Z

ψ

i

P(φ

f

|φ

i

, γ

i

) − Z

f

˜

Z

φ

f

P(ψ

i

|ψ

f

, γ

f

)| ≤ ||

f

SB

||

1

+ ||

i

SB

||

1

:

=  .

(62)

Accepted in Quantum 2018-02-12, click title to verify 22

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 6= exp(−iH

SB

t/~). 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

= |C

i

ihC

i

| ⊗ H

i

S

⊗

B

+ |C

f

ihC

f

| ⊗ H

f

S

⊗

B

+ V

⊥

CSB

+

C

⊗

S

⊗ H

B

(63)

where these control states are orthogonal, hC

f

|C

i

i = 0, and V

⊥

CSB

has orthogonal support to both |C

i

i and

|C

f

i, i.e. V

⊥

CSB

(|C

i

ihC

i

|⊗X

S

⊗X

B

) = V

⊥

CSB

(|C

f

ihC

f

|⊗X

S

⊗X

B

) = 0 for any system and battery operators

X

S

and X

B

. A change in system Hamiltonian H

i

S

to H

f

S

is induced by applying a unitary that evolves the

control from the state |C

i

i to |C

f

i. 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 |C

i

i, or the state |C

f

i, 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

i

S

= H

i

s

⊗ H

b

and H

f

S

= H

f

s

⊗ 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

i

s

⊗ H

b

⊗ Π

i

B

+ H

f

s

⊗ H

b

⊗ Π

f

B

+ V

⊥

sbB

+

sb

⊗ 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

P(φ

f

|φ

i

, γ

i

)

:

= hφ

f

|Tr

sb

h

U

sbB

(γ(H

f

s

) ⊗ γ(H

b

) ⊗ |φ

i

ihφ

i

|)U

†

sbB

i

|φ

f

i and

P(ψ

i

|ψ

f

, γ

f

)

:

= hψ

i

|Tr

sb

h

U

sbB

(γ(H

f

s

) ⊗ γ(H

b

) ⊗ |ψ

f

ihψ

f

|)U

†

sbB

i

|ψ

i

i .

(65)

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

˜

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

˜

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,

˜

E(ψ, H, T )

:

= −k

B

T ln



hψ|exp



−

H

k

B

T



|ψi



, (66)

Accepted in Quantum 2018-02-12, click title to verify 23

to enable us to discuss the properties of the function

˜

E more generally. An analysis of

˜

E provides some

physical support for our interpretation of ∆

˜

E as a quantum generalisation of the energy ﬂow between the

system and the battery. In particular, the following properties hold:

1.

˜

E(ψ, H + δ, T ) =

˜

E(ψ, H, T ) + δ.

2.

˜

E(ψ, λH, T ) = λ

˜

E(ψ, H, λT ).

3. For an energy eigenstate, |E

k

i,

˜

E is simply the associated eigenstate energy,

˜

E(E

k

, H, T ) = E

k

.

4. In the high temperature limit

˜

E tends to the average energy of the state, lim

T →∞

˜

E(ψ, H, T ) = hψ|H |ψi.

5. In the general case

˜

E is less than the average energy:

˜

E(ψ, H, T ) ≤ hψ|H |ψi.

6.

˜

E is independent of the the phase of the state

˜

E(ψ, H, T ) =

˜

E



ψe

iφ

, H, T



∀ φ ∈ <.

Properties 1 and 2 verify that

˜

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,

˜

E is constant in time which is a desirable condition for a statistical estimate of the energy of a

quantum state.

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

= ~ω



a

†

a +

1

2



=

P

n

~ω



n +

1

2



|nihn|, and transition probabilities between coherent states. The measured states |φ

f

i and

|ψ

i

i are set to the coherent states |α

f

i and |α

∗

i

i respectively,

|φ

f

i = |α

f

i and

|ψ

i

i = |α

∗

i

i ,

(67)

where |αi

:

= exp(− |α|

2

/2)

P

∞

n=0

α

n

n!

