Jenny Hide, Vlatko Vedral
We present a method for detecting the entanglement of a state using non-equilibrium processes. A comparison of relative entropies allows us to construct an entanglement witness. The relative entropy can further be related to the quantum Jarzynski equality, allowing non-equilibrium work to be used in entanglement detection. To exemplify our results, we consider two different spin chains.
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We derive a general information-theoretic equality for a system undergoing two projective measurements separated by a general temporal evolution. The equality implies the non-negativity of the mutual information between the measurement outcomes of the earlier and later projective measurements. We show that it also contains the Jarzynski relation between the average exponential of the thermodynamical work and the exponential of the difference between the initial and final free energy. Our result elucidates the information-theoretic underpinning of thermodynamics and explains why the Jarzynski r…
Modified Jarzynski equality in a microcanonical ensemble
本研究针对微正则系综中传统Jarzynski等式不成立的问题，推导了依赖功协议路径逆温度的修正等式，可改进近等温过程的功界，还验证了其在受驱二能级自旋系统中的有效性。
The past two decades witnessed important developments in the field of non-equilibrium statistical mechanics. Among these developments, the Jarzynski equality, being a milestone following the landmark work of Clausius and Kelvin, stands out. The Jarzynski equality relates the free energy difference between two equilibrium states and the work done on the system through far from equilibrium processes. While experimental tests of the equality have been performed in classical regime, the verification of the quantum Jarzynski equality has not yet been fully demonstrated due to experimental challenge…
We show that the quantum Jarzynski equality generalizes to $\mathcal{PT}$-symmetric quantum mechanics with unbroken $\mathcal{PT}$-symmetry. In the regime of broken $\mathcal{PT}$-symmetry the Jarzynski equality does not hold as also the $\mathcal{CPT}$-norm is not preserved during the dynamics. These findings are illustrated for an experimentally relevant system -- two coupled optical waveguides. It turns out that for these systems the phase transition between the regimes of unbroken and broken $\mathcal{PT}$-symmetry is thermodynamically inhibited as the irreversible work diverges at the cri…
The well-known Jarzynski equality, often written in the form $e^{-βΔF}=\langle e^{-βW}\rangle$, provides a non-equilibrium means to measure the free energy difference $ΔF$ of a system at the same inverse temperature $β$ based on an ensemble average of non-equilibrium work $W$. The accuracy of Jarzynski's measurement scheme was known to be determined by the variance of exponential work, denoted as ${\rm var}\left(e^{-βW}\right)$. However, it was recently found that ${\rm var}\left(e^{-βW}\right)$ can systematically diverge in both classical and quantum cases. Such divergence will necessarily po…
Jarzynski equality and related fluctuation theorems can be formulated for various setups. Such an equality was recently derived for nonunitary quantum evolutions described by unital quantum operations, i.e., for completely positive, trace-preserving maps, which preserve the maximally mixed state. We analyze here a more general case of arbitrary quantum operations on finite systems and derive the corresponding form of the Jarzynski equality. It contains a correction term due to nonunitality of the quantum map. Bounds for the relative size of this correction term are established and they are app…
We consider a time-dependent quantum linear oscillator coupled to a bath at an arbitrary strength. We then introduce a generalized Jarzynski equality (GJE) which includes the terms reflecting the system-bath coupling. This enables us to study systematically the coupling effect on the linear oscillator in a non-equilibrium process. This is also associated with the second law of thermodynamics beyond the weak-coupling limit. We next take into consideration the GJE in the classical limit. By this generalization we show that the Jarzynski equality in its original form can be associated with the se…
A superconducting cavity model was proposed as a way to experimentally investigate the work performed in a quantum system. We found a simple mathematical relationship between the free energy variation and visibility measurement in quantum cavity context. If we consider the difference of Hamiltonian at time $t_0$ and $t_λ$ (protocol time) as a quantum work, then the Jarzynski equality is valid and the visibility can be used to determine the work done on the cavity.
