Operational models of temperature superpositions

Carolyn E. Wood,1 Harshit Verma,1 Fabio Costa,1 and Magdalena Zych1

1Australian Research Council Centre for Engineered Quantum Systems,School of Mathematics and Physics, The University of Queensland, St Lucia, QLD 4072, Australia

When thermodynamical quantities are associated with quantum systems a question arises howto treat scenarios where the notion of temperature could exhibit some quantum features. It isknown that the temperature of a gas in thermal equilibrium is not constant in a gravitationalfield, but it is not known how a delocalised quantum system would thermalise with such a bath.In this theoretical work we demonstrate two scenarios in which the notion of a ‘superposition oftemperatures’ arises. First: a probe interacting with different baths dependent on the state ofanother quantum system (control). Second: the probe interacting with a bath in a superposition ofpurified states, each associated with a different temperature. We show that these two scenarios arefundamentally different and can be operationally distinguished. Moreover, we show that the probedoes not in general thermalise even when the involved temperatures of the baths or purificationsare equal. Furthermore, we show the final probe state depends on the specific realisation of thethermalising channels, being sensitive to the particular Kraus representations of the channels. Thispoint appears to explain recent results obtained in the context of quantum interference of relativisticparticle detectors thermalising with Unruh or Hawking radiation. Finally, we show that these resultsare reproduced in partial and pre-thermalisation processes, and thus our approach and conclusionsalso generally apply beyond the idealised scenarios, where thermalisation is not exact.

I. INTRODUCTION

Temperature is well understood as a classical, macro-scopic property of the average kinetic energy of an ensem-ble of particles. In quantum mechanics it features as a pa-rameter defining Gibbs states of a system. Understand-ing the implications of quantum theory for the notion oftemperature is of foundational interest for understandingquantum thermodynamic processes (e.g. heat engines)and the laws of thermodynamics at the intersection withquantum theory [1]. Thus far the main focus has been onthe implications of using quantum systems for tempera-ture sensing (as thermometers) where both limitationsand advantages were identified [2]; on the complemen-tarity relation between temperature and energy fluctua-tions [3], which was proven to hold universally even forstrongly interacting quantum systems; and an argumentwas even laid down for temperature to be described byan operator in quantum theory [4]. Recent work has alsocombined indefinite causal order and quantum thermody-namics, exploring coherent quantum control over the or-der of application of quantum channels [5–9], with claimsthey offer advantages in engine/cycle performances, er-gotropy, etc.

However, there are scenarios arising at the intersec-tion of quantum theory and general and special relativitywhich call for an understanding of the quantum aspectsof temperature and thermalisation processes, and whichare not addressed by approaches considered thus far.

For example, seminal works of Tolman and Ehren-fest [10, 11] showed that in a column of gas in ther-mal equilibrium in a gravitational field, locally mea-sured temperature actually varies with the position inthe gas, which can be attributed to the gravitationalredshift/time dilation. This effect raises a question ofhow a quantum system thermalises with such a bath: A

quantum system cannot be infinitely well-localised andso quantum theory and general relativity together implythat, fundamentally, any system used as a thermometerwill be coherently spread over baths with different lo-cal temperatures. How the thermalisation process looksin such scenarios and whether full thermalisation can beachieved are open questions.

The notion of temperature will also require a quantumdescription in the context of thermalisation arising frommatter coupled to radiation, such as in the Unruh andHawking effects—where internal states of a probe system(called an Unruh-deWitt detector) in non-inertial motionthermalise to a temperature given by the detector’s ac-celeration. Again taking into account that classical tra-jectories are just an approximation, fundamentally onedeals with a system which is coherently spread over tra-jectories with different local accelerations each leading toa different thermal state of the probe.

The scenarios above call for a sharper understandingof thermal processes involving quantum systems, wheretemperature does not take a fixed, classical value. Inother words, the question arises whether temperature canbe ‘in superposition’ in some appropriate sense. Apartfrom the foundational motivations, the question is alsorelevant in the context of applied quantum technologies,where quantum-coherent systems can interact with ther-mal environments in a non-trivial way.

In this work, we consider operational scenarios where asystem is subject to a thermalisation process, but wherethe degrees of freedom responsible for, or subject to, ther-malisation have some form of coherence. We identify twogeneral scenarios that embody different intuitions of ‘su-perposition of temperatures’. The first is best visualisedas a probe system (or ‘thermometer’) prepared in a su-perposition of different locations, where each location in-teracts with an independent thermal bath. In the second

arX

iv:2

112.

0786

0v1

[qu

ant-

ph]

15

Dec

202

1

2

scenario there is a single bath, which is itself in a superpo-sition of states corresponding to different temperatures.

We find that, in both scenarios, the probe system canretain some degree of coherence after thermalisation, con-forming with the notion of ‘superposition’. Surprisingly,we find that, even if the two temperatures involved arethe same, the probe does not necessarily end up in athermal state. Despite several similarities, we find thatthe two scenarios present measurable differences. Thisprovides a route to further insight into the physics ofparticular scenarios through the appropriate choice of aquantum thermal probe. For example, we show that ourapproach can explain some counter-intuitive effects thatarise from Unruh-DeWitt detectors in quantum states ofmotion [12–14].

Our results are first discussed under the usual idealisa-tion where a unitary interaction between the probe anda thermal bath results in a thermal state of the probe atthe same temperature as that of the bath. In order toaddress the question of how much this idealisation im-pacts our results, we also study scenarios of partial orpre-thermalisation by introducing a model of thermali-sation as sequential interactions between the probe andthe bath(s), where the number of interactions controlsthe degree of thermalisation of the probe. Such a pre-thermalisation regime is a more realistic scenario of aquantum system far from equilibrium interacting withsome environment for a finite time. Crucially, we findthat the key features of the behaviour of the probe andcoherence between temperatures are reproduced in thepre-thermalisation regime and thus our results do notdepend on the probe reaching full thermalisation.

II. THERMALISATION AS A CHANNEL ANDPURIFICATIONS OF THERMAL STATES

We are considering scenarios where a probe systeminteracts with some environment(s). Our results willfrequently be general, not restricted to a particular in-teraction or state of the environment, but since ourmain focus is the process of thermalisation we refer tothe environment hereafter as a bath. We consider sce-narios where the probe and the bath begin in a prod-uct state ρin = ρB ⊗ ρS and evolve to a final state

ρout = UBSρinU†BS , where UBS is a unitary operator.

