Jarzynski Relations for Quantum Systems and Some Applications

Hal Tasaki

REVIEW

Jarzynski Relations for Quantum Systems and Some Applications

Hal Tasaki

Statistical Mechanics · Submitted 17 Sep 2000 · 12 references

Pith's verdict: Correct, elementary algebra; the one new claim (compound-system entropy increase) is real but conditional on a final-equilibration premise the paper itself flags, and the conditionality runs in the safe direction.

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arXiv IDcond-mat/0009244 (version 2)

Pith ID6SANYA6S, Pith's permanent record of this paper and its review

CitationHal Tasaki. "Jarzynski Relations for Quantum Systems and Some Applications." arXiv:cond-mat/0009244 (2000). https://pith.science/paper/6SANYA6S

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Editor's note

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Short version: the algebra is right, and the main new claim is narrower than the abstract makes it sound — but the narrowing runs in the safe direction, so this is a labeling issue, not an error. Section 2 gives the quantum Jarzynski equality (2.7) as a one-line unitarity computation plus the Jensen inequality (2.9). Tasaki says up front that [5] and [6] already had this, and the 9/25 remark even concedes that the inequality follows from positivity of relative entropy — so the equality is the real content, and he is honest about that. Section 3 is the claimed new piece: the same computation on a tensor product of Gibbs states at different beta_k, giving (3.15), then (3.18), then the compound-system entropy increase (3.19). Section 4 is a two-system heat-flow application, which he openly says is essentially in Jarzynski [3]. Where it is soft: the step from (3.18) to (3.19) minimizes each beta-tilde_l and calls the result S'_l. That is the equilibrium Legendre transform, and it equals the true entropy of subsystem l only if the final state is an equilibrium state. Section 3.2 assumes H(t) sits at H' long enough for each subsystem to equilibrate, and Section 3.3 and footnote 4 admit the restriction. So the derivation is conditional on a physical premise, not derived. But check the direction: min over beta of beta{ - F(beta)} is a lower bound on the true entropy, so what is actually proved is sum S_k <= sum S'_l^{Gibbs} <= sum S'_l^{true}. The law still follows; the paper just proves a slightly stronger-looking statement than it labels. That is worth one sentence in a referee report, not a rejection. The other honest limit is Section 3.3: only products of exact Gibbs states, and he says extending it 'seems rather difficult for the moment,' which is fair. Reference [12] is his own preprint, but it is used only in remarks about equilibration, not as load-bearing support. Who should read it: anyone working on quantum Jarzynski or fluctuation relations, or on foundations of the second law, who wants a clean self-contained statement of the elementary core. Not for anyone hoping for physics beyond [5], [6], and [3]. Recommendation: worth a serious referee if it came in today, mainly to force the (3.18) to (3.19) step to be stated as conditional. As a 2000 technical note it is fine as it stands.

Referee report

Major objections (2)

[§2.3, Eq. (2.14)-(2.16); §3.2, Eq. (3.18)-(3.19)] The identification of min_beta beta{<H'>_fin - F'(beta)} with the entropy of the final state is valid only if the final state is an equilibrium state at the corresponding temperature. The Legendre transform gives the maximum entropy at fixed mean energy, which is an upper bound on the actual entropy. The paper assumes equilibration in §2.3 and §3.2, but the sentence in §2.3 ('the minimum over beta-tilde of the right-hand side is the entropy S' of the final state') reads as if it were an identity for any state. The paper should state explicitly that the Legendre transform is the maximum entropy at fixed mean energy and that the equilibration premise (plus the thermodynamic limit, footnote 4) is what makes it the actual entropy. Without this, a reader could take (3.19) as a derivation of the second law from quantum mechanics alone, which it is not.
[§3.3] The paper acknowledges that the derivation works only for products of exact Gibbs states and that extending to a wider class of initial states 'seems rather difficult for the moment.' This is a fair acknowledgment, but it limits the scope of the claim in the abstract that the paper derives 'the law of entropy increase for general compound systems.' The paper should make this scope explicit in the abstract or introduction: the derivation is for compound systems whose initial state is a product of exact Gibbs states, and whose final subsystems have equilibrated.