We propose a new thermodynamic equality and several inequalities concerning the relationship between work and information for an isothermal process with Maxwell's demon. Our approach is based on the formulation a la Jarzynski of the thermodynamic engine and on the quantum information-theoretic characterization of the demon. The lower bound of each inequality, which is expressed in terms of the information gain by the demon and the accuracy of the demon's measurement, gives the minimum work that can be performed on a single heat bath in an isothermal process. These results are independent of th…
The Jarzynski equality (JE) is a remarkable statement relating transient irreversible processes to infinite-time free energy differences. Although twenty years old, the JE remains unfamiliar to many; nevertheless it is a robust and powerful law. We examine two of Einstein's most simple and well-known discoveries, one classical and one quantum, and show how each of these follows from the JE. Our first example is Einstein's relation between the drag and diffusion coefficients of a particle in Brownian motion. In this context we encounter a paradox in the macroscopic limit of the JE which is fasc…
The Jarzynski equality equates the mean of the exponential of the negative of the work (per fixed temperature) done by a changing Hamiltonian on a system, initially in thermal equilibrium at that temperature, to the ratio of the final to the initial equilibrium partition functions of the system at that fixed temperature. It thus relates two thermal equilibrium quantum states. Here a generalization is given that relates any two quantum states of a system.
The fluctuation theorems, and in particular, the Jarzynski equality, are the most important pillars of modern non-equilibrium statistical mechanics. We extend the quantum Jarzynski equality together with the Two-Time Measurement Formalism to their ultimate range of validity -- to quantum field theories. To this end, we focus on a time-dependent version of scalar phi-four. We find closed form expressions for the resulting work distribution function, and we find that they are proper physical observables of the quantum field theory. Also, we show explicitly that the Jarzynski equality and Crooks …
Jarzynski-like Equality of Nonequilibrium Information Production Based on Quantum Cross Entropy
针对两次测量方案在随机变量关联 von Neumann 熵时失效的问题，本文采用一次测量方案并基于量子交叉熵定义信息产生分布，导出类 Jarzynski 等式，为研究其在量子通信、机器学习和热力学中的作用提供基础。
We obtain the Crooks and the Jarzynski non-equilibrium fluctuation relations using a direct quantum-mechanical approach for a finite system that is either isolated or coupled not too strongly to a heat bath. These results were hitherto derived mostly in the classical limit. The two main ingredients in the picture are the time-reversal symmetry and the application of the first law to the case where an agent performs work on the system. No further assumptions regarding stochastic or Markovian behavior are necessary, neither a master equation or a classical phase-space picture are required. The s…
We show a practical application of the Jarzynski equality in quantum computation. Its implementation may open a way to solve combinatorial optimization problems, minimization of a real single-valued function, cost function, with many arguments. We consider to incorpolate the Jarzynski equality into quantum annealing, which is one of the generic algorithms to solve the combinatorial optimization problem. The ordinary quantum annealing suffers from non-adiabatic transitions whose rate is characterized by the minimum energy gap $Δ_{\rm min.}$ of the quantum system under consideration. The quantum…
Quantum Rényi-Jarzynski Equality
本文推导了非破坏性量子Jarzynski等式即Rényi-Jarzynski等式，将其用作量子最优控制的可调代价函数，最小化有限量子系统的浴偏离与串扰，揭示了Rényi阶数对浴分布区域的敏感性及相关纠缠要求。
We combine the formalisms of diagonal entropy and Jarzynski Equality to study the thermodynamic properties of closed quantum systems. Applying this approach to a quantum harmonic oscillator, the diagonal entropy offers a notion of temperature for closed systems away from equilibrium, and allows computing free-energy profiles. We also apply this approach to a hard-core boson lattice model, and discuss measures how to estimate temperature, entropy and measure work distribution functions. This technique offers a path to investigate the non-equilibrium quantum dynamics by means of performing work …
A result of great theoretical and experimental interest, Jarzynski equality predicts a free energy change $ΔF$ of a system at inverse temperature $β$ from an ensemble average of non-equilibrium exponential work, i.e., $\langle e^{-βW}\rangle =e^{-βΔF}$. The number of experimental work values needed to reach a given accuracy of $ΔF$ is determined by the variance of $e^{-βW}$, denoted ${\rm var}(e^{-βW})$. We discover in this work that ${\rm var}(e^{-βW})$ in both harmonic and an-harmonic Hamiltonian systems can systematically diverge in non-adiabatic work protocols, even when the adiabatic prot…
