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IEEE Conference on Decision and Control (CDC)

Indirect control invariance of Decoherence-Splitting Manifold (DSM)

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

Published Dec 1, 2014• 2 citations
Publisher

Abstract

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.

DOI
10.1109/cdc.2014.7040296

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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.
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).
When a quantum system interacts with its environment, the so-called decoherence effect will normally destroy the coherence in the quantum state and the entanglement between its subsystems. We propose a feedback control strategy based on quantum weak measurements to protect coherence and entanglement of the quantum state against environmental disturbance. For a one-qubit quantum system under amplitude damping and dephasing decoherence channels, our strategy can preserve the coherence based on the measured information about the population difference between its two levels. For a two-qubit quantum system disentangled by independent amplitude damping and dephasing decoherence channels, the designed feedback control can preserve coherence between the ground state and the highest excited states by tuning the coupling strength between the two qubits, and at the same time minimize the loss of entanglement between the two qubits. As a consequence of dynamic symmetry, the generalization of these results derives the concept of control-induced decoherence-free observable subspace, for which several criteria are provided.
To maintain state coherence in an open quantum systems, the decoherence-free subspace (DFS) is an effective method. The target state is usually needed to be in DFS. In the case the target state is not in DFS, we first design external control field I to construct a DFS included target state. Then another control field II is used to transfer the state of the system to be controlled to the expected target state. As is well known, invariant set contains all stable states of the system. Control Hamiltonian is designed to shrink invariant set and increase the transition probability from the system state to target state. In such a way, the state transfer and coherence maintain are completed. Numerical simulation is given for a four-level energy open quantum system to demonstrate the effectiveness of the control strategy proposed.
Known as decoherence, the unavoidable interaction of a quantum system with its surrounding environment is usually considered to be detrimental for quantum information processing. In this thesis the coherent, open loop control of such open systems is studied. Concepts from quantum control theory and the theory of open quantum system are adopted in order to fight decoherence and implement quantum gates in a noiseless manner. In particular, Lie algebraic methods and numerical optimization tools are used to investigate the control properties of a single spin interacting with a spin environment. We show that, independent of the size of the environment, every unitary transformation can be implemented on the system spin through a single control field. We proceed by investigating dynamical decoupling, a method to suppress the interactions with the environment, for finite- and for in- finite dimensional systems. We prove that every finite dimensional system can be protected from decoherence, even if the environment is infinite dimensional, whereas for noise described by a Lindblad master equation dynamical decoupling will never succeed. This will lead to a new method to distinguish decoherence from intrinsic noise terms. We further prove that not every infinite dimensional system can be protected from decoherence through dynamical decoupling. Afterwards we investigate dynamical decoupling of systems that are described by quadratic Hamiltonians, showing that such interactions can always be suppressed with two simple operations. In the last part we investigate the coherent control of a Lindblad master equation. We show that a strong noise process exhibiting a decoherence free subspace can substantially increase the number of unitary operations that can be implemented, allowing us to fully control parts of the system. Afterwards we develop a scheme to make Hamiltonians and Lindbladians commutative by adding an auxiliary system. The old, possibly non-commutative dynamics, is recovered through a non-selective measurement.
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.