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Lost and found charge in quantum batteries

Debanjan Dey Sarkar, Mallika Mondal, Preeti Parashar, Tamal Guha

TL;DR

The paper investigates recycling leaked energy from quantum batteries (QBs) interacting with a thermal bath by introducing two levels of environment-assisted retrieval: weak (bath-only) and strong (bath plus a purifying reference). It formalizes thermal operations via energy-preserving isometries and shows that the weak retrieval is generically bounded above by $E(\sigma_s)-\frac{1}{\beta}E_f(\sigma_{sR})$, while strong retrieval can reach the total QB energy, with the gap governed by entanglement of formation. A key finding is that the difference between strong and weak retrieval quantifies the entanglement generated by the thermal operation, and in the qubit case an explicit SWAP-like isometry clarifies how the retrieval depends on system parameters; zero-temperature limits further align weak and strong retrieval. Overall, the work establishes fundamental limits and concrete protocols for environmental recycling of QB energy and illuminates the role of entanglement in optimally reclaiming stored work under thermal interactions.

Abstract

Quantum batteries are prone to loosing their stored charge, when interacting with a thermal environment. However, getting a limited assistance from the thermal environment, is it possible to recover the charge back, in a reusable form? Here we answer this question affirmatively, leveraging a non-trivial usage of the seemingly useless thermal environment to recycle the quantum batteries. The framework involves two different kind of assistance from thermal environment - one by accessing only the thermal particle, actively participating in the interaction; and the other, involving assistance from an additional purifying subsystem for the thermal environment, bearing a passive role to the interaction. Interestingly, we report that the difference between the retrieved charge between these two degrees of assistance characterizes the amount of entanglement generated by the thermal operation between the quantum battery and the purifying subsystem for the thermal environment.

Lost and found charge in quantum batteries

TL;DR

The paper investigates recycling leaked energy from quantum batteries (QBs) interacting with a thermal bath by introducing two levels of environment-assisted retrieval: weak (bath-only) and strong (bath plus a purifying reference). It formalizes thermal operations via energy-preserving isometries and shows that the weak retrieval is generically bounded above by , while strong retrieval can reach the total QB energy, with the gap governed by entanglement of formation. A key finding is that the difference between strong and weak retrieval quantifies the entanglement generated by the thermal operation, and in the qubit case an explicit SWAP-like isometry clarifies how the retrieval depends on system parameters; zero-temperature limits further align weak and strong retrieval. Overall, the work establishes fundamental limits and concrete protocols for environmental recycling of QB energy and illuminates the role of entanglement in optimally reclaiming stored work under thermal interactions.

Abstract

Quantum batteries are prone to loosing their stored charge, when interacting with a thermal environment. However, getting a limited assistance from the thermal environment, is it possible to recover the charge back, in a reusable form? Here we answer this question affirmatively, leveraging a non-trivial usage of the seemingly useless thermal environment to recycle the quantum batteries. The framework involves two different kind of assistance from thermal environment - one by accessing only the thermal particle, actively participating in the interaction; and the other, involving assistance from an additional purifying subsystem for the thermal environment, bearing a passive role to the interaction. Interestingly, we report that the difference between the retrieved charge between these two degrees of assistance characterizes the amount of entanglement generated by the thermal operation between the quantum battery and the purifying subsystem for the thermal environment.

Paper Structure

This paper contains 10 sections, 10 theorems, 81 equations, 1 figure.

Table of Contents

  1. Proof of Proposition \ref{['p1']}
  2. Proof of Theorem \ref{['t1']}
  3. Proof of Theorem \ref{['t2']}
  4. Proof of Theorem \ref{['t3']}
  5. Proof of part (i)
  6. Proof of part (ii)
  7. Proof of Proposition \ref{['p2']}
  8. Detailed analysis of the charge retrieval for the qubit battery
  9. Case 1.
  10. Case 2.

Key Result

Proposition 1

For any thermal operation $\Lambda_{\beta}$ and an arbitrary input quantum state $\rho_s$, we have the following relation: where, $\sigma_s=:\Lambda_{\beta}(\rho_s)$ and $H_s$ is the governing system Hamiltonian, as mentioned earlier.

Figures (1)

  • Figure 1: (Color online) A schematic diagram of charge retrieval in a quantum battery, assisted by the thermal environment. (a) The free energetic charge stored in a quantum battery inevitably degraded due to the interaction with a thermal environment of inverse temperature $\beta$. (b) Retrieval of the lost charge in the QB under weak assistance. Here the agent using the QB classically aided by the information extracted from the bath qudit after performing the POVM $\{\Pi^k_b\}_k$ on it. (C) Strong retrieval of the lost charge in the QB. A suitable local POVM $\{\Pi^k_b\otimes \Pi^l_R\}$ is performed on the bath and the reference qudit individually and informed to the battery agent. Accordingly the free-energetic charge enhances in the updated QB.

Theorems & Definitions (16)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Proposition 2
  • Lemma 1
  • proof
  • ...and 6 more