Spins-based Quantum Otto Engines and Majorisation Sachin Sonkarand Ramandeep S. Johaly Department of Physical Sciences Indian Institute of Science Education and Research Mohali

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Spins-based Quantum Otto Engines and Majorisation

Sachin Sonkar∗and Ramandeep S. Johal†

Department of Physical Sciences, Indian Institute of Science Education and Research Mohali,

Sector 81, S.A.S. Nagar, Manauli PO 140306, Punjab, India

The concept of majorisation is explored as a tool to characterize the performance of a quantum

Otto engine in the quasi-static regime. For a working substance in the form of a spin of arbitrary

magnitude, majorisation yields a necessary and suﬃcient condition for the operation of the Otto

engine, provided the canonical distribution of the working medium at the hot reservoir is majorised

by its canonical distribution at the cold reservoir. For the case of a spin-1/2 interacting with an

arbitrary spin via isotropic Heisenberg exchange interaction, we derive suﬃcient criteria for positive

work extraction using the majorisation relation. Finally, local thermodynamics of spins as well as

an upper bound on the quantum Otto eﬃciency is analyzed using the majorisation relation.

I. INTRODUCTION

The fast-growing ﬁeld of quantum thermodynamics brings together methods and tools from a

variety of research areas ranging from quantum information, open quantum systems, quantum optics,

non equilibrium thermodynamics, theory of ﬂuctuations, estimation theory and so on [1–24]. The

quantum extensions of the concepts of heat, work, and entropy have in turn led to generalizations

of the classical heat cycles. The so-called quantum thermal machines exploit new thermodynamic

resources such as quantum entanglement, coherence, quantum interactions and quantum statistics

[25–32]. For instance, quantum Otto engine (QOE) based on various platforms has been widely

studied for its possible quantum advantages—both in its quasi-static formulations as well as the

ones based on time-dependent constraints [33–46]. A QOE oﬀers conceptual simplicity by virtue of

a clear separation of heat and work steps in its heat cycle. The quantum working medium used in

these models may be taken in the form of spins, quantum harmonic oscillator, interacting systems and

so on [47–54]. Further, theoretical models have motivated experimental realizations which promise

to be a boost for future applications in devices [55,56].

One of the prominent analytical tools guiding the theoretical developments is the notion of majori-

sation [57–64] and its generalizations [65–67]. The concept, originally perhaps from matrix analysis,

ﬁnds wide applications in various areas of science, mathematics, economics, social sciences, including

modern applications to entanglement theory and thermodynamic resource theories. The majorisa-

tion partial order was developed to quantify the notion of disorder, in a relative sense, when com-

paring probability distributions. Transformations between pure bipartite states by means of local

operations and classical communication can be determined in terms of majorisation of the Schmidt

coeﬃcients of the states [68–71]. Majorisation has been shown to determine the possibility of state

transformations in the resource theories of entanglement, coherence and purity. It also provides the

ﬁrst complete set of necessary and suﬃcient conditions for arbitrary quantum state transformations

under thermodynamic processes [72–74], which rigorously accounts for quantum coherence among

other quantum mechanical properties.

A majorisation relation may be deﬁned in one of the equivalent ways, as follows. Suppose x↓=

(x1, ..., xn) and y↓= (y1, ..., yn) are two real n-dimensional vectors, where x↓and y↓indicate that

the elements are taken in the descending order. Then, the vector xis said to be majorised by the

∗e-mail: sachisonkar@gmail.com

†e-mail: rsjohal@iisermohali.ac.in

arXiv:2210.02402v1 [quant-ph] 5 Oct 2022

vector y, denoted as x≺y, if, for each m= 1, . . . , n, we have

k=m

xk≥

k=m

yk,(1)

with equality holding for m= 1. The notion of majorisation can be readily applied to compare how

two probability distributions0 deviate from a uniform distribution u= (1/n, . . . , 1/n). Thus, x≺y

implies that the distribution yis more ordered than x. An important consequence of this relation is

the inequality: Pn

k=1 f(xk)≥Pn

k=1 f(yk), where fis any continuous, real-valued concave function.

