Bhatia-Davis formula in the quantum speed limit

TL;DR

This paper introduces the Bhatia-Davis formula as a new upper bound for the quantum speed limit, linking variance bounds to quantum evolution times, and proves its validity in various quantum system scenarios.

Contribution

The paper formulates the Bhatia-Davis formula for quantum speed limits and proves its bounds in specific quantum systems, connecting variance bounds to evolution times.

Findings

Bhatia-Davis formula bounds quantum speed limit from above.

Validates the formula as a lower bound in symmetric energy structures.

Applicable to two-level and certain three-level quantum systems.

Abstract

The Bhatia-Davis theorem provides a useful upper bound for the variance in mathematics, and in quantum mechanics, the variance of a Hamiltonian is naturally connected to the quantum speed limit due to the Mandelstam-Tamm bound. Inspired by this connection, we construct a formula, referred to as the Bhatia-Davis formula, for the characterization of the quantum speed limit in the Bloch representation. We first prove that the Bhatia-Davis formula is an upper bound for a recently proposed operational definition of the quantum speed limit, which means it can be used to reveal the closeness between the timescale of certain chosen states to the systematic minimum timescale. In the case of the largest target angle, the Bhatia-Davis formula is proved to be a valid lower bound for the evolution time to reach the target when the energy structure is symmetric. Regarding few-level systems, it is also proved to be a valid lower bound for any state in two-level systems with any target, and for most mixed states with large target angles in equally spaced three-level systems.