For a particular case of full thermalisation, the initialbath state and the final probe state are each Gibbs states

at the same temperature, i.e. ρβB(S) = e−βHB(S)

Zβ , where

β = 1/kBT is the inverse temperature with kB the Boltz-mann constant and T the temperature. The partition

function is Zβ = Tr e−βHB(S) , and HB(S) is the Hamil-tonian of the bath (probe) system.

The state of the probe post interaction is conve-niently expressed in the operator-sum (Kraus) represen-

tation [15]

E(ρS) = TrB

{UBSρinU

†BS

}≡∑k,l

cβlMklρSM†kl (1)

where we choose for the bath a basis where ρB is diagonal

ρB =∑l cβl |l〉 〈l| and the Kraus operators for the process

E(·) arising from UBS read Mkl = 〈k|UBS |l〉.The Kraus represention of any given process is not

unique: Any M ′kl =∑sMsluks, where uks is an isometry,

is another Kraus representiation of E(·). This isometrycan always be extended to a unitary matrix, which canthen be interpreted as a local unitary on the bath—wethus refer to it hereafter as a local bath unitary.

We will frequently refer to the purification of a mixedstate, in particular for thermal states of a bath. For athermal state ρβ , such a purified state can be written inthe energy eigenbasis as follows:

|θβ〉 =∑n

√cβn |n, n〉 , (2)

where cβn := e−Enβ

Zβ with En, n = 0, 1, ... the spec-trum of the bath Hamiltonian, and the term m in |n,m〉refers to an ancillary system required for the purifica-tion. Such purifications are known as thermofield doublestates in the context of finite-temperature quantum fieldtheory [16], and are frequently used to represent the de-composition of a pure global state of a relativistic quan-tum field into local modes in the presence of a horizonor into modes associated with non-inertial coordinates,such as Rindler modes for uniform acceleration [17].

For modelling the process of pre-thermalisationthrough a unitary interaction between the bath and theprobe, we resort to the theory of thermal attenuators(TA) [18]. In general, TAs represent the effect of a (ther-mal) lossy environment acting on a probe. In the mostgeneric case, this environment could be constituted bybosonic degrees of freedom in a thermal state. However,for studying pre-thermalisation it will be sufficient forus to consider a qubit probe interacting with an envi-ronment also constituted by qubit(s). In this case, aTA channel is a generalized amplitude damping channel(GADC) on the probe.

The unitary interaction between the probe and bathwhich yields a GADC is of the form:

UηBS =

1 0 0 00√

1− η √η 0

0 −√η√

1− η 00 0 0 1

, (3)

where η parametrises the strength of the interaction. Theaction of a GADC on a probe can also be expressed inthe form of Kraus operators, Eq. (1). We note that inthat case the purified bath state will consist of two qubits(one of them being the ancilla for purification). Such apurified state of the bath, constituted by qubits, has theform of Eq. (2) with n = 0, 1 and Zβ = e−E0β + e−E1β .

3

The unitary interaction between the (purified) bath andthe probe is then given as follows:

UBS = UηBS ⊗ I , (4)

where UηBS acts on the probe and the bath qubit (and thefull interaction is an identity on the purification ancilla).

III. TWO-BATH SCENARIO

A. Quantum-controlled channels

The first scenario in which we consider ‘superpositionof temperatures’ consists of an interaction between theprobe and a composite bath comprising two subsystemsin a product state ρB = ρB0

⊗ρB1. This can arise, e.g., if

the subsystems are separated enough to be thermally iso-lated from each other at least over the time-scale of theinteraction with the probe. Next, we introduce a quan-tised control degree of freedom (DoF) which will dictatethe subsystem with which the probe thermalises. Theprobe, the baths and the control system are thus ini-tially in a product state ρ = ρB ⊗ ρC ⊗ ρS , where ρCis the state of the control. We generalize the interactionEq. (1) between the probe and the baths to also includethe control DoF, which gives

U = UB0S ⊗ IB1⊗ |0〉 〈0|C + IB0

⊗ UB1S ⊗ |1〉 〈1|C , (5)

where the control states |i〉C , i = 0, 1 are orthonormal.We refer to this as a quantum-controlled interaction asEq. (5) entails that, depending on the state |i〉C of thecontrol system, the probe interacts with the correspond-ing bath Bi through the unitary UBiS .

An example physical realisation of such a scenario isa Mach-Zehnder interferometer, Fig. 1, where the twostates |i〉C of the control are identified with the two pathsthrough the interferometer. The probe is then an inter-nal DoF of the interfering system (e.g. of a particle) andthermalises with one of the bath subsystems, B0 or B1,depending on the path taken through the setup.

Taking the initial state of the control to be ρC =|+〉 〈+|, with |+〉 = 1√

2(|0〉 + |1〉) and measuring it

in the superposition |φ〉 = 1√2

(|0〉+ eiφ |1〉

), where

the relative phase φ is assumed to be fully control-lable, the conditional output state of the probe ρS(φ) =TrBC

{Uρ(0)U† |φ〉 〈φ|

}reads:

ρS(φ) =1

4TrB

{UB0SρBSU

†B0S

+ UB1SρBSU†B1S

+eiφUB0SρBSU†B1S

+ e−iφUB1SρBSU†B0S

}. (6)

The first two terms in Eq. (6) are each a channel aris-ing from the unitary interaction UBiS between the sys-tem and bath Bi, and have the usual Kraus operatorform as in Eq. (1). The other two are “cross-terms” be-

tween channels and take the form TrB{UB0SρBSU

†B1S

}=

FIG. 1. A Mach-Zehnder interferometer for the two-bath case.Input probe state enters from the left, is placed in a superposi-tion, and travels along the two arms of the interferometer, oneach of which is a bath thermalised to some temperature. Thepaths recombine at the second beamsplitter, and the outputstate is then detected at D+, D−.

