Exact Entanglement-Depth Speed Frontier for Complete Quantum Charging

TL;DR

This paper derives an exact speed limit frontier for quantum charging, linking the charging rate to the minimum entanglement depth required, and shows that faster charging certifies genuine multipartite entanglement.

Contribution

It provides an exact solution to the speed limit problem under entanglement constraints and introduces a mechanism connecting charging speed to entanglement depth certification.

Findings

The maximum normalized charging rate is (k)= N/k^{-1/2}.

An observed rate  certifies entanglement depth at least  N/.

Crossing the threshold >1/2 certifies genuine N-partite entanglement for N>1.

Abstract

Complete quantum charging provides a sharp setting in which to ask how much multipartite entanglement is forced by speed itself. For a closed \(N\)-qubit battery evolving from \(\ket{\downarrow}^{\otimes N}\) to \(\ket{\uparrow}^{\otimes N}\) under a time-independent Hamiltonian, we exactly solve the pure-state depth-constrained speed problem. If the realized trajectory has entanglement depth at most \(k\), then the largest possible QSL-normalized rate \(\eta=\tau_{\rm QSL}/T\) is \(\eta_{\max}(k)=\lceil N/k\rceil^{-1/2}\). Conversely, an observed rate \(\eta\) certifies trajectory entanglement depth at least \(\bigl\lceil N/\lfloor \eta^{-2}\rfloor\bigr\rceil\). The mechanism is block orthogonalization: under a fixed product partition, complete charging forces all blocks to orthogonalize simultaneously, and the quantum speed limit converts this counting constraint into the speed bound. Balanced cluster-flip evolutions saturate the bound, establishing an exact integer staircase frontier. Thus fast complete charging cannot be explained by many small independently charging blocks; in particular, crossing the threshold \(\eta>1/\sqrt2\) certifies, for \(N>1\), the generation of genuine \(N\)-partite entanglement.