|ni. We calculate |φ

i

i and |ψ

f

i using the relation between states |φ

i,f

i

and |ψ

i,f

i as speciﬁed in Eq. (7). This requires calculating the eﬀect of applying the temperature dependent

operation exp



−

H

B

2k

B

T



= exp



−χ



a

†

a +

1

2



where χ =

~ω

2k

B

T

, to a coherent state. Using operator algebra

we ﬁnd that

|φ

i

i =

1

q

˜

Z

α

i

exp(−|α

i

|

2

/2)

X

n

exp



−χ



a

†

a +

1

2



α

n

i

n!

|ni = |α

i

exp (−χ)i and

|ψ

f

i = T





1

q

˜

Z

α

f

exp(−|α

f

|

2

/2)

X

n

exp



−χ



a

†

a +

1

2



α

n

f

n!

|ni





= |α

∗

f

exp (−χ)i .

(68)

Additionally, given the deﬁnition of ∆

˜

E in Eq. (9), it can be shown using operator algebra that

exp

∆

˜

E

k

B

T

!

=

˜

Z

α

f

˜

Z

α

i

=



exp

(

−|α

f

|

2

/2

)

exp(−|α

f

exp(−χ)|

2

/2)



2



exp(−|α

i

|

2

/2)

exp(−|α

i

exp(−χ)|

2

/2)



2

= exp



(|α

f

|

2

− |α

i

|

2

)(exp(−2χ) − 1)



.

(69)

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

AQC, Eq. (18).

Accepted in Quantum 2018-02-12, click title to verify 24

To compare the coherent state Crooks equality to the classical Crooks equality, Eq. 1, we rewrite ∆

˜

E in

terms of the diﬀerence between the average energy change of the battery in the forwards and reverse processes,

W

q

:

= (∆E

+

− ∆E

−

)/2, with

∆E

+

:

= hφ

i

|H

B

|φ

i

i − hφ

f

|H

B

|φ

f

i and

∆E

−

:

= hψ

f

|H

B

|ψ

f

i − hψ

i

|H

B

|ψ

i

i .

(70)

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 |ψ

i,f

i = |E

i,f

i then it follows from

Eq. (45) that |φ

i,f

i = |E

i,f

i and therefore

W

q

=

1

2

((E

i

− E

f

) − −(E

i

− E

f

)) ≡

1

2

(W − (−W )) = W (71)

where (E

i

− E

f

) ≡ 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 ∆E

+

6= −∆E

−

).

By calculating ∆E

+

and ∆E

−

for coherent states,

∆E

+

:

= (exp(−2χ)|α

i

|

2

− |α

f

|

2

)~ω (72)

∆E

−

:

= (exp(−2χ)|α

f

|

2

− |α

i

|

2

)~ω , (73)

we can rewrite ∆

˜

E for coherent states, Eq. 69, in terms of W

q

,

∆

˜

E = k

B

T (|α

i

|

2

− |α

f

|

2

)(1 − exp(−2χ)) =

1

χ

1 − exp(−2χ)

1 + exp(−2χ)

1

2

(∆E

+

− ∆E

−

) =

1

χ

1 − exp(−2χ)

1 + exp(−2χ)

W

q

.

(74)

Thus the coherent state AQC can be written as

P (α

f

|α

i

exp(−χ), γ

i

)

P



α

∗

i

|α

∗

f

exp(−χ), γ

f



= exp



−

∆F

k

B

T



exp



q (χ)

W

q

k

B

T



,

(75)

where we have deﬁned the quantum prefactor,

q (χ)

:

=

1

χ

1 − exp(−2χ)

1 + exp(−2χ)

=

1

χ

tanh(χ) . (76)

The quantum prefactor, Eq. (76), can be related to the average energy of a thermal battery hH

B

i

γ(H

B

)

:

=

Tr[H

B

γ(H

B

)]. We calculate the explicit algebraic form of hH

B

i

γ(H

B

)

and ﬁnd that

hH

B

i

γ(H

B

)

=

X

n

~ω(n + 1/2)

exp



−

~ω(n+

1

2

)

k

B

T



Z

H

B

=

1

2

~ω



1 + exp(−2χ)

1 − exp(−2χ)



=

k

B

T

q(χ)

.

