2012/10/19 by Domenico D'Alessandro, Domenico D’Alessandro, D'Alessandro, Domenico +4
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #FOS: Physical sciences #Mechanical and Optical Resonators #Quantum Information and Cryptography #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph) #Receptor Mechanisms and Signaling #quant-ph
paper · pdf · doi:10.48550/arxiv.1210.5449
Expanded version of the previously posted paper where what was before mentioned as a conjecture is proved
openalex publication_date 2012/10/19 · arxiv created 2013/09/16 · arxiv updated 2013/09/18 · openalex created_date 2022/10/07 · openalex updated_date 2026/08/04
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 nA the dimension of such a minimal A, which we assume to be fully controllable. We show that, if nA 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(nSnA) or L=u(nSnA), where nS denotes the dimension of the system S. If nA=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 LS which describes the evolution on the system S only. We prove that indirect controllability is verified if and only if LS=u(nS), and the initial state of the auxiliary system A is pure.