Quantum work statistics differ from classical ones because initial energy coherence matters. The standard two-point measurement (TPM) gives a positive distribution but erases phase information. Coherence-retaining endpoint-work quasistatistics provide a compact probe of shortcut-to-adiabaticity performance. For work defined with respect to a reference Hamiltonian, an exact counterdiabatic shortcut pulls the final reference Hamiltonian back to an operator diagonal in the initial energy basis. Endpoint Kirkwood-Dirac or Margenau-Hill quasistatistics then lose sensitivity to initial coherence and reduce to the TPM result. Imperfect shortcuts restore this sensitivity: a non-commuting control error produces off-diagonal pulled-back Hamiltonian elements at first order in the error amplitude, whereas population-only transition probabilities change only at second order. Harmonic-oscillator and qubit benchmarks confirm this linear-versus-quadratic contrast. The result complements inclusive work-cost analyses: it does not measure the auxiliary field's energetic cost, but provides a phase-sensitive endpoint diagnostic of residual nonadiabaticity.
Quantum work statistics differ from classical ones because initial energy coherence matters. The standard two-point measurement (TPM) gives a positive distribution but erases phase information. Coherence-retaining endpoint-work quasistatistics provide a compact probe of shortcut-to-adiabaticity performance. For work defined with respect to a reference Hamiltonian, an exact counterdiabatic shortcut pulls the final reference Hamiltonian back to an operator diagonal in the initial energy basis. Endpoint Kirkwood-Dirac or Margenau-Hill quasistatistics then lose sensitivity to initial coherence and reduce to the TPM result. Imperfect shortcuts restore this sensitivity: a non-commuting control error produces off-diagonal pulled-back Hamiltonian elements at first order in the error amplitude, whereas population-only transition probabilities change only at second order. Harmonic-oscillator and qubit benchmarks confirm this linear-versus-quadratic contrast. The result complements inclusive work-cost analyses: it does not measure the auxiliary field's energetic cost, but provides a phase-sensitive endpoint diagnostic of residual nonadiabaticity.
In quantum thermodynamics, the two-projective-measurement (TPM) scheme provides a successful description of stochastic work only in the absence of initial quantum coherence. Extending the quantum work distribution to quasiprobability is a general way to characterize work fluctuation in the presence of initial coherence. However, among a large number of different definitions, there is no consensus on the most appropriate work quasiprobability. In this article, we list several physically reasonable requirements including the first law of thermodynamics, time-reversal symmetry, positivity of second-order moment, and a support condition for the work distribution. We prove that the only definition that satisfies all these requirements is the Margenau-Hill (MH) quasiprobability of work. In this sense, the MH quasiprobability of work shows its advantages over other definitions. As an illustration, we calculate the MH work distribution of a breathing harmonic oscillator with initial squeezed states and show the convergence to classical work distribution in the classical limit.
Abstract To investigate the role of initial quantum coherence in work-probability distribution, it is necessary to consider an incomplete or partial measurement, in which the energy cannot be fully discriminated by the detector. In this paper, we use a harmonic oscillator with a coherent or squeezing state to realize this incomplete or partial measurement, and propose a unified framework of quantum work statistics for a closed system with an arbitrary initial state. We find that work is proportional to the change of the real part of the coherent state parameter, i.e., quantum work can be estimated by the coherent state parameter. The resulting work-probability distribution includes the initial quantum coherence, and can be reduced to the result of the traditional two projective energy measurement scheme (TPM) by squeezing the state of the harmonic oscillator. Our measurement scheme reveals the fundamental connections between measurement error and coherent work. By introducing a ‘coherent work-to-noise ratio’, we find the optimal measurement error, which is determined by the energy difference between the superposed energy levels. As an application, we consider a driven two-level system and investigate the effects of driving velocity on work statistics. We find that only when the driving velocity matches the transition frequency of the system can initial quantum coherence play an important role.
