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IEEE Conference on Decision and Control (CDC)

Controllable quantities for bilinear quantum systems

Gabriel Turinici

Published Nov 11, 2002• 27 citations
Publisher

Abstract

The paper is dedicated to the search of tailored controllability concepts for quantum systems interacting with lasers. A negative result for infinite dimensional spaces serves as motivation for a finite dimensional analysis. We show that under physically reasonable hypothesis we can locally control sets of observables. As a remarkable particular case global exact controllability is proven for the population of the eigenstates.

DOI
10.1109/cdc.2000.912047

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The paper is dedicated to the search of tailored controllability concepts for quantum systems interacting with lasers. A negative result for infinite dimensional spaces serves as motivation for a finite dimensional analysis. We show that under physically reasonable hypothesis we can locally control sets of observables. As a remarkable particular case global exact controllability is proven for the population of the eigenstates.
Quantum phenomena of interest in connection with applications to computation and communication often involve generating specific transfers between eigenstates, and their linear superpositions. For some quantum systems, such as spin systems, the quantum evolution equation (the Schrödinger equation) is finite-dimensional and old results on controllability of systems defined on on Lie groups and quotient spaces provide most of what is needed insofar as controllability of non-dissipative systems is concerned. However, in an infinite-dimensional setting, controlling the evolution of quantum systems often presents difficulties, both conceptual and technical. In this paper we present a systematic approach to a class of such problems for which it is possible to avoid some of the technical issues. In particular, we analyze controllability for infinite-dimensional bilinear systems under assumptions that make controllability possible using trajectories lying in a nested family of pre-defined subspaces. This result, which we call the Finite Controllability Theorem, provides a set of sufficient conditions for controllability in an infinite-dimensional setting. We consider specific physical systems that are of interest for quantum computing, and provide insights into the types of quantum operations (gates) that may be developed.
The systems-theoretic concept of controllability is elaborated for quantum-mechanical systems, sufficient conditions being sought under which the state vector ψ can be guided in time to a chosen point in the Hilbert space ℋ of the system. The Schrödinger equation for a quantum object influenced by adjustable external fields provides a state-evolution equation which is linear in ψ and linear in the external controls (thus a bilinear control system). For such systems the existence of a dense analytic domain 𝒟ω in the sense of Nelson, together with the assumption that the Lie algebra associated with the system dynamics gives rise to a tangent space of constant finite dimension, permits the adaptation of the geometric approach developed for finite-dimensional bilinear and nonlinear control systems. Conditions are derived for global controllability on the intersection of 𝒟ω with a suitably defined finite-dimensional submanifold of the unit sphere Sℋ in ℋ. Several soluble examples are presented to illuminate the general theoretical results.
In the framework of bilinear control of the Schrödinger equation with bounded control operators, it has been proved that the reachable set has a dense complemement in S∩H2{\cal S}\cap {\cal H}^{2}S∩H2. Hence, in this setting, exact quantum control in infinite dimensions is not possible. On the other hand it is known that there is a simple choice of operators which, when applied to an arbitrary state, generate dense orbits in Hilbert space. Compatibility of these two results is established in this paper and, in particular, it is proved that the closure of the reachable set of bilinear control by bounded operators is dense in S∩H2{\cal S}\cap {\cal H}^{2}S∩H2. The requirements for controllability in infinite dimensions are also related to the properties of the infinite dimensional unitary group.
Quantum phenomena of interest in connection with computation and communication often deal with transfers between eigenstates. For systems having only a finite number of states, the quantum evolution equation is finite-dimensional and the results on controllability on Lie groups as worked out decades ago provide most of what is needed in so far as controllability of dissipationless systems is concerned. However, for infinite-dimensional evolution of quantum systems, many difficulties, both conceptual and technical, remain. In this paper we organize some recent results from the physics literature in control theoretic terms and emphasize the type of analysis needed to go beyond what basic differential geometry can provide. In particular, we analyze from a controllability point of view the important results of Law and Eberly, and discuss the problem of controllability of quantum systems subject to the constraint that the trajectories must lie in pre-defined subspaces. The study of this problem is motivated by recent work on the design of quantum computers that avoid loss of coherence by operating in decoherence free subspaces.