Virtual temperatures as a key quantifier for passive states in quantum thermodynamic processes

TL;DR

This work develops a framework to quantify non-equilibrium passive quantum states using virtual temperatures and majorization. It defines a mean virtual temperature $\tilde{T}$ and extremal values $T_{\min}$, $T_{\max}$, and proves that extrema arise on adjacent level pairs, enabling tight bounds on energy exchange and heat flow under thermal contact. A spectrum-dependent bound for the quantum Otto efficiency is derived as $\eta_{\rm ub} = 1 - \min_i (\omega_i'/\omega_i)$, equivalently $\eta_{\rm ub} = 1 - T_{\min}/T_h$, with connections to $T_{\max}'$ and $T_c$; the bound is illustrated with a two-qubit XY model. The results illuminate how virtual temperatures bridge passivity, majorization, and performance limits of quantum thermal machines, and point to extensions to other cycles and finite-time operation.

Abstract

We analyze the role of virtual temperatures for passive quantum states through the lens of majorization theory. A mean temperature over the virtual temperatures of adjacent energy levels is defined to compare the passive states of the system resulting from isoenergetic and isoentropic transformations. The role of the minimum and the maximum (min-max) values of the virtual temperatures in determining the direction of heat flow between the system and the environment is argued based on majorization relations. We characterize the intermediate passive states in a quantum Otto engine using these virtual temperatures and derive an upper bound for the Otto efficiency that can be expressed in terms of the min-max virtual temperatures of the working medium. An explicit example of the coupled-spins system is worked out. Moreover, virtual temperatures serve to draw interesting parallels between the quantum thermodynamic processes and their classical counterparts. Thus, virtual temperature emerges as a key operational quantity linking passivity and majorization to the optimal performance of quantum thermal machines.

Figures (3)

• Figure 1: (a) An active state $\rho$, with Hamiltonian $H$, may be transformed into a passive state $\sigma^p$ via an isoenergetic process ($\Delta U=0$) using a unital CPTP map $\mathcal{T}(\rho) = \sigma^p$, or into a passive state $\rho^p$ via a cyclic unitary, $\mathcal{U} \rho\; \mathcal{U^{\dagger}} = \rho^p$, which preserves the von Neumann entropy ($\Delta \mathcal{S} = 0$). For simplicity, we take the initial active state with zero coherence. The majorization relation $\rho^p \succ \sigma^p$ implies that the corresponding mean virtual temperatures satisfy: $\widetilde{T}_S > \widetilde{T}_P$. (b) The classical analogue where an initial nonequilibrium state, modelled as two subsystems at unequal temperatures $T_1$ and $T_2$, reaches the final equilibrium state at a specific temperature, via an isoenergetic process (final temperature $T_F$) or an isoentropic process (final temperature $T_f$), satisfying $T_F > T_f$.
• Figure 2: (a) Schematic of a quantum Otto engine. A quantum working medium with Hamiltonian $H(\lambda_1)$ is prepared in the thermal state $\rho(T_h)$ with a hot reservoir. The first quantum adiabatic process ($1 \rightarrow 2$) transforms the system into a passive state $\rho^p$ with Hamiltonian $H(\lambda_2)$. The system thermalizes with the cold reservoir attaining the state $\rho(T_c)$. The second quantum adiabatic process ($3 \rightarrow 4$) transforms the system into the passive state $\bar{\rho}^p$ with Hamiltonian $H(\lambda_1)$, finally attaining the state $\rho(T_h)$. $T_{\min}$ and $T_{\max}'$ are the minimum and maximum virtual temperatures associated with $\rho^p$ and $\bar{\rho}^p$, respectively, which can be used to fix the direction of heat flows $Q_{h,c}$ between the system and the reservoir, as well as to upper bound the efficiency of the cycle. (b) shows the classical Otto cycle on a $TS-$plane, with a well-defined temperature $T_2$ ($T_4$) after the first (second) adiabatic stroke, respectively.
• Figure 3: The quantum Otto efficiency ($\eta$) compared with its upper bound ($\eta_{\rm ub}$) versus the dimensionless anisotropy parameter $\gamma$ in the XY Heisenberg interaction model, with $J=0.5$. The inset shows $\eta$ and $\eta_{\rm ub}$ versus $J$, with $\gamma=0.4$. For both figures, $B_1=2.8,B_2=2,T_h=1,T_c=0.5$.