Quantum Rényi-Jarzynski Equality

Abstract: The Jarzynski equality provides a strict link between nonequilibrium work and equilibrium free energy changes. Its typical quantum formulations, however, rely on measurement protocols that destroy coherence. In this Letter, we use the resource-theoretic approach to derive a non-destructive quantum Jarzynski equality conditioned on the outcomes of an arbitrary bath observable. This yields the Rényi-Jarzynski equality, which quantifies a finite bath's drift from equilibrium under a non-adiabatic drive via the Rényi $k$-divergence. We further demonstrate that the Rényi-Jarzynski equality provides a tunable cost function for quantum optimal control problems where minimizing bath drift is desired, such as state preparation and gate design, enabling the minimization of cross-talk in finite quantum systems. Our toy model exhibits a transition between competing minima for some critical value of $k$, illustrating how the Rényi order tunes sensitivity to different regions of a bath distribution. Strikingly, when drive parameters vary across bath energy levels, minimizing bath drift requires generating system-bath entanglement.

• The paper derives an exact non-destructive fluctuation theorem conditioned on arbitrary bath measurements, replacing energy-projective TPM protocols with a resource-theoretic system–bath–weight framework.
• The equality connects kth work moments to the Rényi divergence between initial and final bath-outcome distributions, recovering the ordinary Jarzynski equality when bath drift vanishes and yielding Rényi-resolved second laws.
• The paper shows how Rényi order can tune quantum-control objectives, including a qudit transition near k≈6.18 and cases where minimizing bath disturbance requires deliberate system–bath entanglement.

Overview and main result

The Jarzynski equality (JE), $$\mathbb{E}[e^{-\beta W}]=e^{-\beta\Delta F_\mathcal{S}}$$, links nonequilibrium work statistics to equilibrium free energy differences. Its standard quantum formulations, however, rely on the two-point measurement (TPM) protocol, which projects the system into its energy eigenbasis and destroys the very coherences that characterize quantum dynamics. Bobell, Okyay, and Nandkishore derive a fluctuation theorem that avoids this destructive step: working within the resource-theoretic framework of thermodynamics—where work is recorded by an ideal weight whose Hamiltonian is its position operator—they obtain an exact Jarzynski-type equality conditioned on the outcomes of an arbitrary bath observable $\mathcal{O}_\mathcal{B}$ (2608.19320).

The central result is the quantum Rényi-Jarzynski equality (RJE). In quasi-classical form,

$$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$

where $P_0(b)$ and $P_F(b)$ are the initial and final distributions over bath measurement outcomes and $\mathrm{D}_k$ is the Rényi $k$-divergence. This relation quantifies exactly how far a finite bath is driven from equilibrium by a non-adiabatic protocol on the system: for a large or weakly coupled bath, $P_F \to P_0$, $\mathrm{D}_k \to 0$, and the expression collapses to $k$ copies of the ordinary JE. The authors state that Result 4 is the first exact thermodynamic constraint placed on higher moments of projected ensembles in fully nonequilibrium quantum thermodynamics—a strong claim that rests on their use of the weight formalism rather than TPM.

Setup and single-copy fluctuation theorem

The framework uses a tripartite system–bath–weight ($\mathcal{O}_\mathcal{B}$0) construction with energy-conserving unitaries commuting with weight translations, following Åberg and Alhambra et al. A driving protocol taking $\mathcal{O}_\mathcal{B}$1 is implemented via an auxiliary qubit whose flip is enforced by a thermodynamic operation $\mathcal{O}_\mathcal{B}$2. After the drive, the bath is measured in the eigenbasis of $\mathcal{O}_\mathcal{B}$3, defining a trace-normalized CP map $\mathcal{O}_\mathcal{B}$4. The key technical device is the "twirl" channel $\mathcal{O}_\mathcal{B}$5, which symmetrically reweights configurations by exponentiated energies; sandwiching the drive between twirls preserves hermiticity and positivity of non-commuting operators.

Result 1 shows that the composed map $\mathcal{O}_\mathcal{B}$6 sends the thermal state to $\mathcal{O}_\mathcal{B}$7. From this, the authors derive a conditional bath-measurement Jarzynski equality,

$\mathcal{O}_\mathcal{B}$8

which reduces to the standard JE upon averaging over outcomes, and yields via Jensen's inequality a generalized second law, $\mathcal{O}_\mathcal{B}$9, consistent with known results on intraenvironment correlations (2608.19320).