(77)

Rearranging we are left with

q (χ) =

k

B

T

hHi

γ(H

B

)

:

=

k

B

T

~ω

T

. (78)

where we have implicitly deﬁned the ‘thermal frequency’ ω

T

as the average frequency of a harmonic oscillator

in a thermal state. The rewritten coherent state Crooks equality in the main text, Eq. (15), follows from

Eq. (75) and Eq. (78).

Accepted in Quantum 2018-02-12, click title to verify 25

Comment on classical limit. For the coherent state Crooks equality, the classical limit corresponds to

χ =

~ω

2k

B

T

→ 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, |Cati ∝ (|αi+ |βi). 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 |αi and |−αi. To derive a cat state AQC the measured states |φ

f

i

and |ψ

i

i are set to

|φ

f

i = |Cat

f

i

:

=

1

q

2 + 2<(exp(−

1

2

(|α

f

|

2

+ |β

f

|

2

− 2β

∗

f

α

f

)))

(|α

f

i + |β

f

i) and

|ψ

i

i = |Cat

∗

i

:

=

1

q

2 + 2<(exp(−

1

2

(|α

i

|

2

+ |β

i

|

2

− 2β

∗

i

α

i

)))

(|α

∗

i

i + |β

∗

i

i) ,

(79)

where the normalisation term is calculated using the fact that the overlap between two coherent states |αi

and |βi is given by hβ|αi = exp(−

1

2

(|α|

2

+ |β|

2

− 2β

∗

α)). The prepared states |φ

i

i and |ψ

f

i are calculated

using the relation between states |φ

i,f

i and |ψ

i,f

i as speciﬁed in Eq. (7). Using operator algebra we ﬁnd that

|φ

i

:

= exp



−χ



a

†

a +

1

2



|Cat

i

i ∝



exp



−χa

†

a



|α

i

i + exp



−χa

†

a



|β

i



=

1

q

˜

Z

Cat

i



η

|α

i

|

2

χ

|exp(−χ)α

i

i + η

|β

i

|

2

χ

|exp(−χ)β

i



and

|ψ

f

i =

1

q

˜

Z

Cat

f



η

|α

f

|

2

χ

|exp(−χ)α

∗

f

i + η

|β

f

|

2

χ

|exp(−χ)β

∗

f

i



(80)

where we have deﬁned

η

χ

:

= exp



−

1

2

(1 − exp(−2χ))



(81)

and the normalisation term

˜

Z

Cat

i

(and equivalently for

˜

Z

Cat

f

) takes the form

˜

Z

Cat

i

=

η

2|α

i

|

2

χ

+ η

2|β

i

|

2

χ

+ η

|α

i

|

2

+|β

i

|

2

χ

2<



exp



−

1

2

exp(−2χ)(|β

i

|

2

+ |α

i

|

2

− 2β

∗

i

α

i

)



2 + 2<(exp(−

1

2

(|α

i

|

2

+ |β

i

|

2

− 2β

∗

i

α

i

)))

.

(82)

The normalisation terms

˜

Z

Cat

i

and

˜

Z

Cat

f

also provide the exp



∆

˜

E

k

B

T



term. It follows from the deﬁnition of

∆

˜

E in Eq. (9) that

exp

∆

˜

E

k

B

T

!

=

hφ

f

|exp (−2χ) |φ

f

i

hψ

i

|exp (−2χ) |ψ

i

=

˜

Z

Cat

f

˜

Z

Cat

i

.

(83)

As such, the cat state AQC follows from Eq.(B12 -B16). (We do not state the explicit form of exp



∆

˜

E

k

B

T



in

terms of α

i,f

, β

i,f

, χ and η

χ

as the expression does not simplify.) The error-bounded cat state Crooks equality

follows from the error bounded autonomous Crooks equality, Eq. (62), with

˜

Z

i,f

deﬁned in Eq. (82).

