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Trade-off relations and enhancement protocol of quantum battery capacities in multipartite systems

Yiding Wang, Xiaofen Huang, Shao-Ming Fei, Tinggui Zhang

TL;DR

This work addresses how energy storage capacity can be distributed across subsystems of quantum batteries without diminishing the total capacity. It proves a universal trade-off relation for two-qubit states under Ising/XXZ/XX/XXX Hamiltonians and introduces a residual-capacity measure that splits into incoherent and coherent parts, along with an incoherent-unitary protocol to shift capacity into subsystems. A sufficient condition is established for subsystem-capacity gain, supported by explicit examples, and the results are extended to three-qubit states. The minimal time to implement single incoherent operations is derived via Cartan decomposition, yielding explicit time bounds for common spin-model gates. These findings offer design principles for quantum energy storage and illuminate how coherence and diagonal-structure manipulations can redistribute stored energy while preserving total capacity.

Abstract

First, we investigate the trade-off relations of quantum battery capacities in two-qubit system. We find that the sum of subsystem battery capacity is governed by the total system capacity, with this trade-off relation persisting for a class of Hamiltonians, including Ising, XX, XXZ and XXX models. Then building on this relation, we define residual battery capacity for general quantum states and establish coherent/incoherent components of subsystem battery capacity. Furthermore, we introduce the protocol to guide the selection of appropriate incoherent unitary operations for enhancing subsystem battery capacity in specific scenarios, along with a sufficient condition for achieving subsystem capacity gain through unitary operation. Numerical examples validate the feasibility of the incoherent operation protocol. Additionally, for the three-qubit system, we also established a set of theories and results parallel to those for two-qubit case. Finally, we determine the minimum time required to enhance subsystem battery capacity via a single incoherent operation in our protocol. Our findings contribute to the development of quantum battery theory and quantum energy storage systems.

Trade-off relations and enhancement protocol of quantum battery capacities in multipartite systems

TL;DR

This work addresses how energy storage capacity can be distributed across subsystems of quantum batteries without diminishing the total capacity. It proves a universal trade-off relation for two-qubit states under Ising/XXZ/XX/XXX Hamiltonians and introduces a residual-capacity measure that splits into incoherent and coherent parts, along with an incoherent-unitary protocol to shift capacity into subsystems. A sufficient condition is established for subsystem-capacity gain, supported by explicit examples, and the results are extended to three-qubit states. The minimal time to implement single incoherent operations is derived via Cartan decomposition, yielding explicit time bounds for common spin-model gates. These findings offer design principles for quantum energy storage and illuminate how coherence and diagonal-structure manipulations can redistribute stored energy while preserving total capacity.

Abstract

First, we investigate the trade-off relations of quantum battery capacities in two-qubit system. We find that the sum of subsystem battery capacity is governed by the total system capacity, with this trade-off relation persisting for a class of Hamiltonians, including Ising, XX, XXZ and XXX models. Then building on this relation, we define residual battery capacity for general quantum states and establish coherent/incoherent components of subsystem battery capacity. Furthermore, we introduce the protocol to guide the selection of appropriate incoherent unitary operations for enhancing subsystem battery capacity in specific scenarios, along with a sufficient condition for achieving subsystem capacity gain through unitary operation. Numerical examples validate the feasibility of the incoherent operation protocol. Additionally, for the three-qubit system, we also established a set of theories and results parallel to those for two-qubit case. Finally, we determine the minimum time required to enhance subsystem battery capacity via a single incoherent operation in our protocol. Our findings contribute to the development of quantum battery theory and quantum energy storage systems.

Paper Structure

This paper contains 11 sections, 5 theorems, 83 equations, 3 figures, 1 table.

Table of Contents

  1. I. Introduction
  2. II. Trade-off relations of quantum battery capacity
  3. III. The minimal time to achieve subsystem capacity gain
  4. IV. Discussion and Conclusion
  5. Appendix A: Proof of Theorem 1
  6. Appendix B: Theorem 1 for the Ising ($J>0$, $\alpha=0$, $\beta=1$), XXZ ($J>0$, $\alpha\neq0$, $\beta=1$), XX ($J>0$, $\alpha\neq0$, $\beta=0$) and XXX models ($J>0$, $\alpha=\beta=1$)
  7. Appendix C: Proof of Theorem 2
  8. Appendix D: State (\ref{['exam']}) for the transverse and longitudinal-field Ising models
  9. Appendix E: Proof of Theorem 3
  10. Appendix F: Incoherent operation protocol in three-qubit systems
  11. Appendix G: Proof of Theorem 5

Key Result

Theorem 1

For any two-qubit quantum state $\rho$, the following trade-off relation holds: for any Hamiltonian $H=H_0+H_{int}$ such that $\mathcal{C}(\rho;H)\geq\mathcal{C}(\rho;H_0)$, where $\rho_A$ and $\rho_B$ are reduced density matrices of $\rho$.

Figures (3)

  • Figure 1: Our approach involves applying global unitary operations to reduce the value of $\triangle(\rho,H)$, thereby increasing the value of $Sub(\rho)$, since $\mathcal{C}(\rho;H)$ does not change under the global unitary operation.
  • Figure 2: Process of incoherent operation protocol.
  • Figure 3: Panel (a) displays the residual battery capacity of the transverse-field Ising model before and after incoherent operation. Panels (c) and (d) shows the corresponding comparison for the longitudinal-field Ising model. The dashed lines represent $\triangle(\tilde{\rho}_a,H)$, while the solid lines represent $\triangle(\rho_a,H)$. Panel (b) presents the changes of the subsystem battery capacity before and after incoherent operations for both transverse-field and longitudinal-field Ising models.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • proof
  • proof
  • Theorem 3
  • Theorem 4
  • Theorem 5