Generalized Entropic Quantum Speed Limits

TL;DR

This paper introduces a new class of quantum speed limits based on $ ext{α-}z$-Rényi relative entropy, applicable to various quantum systems and states, with potential applications in quantum information and metrology.

Contribution

It develops generalized entropic quantum speed limits using $ ext{α-}z$-Rényi entropy, covering both unitary and nonunitary evolutions with low computational cost.

Findings

QSL depends on eigenvalues of states and system dynamics.

QSL scales with inverse energy fluctuations for unitary evolution.

Results demonstrated on single-qubit and two-qubit systems.

Abstract

We present a class of generalized entropic quantum speed limits based on $\alpha$-$z$-R\'{e}nyi relative entropy, a real-valued, contractive, two-parameter family of distinguishability measures. The quantum speed limit (QSL) falls into the class of Mandelstam-Tamm bounds, and applies to finite-dimensional quantum systems that undergo a general physical process, i.e., their effective dynamics can be modeled by unitary or nonunitary evolutions. The results cover pure or mixed, separable, and entangled probe quantum states. The QSL time depends on the smallest and largest eigenvalues of the probe and instantaneous states of the system, and its evaluation requires low computational cost. In addition, it is inversely proportional to the time-average of the Schatten speed of the instantaneous state, which in turn is fully characterized by the considered dynamics. We specialize our results to the case of unitary and nonunitary evolutions. In the former case, the QSL scales with the inverse of the energy fluctuations, while the latter depends on the Schatten $1$-norm of the rate of change of the quantum channel Kraus operators. We illustrate our findings for single-qubit and two-qubit states, and unitary and nonunitary evolutions. Our results may find applications in the study of entropic uncertainty relations, quantum metrology, and also entanglement entropies signaled by generalized entropies.