∑l cβ0

l M0llρS

∑l′ c

β1

l′ M1†l′l′ . Hence the final state has the

following representation in terms of Kraus operators:

ρS(φ) =1

4

∑k,l

(cβ0

l M0klρSM

0†kl + cβ1

l M1klρSM

1†kl

eiφcβ0

k cβ1

l M0kkρSM

1†ll + e−iφcβ1

k cβ0

l M1kkρSM

0†ll

). (7)

Unlike the first two terms, the cross terms are not invari-ant under different Kraus representations and thus theresulting process is sensitive to the actual Kraus decom-position of the channels [19].

B. Quantum-controlled thermalisation

To investigate quantum coherence of temperaturesarising from the above approach we now consider specif-ically thermalisation channels in Eqs (6), (7). Notably,due to the dependence of the full process on the Kraus de-composition of the individual channels, one cannot sim-ply pick a particular Kraus decomposition without losinggenerality. However, since any two representations of agiven channel are related by the local bath unitaries in-troduced in Sec. II (see e.g. Theorem 8.2 in [20]) we firstfind a particular probe-bath interaction leading to a spe-cific Kraus representation of a thermalisation channel,and then recover a fully general treatment by includingarbitrary unitary transformations on the bath subsys-tems B0 and B1.

We thus begin by finding a unitary operator UBiS thatdescribes thermalisation of the probe with bath Bi, re-

quiring that TrBi{UBiSρBiSU

†BiS

}= ρβiS for an arbitrary

initial probe state and bath in a thermal state: ρBiS =

ρS ⊗ ρβiBi . A unitary which yields this result is, for exam-

4

ple

UBiS =∑k,l

|k〉S 〈l|S |l〉B 〈k|B , (8)

assuming spec{HS} = spec{HBi}. Associated Kraus op-erators read Mkl = |l〉S 〈k|S .

The final probe state, Eq. (7), for baths at inverse tem-peratures β0, β1 and the probe-bath interaction given byEq. (8) can now be written compactly as

ρS(φ) =1

4

(ρβ0

S + ρβ1

S + eiφρβ0

S ρSρβ1

S + e−iφρβ1

S ρSρβ0

S

),

(9)

where ρβiS =∑k c

βik |k〉S 〈k|S is a thermal state of the

probe under the above assumption about the bath andprobe spectra.

Now introducing the arbitrary unitary matrices ui,with matrix elements uikl, we obtain new Kraus opera-

tors M ′ikl =

∑s u

iks |l〉S 〈s|S , notably different for each

of the two bath subsystems. This yields a final state ofthe probe subject to a quantum superposition of ther-malisations,

ρS(φ) =1

4

(ρβ0

S + ρβ1

S +eiφρβ0

S u0ρSu

1†ρβ1

S

+ e−iφρβ1

S u1ρSu

0†ρβ0

S

). (10)

The final probe state consists of two terms which arethermal states at the temperature of the respective baths(as expected, these terms are insensitive to any local bathdynamics as they result from the usual action of individ-ual thermalisation channels), as well as the two cross-terms each of which depend on the states of both baths,the initial state of the system, and the local bath uni-taries. Such a form of the final state has crucial implica-tions for the visibility, as we will discuss in the followingsection.

One might have thought that if both baths are at thesame temperature, the system simply thermalises to thattemperature. This is, however, not the case, as explicitlyseen from Eq. (10) for β0 = β1 ≡ β.

Indeed, for the final state to be thermal we need

ρβSu1ρSu

0†ρβS = eiαρβS , where α ∈ R is a possible addi-tional phase. Taking matrix elements in the energy eigen-

basis, the above requires eiαcβkδk,l = cβkcβl 〈k|u1ρSu

0† |l〉,for all k, l. However, due to the non-negativity of the

thermal weights cβk , state normalisation, and the unitar-ity of ui, the modulus of the right-hand-side is strictly

less than cβk . That is, unless cβk = δk,0 (indicating bothbaths are at zero temperature), the initial probe state ρSis pure, and the unitaries ui rotate the probe state to theenergy ground state and are thus equal (up to a globalphase α).

The following section will also show how the aboveresult can be understood through complementarity be-tween quantum coherence and which-path information

using the analogy with the Mach-Zehnder interferometerin Fig. 1.

C. Interferometric visibility of the control DoF

In an interferometric scenario, like in Fig. 1, the vis-ibility of the final interference pattern contains relevantinformation about the dynamics and correlations devel-oped between the involved systems. In this section, wethus derive and discuss properties of the visibility asso-ciated with the present (two-bath) scenario.

Recall that visibility is quantified by the magnitude ofthe off-diagonal elements of the final state, here of thecontrol [21, 22]. Equivalently, it is the contrast of theinterference pattern obtained by measuring the controlin a superpostion basis. The probability of measuringthe control in the state |φ〉 is P(φ) = TrρS(φ), explicitly

P(φ) =1

2+

1

2

∣∣∣Tr{u0†ρβ0

S ρβ1

S u1ρS

}∣∣∣ cos(φ+ ψ), (11)

where ψ is a phase defined via TrS{u0†ρβ0

S ρβ1

S u1ρS} ≡

|TrS{u0†ρβ0

S ρβ1

S u1ρS}|e−iψ, and so the visibility is

V =∣∣∣Tr{u0†ρβ0

S ρβ1

S u1ρS

}∣∣∣ . (12)

Crucially, it depends not just on the temperatures of thetwo baths, but also the local unitaries.

Note also that the visibility can never reach its max-imum value of 1, even if β0 = β1, except in the specialcase already discussed in the preceding section, i.e. whereboth baths are at zero temperature, the system is in apure state and the unitaries ui rotate it to the groundstate.

This directly shows that the interactions between theprobe and the baths in general influence the coherenceof the control even for equal bath temperatures. Thisis because the final state of either B0 or B1 depends onthe state of the probe—which itself depends on the stateof the control (in the Mach-Zehnder analogy in Fig. 1, itdepends on the path through the interferometer). Hence,even for identical local unitaries and the same bath tem-peratures, the bath subsystems still store which-way in-formation [22] about the control, except in the aforemen-tioned special case where all systems are in their groundstates.