Most non-equilibrium processes in thermodynamics are quantified only by inequalities, however the Jarzynski relation presents a remarkably simple and general equality relating non-equilibrium quantities with the equilibrium free energy, and this equality holds in both classical and quantum regimes. We report a single-spin test and confirmation of the Jarzynski relation in quantum regime using a single ultracold $^{40}Ca^{+}$ ion trapped in a harmonic potential, based on a general information-theoretic equality for a temporal evolution of the system sandwiched between two projective measurement…
Many studies of quantum-size heat engines assume that the dynamics of an internal system is unitary and that the extracted work is equal to the energy loss of the internal system. Both assumptions, however, should be under scrutiny. In the present paper, we analyze quantum-scale heat engines, employing the measurement-based formulation of the work extraction recently introduced by Hayashi and Tajima~[arXiv:1504.06150]. We first demonstrate the inappropriateness of the unitary time evolution of the internal system (namely the first assumption above) using a simple two-level system; we show that…
Symmetry-Enforced Non-Hermitian Jarzynski Equality in an SU(2)-Rotated Family of Hybrid $\mathcal{PT}$--$\mathcal{APT}$ Systems
本文研究了在非厄米特动力学中，基于对称性交换的条件非厄米特Jarzynski等式成立的条件，通过构造混合PT-APT双能级哈密顿量家族，验证了对称性在SU(2)旋转轨道中的持续性，并通过实验验证了在特定点的工作分布符合预测的对称性准则。
The out-of-time-ordered correlator (OTOC) diagnoses quantum chaos and the scrambling of quantum information via the spread of entanglement. The OTOC encodes forward and reverse evolutions and has deep connections with the flow of time. So do fluctuation relations such as Jarzynski's Equality, derived in nonequilibrium statistical mechanics. I unite these two powerful, seemingly disparate tools by deriving a Jarzynski-like equality for the OTOC. The equality's left-hand side equals the OTOC. The right-hand side suggests a protocol for measuring the OTOC indirectly. The protocol is platform-nons…
We show a practical application of an well-known nonequilibrium relation, the Jarzynski equality, in quantum computation. Its implementation may open a way to solve combinatorial optimization problems, minimization of a real single-valued function, cost function, with many arguments. It has been disclosed that the ordinary quantum computational algorithm to solve a kind of hard optimization problems, has a bottleneck that its computational time is restricted to be extremely slow without relevant errors. However, by our novel strategy shown in the present study, we might overcome such a difficu…
Projection hypothesis in the setting for the quantum Jarzynski equality
本文结合投影量子测量中投影假设的哈密顿过程实现与事件读取所需功的已有结果，提出一个量子热力学方案，以同时实验检验这两个相互独立的理论结果。
In open quantum systems, a clear distinction between work and heat is often challenging, and extending the quantum Jarzynski equality to systems evolving under general quantum channels beyond unitality remains an open problem in quantum thermodynamics. In this Letter, we introduce well-defined notions of guessed quantum heat and guessed quantum work, by exploiting the one-time measurement scheme, which only requires an initial energy measurement on the system alone. We derive a modified quantum Jarzynski equality and the principle of maximum work with respect to the guessed quantum work, which…
Variation on the theme of Jarzynski's inequality
本文探讨了Jarzynski不等式的变体及其在多体量子系统中的扩展，分析了其与最大功定理和波动理论的关联。
It has been established that the inclusive work for classical, Hamiltonian dynamics is equivalent to the two-time energy measurement paradigm in isolated quantum systems. However, a plethora of other notions of quantum work has emerged, and thus the natural question arises whether any other quantum notion can provide motivation for purely classical considerations. In the present analysis, we propose the conditional stochastic work for classical, Hamiltonian dynamics, which is inspired by the one-time measurement approach. This novel notion is built upon the change of expectation value of the e…
We propose a scheme to investigate the nonequilibrium work distribution of a quantum particle under well controlled transformations of the external potential, exploiting the versatility of a single ion in a segmented linear Paul trap. We describe in detail how the motional quantum state of a single ion can be prepared, manipulated and finally read out to fully determine the free energy difference in both harmonic and anharmonic potentials. Uniquely to our system, we show how an ion may be immersed in an engineered laser--field reservoir. Trapped ions therefore represent an ideal tool for inves…
