Minimal time required to charge a quantum system

Ju-Yeon Gyhm, Dario Rosa, and Dominik Šafránek

Phys. Rev. A 109, 022607 – Published 15 February 2024

Abstract

We introduce a quantum charging distance as the minimal time that it takes to reach one state (charged state) from another state (depleted state) via a unitary evolution, assuming limits on the resources invested into the driving Hamiltonian. For pure states it is equal to the Bures angle, while for mixed states its computation leads to an optimization problem. Thus, we also derive easily computable bounds on this quantity. The charging distance tightens the known bound on the mean charging power of a quantum battery, it quantifies the quantum charging advantage, and it leads to an always achievable quantum speed limit. In contrast with other similar quantities, the charging distance does not depend on the eigenvalues of the density matrix, it depends only on the corresponding eigenspaces. This research formalizes and interprets quantum charging in a geometric way, and provides a measurable quantity that one can optimize to maximize the speed of charging of future quantum batteries.

Received 21 September 2023
Revised 28 December 2023
Accepted 9 January 2024

DOI:https://doi.org/10.1103/PhysRevA.109.022607

Physics Subject Headings (PhySH)

Batteries Quantum communication, protocols & technology Quantum control Quantum engineering Quantum foundations Quantum information theory Quantum protocols Quantum state transfer Quantum statistical mechanics Quantum thermodynamics

Quantum many-body systems

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalStatistical Physics & Thermodynamics

Authors & Affiliations

Ju-Yeon Gyhm ^1,*, Dario Rosa ^2,3,†, and Dominik Šafránek ^2,‡

¹Department of Physics and Astronomy, Seoul National University, 1 Gwanak-ro, Seoul 08826, Korea
²Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS), Daejeon-34126, Korea
³Basic Science Program, Korea University of Science and Technology (UST), Daejeon-34113, Korea

^*kjy3665@snu.ac.kr
^†dario_rosa@ibs.re.kr
^‡dsafranekibs@gmail.com

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Issue

Vol. 109, Iss. 2 — February 2024

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Images

Figure 1
The quantum charging distance $D (\hat{ρ}, \hat{σ})$ is the length of the geodesic in the space given by a fixed spectrum of the density matrix. It measures the minimal time it takes to evolve one state to another by a unitary evolution generated by a driving Hamiltonian $\hat{V}$ with a fixed operator norm $| | \hat{V} | | = 1$ . Bures angle, on the other hand, is always smaller than the charging distance. This is because the Bures angle $θ_{B} (\hat{ρ}, \hat{σ})$ is equal to the quantum charging distance in the purification space [77]. In other words, if one is allowed to apply a unitary operator in an extended Hilbert space in which both $\hat{ρ}$ and $\hat{σ}$ are pure, even a faster charging speed can be achieved. However, this is impossible without access to these unknown degrees of freedom. The unitary operation in the purification space corresponds to a nonunitary operation in the original Hilbert space, creating an additional shortcut for charging. Classical charging allows only local unitary operations applied on the product states, thus it does not create any entanglement in the charging process. As a result, classical charging is even more limited than quantum charging, and the time to charge, as measured by the classical charging distance $D_{C} (\hat{ρ}, \hat{σ})$ , is longer.
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Figure 2
Cut through the Bloch sphere to illustrate the charging distance between two qubits. The charging distance is given by the curve's length on the Bloch sphere's surface, $D (\hat{ρ}, \hat{σ}) = θ / 2$ , connecting the two states. The Bures angle $θ_{B} (\hat{ρ}, \hat{σ}) = θ_{B}^{'} / 2$ is always smaller and it depends on $r$ . For $r = 1$ , they are both equal, while for $r \to 0, θ_{B} \to 0$ while the charging distance stays constant. Objects below the triangle and the dashed lines are for the geometric construction of $θ_{B}^{'}$ .
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Figure 3
The quantum charging distance (diamonds), for 20 randomly generated couples $\hat{ρ}$ and $\hat{σ}$ shown from top to bottom, in a three-dimensional Hilbert space was obtained numerically using Corollary t7. For each realization, we also plot the interval of the lower and the upper bounds $[{max}_{i} θ_{B} ({\hat{ρ}}_{i}, {\hat{σ}}_{i}), π (1 - 1 / 3)]$ , obtained from Corollary t8 and Theorem t5, respectively.
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Figure 4
Exact power of a three-level mixed state charged with an optimal Hamiltonian and three bounds on power, with units $h = V = 1$ . The charging distance power bound is always tight for the optimal charging protocol when charging from the passive to the maximally charged state. For nonoptimal charging protocols, other bounds might be tighter.
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Figure 5
The schematic illustration of eigenvalues living in a complex space for $d = 4$ . By the changing global phase, eigenvalues rotate around zero conserving their relative angle. We always minimize $∥ i ln (U) ∥$ to $(π - Δ ϕ_{max} / 2)$ by this method.
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