Operational Quasiprobability in Quantum Thermodynamics: Work Extraction by Coherence and Non-joint Measurability

TL;DR

The paper introduces operational quasiprobability (OQ) as a work distribution in quantum thermodynamics, showing it reproduces the Jarzynski equality and the classical mean work while capturing coherence contributions to the work statistics. It demonstrates that non-joint measurability of measurements can enhance extractable work beyond classical joint-measurability bounds in a two-level system, and proves that OQ coincides with the real part of KDQ (MHQ) for binary 2D measurements, with positivity tied to joint measurability. In a three-level NV-center system, OQ and MHQ exhibit different negativities yet yield the same work, indicating that negativity magnitude is not a faithful indicator of nonclassical work. The results connect coherence and measurement incompatibility to nonclassical work amplification, provide operational criteria to identify non-joint measurability, and offer experimentally accessible routes to study quantum thermodynamics with simple measurement schemes.

Abstract

We employ the operational quasiprobability (OQ) as a work distribution, which reproduces the Jarzynski equality and yields the average work consistent with the classical definition. The OQ distribution can be experimentally implemented through the end-point measurement and the two-point measurement scheme. Using this framework, we demonstrate the explicit contribution of coherence to the fluctuation, the average, and the second moment of work. In a two-level system, we show that non-joint measurability, a generalized notion of measurement incompatibility, can increase the amount of extractable work beyond the classical bound imposed by jointly measurable measurements. We further prove that the real part of Kirkwood-Dirac quasiprobability (KDQ) and the OQ are equivalent in two-level systems, and they are nonnegative for binary unbiased measurements if and only if the measurements are jointly measurable. In a three-level Nitrogen-vacancy center system, the OQ and the KDQ exhibit different amounts of negativities while enabling the same work extraction, implying that the magnitude of negativity is not a faithful indicator of nonclassical work. These results highlight that coherence and non-joint measurability play fundamental roles in the enhancement of work.

Figures (3)

• Figure 1: Measurement settings to obtain the operational quasiprobability. (a) End-point measurement (EPM) performs the measurement $B$ at time $t_2$. (b) Two-point measurement (TPM) is a consecutive measurement performing the measurement $A$ and $B$ at time $t_1$ and $t_2$ ($t_1<t_2$), respectively. The input state $\hat{\varrho}$ evolves according to the time-dependent Hamiltonian ${H}(t)$ defined in \ref{['eq:Ham']}. The quantum channel $\Phi_H$ represents the time evolution by the Hamiltonian ${H}(t)$.
• Figure 2: The work extraction based on the operational quasiprobability (OQ) in a single-qubit system. (a) Operational quasiprobability $q^\text{OQ}_{if}$, and the tuple $(i,f)$ denotes the outcomes. The OQ is negative for the excitation process corresponding to the outcome $(0,1)$. (b) The black solid line represents the amount of extractable work $w$ by non-joint measurability (non-JM) and the red solid line represents the classical bound given by the jointly measurability (JM). The black solid line is obtained by using the sharp measurement where $\mu=1$. The classical bound is obtained by optimizing the state-independent bound \ref{['eq:bound']} over the measurement sharpness $\mu$. The amount of work obtained by the two-point measurement (TPM) scheme is zero (blue solid line). (c) The landscape of classical upper bound of the extractable work by jointly measurable measurements is illustrated. The white dashed line represents the JM bound in (b).
• Figure 3: The OQ and MHQ obtained by the three-dimensional system of Nitrogen-vacancy (NV) center in diamond. (a) The OQ is constructed by the EPM and TPM scheme, and the weak-TPM scheme is additionally considered to construct MHQ. (b) shows the probabilities obtained by the measurement settings in (a). (c) The OQ and the MHQ exhibit negative values, but they do not coincide. The OQ negativity shows the visibility higher than those of MHQ.

Theorems & Definitions (4)

• Lemma
• Theorem 1
• Theorem 2
• Corollary