Minor objections (7)

[§2.2 heading; §2.3; §2.2 remark] Typos: 'Jarzyinski's equality' in the §2.2 heading; 'Holmholtz' should be 'Helmholtz' in §2.3; 'Neuman' should be 'Neumann' in the remark in §2.2.
[Eq. (3.15)] The definition of Z'(beta-tilde) should be Z'_l(beta-tilde) to match the notation.
[Eq. (2.5)] The paper uses 'classical average' for f(E,E'); this term could be clarified, since it is not a classical average in the usual sense but an average over the joint probability distribution p_{i,j}.
[§2.2 remark] The remark (9/25/2000) in §2.2 is useful but its placement interrupts the flow between (2.9) and §2.3; consider moving it to a footnote or to the end of the section.
[Reference [12]] Reference [12] is a preprint (cond-mat/0009206); since it is cited for the discussion at the end of §3.2 and for the proof of the second law for simple systems, the paper should note that it is unpublished.
[§4, Eq. (4.9)] The paper's Eq. (4.9) proves Delta S(t) >= 0, but the interpretation as 'heat never flows from the colder to the hotter' assumes energy conservation (Delta E1 + Delta E2 = 0) and small temperature changes; the paper acknowledges the latter but not the former explicitly. Consider adding a sentence.
[Eq. (2.14)] The paper says 'beta-bar is the unique inverse temperature at which the minimum is attained.' Uniqueness requires a convexity assumption; at a first-order phase transition the minimum is degenerate. Consider adding a caveat.

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The paper

The paper claims

A quantum version of Jarzynski's equality is used to derive the strongest form of the second law, and to bound heat flow between two systems at different temperatures.

The reading

Tasaki asks whether Jarzynski's nonequilibrium work relation — an exact identity linking a fluctuating quantity to a ratio of partition functions — survives the move to quantum mechanics, and whether it can be pushed far enough to prove the second law in its strongest form. The answer he gives is yes on both counts. The central identity concerns a joint probability of finding the system in energy eigenstate $i$ of the initial Hamiltonian and eigenstate $j$ of the final one after unitary evolution; averaging $e^{βE_i - β̃E'_j}$ over that distribution collapses, by unitarity, to $Z'(β̃)/Z(β)$. Jensen's inequality then converts the equality into a family of linear inequalities in the average energies, one for each choice of $β̃$. The application the author considers new is the compound-system case: several subsystems, each in its own Gibbs state at its own temperature, separated by adiabatic walls, are manipulated together. The same averaging argument, now with one free parameter per final subsystem, yields the statement that the total entropy of the parts can never decrease.

What carries the argument

The load-bearing object is the joint distribution $p_{i,j} = (e^{-βE_i}/Z(β))|⟨ϕ'_j|U|ϕ_i⟩|^2$ — the probability of starting in the $i$-th eigenstate of the initial Hamiltonian and ending in the $j$-th eigenstate of the final one — together with its multi-temperature generalization $p_{i,j} = (∏_k e^{-β_k E^{(k)}_{i_k}}/Z_k(β_k))|⟨Φ'_j|U|Φ_i⟩|^2$. Its virtue is that the exponential weight in the initial Gibbs factor cancels exactly against the $e^{βE}$ inserted into the average, and unitarity then sums the remaining matrix elements to one, leaving only the ratio of partition functions. Two consequences follow mechanically: the equality itself, and, via the convexity of the exponential, an inequality for ordinary quantum expectation values of the subsystem Hamiltonians. The same maneuver, applied to a time-independent two-body Hamiltonian with a product initial state, produces the heat-flow results: $[e^{-s}]_t = 1$ where $s = β_1(E'^{(1)} - E^{(1)}) + β_2(E'^{(2)} - E^{(2)})$, hence $ΔS(t) ≥ 0$, and the sharper term-by-term representation $ΔS(t) = ∑ p_{i,j,ℓ,m}(t) d[s]$ with $d(x) = e^{-x} - 1 + x ≥ 0$.