Moreover, for arbitrary fixed temperatures βi thereexist ui which yield V = 0. Indeed, decomposing theprobe state in its eigenbasis ρS =

∑s ps |s〉 〈s|, where

0 ≤ ps ≤ 1,∑s ps = 1, Eq. (12) reads

V =

∣∣∣∣∣∑s

ps∑k

cβ0

k cβ1

k 〈s|u0† |k〉 〈k|u1 |s〉

∣∣∣∣∣ . (13)

Taking u0 to map each state |s〉 to some energy eigen-state, and u1 to be u0 times a cyclic permutation of

5

energy eigenstates, means that if 〈k|u0 |s〉 = 1 then〈k|u1 |s〉 = 0 and vice versa, resulting in V = 0.

The above means that the minimum visibility over thedifferent physical realisations of the thermalisation chan-nel is zero. We have also seen that the coherence is ingeneral not maximal. The question is therefore: what isthe maximum of the visibility? To answer this questionwe find the maximum of V in Eq. (13) over local bathunitaries ui for arbitrary but fixed temperatures of thebaths and the probe initial state.

We let 〈s|u0† |k〉 = α0k(s) and 〈k|u1 |s〉 = α1

k(s) and

because cβik as well as ps are non-negative, for maximumvisibility, α0

k(s) and α1k(s) must also be real and non-

negative (up to a global phase). Hence Eq. (13) becomes

V =∑s ps

∑k c

β0

k cβ1

k α0k(s)α1

k(s).Starting with a pure state ρS = |s〉 〈s|, using the La-

grange multiplier method (see Appendix for full deriva-tion), the maximum visibility is found to be Vpure

max =

cβ0

0 cβ1

0 . The intuition here is that visibility is maximisedwhen local unitaries effectively rotate the probe state |s〉to the energy ground state (up to a phase), as it has thehighest overlap with a thermal state at any finite temper-ature. Thus for an arbitrary mixed state the maximumvisibility reads

Vmax =∑s

pscβ0s c

β1s , (14)

where without loss of generality the probabilities ps areordered decreasingly, ps ≥ ps+1. This is because thermal

weights are likewise ordered decreasingly, i.e. cβik > cβik+1and due to the orthogonality of the states |s〉 a unitaryui can only rotate one of the states from an orthogonalset to the energy ground state, but the same strategy canbe iterated to find a unitary mapping of the eigenstatesof ρS to progressively higher energy eigenstates.

IV. ONE-BATH SCENARIO

The previous case used quantum control of thermalchannels associated with two baths, treated as two dif-ferent systems, as an operational model of a temperaturesuperposition. Here, we explore a different model wherethe bath is a single system which, in the sense explainedbelow, can itself exhibit superposition of temperatures;see Fig. 2. The procedure has some similarity with the“superposition of thermal states” in optics, considered inRef. [23], although no superpositions of different temper-atures were considered therein.

Specifically, here we take purifications of a form

|θβ(x)〉 =1√2

∑b

e−iφx√cβxb |b, a(b, x)〉 , (15)

where x signifies that the purifications give rise todifferent thermal states at temperature βx and where

FIG. 2. A diagram for the one-bath case in analogy to Fig-ure 1. The bath is placed in a superposition and acquiresa different temperature depending on the arm. The probeinteracts/thermalises with the bath while the bath is in su-perposition.

{|a(b, x)〉}b is an orthonormal basis for the ancillary sys-tem. This can in principle be different for the differentpurifications x = {0, 1}. Phase φx ∈ R.

An unnormalised superposition of such states thenreads |ψ〉 =

∑x=0,1 |θβ(x)〉.

The superposition of purifications can be prepared

using another DoF (a control), such that |ψ〉 =1√2

∑x=0,1 |θβ(x)〉 |x〉C , specifically

|ψ〉 =1√2

∑x=0,1

∑b

e−iφx√cβxb |b, a(b, x)〉 |x〉C , (16)

A conditional state of the bath is prepared bymeasuring the control in the superposition |φ〉C =1√2

(|0〉C + eiφC |1〉C

). Such a conditional state is

ρinB(φ) := 〈φ|C Tr{|ψ〉 〈ψ|} |φ〉C . This results in the in-

put reduced state of the bath:

ρinB(φ) =

1

4

[ρβ0 + ρβ1

+ e−iφ∑b,b′

√cβ0

b cβ1

b′ V01bb′ |b〉 〈b′|+ H.c.

](17)

where φ = φ0 − φ1 − φC , and we define V xx′

bb′ :=

〈a(b′, x′)|a(b, x)〉, for V xx′

bb′ a matrix element of some uni-tary operator transforming between ancilla bases whichare associated with the control states x, x′. This an-cillary unitary has properties V xxbb′ = V x

′x′

bb′ = δbb′ , and

V xx′

= (V x′x)†. If the purifications are the same for x

and x′ this reduces to a delta function δbb′δxx′ .The initial state of the bath and the probe is a tensor

product ρinBS(φ) = ρin

B(φ) ⊗ ρS . The state of the probe

6

post interaction, for a generic probe-bath unitary interac-

tion UBS is ρS(φ) := TrB{UBSρinBS(φ)U†BS} and in terms

of Kraus operators introduced in Sec. II takes the form

ρ(φ)S =

1

4

∑k,b

[cβ0

b MkbρSM†kb + cβ1

b MkbρSM†kb

+ e−iφ∑b′

√cβ0

b cβ1

b′ V01bb′MkbρSM

†kb′ + H.c.

](18)

For the unitary operator Eq. (8) this state reads

ρS(φ) =1

4

[ρβ0

S + ρβ1

S

+ e−iφ∑b,b′

√cβ0

b cβ1

b′ V01bb′ |b〉 〈b′|+ H.c.

](19)

In this scenario, the generalisation of Eq. (18) to arbi-trary Kraus decompositions is again equivalent to intro-ducing the additional local unitaries to the interactionwith the bath, as in the previous section. In this onebath case, however, the unitaries can depend on the con-trol, and can also act on both the bath and the ancillarysystem. Hence, we use the notation uxBA to indicate this.The probe-bath interaction, Eq. (8), extended to includethese local unitaries takes the form

UBACS =∑x=0,1

[(uxBA ⊗ IS) · (UBS ⊗ IA)]⊗ |x〉C 〈x|C .

