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arXiv:1210.4953

Equivalence between indirect controllability and complete controllability for quantum systems

Domenico D’Alessandro

Published Oct 17, 2012• 0 citations
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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.

DOI
10.48550/arxiv.1210.4953

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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.
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.
In coherent 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. In this paper we show how indirect controllability of S is equivalent to complete controllability of the combined system S + A, if the dimension of A is ≥ 3. In the case where the dimension of A is equal to 2, it is possible to have indirect controllability without having complete controllability of S +A and we give sufficient conditions for this to happen. We conjecture that these conditions are also necessary. The results of the paper extend the result of [5] and expand the results of [6] to systems of arbitrary dimensions.
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.
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.
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.
Indirect quantum control aims to manipulate a target system through interventions on an auxiliary controller with which it interacts. We formulate finite-step indirect-control protocols using the language of higher-order quantum operations, representing admissible controller manipulations as deterministic superinstruments. This representation separates the uncontrolled but fixed dynamics from the controllable operations and allows the optimal control problem to be written as a semidefinite program for a chosen figure of merit. The resulting optimum gives an operational benchmark by providing the best performance achievable by a finite sequence of control operations, including protocols with classical or quantum feed-forward. More restricted and experimentally motivated control classes, such as independent unitary controls or memoryless quantum channels, can then be compared against this benchmark. We illustrate the framework with a two-step qubit-purification task, in which an initially mixed target qubit is steered toward a pure state through a fixed interaction with a controller qubit. The example shows regimes where simple unitary strategies saturate the higher-order benchmark, as well as regimes where they are provably suboptimal. This approach provides a systematic way to study the value of control resources in indirect quantum-control schemes.