Extended reading notes

Core claim

The paper's central claim is that the law of entropy increase for a general compound system — the statement that when $n$ simple systems at inverse temperatures $β_1, …, β_n$ are manipulated into $m$ simple systems, the sum of their entropies cannot decrease — follows from a multi-temperature quantum Jarzynski equality. The equality (Eq. 3.15) says that for an initial state that is a product of Gibbs states $⨂_k e^{-β_k H_k}/Z_k(β_k)$, and for any set of final inverse temperatures $β̃_1, …, β̃_m$, the average of $e^{∑_k β_k E^{(k)} - ∑_ℓ β̃_ℓ E'^{(ℓ)}}$ over the joint transition probabilities equals $∏_ℓ Z'_ℓ(β̃_ℓ) / ∏_k Z_k(β_k)$. Jensen's inequality turns this into $∑_k β_k(⟨H_k⟩ - F_k(β_k)) ≤ ∑_ℓ β̃_ℓ(⟨H'_ℓ⟩ - F'_ℓ(β̃_ℓ))$ for every choice of the $β̃_ℓ$; minimizing the right side over those parameters identifies it with the final entropy, and the left side is the initial entropy. The author notes that the ordinary, single-system entropy increase is a special case, and that the compound-system version is the one with real content, since it constrains operations that redistribute energy among parts rather than merely among states of one part.

Load-bearing premise

The step from the inequality to a statement about entropy requires that each subsystem at the end of the operation has actually settled into equilibrium at some temperature, so that its entropy can be read off as the Legendre transform of its free energy; the author assumes the final Hamiltonian is held fixed long enough for this to happen and notes separately that the derivation works only when the initial state is a product of exact Gibbs states.

Editorial extensions

If this is right

The minimum work needed to drive a quantum system between two Hamiltonians is bounded below by the free-energy difference, with the bound following from the equality rather than assumed.
The entropy of a compound system cannot decrease under any adiabatic operation that starts from a product of Gibbs states, even when walls are inserted or removed and the number of subsystems changes.
Heat flow between two quantum systems initially at different temperatures is one-directional in the sense that $ΔS(t) ≥ 0$ at every time, not merely in the long-time limit.
Because $ΔS(t)$ decomposes into a sum of individually nonnegative terms, a rigorous lower bound on heat flow can be obtained by discarding any unwanted terms and keeping only those that can be controlled in a specific model.
The distribution of the entropy-production-like variable $s$ obeys $e^{-s}P_t(s) = P_{-t}(-s)$, and reduces to $e^{-s}P_t(s) = P_t(-s)$ when the dynamics is time-reversal symmetric.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The term-by-term positivity of $ΔS(t)$ is the more useful half of the heat-transfer result: it suggests a general recipe for proving finite heat flow in concrete lattice or spin-bath models, where one would keep only the transitions between a few low-lying levels and bound $d(x)$ from below by a quadratic.
The author's own remark that the inequalities follow directly from positivity of quantum relative entropy suggests the equality is doing work only where the inequality is not enough — namely in the fluctuation-type symmetries and in the exact partition-function ratios, so a fair test of the method's value is whether those stronger statements get used elsewhere.
If the derivation extends to initial states that are only approximately Gibbs, the compound-system result would inherit the robustness that the author notes already exists for simple systems, which would make the entropy principle independent of the exact-canonical assumption.
The two-system heat-flow setup is a natural testbed for asking whether the bound degrades with system size in a way that matches the usual linear-response prediction for small temperature differences.

Original abstract

We derive quantum analogues of Jarzynski's relations, and discuss two applications, namely, a derivation of the law of entropy increase for general compound systems, and a preliminary analysis of heat transfer between two quantum systems at different temperatures. We believe that the derivation of the law of entropy increase is new and of importance.