Accepted in Quantum 2018-02-12, click title to verify 26

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 |φ

f

i and |ψ

i

i are set to the squeezed displaced

states |α

f

, r

f

i and |α

∗

i

, r

i

i respectively, i.e.

|φ

f

i = |α

f

, r

f

i = D(α

f

)S(r

f

) |0i and

|ψ

i

i = |α

∗

i

, r

i

i = D(α

∗

i

)S(r

i

) |0i ,

(84)

where we have introduced the displacement operator D(α) = exp(αa

†

− α

∗

a), the squeeze operator S(r) =

exp(

r

2

(a

2

− a

†

2

)), and for simplicity we assume from the outset that the squeezing parameters r

i

and r

f

are

real. We calculate |φ

i

i and |ψ

f

i using the relation between states |φ

i,f

i and |ψ

i,f

i as speciﬁed in Eq. (7). This

amounts to calculating the eﬀect of applying the operator exp



−χa

†

a



to a squeezed displaced state, i.e.

|φ

i

i =

1

q

˜

Z

α

i

,r

i

exp



−χ



a

†

a +

1

2



D(α

i

)S(r

i

) |0i ≡ D(µ

i

)S(t

i

) |0i = |µ

i

, t

i

i and

|ψ

f

∗

i =

1

q

˜

Z

α

f

,r

f

exp



−χ



a

†

a +

1

2



D(α

∗

f

)S(r

f

) |0i ≡ D(µ

∗

f

)S(t

f

) |0i = |µ

∗

f

, t

f

i .

(85)

We introduce µ

i,f

and t

i,f

to denote the displacement and squeezing parameters respectively of the prepared

states. (That the states |φ

i

i and |ψ

∗

f

i are also squeezed displaced states is a non-trivial result of the calcula-

tion.) To derive the AQC we will also need to calculate the exp



∆

˜

E

k

B

T



term. It follows from the deﬁnition of

∆

˜

E in Eq. (9) that this can be written as

exp

∆

˜

E

k

B

T

!

=

˜

Z

α

f

,r

f

˜

Z

α

i

,r

i

=

(exp(−χa

†

a)D(α

f

)S(r

f

) |0i)

†

exp(−χa

†

a)D(α

f

)S(r

f

) |0i

(exp(−χa

†

a)D(α

∗

i

)S(r

i

) |0i)

†

exp(−χa

†

a)D(α

∗

i

)S(r

i

) |0i

.

(86)

To derive the AQC we thus need to calculate µ, s and

˜

Z

α,r

.

B.3.1 Useful Identities

To calculate µ, s and N

α,r

we will make use of the following equalities:

1. exp(ma

†

a) exp(na

†

) exp(−ma

†

a) = exp(n exp(m)a

†

)

Proof. This is derived using the Taylor expansion of the exponential followed by the Hadamard lemma [69],

exp(ma

†

a) exp(na

†

) exp(−ma

†

a) =

X

k

(n exp(ma

†

a)a

†

exp(−ma

†

a))

k

k!

=

X

k

(na

†

exp(m))

k

k!

= exp(n exp(m)a

†

) .

2. exp(ma) exp(na

†

) = exp(mn) exp(na

†

) exp(ma)

Proof. This is derived using the Baker-Campbell-Hausdorﬀ formula [69] followed by the Zassenhaus formula [69],

exp(ma) exp(na

†

) = exp(ma + na

†

) exp



mn

2



= exp(na

†

) exp(ma) exp(mn) .

3. exp(ma

2

) exp(na

†

) = exp(mn

2

) exp(na

†

) exp(ma

2

) exp(2mna)

Proof. This is again derived using the Baker-Campbell-Hausdorﬀ formula followed by the Zassenhaus formula,

exp(ma

2

) exp(na

†

) = exp



mn

2

6



exp(ma

2

+ mna + na

†

) = exp(mn

2

) exp(na

†

) exp(ma

2

) exp(2mna)

Accepted in Quantum 2018-02-12, click title to verify 27

4. exp



1

2

(ma

2

+ na

†

2

)



=

1

p

cos

(

√

mn

)

exp



1

2

p

n

m

tan(

√

mn)a

†

2



exp



−ln (cos(

√

mn)) a

†

a



exp



1

2

p

m

n

tan(

√

mn)a

2



Proof. To derive this equality it is helpful to ﬁrst introduce the operators K

+

:

=

1

2

a

†

2

, K

−

:

=

1

2

a

2

, K

z

:

=

1

4

(aa

†

+ a

†

a). These operators obey the commutation relations [K

+

, K

−

] = −2K

z

and [K

z

, K

±

] = ±K

±

. Next,

we introduce the ansatz

f

:

= exp((mK

+

+ nK

−

)t) = exp(pK

+

) exp(qK

z

) exp(sK

−

) .

