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arXiv:2104.09364 · Quantum

Energy conservation and fluctuation theorem are incompatible for quantum work

Karen V. Hovhannisyan, Alberto Imparato

Published May 6, 2024• 12 citations
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Abstract

Characterizing fluctuations of work in coherent quantum systems is notoriously problematic. Here we reveal the ultimate source of the problem by proving that ( A ) energy conservation and ( B ) the Jarzynski fluctuation theorem cannot be observed at the same time. Condition A stipulates that, for any initial state of the system, the measured average work must be equal to the difference of initial and final average energies, and that untouched systems must exchange deterministically zero work. Condition B is only for thermal initial states and encapsulates the second law of thermodynamics. We prove that A and B are incompatible for work measurement schemes that are differentiable functions of the state and satisfy two mild structural constraints. This covers all existing schemes and leaves the theoretical possibility of jointly observing A and B open only for a narrow class of exotic schemes. For the special but important case of state-independent schemes, the situation is much more rigid: we prove that, essentially, only the two-point measurement scheme is compatible with B .

DOI
10.22331/q-2024-05-06-1336

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Energy conservation and Jarzynski equality are incompatible for quantum work

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Measuring the fluctuations of work in coherent quantum systems is notoriously problematic. Aiming to reveal the ultimate source of these problems, we demand of work measurement schemes the sheer minimum and see if those demands can be met at all. We require ($\mathfrak{A}$) energy conservation for arbitrary initial states of the system and ($\mathfrak{B}$) the Jarzynski equality for thermal initial states. By energy conservation we mean that the average work must be equal to the difference of initial and final average energies, and that untouched systems must exchange deterministically zero work. Requirement $\mathfrak{B}$ encapsulates the second law of thermodynamics and the quantum--classical correspondence principle. We prove that work measurement schemes that do not depend on the system's initial state satisfy $\mathfrak{B}$ if and only if they coincide with the famous two-point measurement scheme, thereby establishing that state-independent schemes cannot simultaneously satisfy $\mathfrak{A}$ and $\mathfrak{B}$. Expanding to the realm of state-dependent schemes allows for more compatibility between $\mathfrak{A}$ and $\mathfrak{B}$. However, merely requiring the state-dependence to be continuous still effectively excludes the coexistence of $\mathfrak{A}$ and $\mathfrak{B}$, leaving the theoretical possibility open for only a narrow class of exotic schemes.

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Work is an important quantity in thermodynamics. In a closed quanutm system, the two-point energy measurements can be applied to measure the work but cannot be utilized in an open quantum system. With the two-point energy measurements, it has been shown that the work fluctuation satisfies the Jarzynski equality. We propose a scheme to measure the work in an open quantum system through the technique of reservoir engineering. Based on this scheme, we show that the work fluctuation in open quantum system may violate the Jarzynski equality. We apply our scheme to a two-level atom coupled to an engineered reservoir and numerically justify the general results, especially demonstrating that the second law of thermodynamics can be violated.