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We derive quantum analogues of Jarzynski's relations, and discuss two applications, namely, a derivation of the law of entropy increase for general compound systems, and a preliminary analysis of heat transfer between two quantum systems at different temperatures. We believe that the derivation of the law of entropy increase is new and of importance.

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s0	We derive quantum analogues of Jarzynski's relations, and discuss two applications, namely, a derivation of the law of entropy increase for general compound systems, and a preliminary analysis of heat transfer between two quantum systems at different temperatures.	contribution 1.00	0.96	0.50	asserted 1.00
s1	We believe that the derivation of the law of entropy increase is new and of importance.	interpretation 0.78	0.71	0.22	hedged possibility 0.35

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References

Works cited (12)

[5]

B. Piechocinska, Information erasure , Phys. Rev. A 61, 062314 (2000)

View on Pith 2000
[6]

J. Kurchan, A Quantum Fluctuation Theorem , cond-mat/0007360

View on Pith Pith review arXiv
[3]

C. Jarzynski, Hamiltonian derivation of a detailed ﬂuctuation theorem , J. Stat. Phys. 98, 77 (2000), cond-mat/9908286

View on Pith Pith review arXiv 2000
[12]

H. Tasaki, Statistical mechanical derivation of the second law of ther modynamics, preprint, cond-mat/0009206. 11

View on Pith Pith review arXiv
[2]

C. Jarzynski, Microscopic analysis of Clausius-Duhem processes , J. Stat. Phys. 96, 415 (1999), cond-mat/9802249

View on Pith Pith review arXiv 1999
[1]

C. Jarzynski, A nonequilibrium equality for free energy diﬀerences , Phys. Rev. Lett. 78, 2690 (1997), cond-mat/9610209

View on Pith Pith review arXiv 1997
[4]

G. E. Crooks, The Entropy Production Fluctuation Theorem and the Nonequi librium Work Relation for Free Energy Diﬀerences , Phys. Rev. E 60, 2721 (1999), cond- mat/9901352

View on Pith 1999
[7]

S. Yukawa A Quantum Analogue of the Jarzynski Equality , J. Phys. Soc. Jpn. 69, 2367 (2000), cond-mat/0007456

View on Pith Pith review arXiv 2000
[8]

A. Wehrl, General properties of entropy , Rev. Mod. Phys. 50, 221–260 (1978)

View on Pith 1978
[9]

C. Maes, The Fluctuation Theorem as a Gibbs Property , J. Stat. Phys. 95, 367–392 (1999)

View on Pith 1999
[10]

J. L. Lebowitz and H. Spohn, A Gallavotti-Cohen-Type Symmetry in the Large De- viation Functional for Stochastic Dynamics , J. Stat. Phys. 95, 333–365 (1999)

View on Pith 1999
[11]

E. H. Lieb and J. Yngvason, The Physics and Mathematics of the Second Law of Thermodynamics, Phys. Rep. 310, 1 (1999), cond-mat/9708200; A Fresh Look at Entropy and the Second Law of Thermodynamics , Physics Today, April 2000, math- ph/0003028

View on Pith Pith review arXiv 1999

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How later papers used it

281 later papers cite it

Evidence: substantial. 280 of the citing papers share no author with it, and 39 of those carry a confident label from Pith's citation classifier.

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4 later papers by other authors are listed below: 1 built on it and 3 used its method or tool.

Builds on it 1 of 8 papers with this label

Extended related results of this paper in its following analysis.

Non-equilibrium equalities with unital quantum channels (2013) · 0.99 · paraphrased: the licence does not allow quoting

Uses its method or tool 3 of 18 papers with this label

To obtain Eq. [ref] by similar means, we follow Tasaki [ref] and write explicit expressions for the forward and reverse work distributions (corresponding to piston expansion and compression, respectively): [equation] For every realization $m^A →n^B$ that gives a particular work value during the forward process, there is a corresponding realization $n^B →m^A$ that gives the opposite work value during the reverse process.