(87)

To ﬁnd the coeﬃcients p, q and s we diﬀerentiate f with respect to t,

f

0

= (mK

+

+ nK

−

)f = (p

0

K

+

+ q

0

exp(pK

+

)K

z

exp(−pK

+

) + s

0

exp(pK

+

) exp(qK

z

)K

−

exp(−qK

z

) exp(−pK

+

)) f.

(88)

To simplify this expression we use the Hadamard lemma to obtain the following relations,

exp(pK

+

)K

z

exp(−pK

+

) = K

z

− pK

+

and

exp(pK

+

) exp(qK

z

)K

−

exp(−qK

z

) exp(−pK

+

) = exp(pK

+

) exp(−q)K

−

exp(−pK

+

) = exp(−q)(K

−

− 2pK

z

+ p

2

K

+

) .

(89)

Eq. (89) can be substituted into the right hand side of Eq. (88), from which it then follows that

mK

+

+ nK

−

= p

0

K

+

+ q

0

(K

z

+ pK

+

) + s

0

exp(−q)(K

−

− 2pK

z

+ p

2

K

+

) .

(90)

Equating coeﬃcients in Eq. (90) we obtain a set of diﬀerential equations,

p

0

− q

0

p + s

0

exp(−q)p

2

= m

q

0

− 2ps

0

exp(−q) = 0

s

0

exp(−q) = n ,

which can be solved, subject to the constraint that p(0) = q(0) = s(0) = 0, to ﬁnd that

p =

m

n

tan



√

mnt



q = −2ln



cos



√

mnt



s =

n

m

tan



√

mnt



.

(91)

Identity 4 is obtained by substituting this solution, Eq. (91), back into the ansatz, Eq. (87).

5. exp



1

2



ma

2

+ na

†

2



=

p

cos(

√

mn) exp



1

2

p

m

n

tan(

√

mn)a

2



exp



ln (cos(

√

mn)) a

†

a



exp



1

2

p

n

m

tan(

√

mn)a

†

2



Proof. The proof here follows the same method as for identity 4. Using the ansatz f = exp((mK

+

+ nK

−

)t) =

exp(pK

−

) exp(qK

z

) exp(sK

+

) it is possible to derive the same diﬀerential equations as in Eq. (91) but with

q → −q.

6. exp



ma

2



exp



na

†

2



=

1

√

1−4mn

exp



n

1−4mn

a

†

2



exp



ln



1

1−4mn

a

†

a



exp



m

1−4mn

a

2



Proof. This is proven using identities 4 and 5. Let p =

1

2

p

m

n

tan(

√

mn), q =

1

2

p

n

m

tan(

√

mn) and exp(−2M) =

cos(

√

mn)

2

=

1

1−4pq

, it follows that

√

M exp(pa

2

) exp(ln(M )a

†

a) exp(qa

†

2

) =

1

√

M

exp(qa

†

2

) exp(−ln(M )a

†

a) exp(pa

2

) ,

which can be rearranged to give

exp



−pa

2



exp



qa

†

2



= M exp(ln(Ma

†

a) exp(qa

†

2

) exp(−pa

2

) exp(ln(M )a

†

a)

= M exp(q exp(−2M)a

†

2

) exp(2ln(M a

†

a) exp(−p exp(−2M )a

2

) .

The substitution p ↔ −p then gives identity 6.

Accepted in Quantum 2018-02-12, click title to verify 28

B.3.2 Main Derivation

We are now in a position to calculate the eﬀect of the operator exp



−χa

†

a



on a squeezed displaced state,

exp



−χa

†

a



D(α)S(r) |0i = exp



−χa

†

a



exp(αa

†

− α

∗

a) exp



r

2



a

2

− a

†

2





|0i . (92)

We ﬁrst use the standard factorised form of the displacement and squeeze operators [70],

D(α)S(r) |0i =

exp



−

1

2

|α|

2



p

cosh(r)

exp



αa

†



exp (−α

∗

a) exp



−

1

2

tanh(r)a

†

2



|0i (93)

to rewrite Eq. (92),

= exp



−χa

†

a



exp



−

1

2

|α|

2



p

cosh(r)

exp



αa

†



exp (−α

∗

a) exp



−

1

2

tanh(r)a

†

2



|0i .

