Equivalent Hamiltonians for State to State Transfer in the Case of Partial Quantum Control

TL;DR

This paper characterizes all Hamiltonians that replicate a desired quantum state transfer within a given time, even when the target Hamiltonian is outside the controllable set, expanding understanding of quantum control possibilities.

Contribution

It provides a complete characterization of Hamiltonians achieving the same state transfer, including cases outside the control Lie algebra, with practical examples.

Findings

Identifies Hamiltonians equivalent to a target for state transfer

Includes cases where the target Hamiltonian is outside the control Lie algebra

Provides explicit example demonstrating the theory

Abstract

Given a fixed initial state, a desired Hamiltonian, and an amount of time, we provide a complete characterization of the set of Hamiltonians which perform the same action as the desired Hamiltonian on the state of interest. An example is provided where the desired Hamiltonian lies outside of the control Lie algebra, but implementable Hamiltonians exist within the characterization.