Validity of nonequilibrium work relations for the rapidly expanding quantum piston (2011) · Introduction · 0.99
Used the definition due to another work and this paper, which has a transparent physical meaning as the energy change in the system.

From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics (2015) · 0.95 · paraphrased: the licence does not allow quoting
Followed this paper to define the corresponding average over the ensemble of possible pairs of measurement outcomes.

Jarzynski equality for quantum stochastic maps (2013) · 0.91 · paraphrased: the licence does not allow quoting

How the literature uses it

280 arXiv papers cite this in 495 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 275 sentences without a confident label

Contests it 1 paper

This result demands us to change the derivation of the quantum Jarzynski equality from the previous ones [ref] based on the conventional approach, which employed the internal unitary time evolution and regarded the measured value of the energy loss of the internal system as the “work”, whereas the energy gain of the external system is the actual work that we can use.

Quantum Jarzynski equality of measurement-based work extraction · Introduction · contest 0.81

Builds on it 7 papers

Specifically, we present generalised versions of the Tasaki-Crooks relation (TCR) [ref] and the quantum Jarzynski equality (QJE) [ref] suitable to describe fluctuations in systems with an arbitrary, and possibly variable, number of charges.

Revealing missing charges with generalised quantum fluctuation relations · Introduction · 1.00
Assuming unitary evolution, the relation ([ref]) has been derived by Tasaki [ref] .

Non-equilibrium equalities with unital quantum channels · Jarzynski's equality with unital quantum channels · 0.90
Based on the discussion in Sec. 2, we now consider how the quantum Jarzynski equality [ref] for the combined measured system $S$ would be modified by including the event reading process.

Work Required for Selective Quantum Measurement · Thermodynamics of the Measured System · 0.91

Uses its method or tool 18 papers

At this time, we invoke the quantum version of the Jarzynski equality [ref] .

Work Required for Selective Quantum Measurement · Introduction · 0.98
Based on the two-point measurement scheme [ref] , the probability distribution function of a quantum work at time $t$ is written as P(W, t) = ∑_nkδ(W-δE_kn(t))P_kn(t)

Experimental demonstration of work fluctuations along a shortcut to adiabaticity with a superconducting Xmon qubit · Statistics of STA Work · 0.89
In this paper, we adopt the point of view of stochastic quantum thermodynamics, so that the entropy production is defined as a stochastic variable, whose probability distribution and statistical moments can be computed by resorting to the Tasaki-Crooks formalism [ref] .

Stochastic entropy production: Fluctuation relation and irreversibility mitigation in non-unital quantum dynamics · Introduction · 0.98

Compares with it 10 papers

Using notation ([ref]) for a function of two eigenvalues, we then obtain [equation] This result was recently derived in Ref. [ref] and earlier by Tasaki [ref] under a weaker assumption of a unitary evolution.

Jarzynski equality for quantum stochastic maps · Jarzynski equality for bistochastic maps · 0.84
The two central results in the paper are: the observation of intermittency between dynamical phases of distinct activity, itself a consequence of the reducibility of the dynamics in an appropriate limit due to the collective jump operators; and the existence of a Gallavotti-Cohen symmetry in LD functions associated to the time-asymmetric order parameter, analogous to that found in driven classical systems [ref] , which gives rise to a fluctuation theorem [ref] relating to the quantum jump rate.

Dynamical symmetries and crossovers in a three-spin system with collective dissipation · Introduction · 0.84
Hence, we obtain the same form as the previous derivations [ref] of the Jarzynski equality under the cyclic process.

Quantum Jarzynski equality of measurement-based work extraction · Cyclic process · 0.95

Cites it as support 3 papers

However, it can be shown that if one imposes that the statistics of work is consistent with the standard fluctuation theorems, the distribution of work should be defined by the two-point measurement protocol [ref] .

A Wigner quasiprobability distribution of work · Work Statistics · 0.97
In fact, without invoking the fluctuation theorem, we can directly derive the second law by using the positivity of the relative entropy [ref] .