For example, the relation x≺yimplies that the corresponding Shannon entropies are related as:

S(x)≥S(y), where S(x) = −Pn

k=1 xkln xk. Further, there are many equivalent characterizations

of the majorisation relation. For instance, xis majorised by yonly when ycan be obtained from x

by the action of a bistochastic matrix [57].

In the present work, we characterize the operation of a QOE through the notion of majorisation.

We show that a spin-based quantum working substance provides a natural platform by which the

majorisation conditions characterize the operation of a QOE. Thus, majorisation provides suﬃcient

criteria for the operation of a spin-based Otto engine. In fact, the analysis can be extended to a model

of two spins coupled via Heisenberg exchange interactions. Further, majorisation provides insight

into the local thermodynamics of individual spins in the coupled model. Using the majorisation

conditions, we also validate an upper bound for Otto eﬃciency in the coupled case, which is tighter

than the Carnot value.

The paper is organized as follows. In Section II, we describe the quantum Otto cycle and its

various stages based on a quantum working substance. In Section III, we express the work output

in terms of the relative entropy between the two equilibrium distributions corresponding to hot and

cold reservoirs, and we show that a greater value of the Shannon entropy of the system at the hot

reservoir (as compared to the cold reservoir) does not ensure that net work may be extracted in the

Otto cycle. In Section IV, we show how the majorisation relation, P≺P0, between the hot and

cold reservoir equilibrium distributions lead to positive work condition for the QOE. This is shown

in Section IV.A for a single spin-s. In Section V, we analyze the coupled spins model, showing our

main results for a special case of (1/2,1) system. In Section VI, local work by individual spins is

analyzed based on global conditions. Lastly, Section VII shows proves the enhancement in Otto

eﬃciency based on majorisation conditions. We end our paper by summarizing our main results in

Section VIII. The derivations of various results are presented in Appendix.

II. QUANTUM OTTO ENGINE (QOE)

The classical Otto cycle is a textbook example of a four-step heat cycle in which a classical working

medium, in the form of a gas or air-fuel mixture, undergoes two adiabatic and two isochoric steps

[75]. A QOE is based on a quantum generalization of the classical cycle, in which the quantum

working substance (referred to below as the system) undergoes two quantum adiabatic steps and

two isochoric steps. Here, we are interested in a quasi-static Otto cycle in which each of the steps can

take an arbitrarily long time. A quantum adiabatic process, either in the compression or expansion

stage to be explained below, is performed by varying an externally controllable parameter. Secondly,

for such a process, the quantum adiabatic theorem [76] is assumed to hold so the process does not

cause any transitions between the energy levels, thus preserving their occupation probabilities. The

remaining two steps are the isochoric heating and cooling processes which involve thermal interaction

of the system with the hot or cold reservoir. Here, the system gets enough time to reach thermal

equilibrium with the corresponding reservoir. The heat cycle is described in a more quantitative

detail as follows.

FIG. 1: Schematic of a quantum Otto cycle consisting of the stages 1 →2→3→4→1. Steps 1 →2 and 3 →4

respectively denote quantum adiabatic expansion and compression processes, which preserve occupation probability

and involve work W1→2<0 and W3→4>0. The net work extracted in one cycle is W=W1→2+W3→4<0. In step

4→1, heat Q1>0 enters the working substance from a hot reservoir, while in step 2 →3, heat Q2<0 is rejected

to the cold reservoir. Conservation of energy implies, Q1+Q2+W= 0.

