Lower bound of minimal time evolution in quantum mechanics

TL;DR

This paper establishes a fundamental lower bound on the total evolution time for quantum states returning to their initial state, regardless of whether the Hamiltonian is Hermitian or non-Hermitian, highlighting intrinsic temporal constraints in quantum dynamics.

Contribution

It demonstrates that the minimal total evolution time for a quantum state to undergo a round trip on the Bloch sphere is fixed and independent of Hamiltonian Hermiticity, providing a new understanding of quantum speed limits.

Findings

The minimal total evolution time is the same for Hermitian and non-Hermitian Hamiltonians.

The minimal time condition relates to the orthogonality of polarization vectors in quantum states.

Hamiltonian parameters can be reduced to two independent variables for analysis.

Abstract

We show that the total time of evolution from the initial quantum state to final quantum state and then back to the initial state, i.e., making a round trip along the great circle over S^2, must have a lower bound in quantum mechanics, if the difference between two eigenstates of the 2\times 2 Hamiltonian is kept fixed. Even the non-hermitian quantum mechanics can not reduce it to arbitrarily small value. In fact, we show that whether one uses a hermitian Hamiltonian or a non-hermitian, the required minimal total time of evolution is same. It is argued that in hermitian quantum mechanics the condition for minimal time evolution can be understood as a constraint coming from the orthogonality of the polarization vector \bf P of the evolving quantum state \rho={1/2}(\bf 1+ \bf{P}\cdot\boldsymbol{\sigma}) with the vector \boldsymbol{\mathcal O}(\Theta) of the 2\times 2 hermitian Hamiltonians H ={1/2}({\mathcal O}_0\boldsymbol{1}+ \boldsymbol{\mathcal O}(\Theta)\cdot\boldsymbol{\sigma}) and it is shown that the Hamiltonian H can be parameterized by two independent parameters {\mathcal O}_0 and \Theta.