The quantum Jarzynski equality is an important theorem of modern quantum thermodynamics. We show that the Jarzynski equality readily generalizes to relativistic quantum mechanics described by the Dirac equation. After establishing the conceptual framework we solve a pedagogical, yet experimentally relevant, system analytically. As a main result we obtain the exact quantum work distributions for charged particles traveling through a time-dependent vector potential evolving under Schrödinger as well as under Dirac dynamics, and for which the Jarzynski equality is verified. Special emphasis is pu…
We present quantum versions of the Jarzynski equality for the energy costs of information processes, namely the measurement and the information erasure. We also obtain inequalities for the energy costs of the information processes, using the Jensen inequality. The inequalities include Sagawa and Ueda's inequalities [Phys. Rev. Lett. 102, 250602 (2009)] as a special case.
Quantum many-body Jarzynski equality and dissipative noise on a digital quantum computer
该研究在囚禁离子和超导量子计算机上对最多十六个相互作用自由度的系统验证量子多体 Jarzynski 等式与 Crooks 关系，并揭示快速幺正驱动可补偿能量耗散。
Since the first suggestion of the Jarzynski equality many derivations of this equality have been presented in both, the classical and the quantum context. While the approaches and settings greatly differ from one to another, they all appear to rely on the initial state being a thermal Gibbs state. Here, we present an investigation of work distributions in driven isolated quantum systems, starting off from pure states that are close to energy eigenstates of the initial Hamiltonian. We find that, for the nonintegrable system in quest, the Jarzynski equality is fulfilled to good accuracy.
A system--bath (SB) model is considered to examine the Jarzynski equality in the fully quantum regime. In our previous paper [J. Chem. Phys. 153 (2020) 234107], we carried out "exact" numerical experiments using hierarchical equations of motion (HEOM) in which we demonstrated that the SB model describes behavior that is consistent with the first and second laws of thermodynamics and that the dynamics of the total system are time irreversible. The distinctive quantity in the Jarzynski equality is the "work characteristic function (WCF)", $\langle \exp[-βW] \rangle$, where $W$ is the work perfor…
We present a generalization of Jarzynski's Equality, applicable to quantum systems, relating discretized mechanical work and free-energy changes. The theory is based on a step-wise pulling protocol. We find that work distribution functions can be constructed from fluctuations of a reaction coordinate along a reaction pathway in the step-wise pulling protocol. We also propose two sets of equations to determine the two possible optimal pathways that provide the most significant contributions to free-energy changes. We find that the transitions along these most optimal pathways, satisfying both s…
Experimental test of the Jarzynski equality in a single spin-1 system using high-fidelity single-shot readouts
本研究在高维单自旋1系统中通过级联高保真单次读出和两点测量协议直接测量功分布，实验检验了Jarzynski等式，验证了其在不同条件下的有效性，并确立了NV中心系统在随机量子热力学实验中的重要地位。
A general tool for description of open quantum systems is given by the formalism of quantum operations. Most important of them are trace-preserving maps also known as quantum channels. We discuss those conditions on quantum channels under which the Jarzynski equality and related fluctuation theorems hold. It is essential that the representing quantum channel be unital. Under the mentioned condition, we first derive the corresponding Jarzynski equality. For bistochastic map and its adjoint, we further formulate a theorem of Tasaki-Crooks type. In the context of unital channels, some notes on he…
We study the thermodynamics of quantum projective measurements by using the set up for the Jarzynski equality. We prove the fluctuations of energy change induced by measurements satisfy the Jarzynski equality, revealing that the quantum projective measurements perform a work on a measured system. When the system is brought into a thermal contact with a reservoir after measurements, the work done by projective measurements is totally dissipated in the form of heat from the system into the reservoir. Explicitly showing that the measurements always increase the total entropy, we prove the link be…
We investigate non-equilibrium nature of fluctuations of black hole horizons by applying the fluctuation theorems and the Jarzynski equality developed in the non-equilibrium statistical physics. These theorems applied to space-times with black hole horizons lead to the generalized second law of thermodynamics. It is also suggested that the second law should be violated microscopically so as to satisfy the Jarzynski equality.