Entropy•Jan 25, 2022•1 citation
Characterizing fluctuations of work in coherent quantum systems is notoriously problematic. Here we reveal the ultimate source of the problem by proving that ( A ) energy conservation and ( B ) the Jarzynski fluctuation theorem cannot be observed at the same time. Condition A stipulates that, for any initial state of the system, the measured average work must be equal to the difference of initial and final average energies, and that untouched systems must exchange deterministically zero work. Condition B is only for thermal initial states and encapsulates the second law of thermodynamics. We prove that A and B are incompatible for work measurement schemes that are differentiable functions of the state and satisfy two mild structural constraints. This covers all existing schemes and leaves the theoretical possibility of jointly observing A and B open only for a narrow class of exotic schemes. For the special but important case of state-independent schemes, the situation is much more rigid: we prove that, essentially, only the two-point measurement scheme is compatible with B .
Measuring the fluctuations of work in coherent quantum systems is notoriously problematic. Aiming to reveal the ultimate source of these problems, we demand of work measurement schemes the sheer minimum and see if those demands can be met at all. We require (A\mathfrak{A}A) energy conservation for arbitrary initial states of the system and (B\mathfrak{B}B) the Jarzynski equality for thermal initial states. By energy conservation we mean that the average work must be equal to the difference of initial and final average energies, and that untouched systems must exchange deterministically zero work. Requirement B\mathfrak{B}B encapsulates the second law of thermodynamics and the quantum--classical correspondence principle. We prove that work measurement schemes that do not depend on the system's initial state satisfy B\mathfrak{B}B if and only if they coincide with the famous two-point measurement scheme, thereby establishing that state-independent schemes cannot simultaneously satisfy A\mathfrak{A}A and B\mathfrak{B}B. Expanding to the realm of state-dependent schemes allows for more compatibility between A\mathfrak{A}A and B\mathfrak{B}B. However, merely requiring the state-dependence to be continuous still effectively excludes the coexistence of A\mathfrak{A}A and B\mathfrak{B}B, leaving the theoretical possibility open for only a narrow class of exotic schemes.
In this Letter we investigate the concept of quantum work and its measurability from the viewpoint of quantum measurement theory. Very often, quantum work and fluctuation theorems are discussed in the framework of projective two-point measurement (TPM) schemes. According to a well-known no-go theorem, there is no work observable which satisfies both (i) an average work condition and (ii) the TPM statistics for diagonal input states. Such projective measurements represent a restrictive class among all possible measurements. It is desirable, both from a theoretical and experimental point of view, to extend the scheme to the general case including suitably designed unsharp measurements. This shifts the focus to the question of what information about work and its fluctuations one is able to extract from such generalized measurements. We show that the no-go theorem no longer holds if the observables in a TPM scheme are jointly measurable for any intermediate unitary evolution. We explicitly construct a model with unsharp energy measurements and derive bounds for the visibility that ensure joint measurability. In such an unsharp scenario a single work measurement apparatus can be constructed that allows us to determine the correct average work and to obtain free energy differences with the help of a 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.
An open question of fundamental importance in thermodynamics is how to describe the fluctuations of work for quantum coherent processes. In the standard approach, based on a projective energy measurement both at the beginning and at the end of the process, the first measurement destroys any initial coherence in the energy basis. Here we seek extensions of this approach which can possibly account for initially coherent states. We consider all measurement schemes to estimate work and require that (i) the difference of average energy corresponds to average work for closed quantum systems and that (ii) the work statistics agree with the standard two-measurement scheme for states with no coherence in the energy basis. We first show that such a scheme cannot exist. Next, we consider the possibility of performing collective measurements on several copies of the state and prove that it is still impossible to simultaneously satisfy requirements (i) and (ii). Nevertheless, improvements do appear, and in particular, we develop a measurement scheme that acts simultaneously on two copies of the state and allows us to describe a whole class of coherent transformations.
Abstract Work is a process-based quantity, and its measurement typically requires interaction with a measuring device multiple times. While classical systems allow for non-invasive and accurate measurements, quantum systems present unique challenges due to the influence of the measuring device on the final value of work. As recent studies have shown, among these challenges is the impossibility of formulating a universal definition of work that respects energy conservation for coherent quantum systems and is compatible with the Jarzynski equality—a fluctuation relation linking the equilibrium free energy difference to the non-equilibrium work. Here, we overcome this challenge by introducing a genuinely quantum, positive correction to the Jarzynski equality stemming from imposing energy conservation. When sufficiently large, this correction forces quantum work to violate the second law more often. Moreover, we construct modified two-point measurement (TPM) schemes for work, along with circuit implementations for them. These measurement schemes correctly certify energy conservation and remain consistent with our quantum-corrected fluctuation relation.
Work is an important quantity in thermodynamics. In a closed quanutm system, the two-point energy measurements can be applied to measure the work but cannot be utilized in an open quantum system. With the two-point energy measurements, it has been shown that the work fluctuation satisfies the Jarzynski equality. We propose a scheme to measure the work in an open quantum system through the technique of reservoir engineering. Based on this scheme, we show that the work fluctuation in open quantum system may violate the Jarzynski equality. We apply our scheme to a two-level atom coupled to an engineered reservoir and numerically justify the general results, especially demonstrating that the second law of thermodynamics can be violated.