(20)Using Eq. (16) for the bath, we begin with bath, an-

cillary system, and control in that initial state:

ρinBAC =

1

2

∑x,x′

=0,1

∑b,b′

e−i(φx−φx′ )√cβxb c

βx′b′

|a(b, x)〉 |b〉 |x〉 〈x′|C 〈a(b′x′)| 〈b′| (21)

The final probe state, where we condition on the

control |φ〉C = 1√2

(|0〉C + eiφC |1〉C

), is ρ

(φ)S =

〈φ|C TrBA

{UBACSρ

inBAC ⊗ ρSU

†BACS

}|φ〉C .

For Eq. (8) this is, explicitly

ρ(φ)S =

1

4

[ρβ0 + ρβ1

+ e−iφ∑b,b′

√cβ0

b cβ1

b′ W10bb′ |b〉 〈b′|+ H.c.

](22)

where W xx′

bb′ := 〈a(b′, x′)|TrS{u1†ASu

0ASρS} |a(b, x)〉.

We immediately see that, in this model too, the finalprobe state still depends on the temperatures of bothbaths, the initial state of the probe and on the local uni-taries, but there are marked differences with the modelof the previous section.

Unlike in the two-bath case, the probe can thermalise.This is possible when both amplitudes of the purifica-tion (or superposed purifications) correspond to the samethermal state of the bath, and when the purification ba-sis is the same for both amplitudes (in which case thematrix V 01

bb′ is the identity).In this particular case, for equal temperatures β0 =

β1 ≡ β, the probe thermalises to the common tempera-ture β for arbitrary local unitaries and initial probe state.

As in the previous model, we now look at the visibilityof the interference between the amplitudes of the controlDoF. The probability to measure the control in |φ〉C is

P (φ) =1

4

[2 + e−iφ

∑b

√cβ0

b cβ1

b W01bb + H.c.

]. (23)

The visibility is

V =

∣∣∣∣∣∑b

√cβ0

b cβ1

b W01bb

∣∣∣∣∣ . (24)

As in the two-bath case in Section III, it is also thecase here that there always exist uiAS for which the visi-bility vanishes, which follows from the same argument asin Sec. III: Considering eigenstates |s〉 of ρS it suffices totake uiAS which map each |s〉 to different energy eigen-states of the probe, i.e. such that u1

AS |s〉 is orthogonal tou0AS |s〉 for every s.Let us now look at the maximum visibility. It is easy

to see that∣∣W 01bb′

∣∣ =∣∣∣〈a(b′, x′)|TrS{u0†

ASu1ASρS} |a(b, x)〉

∣∣∣ ≤ 1,

(25)where this inequality is saturated—thus maximising the

visibility—when u0†ASu

1AS = eiϕI for arbitrary ϕ ∈ R.

Therefore, the maximum visibility is:

Vmax =∑b

√cβ0

b cβ1

b , (26)

once again different to the Section III case. Interest-ingly, this expression coincides with the fidelity between

the two thermal states, Vmax = Tr

√√ρβ0ρβ1

√ρβ0 =

Tr√ρβ0ρβ1 , where the latter simplification is possible be-

cause the two states commute [24].

When the ancillary unitary is equal to I, indicatingboth bath purifications are the same, the inequality abovebecomes an equality. However, when V 01

bb 6= 1, even forβ0 = β1, the visibility in Eq. (24) will be less than one.

A further particular case of interest is when the inter-action between the probe and the bath is independent ofthe control. This renders uxBA independent of x, and the

visibility is simply: V =∑b

√cβ0

b cβ1

b . This corresponds

to a situation where the probe interacts with the bath af-ter it has been prepared in a superposition, namely aftera particular state of the control has been detected post

7

its interference. We will consider this particular case inSec. V in the pre-thermalisation regime.

In summary, we generally find that in the two-bathscenario, full thermalisation cannot be attained, while inthis single-bath scenario it can, with particular condi-tions.

See Table I for a side-by-side look at the one- and two-bath cases.

V. PARTIAL THERMALISATION

The two scenarios considered in the previous sectiondealt with full thermalisation, which can be seen as theasymptotic future of any physical process taking, in prac-tise, finite time. In this section we thus look at partial,or pre-, thermalisation processes corresponding to ourtwo models, proposing that the general features identi-fied therein also hold for finite times where exact ther-malisation is not reached.

As highlighted before, the process of thermalisation ofa finite dimensional quantum system can be modelled bythe application of a GADC. If the interaction parameteris set to be the maximum (η = 1), the probe thermalisesafter a single interaction with the bath, or equivalentlywith a single application of the GADC, thus here we setη < 1. As such, after a single interaction with the bath,the probe acquires an intermediate state between the ini-tial state and a thermal one.

In this section, we devise a collisional model [25],wherein we consider identical subsystems of the bath inthe initial state |θβ〉. The probe interacts with each ofthe subsystems individually and successively through theunitary operator USB . Notably, this corresponds to mul-tiple consecutive applications of the GADC on the probesuch that each application represents a time step. Wegenerally expect that after a sufficient number of appli-cations of the GADC, the probe assumes a thermal stateand continues to remain in that state even if the processis continued.

A. Collisional model with generalized amplitudedamping channel (GADC)

In this model of gradual thermalisation the state ofa complete bath is of the form |θβ〉⊗M , where M isthe number of bath subsystems. As apparent, we haveconsidered each of the bath subsystems to be in a purifiedstate. We call the successive operation of the unitary USB(on the probe and purified bath subsystem) as collisionsand assign a ‘collision number’ to each such operation,see Fig. (3).