Stage 1. Consider an n-level quantum system with Hamiltonian H(B1) whose eigenvalues can be

arranged (in descending order) as: ε↓= (εn, . . . , ε1). The system is in thermal equilibrium with the

hot reservoir at temperature T1. The canonical occupation probabilities for diﬀerent energy levels,

Pk=e−εk/T1/Pke−εk/T1, are arranged as P↓= (P1, . . . , Pn), where we have set the Boltzmann

constant kBequal to unity. The energy eigenstates are represented by {|ψki|k= 1, ...n}. Thus, the

density matrix representing the thermal state of the system is given by:

ρ=

k=1

Pk|ψkihψk|,(2)

Stage 2. The system is detached from the hot reservoir and undergoes a quantum adiabatic process

in which the external ﬁeld strength is lowered from B1to B2. Here, the quantum adiabatic theorem

ensures that no transitions are induced between the energy levels in the change from εkto ε0

Suppose that the energies after the ﬁrst adiabatic process are given by: ε0↓= (ε0

n, . . . , ε0

1), where we

assume no level-crossing as the Hamiltonian changes from H(B1) to H(B2).

Stage 3. The system is brought in thermal contact with the cold reservoir at temperature T2(<

T1). The energy eigenvalues remain at ε0

kwhile the occupation probabilities change from Pkto

k=e−ε0

k/T2/Pke−ε0

k/T2, which are ordered as: P0↓ = (P0

1, . . . , P 0

n). Thus, the density matrix of the

system at the end of Stage-3 is given by:

ρ0=

k=1

k|ψkihψk|.(3)

Stage 4. The system is detached from the cold reservoir and the ﬁeld strength is changed back to

B1. The occupation probabilities {P0

k}remain unchanged, while the energy levels change back from

{ε0

k}to {εk}.

Finally, the system is attached to the hot reservoir again whereby the initial state (ρ) is recovered,

thus completing one heat cycle. Note that only heat is exchanged between the system and the

reservoir during an isochoric process, which is given by the diﬀerence between the ﬁnal and initial

mean energies of the system in that process. Thus, in Stage-1 and Stage-3, the heat exchanged is

given respectively as:

Q1=

k=1

εk(Pk−P0

k), Q2=

k=1

ε0

k(P0

k−Pk).(4)

On the other hand, only work is performed during the adiabatic branches of the quantum Otto cycle.

Let Wbe the net work performed in one cycle. Applying the law of conservation of energy to the

cyclic process, we have: Q1+Q2+W= 0. The operation of a heat engine requires that heat is

absorbed (rejected) by the system at the hot (cold) reservoir, while net work is extracted from the

system by the end of the cycle. These conditions can be satisﬁed by choosing the sign convention:

Q1>0, Q2<0 and W < 0. The net work performed by the QOE can then be written as

|W|=Q1+Q2=

k=1

(εk−ε0

k)(Pk−P0

k).(5)

We denote |W| ≥ 0 as the positive work condition (PWC) of our engine. The eﬃciency of the QOE

is deﬁned as η=|W|/Q1= 1 + Q2/Q1.

III. RELATIVE ENTROPY AND QOE

In this section, we cast the thermodynamic quantities for a QOE in terms of the relative en-

tropy which is deﬁned as D(x||y)≡Pkxk(ln xk−ln yk)≥0. Also known as the Kullback-Leibler

divergence[77], this quantity is a measure of the ’distance’ between two discrete probability distri-

butions, and vanishes only when the distributions xand yare identical.

Now, the expressions for canonical probabilities may be inverted as: εk≡ −T1ln (PkPje−εj/T1)

and ε0

k≡ −T2ln (P0

kPje−ε0

j/T2). Substituting these expressions in Eq. (4), and after some algebra

(Appendix A), the heat exchanged with each the reservoir is expressed as:

Q1=T1(S1−S2)−T1D(P0||P),(6)

Q2=−T2(S1−S2)−T2D(P||P0),(7)

where S1=−PkPkln Pkand S2=−PkP0

kln P0

kare the Shannon entropies of the system in

equilibrium with hot and cold reservoirs, respectively. as we have let kB= 1 Shannon entropy is

equal to canonical entropy.

Thus, it can be seen that D(P0||P) is equal to the entropy generated in the hot isochoric step.

