Quantum Brachistochrone Problem and the Geometry of the State Space in Pseudo-Hermitian Quantum Mechanics

TL;DR

This paper explores how non-Hermitian Hamiltonians with real spectra can define unitary quantum systems through inner product adjustments, analyzing the geometry of state space and the limits of evolution speed.

Contribution

It provides a detailed geometric analysis of pseudo-Hermitian quantum mechanics and clarifies the limitations of non-Hermitian Hamiltonians in achieving faster quantum evolutions.

Findings

Non-Hermitian Hamiltonians with real spectra can define unitary systems with appropriate inner products.

The quantum Brachistochrone problem cannot be circumvented using PT-symmetric or non-Hermitian Hamiltonians.

Faster quantum evolutions are not achievable beyond Hermitian Hamiltonian limits.

Abstract

A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining inner product of the physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.