Such modelling of the thermalisation process enablesus to be in the pre-thermalisation regime by controllingthe strength and/or the number of collisions. Fig. (3),shows the trace distance between the state of the probe,initialized as |0〉 (the ground state), and a thermal state

Input State

1 ……………… 2 …………..… M

Output State

Bath

Bath subsystems

Collision number :

(a)

1 2 3 4 5 6r

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

||S

=1 ||

= 0.75= 0.8= 0.85= 0.9

(b)

FIG. 3. (a) Schematic diagram showing the collisional modelwith each step consisting of the GADC unitary (UηSB) actingon the probe and the bath, each of whose subsystems are in aGibbs’ state at a fixed temperature. Equivalently, one couldconsider the unitary interaction: USB = UηSB ⊗ I2 betweenone of the bath subsystems (purified) and the probe at each“collision”. (b) Trace distance between the state of the probeand a thermal state as a function of the collision number in thecollisional GADC model. The probe is initialized in the state|0〉, whereas the initial state of each of the bath subsystemsis defined as |θβ=1〉 which corresponds to temperature T=1.∆E = 1 for both probe and bath subsystems.

at a temperature T = 1, as a function of the number ofcollisions. This distance approaches 0 as the the num-ber of collisions increase. The number of collisions af-ter which the trace norm falls below a fixed threshold,||ρS − ρβ=1|| < ε —signalling that the probe approachesthermalisation—depends on the interaction strength andon the initial state of the probe.

In the following subsection, we use the above modelto illustrate the drop in visibility in the pre/post ther-malisation regime and the key differences therein whilefocusing on the two scenarios identified in the previoussections—the one-bath and two-bath cases. In all in-stances shown here, we consider the local unitaries actingon the bath(s), identified as u0 and u1, to be the identity,i.e., the bath does not have an intrinsic dynamics of itsown.

8

TABLE I. Comparison of two-bath and one-bath scenarios. Final probe states are conditioned on |φ〉C = 1√2(|0〉+eiφ |1〉),

ρβj =∑b cβb |b〉 〈b| denotes a thermal state, and the probe’s initial state is ρS =

∑b pb |b〉 〈b|. Note that the local bath unitaries

in the one bath case act on both the probe and the ancilla. The visibilities are for arbitrary bath temperatures and initialprobe state, for the control DoF. The visibilities are maximised over the local unitaries. In both cases there are always localunitaries for which the respective visibilities vanish.

Two bath case One bath case

Final probe state ρS(φ) =1

4

(ρβ0S + ρβ1S

+ eiφρβ0S u0ρSu

1†ρβ1S + H.c.

) ρS(φ) =1

4

(ρβ0S + ρβ1S

+ e−iφ∑b,b′

√cβ0b c

β1b′ W

10bb′ |b〉 〈b′|+ H.c.

)Visibility V =

∣∣∣〈u0†ρβ0S ρβ1S u

1〉ρS∣∣∣ V =

∣∣∣∑b

√cβ0b c

β1b W

10bb′

∣∣∣Max Visibility Vmax =

∑b

pbcβ0b c

β1b Vmax =

∑b

√cβ0b c

β1b

B. Two bath case

In the two-bath case of Sec. III the probe interacts withtwo separate bath subsystems—for example, localised inthe arms of a MZ interferometer Fig. 1. For the study ofpre-thermalisation in this model it will be convenient touse purified states of the baths, by which the initial stateof the entire bath is given as follows:

|B〉 = |θβ1〉⊗M ⊗ |θβ2〉⊗M , (27)

where β1 and β2 correspond to the effective temperaturesof the baths: T1 and T2, respectively. The interactionbetween the probe, the control, and the baths takes theform of Eq. (5) where the probe-bath unitaries are nowadditionally split into a tensor product of terms actingat subsequent collisions, and so for the rth collision wehave the following:

UrB0S = (USB)r ⊗ (I2 ⊗ I2)⊗r ,

UrB1S = (USB)r ⊗ (I2 ⊗ I2)⊗r , (28)

where and USB denotes the unitary operator in Eq. (4),and r refers to the remaining subsystems from M. Thesuperscript r indicates the scope of the unitary beingconfined to the rth subsystem alone.

We can now compare the pre-and post-thermalisationvisibilities for the two-bath case (cf Sec. III C), by chang-ing the number of collisions. The results are summarisedin Fig. 4. We keep the interaction strength constant andtake the energy ground state as the initial state of theprobe. Notably, we see a very good agreement betweenfull- and pre-thermalisation scenarios. In particular, wefind that in the pre-thermalisation case, even if the tem-peratures of the two baths are equal, the visibility issignificantly reduced, and this reduction is stronger forhigher temperatures. This is in contrast with the one-bath case, which is discussed later, wherein for equal tem-peratures the visibility is maximal. Moreover, the visi-bility generally falls with the number of collisions while

the high-visibility parameter region shrinks. These re-sults are in agreement with the analytical results for fullthermalisation.

C. One bath case

As outlined in Sec. IV, in the one bath case the purifiedstate of the bath is entangled with the control DoF, andits initial state is given as follows:

|B〉 = |0〉C∣∣θβ1

⟩⊗M+ |1〉C

∣∣θβ2⟩⊗M

. (29)

Here, we assume that the purification of the bath is donein the same basis in both the arms of the interferometersuch that V 01

bb′ is an identity. Note that in the rth collisionthe probe interacts with the rth subsystem of the bathand thus the unitary operation is of the form: (USB)r ⊗(I2 ⊗ I2)⊗r Here USB denotes the unitary operator fromEq. (4), and the superscript r refers to the fact that it isacting between the probe and the rth subsystem of thebath; r refers to the remaining subsystems from M.

The comparison between the visibilities obtained forpartially thermalising maps and completely thermalis-ing maps is shown in Fig. 5. These cases have beenobtained by changing the number of subsystems withinthe bath (no. of collisions), once again keeping the in-teraction strength constant and taking the probe in itsenergy ground state as the initial state. We find that thevisibility is the greatest for similar temperatures of thebaths and decreases when the temperatures are different.This decrease is steeper for higher number of collisionsand therefore, depends on the fact that probe is not fullythermalised. Also, the parameter space of T1 and T2,where the visibility is close to its maximum, shrinks asthe number of collisions increases, signalling an increasein the distinguishability (loss of coherence).

9

(a)

(b)

FIG. 4. Heatmap of the visibility of the control as a func-tion of the temperatures of the baths in the two-bath case(manifested in their respective initial states) for (a) pre-thermalisation, M = 3 and (b) post-thermalisation, M = 5.The interaction parameter is set: η = 0.8 to allow for partialthermalisation.