The standard definition of quantum fluctuating work is based on the two-projective energy measurement, which however does not apply to systems with initial quantum coherence because the first projective energy measurement destroys the initial coherence, and affects the subsequent evolution of the system. To address this issue, several alternative definitions, such as those based on the full counting statistics and the Margenau-Hill distribution, have been proposed recently. These definitions seem ad hoc because justifications for them are still lacking. In the current study, by utilizing the quantum Feynman-Kac formula and the phase space formulation of quantum mechanics, we prove that the leading order of work distributions is equal to the classical work distribution. Thus we prove the validity of the quantum-classical correspondence of work distributions for initial states with quantum coherence, and provide some justification for those definitions of work. We use an exactly solvable model of the linearly dragged harmonic oscillator to demonstrate our main results.
For a quantum system undergoing a unitary, process work is commonly defined based on the two projective measurement protocols, which measures the energies of the system before and after the process. However, it is well known that projective measurements disregard quantum coherences of the system with respect to the energy basis, thus removing potential quantum signatures in the work distribution. Here we consider weak measurements of the system's energy difference and establish corrections to work averages arising from initial system coherences. We discuss two weak measurement protocols that couple the system to a detector, prepared and measured either in the momentum or the position eigenstates. Work averages are derived for when the system starts in the proper thermal state versus when the initial system state is a pure state with thermal diagonal elements and coherences characterized by a set of phases. We show that by controlling only the phase differences between the energy eigenstate contributions in the system's initial pure state, the average work done during the same unitary process can be controlled. By changing the phases alone, one can toggle from regimes where the system absorbs energy, i.e., a work cost, to the ones where it emits energy, i.e., work can be drawn. This suggests that the coherences are additional resources that can be used to manipulate or store energy in a quantum system.
A central challenge in quantum thermodynamics is to access work fluctuations in coherent processes without distorting the energetics of the unmeasured evolution. In standard two-point schemes, the initial energy measurement dephases coherent inputs, causing the measured average work to differ from that of the unmeasured evolution. Here, we develop an operational scheme for accessing work statistics for closed quantum systems based on the abstract notion of variation in the Heisenberg picture Hamiltonian. This scheme preserves energetically relevant coherences, thereby faithfully reproducing unmeasured work, while still producing positive probabilities. We derive modified Jarzynski and Crooks relations, as well as a thermodynamic uncertainty relation, identifying coherence-induced correction terms. Furthermore, we show that this scheme can reliably quantify the performance of a coherent engine in situations where the two-point energy measurement would suppress work output. In addition, the scheme requires only a single measurement and can predict the work associated with a subsequent unitary transformation. We exploit this feature to construct a Maxwell-demon protocol that can outperform energy-based feedback engines for coherent work extraction. Our results establish this scheme as a framework for accessing coherent work fluctuations without erasing the coherence that drives quantum thermodynamic performance.
Homothetic quantum Otto engines---where all populated energy gaps are rescaled by a common factor---provide a reference model in which the quasistatic stochastic efficiency is trajectory-independent while work remains fluctuating. For arbitrary finite homothetic spectra we derive the two-point-measurement work distribution and reduce the first two work moments to endpoint energy moments. Specializing to a uniformly spaced ladder gives closed finite-N expressions for the full work distribution, mean work, variance, and signal-to-width reliability. This ladder connects the qubit and oscillator limits, reveals a finite-N reliability crossover, and demonstrates that the high-temperature and infinite-dimensional limits do not commute. The noncommutation reflects a bounded-versus-unbounded spectral distinction: at fixed finite N the Gibbs state has a normalizable infinite-temperature limit, whereas the oscillator retains an ever-expanding thermal tail. The exact formulas are used to compare standard mean-output prescriptions with work reliability, showing that maximum mean output and maximum dimensionless reliability select different operating points. The benchmark is extended to incomplete diagonal reset and to finite-time unitary strokes described by transition matrices, with a finite-ladder protocol and a harmonic sudden-switch oscillator benchmark as controlled examples. Weak deviations from exact homothety are treated perturbatively, showing how level-dependent gap distortions reintroduce quasistatic efficiency fluctuations and modify work reliability. Together, these results separate finite-size, incomplete thermalization, finite-time, and weak spectral-distortion contributions to work unreliability in quantum Otto engines.