Equivalence between indirect controllability and complete controllability for quantum systems

TL;DR

This paper proves that, in finite-dimensional quantum systems, the ability to fully control a combined system is equivalent to controlling only the primary system indirectly through an auxiliary system, using Lie algebraic methods.

Contribution

It establishes the equivalence between complete controllability and indirect controllability for finite-dimensional quantum systems, providing a rigorous mathematical proof.

Findings

Equivalence holds under appropriate conditions and definitions.

Lie algebraic methods are used to prove the result.

The result clarifies the relationship between different controllability notions.

Abstract

We consider a control scheme where a quantum system S is put in contact with an auxiliary quantum system A and the control can affect A only, while S is the system of interest. The system S is then controlled indirectly through the interaction with A. Complete controllability of S +A means that every unitary state transformation for the system S +A can be achieved with this scheme. Indirect controllability means that every unitary transformation on the system S can be achieved. We prove in this paper, under appropriate conditions and definitions, that these two notions are equivalent in finite dimension. We use Lie algebraic methods to prove this result.