Pontryagin-Optimal Control of a non-Hermitian Qubit

TL;DR

This paper explores the control of a non-Hermitian qubit, revealing topological features and developing optimal control strategies that enable high-purity state trajectories despite non-Hermitian effects and detection inefficiencies.

Contribution

It introduces a method to realize arbitrary pure-state trajectories in non-Hermitian qubits and applies Pontryagin's maximum principle to find optimal control paths.

Findings

Able to realize any pure-state trajectory conditioned on no emission

Derived optimal trajectories balancing dynamics and control energy

Maintained high state purity even with inefficient detection

Abstract

Open-system quantum dynamics described by non-Hermitian effective Hamiltonians have become a subject of considerable interest. Studies of non-Hermitian physics have revealed general principles, including relationships between the topology of the complex eigenvalue space and the breakdown of adiabatic control strategies. We study here the control of a single non-Hermitian qubit, similar to recently realized experimental systems in which the non-Hermiticity arises from an open spontaneous emission channel. We review the topological features of the resulting non-Hermitian Hamiltonian and then present two distinct results. First, we illustrate how to realize any continuous and differentiable pure-state trajectory in the dynamics of a qubit that are conditioned on no emission. This result implicitly provides a workaround for the breakdown of standard adiabatic following in such non-Hermitian systems. Second, we use Pontryagin's maximum principle to derive optimal trajectories connecting boundary states on the Bloch sphere, using a cost function which balances the desired dynamics against the controller energy used to realize them. We demonstrate that the latter approach can effectively find trajectories which maintain high state purity even in the case of inefficient detection.