VI. DISCUSSION

We identified two relevant ways to operationally makesense of ‘superposition of temperatures’. In both cases,some system in coherent superposition is subject to ther-malisation processes that, in general, differ for the dif-ferent amplitudes. In one model, there are two indepen-dent thermal baths and a probe system in a superposi-tion of interacting with one or the other. In the othermodel, the probe interacts with a single bath, which isitself in a superposition of purifications of two thermalstates. We found that the two formulations are not phys-ically equivalent, as they lead to observably distinct be-haviours. This indicates that the extension of the notionof ‘superposition’ beyond pure states requires precisifica-tion as to what physical aspects it should capture.

(a)

(b)

FIG. 5. Heatmap of the visibility of the control as a functionof temperatures of the bath in the one-bath case (manifestedin its initial state which is quantum-controlled) for (a) pre-thermalisation, M = 3 and (b) post-thermalisation, M = 5.The interaction parameter is set: η = 0.8 to allow for partialthermalisation.

As a general feature, we have found that systems un-dergoing thermalisation ‘in superposition’ do not in gen-eral end up in a thermal state. We also found that thecoherent thermalisation process depends on the physicalimplementation of the thermalisation channels (e.g., onthe particular type of system-bath interaction). A physi-cal consequence is that experiments probing thermal sys-tems ‘in superposition’ can be sensitive to details of thethermal bath, to which ordinary thermometry would beoblivious.

Technically, the sensitivity to the channel’s implemen-tation manifests itself as a dependence of observableproperties on the particular Kraus representation of thechannel. This is in stark contrast to the standard applica-tion of channels on quantum systems, for which differentKraus decompositions are equivalent. However, this is aknown feature in the general context of ‘superpositionsof quantum channels’—that is, in general scenarios where

10

channels are applied to subspaces, rather than systems orsubsystems [26–30].

Both schemes we have considered feature a controlsystem—typically a path in an interferometer—preparedin a superposition of two orthogonal states, each associ-ated with one of the two temperatures. Intuitively, wethink of different temperatures as distinct, macroscopicvariables and one might expect that knowing the temper-ature might reveal enough information about the controlto completely destroy its coherence. We have seen thatthis is not necessarily the case as, again, the answer de-pends on the details of the probe-bath interaction. Al-though one can always find parameters for which the vis-ibility vanishes, this is possible even for equal tempera-tures, meaning that the which-way information leaking tothe environment is not encoded in temperature alone. Onthe other hand, the maximal visibility remains strictlybelow one when the two temperatures in superpositionare different, giving a meaningful measure of how tem-perature reveals which-way information.

The two scenarios we have considered are highly rel-evant in relativistic quantum thermodynamics. For ex-ample, the first model, where a system is in a superposi-tion of interacting with two different baths, would arisenaturally in extended thermal systems in general rela-tivity. Due to the Tolman-Ehrenfest effect, a bath inthermal equilibrium in a gravitational field has a non-uniform temperature, as measured by a local thermome-ter. Therefore, a probe in a superposition of differentlocations would effectively interact with different bathsat different temperatures.

The second model appears to be closely related to arecently extended model of an Unruh-deWitt (UdW) de-tector where the detector can follow a superposition ofdifferent trajectories [12, 13]. A single trajectory withwell-defined acceleration, as per the Unruh effect, leadsto the internal DoF of the detector thermalising to anacceleration-dependent temperature. The internal statesof the detector play the role of the probe, its position (tra-jectory) plays the role of the control, and the field withwhich it interacts constitutes the bath. Although de-tailed analysis is required to pin down the precise connec-tion between the various aspects of the UdW model andour scenario, the so far obtained results [12–14] are con-sistent with the following conjecture: When the super-posed paths share a common Rindler horizon, the detec-tor interacts with the same set of field modes, with onlythe effective temperature (through the thermal weights

cβxb ) depending on the acceleration. This is similar toEq. (15), in the special case where the purification is in-dependent of the control. On the other hand, when thesuperposed trajectories do not share a horizon, the asso-ciated field modes are in general different. This appearsto correspond to Eq. (15) in the case where, in additionto the temperature, the bases for the purifications do de-pend on the control.

Finally, as shown in Sec. V, our results do not de-pend on full thermalisation of the probe. Thus they

will be relevant for physical, non-idealised scenarios, in-cluding those discussed in the motivation of this work,e.g. a quantum extension of the Tolman-Ehrenfest prob-lem [11], or the Unruh and Hawking effects for particledetectors with quantised centre of mass [12–14].

ACKNOWLEDGMENTS

M.Z and F.C. acknowledge support through Aus-tralian Research Council (ARC) DECRA grantsDE180101443 and DE170100712, and ARC Centre EQuSCE170100009. The authors acknowledge the traditionalowners of the land on which the University of Queenslandis situated, the Turrbal and Jagera people.

Appendix A: Visibility Optimisation with LagrangeMultipliers

To maximise the visibility we use the method of La-grange multipliers to find the optimum values of α0

k andα1k.The Lagrangian L = V which we will optimise is

V = V − λ0

(1−

∑k

∣∣α0k

∣∣2)− λ1

(1−

∑k

∣∣α1k

∣∣2)

where∑k

∣∣α0k

∣∣2 = 1 and∑k

∣∣α1k

∣∣2 = 1.

With dV = 0 as a constraint, we arrive at a set of foursimultaneous equations:

dVdα0

l

= cβ0

l cβ1

l α1l + λ02α0

l = 0 (A1)

dVdα1

l

= cβ0

l cβ1

l α0l + λ12α1

l = 0 (A2)

dVdλ0

= −1 +∑k

∣∣α0k

∣∣2 = 0 =⇒∑k

∣∣α0k

∣∣2 = 1 (A3)

dVdλ1

= −1 +∑k

∣∣α1k

∣∣2 = 0 =⇒∑k

∣∣α1k

∣∣2 = 1 (A4)

Working with equations (A1) and (A2),

α0k =

cβ0

k cβ1

k α1k

2λ0(A5)

α1k =

cβ0

k cβ1

k α0k

2λ1(A6)

Substituting (A6) into (A5),

α0k =

cβ0

k cβ1

k

2λ0

(cβ0

k cβ1

k α0k

2λ1

)

=

(cβ0

k cβ1

k

2

)2α0k

λ0λ1

11

This has the trivial solution α0k = 0, which would imply

α1k = 0 too.

