Quantum Control Landscape of Bipartite Systems

TL;DR

This paper investigates the control landscape of bipartite quantum systems where only one subsystem is accessible, providing conditions for trap-free control and analyzing the role of extended control parameters.

Contribution

It introduces an extended control landscape framework for bipartite quantum systems and derives conditions ensuring trap-free optimization of target unitaries.

Findings

Spectral decomposition simplifies landscape analysis.

Rank condition guarantees trap-free control search.

Extended landscape parameters can compensate for control limitations.

Abstract

The control landscape of a quantum system $A$ interacting with another quantum system $B$ is studied. Only system $A$ is accessible through time dependent controls, while system B is not accessible. The objective is to find controls that implement a desired unitary transformation on $A$, regardless of the evolution on $B$, at a sufficiently large final time. The freedom in the evolution on $B$ is used to define an \emph{extended control landscape} on which the critical points are investigated in terms of kinematic and dynamic gradients. A spectral decomposition of the corresponding extended unitary system simplifies the landscape analysis which provides: (i) a sufficient condition on the rank of the dynamic gradient of the extended landscape that guarantees a trap free search for the final time unitary matrix of system $A$, and (ii) a detailed decomposition of the components of the overall dynamic gradient matrix. Consequently, if the rank condition is satisfied, a gradient algorithm will find the controls that implements the target unitary on system $A$. It is shown that even if the dynamic gradient with respect to the controls alone is not full rank, the additional flexibility due to the parameters that define the extended landscape still can allow for the rank condition of the extended landscape to hold. Moreover, satisfaction of the latter rank condition subsumes any assumptions about controllability, reachability and control resources. Here satisfaction of the rank condition is taken as an assumption. The conditions which ensure that it holds remain an open research question. We lend some numerical support with two common examples for which the rank condition holds.