D(P||P0) has a similar meaning for the cold isochoric step. The net work extracted in an Otto cycle

is given by:

|W|= (T1−T2)(S1−S2)−T1D(P0||P)−T2D(P||P0).(8)

Using, Eq. (6) and Eq. (7) the total entropy generated in the heat cycle, ∆totS=−Q2/T2−Q1/T1,

can be expressed as:

∆totS=D(P||P0) + D(P0||P).(9)

The total entropy generated in a quantum Otto cycle is thus equal to the symmetric sum of the

relative entropies. This quantity is also known as the symmetrized divergence and is distinguished by

the fact that it serves as a metric in the space of probability distributions. Finally, the positivity of the

total entropy generated proves the consistency of the QOE with the second law of thermodynamics,

and hence its eﬃciency is bounded by the Carnot value: η≤1−T2/T1.

Returning to Eq. (8), the positivity of the relative entropy indicates that for T1> T2,S1> S2is

a necessary, but not a suﬃcient condition for |W| ≥ 0. The necessity of the condition S1> S2can

be reasoned due to the fact that heat is absorbed by the system at the hot reservoir, while heat is

rejected by the system at the cold reservoir, and the intermediate, quantum adiabatic processes do

not alter the entropy of the system. These considerations suggest that more general conditions are

desirable to characterize the probability distributions, which not only ensure S1> S2, but also the

PWC or |W| ≥ 0. In this paper, we show that the majorisation relation (P≺P0) provides suﬃcient

conditions for the operation of a spins-based quantum Otto cycle as a heat engine.

IV. MAJORISATION AND QOE

As mentioned earlier, the majorisation relation P≺P0implies the following set of inequalities:

k=m

Pk≥

k=m

k,(m= 1, . . . , n) (10)

with the equality holding for m= 1 owing to the normalization property of each distribution.

Speciﬁcally, we obtain from the above inequalities, for m=n

Pn≥P0

n,(11)

and, for m= 2, along with normalization

1≥P1.(12)

These inequalities may be combined as: P0

1/P 0

n≥P1/Pn, to yield the condition:

ε0

n−ε0

T2≥εn−ε1

.(13)

For T1> T2, the above inequality will yield a nontrivial condition, provided that εn−ε1> ε0

n−ε0

1. In

other words, we must assume that the range of the energy spectrum shrinks during the ﬁrst quantum

adiabatic process. Apart from that, the condition (13) is derived for a generic, non-degenerate

spectrum.

Now, an important question arises regarding the circumstances under which the majorisation

inequalities, Eq. (10), hold. Naturally, this is dependent on the form of Hamiltonian (or the energy

spectrum which enters the expressions for the canonical probabilities). In the following, we show for

a spin system, a set of necessary and suﬃcient conditions to satisfy the majorisation relation.

A. QOE with a single spin-s

Suppose the system is in the form of a quantum spin of magnitude s. The energy eigenvalues in

Stage-1 are: εk= 2(k−s−1)B1, where k= 1,...,2s+ 1. Explicitly, we have

ε1=−2sB1, ε2=−2(s−1)B1, . . . , ε2s= 2(s−1)B1, ε2s+1 = 2sB1.

After the ﬁrst quantum adiabatic step, the energy spectrum is given by: ε0

k= 2(k−s−1)B2, where

k= 1,...,2s+ 1 and B2< B1.

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Spins-basedQuantumOttoEnginesandMajorisationSachinSonkarandRamandeepS.JohalyDepartmentofPhysicalSciences,IndianInstituteofScienceEducationandResearchMohali,Sector81,S.A.S.Nagar,ManauliPO140306,Punjab,IndiaTheconceptofmajorisationisexploredasatooltocharacterizetheperformanceofaquantumOttoengineinthe...

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Spins-based Quantum Otto Engines and Majorisation Sachin Sonkarand Ramandeep S. Johaly Department of Physical Sciences Indian Institute of Science Education and Research Mohali

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