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arXiv:quant-ph/0506268 · IEEE Transactions on Automatic Control

Feedback Stabilization of Isospectral Control Systems on Complex Flag Manifolds: Application to Quantum Ensembles

Claudio Altafini

Published Nov 1, 2007• 81 citations
arXivHTMLPDF

Abstract

The convex set of density operators of an N-level quantum mechanical system foliated as a complex flag manifold, where each leaf is identified with the adjoint unitary orbit of the eigenvalues of a density matrix. For an isospectral bilinear control system evolving on such an orbit, the state feedback stabilization problem admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a ldquoroot-spacerdquo-like structure of the cone of density operators. The converging conditions are time independent but depend on the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing topological obstructions to global stabilizability.

DOI
10.1109/tac.2007.908306

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The convex set of density operators of an N-level quantum mechanical system foliated as a complex flag manifold, where each leaf is identified with the adjoint unitary orbit of the eigenvalues of a density matrix. For an isospectral bilinear control system evolving on such an orbit, the state feedback stabilization problem admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a ldquoroot-spacerdquo-like structure of the cone of density operators. The converging conditions are time independent but depend on the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing topological obstructions to global stabilizability.
In a closed N-level quantum mechanical system, the problem of unitary feedback stabilization of mixed density operators to periodic orbits admits a natural Lyapunov-based time-varying feedback design. A global description of the domain of attraction of the closed-loop system can be provided based on a "root-space"-like structure of the space of density operators. This convex set foliates as a complex flag manifold where each leaf is identified with the adjoint orbit of the eigenvalues of the density operator. The converging conditions are time-independent but depend from the topology of the flag manifold: it is shown that the closed loop must have a number of equilibria at least equal to the Euler characteristic of the manifold, thus imposing obstructions of topological nature to global stabilizability
Quantum coherence of open quantum systems is usually compromised because of the interaction with the ambient environment. A "decoherence-free subspace" (DFS) of the system Hilbert space is defined where the evolution remains unitary. In the absence of a priori existence of such subspaces, it seems natural that utilizing quantum control may help generate and/or retain a DFS. Here, we introduce a time-varying DFS wherein the system's density matrix has a unitarily evolving sub-density corresponding to some given set of its eigenvalues (which we aim to preserve). This subspace is characterized from both topological and algebraic perspectives. In particular, we show that this DFS admits a complex vector bundle structure over a real-analytic manifold (the decoherence-free manifold).
Quantum coherence of open quantum systems is usually compromised because of the interaction with the ambient environment. A ¿decoherence-free subspace¿ (DFS) of the system Hilbert space is defined where the evolution remains unitary. In the absence of a priori existence of such subspaces, it seems natural that utilizing quantum control may help generate and/or retain a DFS. Here, we introduce a time-varying DFS wherein the system's density matrix has a unitarily evolving sub-density corresponding to some given set of its eigenvalues (which we aim to preserve). This subspace is characterized from both topological and algebraic perspectives. In particular, we show that this DFS admits a complex vector bundle structure over a real-analytic manifold (the decoherence-free manifold).
The natural trajectory tracking problem is studied for generic quantum states represented by density operators. A control design based on the Hilbert-Schmidt distance as a Lyapunov function is considered. The control dynamics is redefined on an extended space where the LaSalle invariance principle can be correctly applied even for non-stationary target states. LaSalle's invariance principle is used to derive a general characterization of the invariant set, which is shown to always contain the critical points of the Lyapunov function. Critical point analysis of the latter is used to show that, for generic states, it is a Morse function with n! isolated critical points, including one global minimum, one global maximum and n!-2 saddles. It is also shown, however, that the actual dynamics of the system is not a gradient flow, and therefore a full eigenvalue analysis of the linearized dynamics about the critical points of the dynamical system is necessary to ascertain stability of the critical points. This analysis shows that a generic target state is locally asymptotically stable if the linearized system is controllable and the invariant set is regular, and in fact convergence to the target state (trajectory) in this case is almost global in that the stable manifolds of all other critical points form a subset of measure zero of the state space. On the other hand, if either of these sufficient conditions is not satisfied, the target state ceases to be asymptotically stable, a center manifold emerges around the target state, and the control design ceases to be effective.
A Lyapunov-based method is presented for stabilizing and controlling of closed quantum systems. The proposed method is constructed upon a novel quantum Lyapunov function of the system state trajectory tracking error. A positive-definite operator in the Lyapunov function provides additional degrees of freedom for the designer. The stabilization process is analyzed regarding two distinct cases for this operator in terms of its vanishing or non-vanishing commutation with the Hamiltonian operator of the undriven quantum system. To cope with the global phase invariance of quantum states as a result of the quantum projective measurement postulate, equivalence classes of quantum states are defined and used in the proposed Lyapunov-based analysis and design. Results show significant improvement in both the set of stabilizable quantum systems and their invariant sets of state trajectories generated by designed control signals. The proposed method can potentially be applied for high-fidelity quantum control purposes in quantum computing frameworks.