(94)

Eq. (94) is then rewritten using identity 3,

=

exp



−

1

2

|α|

2



p

cosh(r)

exp



−χa

†

a



exp



αa

†



exp



−

1

2

tanh(r)α

∗

2



exp



α

∗

tanh(r)a

†



exp



−

1

2

tanh(r)a

†

2



|0i ,

(95)

followed by identity 1,

=

exp(−

1

2

(|α|

2

− α

∗

2

tanh(r)))

p

cosh(r)

exp



exp (−χ) (α + α

∗

tanh(r))a

†



exp



−

1

2

exp (−2χ) tanh(r)a

†

2



|0i .

(96)

By comparing Eq. (96) with itself in the limit that χ = 0, i.e. the limit in which exp



−χa

†

a



is not applied

and Eq. (96) is simply a rewritten version of a squeezed displaced state, we identify the following modiﬁed

displacement and squeeze coeﬃcients µ and s,

<(µ) = <(α)

exp (−χ) (1 + tanh(r))

1 + exp (−2χ) tanh(r)

,

=(µ) = =(α)

exp (−χ) (1 − tanh(r)

1 − exp (−2χ) tanh(r))

and

tanh(s) = exp (−2χ) tanh(r) .

(97)

It remains to calculate the exp



∆

˜

E

k

B

T



term. This amounts to calculating,

˜

Z

α,r

=



exp



−χa

†

a



D(α)S(r) |0i



†

exp



−χa

†

a



D(α)S(r) |0i

(98)

The above equation can be immediately simpliﬁed using Eq. (96)

=

exp



−(|α|

2

− <(α)

2

tanh(r))



cosh(r)

h0|exp



ma

2



exp (na) exp



n

∗

a

†



exp



m

∗

a

†

2



|0i

(99)

where

m = −

1

2

exp (−2χ) tanh(r) and

n = exp (−χ) (α

∗

+ α tanh(r)) .

(100)

Accepted in Quantum 2018-02-12, click title to verify 29

Using ‘Useful Identities’ 6, 3, 1, and 2 in succession this can be rewritten to give,

h0|exp



ma

2



exp (na) exp



m

∗

a

†



exp



n

∗

a

†



|0i

=

1

r

h0|exp (na) exp



m

∗

exp(−2r)

a

†

2



exp



−ln (exp(−2r)) a

†

a



exp



m

exp(−2M)

a

2



exp



n

∗

a

†



|0i

=

1

M

exp

2<(n

2

m)

exp(−2M)

!

h0|exp (na) exp(−ln(exp(−2M))a

†

a) exp



n

∗

a

†



|0i

=

1

M

exp

2<(n

2

m)

exp(−2M)

!

h0|exp



n

M

a



exp



n

∗

M

a

†



|0i

=

1

M

exp

2<(n

2

m)

exp(−2M)

!

exp

|n|

2

exp(−2M)

!

(101)

where M =

1

√

1−4|m|

2

. It follows that,

˜

Z

α,r

:

=

exp(−(|α|

2

− <(α

2

) tanh(r)))

cosh(r)

p

1 − tanh(r)

2

exp (−4χ)

exp

exp (−2χ) (|α|

2

(1 + tanh(r)

2

) + 2 tanh(r)<(α

2

)) − exp (−2χ) tanh(r)



<(α

2

) + <(α

2

) tanh(r)

2

+ 2|α|

2

tanh(r)



1 − tanh(r)

2

exp (−4χ)

!

,

(102)

which can be simpliﬁed if we assume that the displacement parameters are real, α

i,f

= <(α

i,f

),

˜

Z

α,r

:

=

exp



−|α|

2

(1 − tanh(r)



cosh(r)

p

1 − tanh(r)

2

exp (−4χ)

exp

|α|

2

(1 + tanh(r))

2

(1 − tanh(r) exp (−2χ)) exp (−2χ)

1 − tanh(r)

2

exp (−4χ)

!

.