Higher moments and operational form

The multi-copy extension defines maps $$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$0 and $$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$1 acting on $$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$2-fold copies of the post-measurement state. Because the copies commute, these factorize into tensor powers of the single-copy maps, and linearity in the outcome probabilities yields fluctuation theorems for the $$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$3-th moment of the projected ensemble—the ensemble $$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$4 generated by bath measurements, whose moments are central to studies of deep thermalization. Operationally, the RJE becomes an expectation value over a $$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$5-fold projected ensemble:

$$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$6

so that sweeping $$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$7 reconstructs the entire family $$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$8 from conditional work statistics alone, without tomography. Applying Jensen's inequality produces a second Rényi-resolved second law, $$\mathbb{E}_{P_F(b)}\big[\mathbb{E}_{P_F(w|b)}[e^{-\beta W}]^k\big] = e^{(k-1)\mathrm{D}_k(P_0||P_F)-k\beta\Delta F_\mathcal{S}},$$9, notable because the roles of $P_0(b)$0 and $P_0(b)$1 as reference and target swap relative to the KL version.

The authors explicitly caution against reading this as a probe of deep thermalization: the joint $P_0(b)$2 need not thermalize even if $P_0(b)$3 alone does, since a drive may scramble information across $P_0(b)$4 while leaving the weight with memory of its initial conditions. Connecting fluctuation theorems to deep thermalization is left open.

Application to quantum optimal control

The practical payoff is a tunable cost function for control problems where preserving a finite auxiliary system matters, such as gate implementation and crosstalk mitigation on NISQ devices. Because large $P_0(b)$5 penalizes underestimating tail regions of $P_0(b)$6 ("mode-covering" behavior) while small $P_0(b)$7 penalizes spurious probability mass ("mode-seeking"), the order $P_0(b)$8 tunes which errors dominate the optimization. At $P_0(b)$9 the divergence reduces to $P_F(b)$0 for diagonal states, recovering fidelity-based costs as a special case.

The proof-of-principle model is a qubit/qudit system driven unitarily, with entanglement between $P_F(b)$1 and $P_F(b)$2 controlled by rotation angles $P_F(b)$3 across two-state sectors. For the two-qubit case, minimizing $P_F(b)$4 yields $P_F(b)$5 independent of $P_F(b)$6, i.e., no entanglement. For qudits of dimension $P_F(b)$7, the optimum acquires weak $P_F(b)$8-dependence, and at $P_F(b)$9 the paper finds a genuine transition: two nearly degenerate minima at $\mathrm{D}_k$0 and $\mathrm{D}_k$1 exchange global-minimum status discontinuously at a critical value $\mathrm{D}_k$2, verified by a sign change in the difference $\mathrm{D}_k$3 across $\mathrm{D}_k$4. For $\mathrm{D}_k$5 the branch protecting low-probability tail events wins. Most strikingly, when drive parameters vary across bath levels ($\mathrm{D}_k$6), the optimum requires $\mathrm{D}_k$7: minimizing bath drift strictly demands generating system-bath entanglement. This is a counterintuitive claim—an operation judged purely by how little it disturbs the bath can be optimal only if it entangles the bath with the system—and follows directly from the structure of the admissible drives rather than from any relaxation of the objective.

Limitations and open questions

Several caveats bear directly on the strength of the results. First, although the equality is fully quantum, the divergence $\mathrm{D}_k$8 compares only classical marginals and is blind to bath coherences. The authors respond by showing it lower-bounds the sandwiched Rényi divergence $\mathrm{D}_k$9 via the data processing inequality applied to dephasing, so the RJE never overestimates drift—but how much coherence contributes cannot be inferred from work statistics alone, and the bound is generically strict unless initial and final bath states commute. Second, the framework strictly requires full-rank thermal initial states; approximately pure targets must be regularized with small $k$0 mixing, introducing an uncontrolled approximation in practice. Third, the toy models are analytically tractable two-level chains with a single control parameter; whether the entanglement requirement and the $k$1 transition persist in realistic multi-qubit hardware remains untested. Fourth, the proposed estimator's efficiency is unresolved: estimating Rényi divergences naively costs $k$2 samples in alphabet size, and while the work-average estimator replaces $k$3 by the work variance, it is well behaved precisely in the near-equilibrium regime where $k$4 is small and uninformative. Finally, incorporating measurement feedback à la Sagawa-Ueda into the $k$5-copy theorem, and the connection to deep thermalization, are both deferred to future work.

Conclusion

This Letter derives a family of exact, non-destructive quantum fluctuation theorems conditioned on arbitrary bath observables, culminating in a Rényi-Jarzynski equality that constrains all higher moments of the projected system-weight ensemble through expectation values of exponentiated work. Beyond its foundational content, the equality functions as a tunable, physically motivated cost function for preserving finite baths under non-adiabatic drives, with the qudit example demonstrating both a critical transition in the optimal protocol as a function of Rényi order and the necessity of entanglement generation for minimal bath disturbance. The framework offers a concrete route toward crosstalk-aware optimal control on near-term quantum devices, contingent on resolving the estimator-efficiency question in the intermediate regime where the divergence is both measurable and informative.