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arXiv:1210.5449

Exact Algebraic Conditions for Indirect Controllability in Quantum Coherent Feedback Schemes

Domenico D’Alessandro, Francesca Albertini, Romano, Raffaele

Published Oct 19, 2012• 1 citation
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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.

DOI
10.48550/arxiv.1210.5449

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In several quantum control schemes, a target quantum system SSS is put in contact with an auxiliary system AAA and the coherent control can directly affect only AAA. The system SSS is controlled indirectly through the interaction with AAA. The system SSS is said to be indirectly controllable if every unitary transformation can be performed on the state of SSS with this scheme. The indirect controllability of SSS will depend on the dynamical Lie algebra L{\cal L}L characterizing the dynamics of the total system S+AS+AS+A and on the initial state of the auxiliary system AAA. In this paper, we describe this characterization exactly. A natural assumption is that the auxiliary system AAA is minimal, which means that there is no part of AAA that is not coupled to SSS, and we denote by nAn_AnA​ the dimension of such a minimal AAA, which we assume to be fully controllable. We show that if nAn_AnA​ is greater than or equal to 333, indirect controllability of SSS is verified if and only if complete controllability of the total system S+AS+AS+A is verified, i.e., L=su(nSnA){\cal L}=su(n_Sn_A)L=su(nS​nA​) or L=u(nSnA){\cal L}=u(n_Sn_A)L=u(nS​nA​), where nSn_SnS​ denotes the dimension of the system SSS. If nA=2n_A=2nA​=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{\cal L}_SLS​ which describes the evolution of the system SSS only. We prove that indirect controllability is verified if and only if LS=u(nS){\cal L}_S=u(n_S)LS​=u(nS​) and the initial state of the auxiliary system AAA is pure.
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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.
A quantum mechanical system S is indirectly controlled when the control affects an ancillary system A and the evolution of S is modified through the interaction with A only. A study of indirect controllability gives a description of the set of states that can be obtained for S with this scheme. In this paper, we study the indirect controllability of quantum systems in the finite dimensional case. After discussing the relevant definitions, we give a general necessary condition for controllability in Lie algebraic terms. We present a detailed treatment of the case where both systems, S and A, are two-dimensional (qubits). In particular, we characterize the dynamical Lie algebra associated with S + A, extending previous results, and prove that complete controllability of S + A and an appropriate notion of indirect controllability are equivalent properties for this system. We also prove several further indirect controllability properties for the system of two qubits, and illustrate the role of the Lie algebraic analysis in the study of reachable states.
In many experiments, a target quantum mechanical system is controlled and-or measured indirectly using an auxiliary quantum system. In the case of control, this means that the control action only affects the auxiliary system while the evolution of the target system is affected indirectly through the interaction with the auxiliary system. In the case of measurement, only the state of the auxiliary system can be measured, providing information on the initial state of the target system. Indirect controllability is the property of the target system of being driven between two arbitrary states while indirect observability refers to the possibility of extracting all the information on the state of the target system by measuring the auxiliary system. This paper has three main goals. First, we summarize recent notions and results introduced by us on the study of indirect controllability. Then, we present some new technical results on indirect controllability. In particular, we present a counterexample to the converse of a criterion for indirect controllability, which shows that the Lie algebraic condition introduced in this criterion is necessary but not sufficient for indirect controllability. This gives an open problem which is how to strengthen this condition to obtain a necessary and sufficient condition for indirect controllability. Lastly, we present a parallel treatment of indirect observability, give the relevant notions and definitions and relate the properties of indirect controllability and observability. Our final result says that strong notions of indirect controllability and observability are equivalent properties to complete controllability of the system. We discuss examples of physical systems which are controlled and-or measured indirectly. The research is motivated by experimental schemes of current interest
In this work, we study operator controllability of finite-dimensional closed quantum systems driven by a single coherent control. Two general Lie-algebraic results on controllability of such systems are proved. The first gives sufficient conditions under which controllability of the system is preserved when controlled energy-level shifts are added. The second gives a sufficient basis-element condition under which the dynamical Lie algebra contains su(N)\mathfrak{su}(N)su(N) and the system is therefore operator controllable. These results, together with known controllability conditions, are applied to obtain a complete classification of operator controllability of all three-level quantum systems in terms of their free and interaction Hamiltonian matrix elements. This classification covers all combinations of energy-level and level-spacing degeneracies, two or three allowed transitions, as well as the presence or absence of controlled energy-level shifts, and gives necessary and sufficient conditions for operator controllability in each case.