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arXiv:1002.1515

Control-Induced Decoherence-Free Manifolds

EDMOND A. JONCKHEERE, Alireza Shabani, A. T. Rezakhani

Published Feb 8, 2010• 0 citations
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Abstract

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).

DOI
10.48550/arxiv.1002.1515

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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).
Quantum coherence of open quantum systems is usually compromised because of the interaction with the ambient environment. Here, we introduce the concept of Decoherence Splitting Manifold (DSM), a submanifold of the space of density operators, wherein the system's density matrix clearly exhibits a splitting between a unitarily evolving subdensity corresponding to some given set of its eigenvalues, which we aim to preserve, and another subdensity, in which eigenvalues are not preserved. The eigenspace of the eigenvalues that are preserved provides the concept of (time-varying) Decoherence Protected Subspace (DPS), a concept that subtlety differs from the (time-varying) Decoherence Free Subspace (DFS) in the way the protected subspace is embedded in the full Hilbert space. With the DSM concept, the DPS is reformulated as a complex vector bundle over the DSM. In the absence of the traditional DFS, looking for less restrictive spaces and/or utilizing quantum control may help generate and/or retain a decoherence-free subevolution. This is achieved by searching for a DSM whose tangent bundle distribution is invariant under indirect control of an auxiliary qubit.
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.
This paper discusses the concept of controllable subspace for open quantum dynamical systems.It is constructively demonstrated that combining structural features of decoherence-free subspaces with the ability to perform open-loop coherent control on open quantum systems will allow decoherence-free subspaces to be controllable.This is in contrast to the observation that open quantum dynamical systems are not open-loop controllable.To a certain extent, this paper gives an alternative control theoretical interpretation on why decoherence-free subspaces can be useful for quantum computation.