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arXiv:2608.19320

Quantum Rényi-Jarzynski Equality

Benjamin Bobell, Mert Okyay, Rahul Nandkishore

cond-mat.stat-mechphysics.chem-phquant-phSubmitted Aug 19, 2026• Announced Aug 21, 2026• 0 citations
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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.

Comments
6 pages, 1 figure, comments welcome

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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 kkk-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 kkk, 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.
A result of great theoretical and experimental interest, the Jarzynski equality predicts a free energy change $\mathrm{\ensuremath{\Delta}}F$ of a system at inverse temperature $\ensuremath{\beta}$ from an ensemble average of nonequilibrium exponential work, i.e., $\ensuremath{\langle}{e}^{\ensuremath{-}\ensuremath{\beta}W}\ensuremath{\rangle}={e}^{\ensuremath{-}\ensuremath{\beta}\mathrm{\ensuremath{\Delta}}F}$. The number of experimental work values needed to reach a given accuracy of $\mathrm{\ensuremath{\Delta}}F$ is determined by the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, denoted $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$. We discover in this work that $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ in both harmonic and anharmonic Hamiltonian systems can systematically diverge in nonadiabatic work protocols, even when the adiabatic protocols do not suffer from such divergence. This divergence may be regarded as a type of dynamically induced phase transition in work fluctuations. For a quantum harmonic oscillator with time-dependent trapping frequency as a working example, any nonadiabatic work protocol is found to yield a diverging $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ at sufficiently low temperatures, markedly different from the classical behavior. The divergence of $\mathrm{var}({e}^{\ensuremath{-}\ensuremath{\beta}W})$ indicates the too-far-from-equilibrium nature of a nonadiabatic work protocol and makes it compulsory to apply designed control fields to suppress the quantum work fluctuations in order to test the Jarzynski equality.
The classical Jarzynski equality establishes an exact relation between the stochastic work performed on a system driven out of thermal equilibrium and the free energy difference in a corresponding quasistatic process. This fluctuation theorem bears experimental relevance, as it enables the determination of the free energy difference through the measurement of externally applied work in a nonequilibrium process. In the quantum case, the Jarzynski equality only holds if the measurement procedure of the stochastic work is drastically changed: it is replaced by a so-called two-point measurement scheme that requires the knowledge of the initial and final Hamiltonian and therefore lacks the predictive power for the free energy difference that the classical Jarzynski equation is known for. Here, we propose a quantum fluctuation theorem that is valid for externally measurable quantum work determined during the driving protocol. In contrast to the two-point measurement case, the theorem also applies to open quantum systems and the scenario can be realized without knowing the system Hamiltonian. Our fluctuation theorem comes in the form of an inequality and therefore only yields bounds to the true free energy difference. The inequality is saturated in the quasiclassical case of vanishing energy coherences at the beginning and at the end of the protocol. Thus, there is a clear quantum disadvantage.
The quantum Jarzynski equality and the Crooks relation are fundamental laws connecting equilibrium processes with nonequilibrium fluctuations.They are promising tools to benchmark quantum devices and measure free energy differences.While they are well established theoretically and also experimental realizations for few-body systems already exist, their experimental validity in the quantum many-body regime has not been observed so far.Here, we present results for nonequilibrium protocols in systems with up to 16 interacting degrees of freedom obtained on trapped ion and superconducting qubit quantum computers, which test the quantum Jarzynski equality and the Crooks relation in the many-body regime.To achieve this, we overcome present-day limitations in the preparation of thermal ensembles and in the measurement of work distributions on noisy intermediate-scale quantum devices.We discuss the accuracy to which the Jarzynski equality holds on different quantum computing platforms subject to platformspecific errors.The analysis reveals the validity of Jarzynski's equality in a regime with energy dissipation, compensated for by a fast unitary drive.This provides new insights for analyzing errors in many-body quantum simulators.
The central quantity in the celebrated quantum Jarzynski equality is ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, where WWW is work and $\ensuremath{\beta}$ is the inverse temperature. The impact of quantum randomness on the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and hence on the predictive power of the Jarzynski estimator is an important problem. Working on a single nitrogen-vacancy center in diamond and riding on an implementation of the two-point measurement of nonequilibrium work with single-shot readouts, we have conducted a direct experimental investigation of the relationship between the fluctuations of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ and adiabaticity of nonequilibrium work protocols. It is observed that adiabatic processes minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$, thus verifying an early theoretical concept, the so-called principle of minimal work fluctuations. Furthermore, it is experimentally demonstrated that shortcuts-to-adiabaticity control can be exploited to minimize the variance of ${e}^{\ensuremath{-}\ensuremath{\beta}W}$ in fast work protocols. Our work should stimulate further experimental studies of quantum effects on the bias and error in the estimates of free-energy differences based on the Jarzynski equality.
In recent years entanglement measures, such as the von Neumann and the Rényi entropies, provided a unique opportunity to access elusive features of quantum many-body systems. However, extracting entanglement properties analytically, experimentally, or in numerical simulations can be a formidable task. Here, by combining the replica trick and the Jarzynski equality we devise an alternative effective out-of-equilibrium protocol for measuring the equilibrium Rényi entropies. The key idea is to perform a quench in the geometry of the replicas. The Rényi entropies are obtained as the exponential average of the work performed during the quench. We illustrate an application of the method in classical Monte Carlo simulations, although it could be useful in different contexts, such as in quantum Monte Carlo, or experimentally in cold-atom systems. The method is most effective in the quasistatic regime, i.e., for a slow quench. As a benchmark, we compute the Rényi entropies in the Ising universality class in 1+1 dimensions. We find perfect agreement with the well-known conformal field theory predictions.