Quantifying entanglement in quantum thermodynamics via separability constraints

Abstract

The role of quantum entanglement in thermodynamical systems remains elusive. Does entanglement result in thermodynamic advantages or does it impose fundamental limitations? Here, we unambiguously quantify the amount of heat and work in a quantum system that is due to the presence of entanglement. This is achieved by constraining the system's non-equilibrium dynamics to separable states, thereby isolating the impact entanglement has on thermodynamic effects. Unlike thermodynamic entanglement measures, which signify a loose connection between entanglement and thermodynamic properties, imposing a constraint constitutes an active intervention into a system -- answering how much of a system's thermodynamics is caused by (not correlated with) its quantumness. We benchmark our theory by applying the constrained dynamics to several multipartite systems, including quantum batteries and quantum refrigerators.

Figures (6)

• Figure 1: (a) Trajectory governed by a constrained Hamiltonian $H_\mathrm{s}$ contrasted with the freely evolving trajectory governed by the Hamiltonian $H$. (b) Schematic representation of a constraint. In general, the unrestricted evolution takes a state outside the manifold of separable states. For constrained dynamics, at each time step, the evolving state is projected onto the tangent space of separable states. (c) Correlated decay of a doubly-excited state $\ket{ee}$ into a Bell state $\ket{\Psi^+}=(\ket{ge}+\ket{eg})/\sqrt{2}$. While the freely evolving state $\rho$ approaches $\ket{\Psi^+}$, the constrained state $\rho_\mathrm{s}$ remains separable at all times, e.g., $\braket{\Psi^+|\rho_\mathrm{s}|\Psi^+}\leq 1/2$.
• Figure 2: Bloch-sphere representation of the first qubit of the battery when evolving freely (red) and when constrained to separable states (blue). (a) Free evolution of the initial state $\ket{0}$ under $H_{\mathrm{D}}$. Due to entanglement being generated during the evolution, the reduced state moves through the interior of the Bloch sphere. (b) Constrained evolution of the initial state $\ket{0}$. Due to $H_{\mathrm{D}}$ driving orthogonal to the set of separable states (surface of the Bloch sphere), the system remains in the state $\ket{0}$. (c) Free and constrained evolution of the initial state $\ket{\psi(0)} \propto \ket{0}+\varepsilon\ket{1}$ for $\varepsilon=0.1$. The constrained state changes slower than the freely evolving one, and remains on the surface of the Bloch sphere as its global state is separable.
• Figure 3: Ground-state population of each qubit of the quantum refrigerator when coupled to individual thermal baths and for a system initialized in the state $\ket{+++}$. The constrained populations (light) approach the dashed lines, indicating the same populations as if $g=0$. In contrast, the freely evolving populations (dark) approach values above or below these values, which is indicative of cooling or heating beyond the temperature of the bath. Parameters are $T_{\mathrm{w}}=6.33$, $T_{\mathrm{h}}=3.25$, $T_{\mathrm{c}}=2.4$, $\omega_{\mathrm{w}}=1$, $\omega_{\mathrm{c}}=0.687$, $g=0.1$, and $p_i=0.1$. The constrained populations are obtained by the Monte Carlo wave function method averaging over $4\times 10^6$ trajectories Data_set.
• Figure 4: Heat exchanged with the environment for a quantum refrigerator with (blue, $Q_s$) and without (red, $Q$) constraints, for the localized (dashed) and the delocalized (full) models. With the initial state $\ket{+++}$, one observes larger disagreement between the heat flows for the delocalized model than for the localized one. The parameters used are $T_\mathrm{w}=6.33$, $T_\mathrm{h}=3.25$, $T_\mathrm{c}=2.4$, $\omega_\mathrm{w}=1$, $\omega_\mathrm{c}=0.687$, $g=0.1$, and (a) $p_k=0.1$ and (b) $\gamma=0.01$. The plots are the result of averaging $4\times10^6$ (localized) and $5\times10^5$ (delocalized) trajectories Data_set.
• Figure 5: Heat exchange during a correlated dephasing process for the initial state $\ket{\psi_0}\propto (\ket{0}+\lambda\ket{1})^{\otimes 2}$, for $\lambda=1$, $3/4$, $1/2$, $1/4$, and $\omega=\gamma$. The freely evolving system does not accumulate heat, $Q=0$. In contrast, the evolution constrained to separable states experience a non-zero heat exchange, $Q_\mathrm{s}>0$. Results were obtained via the Monte Carlo wave function method averaging $2\cdot 10^4$ constrained trajectories for each $\lambda$Data_set.