Second Law-Like Inequalities with Quantum Relative Entropy: An Introduction · Introduction · 0.96
This retrodictive element can be identified a posteriori in all previously reported fluctuation relations that we checked, including the most famous ones, both classical [ref] and quantum [ref] , that are highlighted in the many available reviews [ref] .

Fluctuation Theorems with Retrodiction rather than Reverse Processes · Processes versus inferences · 0.99

Cites it as background 147 papers

The fluctuation theorem [ref] [ref] [ref] [ref] [ref] is a general result in non-equilibrium thermodynamics which essentially depends only on the statistical reversibility of the underlying system.

Hawking Radiation and Entropic Gravity · Fluctuation Theorem · 0.92
The verification and use of quantum fluctuation relations [ref] requires the design of experimentally feasible strategies for the determination of the work distribution following a process undergone by a system.

Experimental reconstruction of work distribution and verification of fluctuation relations at the full quantum level · Introduction · 0.92
are further constrained by the Jarzynski equality [ref] [equation] Here $⟨·⟩$ denotes trajectory-ensemble average, $β^-1$ is the reservoir temperature, $Z(t)≡Trexp[-βH(t)]$ is the system partition function, and $ΔF≡F(τ)-F(0)$ is the Helmholtz free energy change in time $τ$.

Constraining work fluctuations of non-Hermitian dynamics across the exceptional point of a superconducting qubit · Introduction · 0.92

Discussion

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[1] [5]

B. Piechocinska, Information erasure , Phys. Rev. A 61, 062314 (2000)

View on Pith 2000

[2] [6]

J. Kurchan, A Quantum Fluctuation Theorem , cond-mat/0007360

View on Pith Pith review arXiv

[3] [3]

C. Jarzynski, Hamiltonian derivation of a detailed ﬂuctuation theorem , J. Stat. Phys. 98, 77 (2000), cond-mat/9908286

View on Pith Pith review arXiv 2000

[4] [12]

H. Tasaki, Statistical mechanical derivation of the second law of ther modynamics, preprint, cond-mat/0009206. 11

View on Pith Pith review arXiv

[5] [2]

C. Jarzynski, Microscopic analysis of Clausius-Duhem processes , J. Stat. Phys. 96, 415 (1999), cond-mat/9802249

View on Pith Pith review arXiv 1999

[6] [1]

C. Jarzynski, A nonequilibrium equality for free energy diﬀerences , Phys. Rev. Lett. 78, 2690 (1997), cond-mat/9610209

View on Pith Pith review arXiv 1997

[7] [4]

G. E. Crooks, The Entropy Production Fluctuation Theorem and the Nonequi librium Work Relation for Free Energy Diﬀerences , Phys. Rev. E 60, 2721 (1999), cond- mat/9901352

View on Pith 1999

[8] [7]

S. Yukawa A Quantum Analogue of the Jarzynski Equality , J. Phys. Soc. Jpn. 69, 2367 (2000), cond-mat/0007456

View on Pith Pith review arXiv 2000

[9] [8]

A. Wehrl, General properties of entropy , Rev. Mod. Phys. 50, 221–260 (1978)

View on Pith 1978

[10] [9]

C. Maes, The Fluctuation Theorem as a Gibbs Property , J. Stat. Phys. 95, 367–392 (1999)

View on Pith 1999

[11] [10]

J. L. Lebowitz and H. Spohn, A Gallavotti-Cohen-Type Symmetry in the Large De- viation Functional for Stochastic Dynamics , J. Stat. Phys. 95, 333–365 (1999)

View on Pith 1999

[12] [11]

E. H. Lieb and J. Yngvason, The Physics and Mathematics of the Second Law of Thermodynamics, Phys. Rep. 310, 1 (1999), cond-mat/9708200; A Fresh Look at Entropy and the Second Law of Thermodynamics , Physics Today, April 2000, math- ph/0003028

View on Pith Pith review arXiv 1999