An optimum Hamiltonian for non-Hermitian quantum evolution and the complex Bloch sphere

TL;DR

This paper develops a method to find the optimal non-Hermitian Hamiltonian that transforms a quantum state into another in minimal time, using the complex Bloch sphere representation.

Contribution

It introduces a framework for determining the optimal Hamiltonian for nonunitary quantum evolution based on the complex Bloch sphere geometry.

Findings

Derived a formula for the minimal evolution time τ

Established a relationship between initial and final states on the complex Bloch sphere

Provided a method to construct the optimal Hamiltonian for state transformation

Abstract

For a quantum system governed by a non-Hermitian Hamiltonian, we studied the problem of obtaining an optimum Hamiltonian that generates nonunitary transformations of a given initial state into a certain final state in the smallest time $\tau$. The analysis is based on the relationship between the states of the two-dimensional subspace of the Hilbert space spanned by the initial and final states and the points of the two-dimensional complex Bloch sphere.