Exact Algebraic Conditions for Indirect Controllability in Quantum Coherent Feedback Schemes

TL;DR

This paper provides exact algebraic conditions under which a quantum system can be indirectly controlled via an auxiliary system in coherent feedback schemes, linking controllability to the dynamical Lie algebra structure.

Contribution

It characterizes precisely when indirect controllability is achievable in quantum feedback, especially relating it to the dynamical Lie algebra and auxiliary system dimension.

Findings

For auxiliary dimension ≥ 3, indirect controllability equals total system controllability.

When auxiliary dimension is 2, indirect controllability depends on a specific Lie algebra condition.

Initial pure state of auxiliary system A is necessary for indirect controllability when n_A=2.

Abstract

In coherent quantum feedback control schemes, a target quantum system S is put in contact with an auxiliary system A and the coherent control can directly affect only A. The system S is controlled 'indirectly' through the interaction with A. The system S is said to be indirectly controllable if every unitary transformation can be performed on the state of S with this scheme. The indirect controllability of S will depend on the `dynamical Lie algebra' L characterizing the dynamics of the total system S+A and on the initial state of the auxiliary system A. In this paper we describe this characterization exactly. A natural assumption is that the auxiliary system A is minimal which means that there is no part of A which is uncoupled to S, and we denote by n_A the dimension of such a minimal A, which we assume to be fully controllable. We show that, if n_A is greater than or equal to 3, indirect controllability of S is verified if and only if complete controllability of the total system S+A is verified, i.e., L=su(n_Sn_A) or L=u(n_Sn_A), where n_S denotes the dimension of the system S. If n_A=2, it is possible to have indirect controllability without having complete controllability. The exact condition for that to happen is given in terms of a Lie algebra L_S which describes the evolution on the system S only. We prove that indirect controllability is verified if and only if L_S=u(n_S), and the initial state of the auxiliary system A is pure.