Constraining work fluctuations of non-Hermitian dynamics across the exceptional point of a superconducting qubit
该研究利用耗散超导电路和后选择实验证明，非厄米量子比特跨越例外点时的功涨落仍可满足 Jarzynski 等式，其关键在于区分非厄米动力学与 Floquet Hamiltonian 的能量学及 PT 对称性。
Here we investigate the impact of temporal entanglement on a system's ability to perform thermodynamical work. We show that while the quantum version of the Jarzynski equality remains satisfied even in the presence of temporal entanglement, the individual thermodynamical work moments in the expansion of the free energy are, in fact, sensitive to the genuine quantum correlations. Therefore, while individual moments of the amount of thermodynamical work can be larger (or smaller) quantumly than classically, when they are all combined together into the (exponential of) free energy, the total effe…
We demonstrate with an experiment how molecules are a natural test-bed for probing fundamental quantum thermodynamics. Single-molecule spectroscopy has undergone transformative change in the past decade with the advent of techniques permitting individual molecules to be distinguished and probed. By considering the time-resolved emission spectrum of organic molecules as arising from quantum jumps between states, we demonstrate that the quantum Jarzynski equality is satisfied in this set-up. This relates the heat dissipated into the environment to the free energy difference between the initial a…
Fluctuations of path-dependent thermodynamic quantities in open quantum systems via two-point system-only measurements
本文提出基于动力学依赖热力学可观测量的两步测量方案，精确计算开放量子系统中路径依赖热力学量的波动，并分离Jarzynski等式的修正因子，适用于封闭系统极限情况。
In recent years a quantum information theoretic framework has emerged for incorporating non-classical phenomena into fluctuation relations. Here we elucidate this framework by exploring deviations from classical fluctuation relations resulting from the athermality of the initial thermal system and quantum coherence of the system's energy supply. In particular we develop Crooks-like equalities for an oscillator system which is prepared either in photon added or photon subtracted thermal states and derive a Jarzynski-like equality for average work extraction. We use these equalities to discuss t…
Finite-time bounds on the probabilistic violation of the second law of thermodynamics
本文针对热马尔可夫环境中的驱动系统，利用热力学长度推导出违背热力学第二定律概率的有限时间界限，并证明其收敛到Jarzynski界限的速率慢于$1/\sqrtτ$，揭示了热力学几何与功的高阶统计性质之间的联系。
We present some aspects of the fidelity approach to phase transitions based on lower and upper bounds on the fidelity susceptibility that are expressed in terms of thermodynamic quantities. Both commutative and non commutative cases are considered. In the commutative case, in addition, a relation between the fidelity and the nonequilibrium work done on the system in a process from an equilibrium initial state to an equilibrium final state has been obtained by using the Jarzynski equality.
Experimental study on the principle of minimal work fluctuations
本文基于金刚石单氮-空位中心的两点测量实验，验证了绝热过程最小化 $e^{-βW}$ 方差的最小功涨落原理，并证明绝热捷径控制可在快速功协议中降低方差，为研究量子效应对 Jarzynski 自由能估计的影响提供了实验基础。
Thermodynamics is the phenomenological theory of heat and work. Here we analyze to what extent quantum thermodynamic relations are immune to the underlying mathematical formulation of quantum mechanics. As a main result, we show that the Jarzynski equality holds true for all non-hermitian quantum systems with real spectrum. This equality expresses the second law of thermodynamics for isothermal processes arbitrarily far from equilibrium. In the quasistatic limit however, the second law leads to the Carnot bound which is fulfilled even if some eigenenergies are complex provided they appear in c…
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