(103)

This completes the derivation of the squeezed state AQC. The error-bounded squeezed state Crooks equality

follows from the error bounded AQC, Eq. (18).

C Dynamics of proposal

Analysis of the dynamics of the harmonic oscillator state under the proposed interaction, Eq. (23), veriﬁes

that the oscillator behaves as a battery. The interaction is diagonal in the system energy eigenbasis and thus

the total Hamiltonian can be written in the form

H

SB

= |eihe|⊗ H

e

B

+ |gihg| ⊗ H

g

B

, (104)

where H

e

B

= H

B

+ E(x

B

) and H

g

B

= H

B

− E(x

B

). Consequently, H

SB

does not induce transitions between

the system energy levels. At the start of the forwards protocol the two-level system and oscillator are prepared

in the state

ρ

SB

(0) = (p

e

|eihe|+ p

g

|gihg|) ⊗ |φ

i

ihφ

i

| , (105)

where

p

e

p

g

= exp



−2E

i

k

B

T



. The system and battery then evolve under H

SB

, for some time t, to the state

ρ

SB

(t) = p

e

|eihe|⊗ exp(−itH

e

B

/~) |φ

i

ihφ

i

|exp(itH

e

B

/~) + p

g

|gihg| ⊗ exp(−itH

g

B

/~) |φ

i

ihφ

i

|exp(itH

g

B

/~).

(106)

As such the oscillator state evolves into two non-equally weighted components, |φ

e

i

(t)i

:

= exp(−itH

e

B

/~) |φ

i

and |φ

g

i

(t)i

:

= exp(−itH

g

B

/~) |φ

i

i, correlated with the system being prepared in the excited and ground state

respectively, i.e.

ρ

SB

(t) = p

e

|eihe|⊗ |φ

e

B

(t)ihφ

e

B

(t)| + p

g

|gihg| ⊗ |φ

g

B

(t)ihφ

g

B

(t)| .

(107)

The oscillator state in the forwards protocol is prepared such that its average position is initially in the

region of narrow splitting, i.e. hφ

i

|x

B

|φ

i

i < x

i

. In this case |φ

e

B

(t)ihφ

e

B

(t)| travels up a potential hill and by

energy conservation its average energy decreases by an amount E

f

−E

i

as it travels from x

i

to x

f

. Conversely,

|φ

g

B

(t)ihφ

g

B

(t)| travels down a potential hill and its average energy increases by an amount E

f

− E

i

.

Accepted in Quantum 2018-02-12, click title to verify 30

(a) Initial State (b) Evolved State

Figure 8: These are plots of the Wigner function of the a) initial oscillator state and b) ﬁnal oscillator state after evolution

under our proposed Hamiltonian. In these plots the following parameters are chosen somewhat arbitrarily: m = 1, ~ω = 1,

E

i

= 1, E

f

= 21 and the interaction region extends from x

i

= −2 to x

f

= 2. The oscillator starts in a coherent state

centered around x = −9. These plots correspond to the case in which the system is prepared in the excited state and so

show the component of the wavepacket that travels up the potential hill: |φ

e

i

(t)i

:

= exp(−iH

e

B

t) |exp(−χ)αi. The oscillator

wavepacket is slowed as it evolves through the potential hill and the interaction squeezes the coherent state and leave small

residual ripples trailing behind the bulk of the Wigner function. The Wigner function is negative in the troughs of these

ripples.

Numerical simulations reveal that the oscillator state wavepacket is distorted as it evolves through the

interaction region. In Fig. 8 we see that a coherent oscillator states is squeezed and interference ‘ripples’ are

generated by the interaction. The Wigner function of these ‘ripples’ is in places negative. Negative values of

a quasi probability distribution are hard to explain classically so this is a signature of quantum mechanical

phenomena. Furthermore, numerical simulations indicate that when the ﬁnal state is approximated by a

coherent state with the same average position and momentum as the complete ﬁnal wavepacket the error-

bounded coherent state AQC does not always hold. As such, we see that even when the battery is prepared in

a coherent state, the most classical of the motional quantum states, it does not remain in this approximately

classical state. Moreover, the quantum distortion eﬀects are essential to the coherent state AQC being obeyed.

Accepted in Quantum 2018-02-12, click title to verify 31

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