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. 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.

IEEE Conference on Decision and Control (CDC)Dec 1, 20142 citations

Some Mathematical Problems of Control of Quantum Systems

Alexander Nikolaevich Pechen

Journal of Mathematical SciencesJul 29, 20193 citations

Quantum Affine Algebras and Integrable Quantum Systems

Vyjayanthi Chari, Andrew Pressley

NATO ASI series. Series B : PhysicsJan 1, 19974 citations

Subspace stabilization analysis for a class of non-Markovian open quantum systems

Shikun Zhang, Kun Liu, Daoyi Dong +2 more

Studied in this article is non-Markovian open quantum systems parametrized by Hamiltonian $H$ , coupling operator $L$ , and memory kernel function $\ensuremath{\gamma}$, which is a proper candidate for describing the dynamics of various solid-state quantum information processing devices. We look into the subspace stabilization problem of the system from the perspective of dynamical systems and control. The problem translates itself into finding analytic conditions that characterize invariant and attractive subspaces. Necessary and sufficient conditions are found for subspace invariance based on algebraic computations, and sufficient conditions are derived for subspace attractivity by applying a double integral Lyapunov functional. Mathematical proof is given for those conditions and a numerical example is provided to illustrate the theoretical result.Studied in this article is non-Markovian open quantum systems parametrized by Hamiltonian $H$, coupling operator $L$, and memory kernel function $\ensuremath{\gamma}$, which is a proper candidate for describing the dynamics of various solid-state quantum information processing devices. We look into the subspace stabilization problem of the system from the perspective of dynamical systems and control. The problem translates itself into finding analytic conditions that characterize invariant and attractive subspaces. Necessary and sufficient conditions are found for subspace invariance based on algebraic computations, and sufficient conditions are derived for subspace attractivity by applying a double integral Lyapunov functional. Mathematical proof is given for those conditions and a numerical example is provided to illustrate the theoretical result.

arXiv:1901.10088Physical Review AApr 22, 20202 citations

Quantum Integrable Systems on a Classical Integrable Background

Andrii Nikolaevich Liashyk, Nicolai Reshetikhin, Ivan Andreevich Sechin

arXiv:2405.17865Communications in Mathematical PhysicsDec 8, 20254 citations

Quantum subspace controllability implying full controllability

Francesca Albertini, Domenico D’Alessandro

In the analysis of controllability of finite dimensional quantum systems, subspace controllability refers to the situation where the underlying Hilbert space splits into the direct sum of invariant subspaces, and, on each of such invariant subspaces, it is possible to generate any arbitrary unitary operation using appropriate control functions. This is a typical situation in the presence of symmetries for the dynamics. We investigate whether and when if subspace controllability is verified, the addition of an extra Hamiltonian to the dynamics implies full controllability of the system. Under the natural (and necessary) condition that the new Hamiltonian connects all the invariant subspaces, we show that this is always the case, except for a very specific case we shall describe. Even in this specific case, a weaker notion of controllability, controllability of the state ( Pure State Controllability ) is verified.In the analysis of controllability of finite dimensional quantum systems, subspace controllability refers to the situation where the underlying Hilbert space splits into the direct sum of invariant subspaces, and, on each of such invariant subspaces, it is possible to generate any arbitrary unitary operation using appropriate control functions. This is a typical situation in the presence of symmetries for the dynamics. We investigate whether and when if subspace controllability is verified, the addition of an extra Hamiltonian to the dynamics implies full controllability of the system. Under the natural (and necessary) condition that the new Hamiltonian connects all the invariant subspaces, we show that this is always the case, except for a very specific case we shall describe. Even in this specific case, a weaker notion of controllability, controllability of the state ( Pure State Controllability ) is verified.

Linear Algebra and its ApplicationsNov 7, 20241 citation

Studied in this article is non-Markovian open quantum systems parametrized by Hamiltonian

H

, coupling operator

L

, and memory kernel function $\ensuremath{\gamma}$, which is a proper candidate for describing the dynamics of various solid-state quantum information processing devices. We look into the subspace stabilization problem of the system from the perspective of dynamical systems and control. The problem translates itself into finding analytic conditions that characterize invariant and attractive subspaces. Necessary and sufficient conditions are found for subspace invariance based on algebraic computations, and sufficient conditions are derived for subspace attractivity by applying a double integral Lyapunov functional. Mathematical proof is given for those conditions and a numerical example is provided to illustrate the theoretical result.