Otherwise, the above expression can also be rearrangedfor:

λ0λ1 =

(cβ0

l cβ1

l

2

)2

and

(α0l )

2 =λ1

λ0(α1l )

2

Which implies λ1 = λ0, which in turn implies λ0 =

± cβ0l c

β1l

2 . However, this means that cβ0

l cβ1

l = const, for

all l. That is, cβ0

l = const, cβ1

l = const.

Since λ0λ1 =

(cβ0l c

β1l

2

)2

is the solution in the case

of non-zero α0l , we can only have non-zero alpha as a

solution.With α0

l = ±α1l = ±δl,l′ , then, Vmax = cβ0

l′ cβ1

l′ .

Since for a thermal state cβil > cβil+1,

Vmax = cβ0

0 cβ1

0

[1] S. Vinjanampathy and J. Anders, “Quantumthermodynamics,” Contemporary Physics 57, 545–579(2016).

[2] M. Mehboudi, A. Sanpera, and L. A. Correa,“Thermometry in the quantum regime: recenttheoretical progress,” Journal of Physics A:Mathematical and Theoretical 52, 303001 (2019).

[3] H. J. Miller and J. Anders, “Energy-temperatureuncertainty relation in quantum thermodynamics,”Nature Communications 9, 1–8 (2018).

[4] S. Ghonge and D. C. Vural, “Temperature as aquantum observable,” Journal of Statistical Mechanics:Theory and Experiment 2018, 073102 (2018).

[5] D. Ebler, S. Salek, and G. Chiribella, “EnhancedCommunication with the Assistance of Indefinite CausalOrder,” Phys. Rev. Lett. 120, 120502 (2018).

[6] A. A. Abbott, J. Wechs, D. Horsman, M. Mhalla, andC. Branciard, “Communication through coherentcontrol of quantum channels,” Quantum 4, 333 (2020).

[7] T. Guha, M. Alimuddin, and P. Parashar,“Thermodynamic advancement in the causallyinseparable occurrence of thermal maps,” Phys. Rev. A102, 032215 (2020).

[8] M. Ban, “Two-qubit correlation in two independentenvironments with indefiniteness,” Physics Letters A385, 126936 (2021).

[9] D. Felce and V. Vedral, “Quantum Refrigeration withIndefinite Causal Order,” Phys. Rev. Lett. 125, 070603(2020).

[10] R. C. Tolman, “On the Weight of Heat and ThermalEquilibrium in General Relativity,” Phys. Rev. 35,904–924 (1930).

[11] R. C. Tolman and P. Ehrenfest, “TemperatureEquilibrium in a Static Gravitational Field,” Phys. Rev.36, 1791–1798 (1930).

[12] J. Foo, S. Onoe, and M. Zych, “Unruh-deWitt detectorsin quantum superpositions of trajectories,” Phys. Rev.D 102, 085013 (2020).

[13] L. C. Barbado, E. Castro-Ruiz, L. Apadula, andC. Brukner, “Unruh effect for detectors in superpositionof accelerations,” Phys. Rev. D 102, 045002 (2020).

[14] J. Foo, S. Onoe, R. B. Mann, and M. Zych,“Thermality, causality, and the quantum-controlledUnruh–deWitt detector,” Phys. Rev. Research 3,043056 (2021).

[15] K. Kraus, States, effects and operations, vol. 190 ofLecture Notes in Physics. Springer, Berlin, 1983.

[16] Y. Takahashi and H. Umezawa, “Higher ordercalculation in thermo field theory,” Collectivephenomena 2, 55 (1975).

[17] W. Israel, “Thermo-field dynamics of black holes,”Physics Letters A 57, 107–110 (1976).

[18] M. Rosati, A. Mari, and V. Giovannetti, “Narrowbounds for the quantum capacity of thermalattenuators,” Nature Communications 9, 4339 (2018).

[19] D. K. L. Oi, “Interference of Quantum Channels,”Phys. Rev. Lett. 91, 067902 (2003).

[20] M. A. Nielsen and I. L. Chuang, Quantum computationand quantum information. Cambridge University Press,Cambridge ; New York, 10th anniversary ed. ed., 2010.

[21] L. Mandel, “Coherence and indistinguishability,” Opt.Lett. 16, 1882–1883 (1991).

[22] B.-G. Englert, “Fringe Visibility and Which-WayInformation: An Inequality,” Phys. Rev. Lett. 77,2154–2157 (1996).

[23] H. Jeong and T. C. Ralph, “Quantum superpositionsand entanglement of thermal states at hightemperatures and their applications toquantum-information processing,” Phys. Rev. A 76,042103 (2007).

[24] T. Heinosaari and M. Ziman, The MathematicalLanguage of Quantum Theory: From Uncertainty toEntanglement. Cambridge University Press, 2011.

[25] V. Scarani, M. Ziman, P. Stelmachovic, N. Gisin, andV. Buzek, “Thermalizing Quantum Machines:Dissipation and Entanglement,” Phys. Rev. Lett. 88,097905 (2002).

[26] M. Araujo, A. Feix, F. Costa, and C. Brukner,“Quantum circuits cannot control unknownoperations,” New J. Phys. 16, 093026 (2014).

[27] A. Bisio, M. Dall’Arno, and P. Perinotti, “Quantumconditional operations,” Phys. Rev. A 94, 022340 .

[28] G. Chiribella and H. Kristjansson, “Quantum Shannontheory with superpositions of trajectories,” Proc. R.Soc. A 475, 20180903 .

[29] A. A. Abbott, J. Wechs, D. Horsman, M. Mhalla, andC. Branciard, “Communication through coherentcontrol of quantum channels,” Quantum 4, 333 (2020).

[30] C. Branciard, A. Clement, M. Mhalla, and S. Perdrix,“Coherent Control and Distinguishability of Quantum

12

Channels via PBS-Diagrams,” in 46th InternationalSymposium on Mathematical Foundations of ComputerScience (MFCS 2021), F. Bonchi and S. J. Puglisi, eds.,vol. 202 of Leibniz International Proceedings in

Informatics (LIPIcs), pp. 22:1–22:20. Schloss Dagstuhl– Leibniz-Zentrum fur Informatik, 2021.