Quasiprobability Thermodynamic Uncertainty Relation

TL;DR

The paper derives a quantum thermodynamic uncertainty relation (TUR) in which dynamical fluctuations are quantified by the Terletsky-Margenau-Hill quasiprobability, enabling coherence to contribute without invasive TPM measurements for Markovian GKSL dynamics. The main result is a bound $\dot{\Sigma}(\rho(t)) \geq \frac{2|J_X^{\mathrm{d}}(t)|^2}{m_X(t)}$, where $m_X(t)$ is the short-time TMH fluctuation, providing a quantum extension of short-time TURs. It shows that negativity of the TMH quasiprobability or a non-classical enhanced escape rate is necessary to surpass classical limits, with a basis-independent, coherence-strengthened perspective. An illustrative model demonstrates a dissipationless heat current arising from quantum effects and highlights that high coherence alone with nonnegative TMH cannot yield such behavior, emphasizing the pivotal role of quasiprobability negativity and outlining connections to FCS and TPM approaches in quantum thermodynamics.

Abstract

We derive a quantum extension of the thermodynamic uncertainty relation where dynamical fluctuations are quantified by the Terletsky-Margenau-Hill quasiprobability, a quantum generalization of the classical joint probability. The obtained inequality plays a complementary role to existing quantum thermodynamic uncertainty relations, focusing on observables' change rather than exchange of charges through jumps and respecting initial coherence. Quasiprobabilities show anomalous behaviors that are forbidden in classical systems, such as negativity; we reveal that negativity or a non-classically enhanced escape rate is necessary to increase an output-to-dissipation ratio beyond classical limitations and show that the requirements are basis-independent and stronger than quantum coherence. To illustrate these statements, we employ a model that can exhibit a dissipationless heat current, which would be prohibited in classical systems; we construct a state that has much coherence but does not lead to a dissipationless current due to the absence of anomalous behaviors in quasiprobabilities.

Figures (2)

• Figure 1: (a) In open systems, we cannot reduce two costs simultaneously: irreversibility, quantified by the entropy production rate $\dot{\Sigma}$, and fluctuations. That is represented by a universal trade-off relation called the thermodynamic uncertainty relation (TUR), where the product between $\dot{\Sigma}$ and the dynamical fluctuation $S_X$ of a physical quantity $X$ is bounded by a current strength $J_X$. (b) We prove a quantum TUR with $S_X=m_X$ [Eq. \ref{['eq:qtur']}], where the observable's dynamical fluctuation is quantified by the quasiprobability, a quantum extension of the classical joint probability that may take negative values.
• Figure 2: Hierarchy of conditions. Condition (Q1) or (Q2) is always required for $m_X(\rho)$ to scale anomalously, which is necessary for the output-to-dissipation ratio $Q_X(\rho)$ to grow faster than the classical limit. (a) For classical eigenbases, our necessary conditions are tighter than the condition that the state has more than $O(1)$ coherence. (b) If we take a non-classical eigenbasis, coherence regarding that basis loses its connection to the scaling. In contrast, our two conditions remain relevant. In the example, we examine two characteristic cases, $\rho^+$ and $\rho^-$.