Unified entropies and quantum speed limits for nonunitary dynamics

TL;DR

This paper introduces quantum speed limits based on unified quantum entropies for nonunitary processes, providing bounds that depend on the system's eigenvalues and the type of nonunitary evolution, with applications to quantum channels, non-Hermitian Hamiltonians, and many-body systems.

Contribution

It develops a unified framework for quantum speed limits using ($ ext{α,μ}$)-entropies applicable to various nonunitary dynamics, including channels and non-Hermitian evolutions, and explores their implications in many-body systems.

Findings

QSL depends on the smallest eigenvalue of the evolved state.

QSL becomes loose as the smallest eigenvalue approaches zero.

QSL decreases with increasing system size in the XXZ model.

Abstract

We discuss a class of quantum speed limits (QSLs) based on unified quantum ($\alpha,\mu$)-entropy for nonunitary physical processes. The bounds depend on both the Schatten speed and the smallest eigenvalue of the evolved state, and the two-parametric unified entropy. We specialize these results to quantum channels and non-Hermitian evolutions. In the first case, the QSL depends on the Kraus operators related to the quantum channel, while in the second case the speed limit is recast in terms of the non-Hermitian Hamiltonian. To illustrate these findings, we consider a single-qubit state evolving under the (i) amplitude damping channel, and (ii) the nonunitary dynamics generated by a parity-time-reversal symmetric non-Hermitian Hamiltonian. The QSL is nonzero at earlier times, while it becomes loose as the smallest eigenvalue of the evolved state approaches zero. Furthermore, we investigate the interplay between unified entropies and speed limits for the reduced dynamics of quantum many-body systems. The unified entropy is upper bounded by the quantum fluctuations of the Hamiltonian of the system, while the QSL is nonzero when entanglement is created by the nonunitary evolution. Finally, we apply these results to the XXZ model and verify that the QSL asymptotically decreases as the system size increases. Our results find applications to nonequilibrium thermodynamics, quantum metrology, and equilibration of many-body systems.