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arXiv:1510.01038 · Physical Review A

Dynamical invariants of open quantum systems

S. L. Wu, X. Y. Zhang, X. X. Yi

Published Dec 15, 2015• 12 citations
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Abstract

For a closed quantum system, a dynamical invariant is defined as an operator whose expectation value is a constant. In this paper, we extend the concept of dynamical invariants from closed systems to open systems. A dynamical equation for invariants (the dynamical invariant condition) is derived for Markovian dynamics. Different from dynamical invariants of closed quantum systems, the time evolution of dynamical invariants of open quantum systems is no longer unitary, and eigenvalues of any invariant are time dependent in general. Since any Hermitian operator which can fulfill the dynamical invariant condition is a dynamical invariant, we construct a type of special dynamical invariant of which a part of the eigenvalues is still constant. The dynamical invariants in the subspace spanned by these eigenstates thus evolve unitarily.

DOI
10.1103/physreva.92.062122

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For a closed quantum system, a dynamical invariant is defined as an operator whose expectation value is a constant. In this paper, we extend the concept of dynamical invariants from closed systems to open systems. A dynamical equation for invariants (the dynamical invariant condition) is derived for Markovian dynamics. Different from dynamical invariants of closed quantum systems, the time evolution of dynamical invariants of open quantum systems is no longer unitary, and eigenvalues of any invariant are time dependent in general. Since any Hermitian operator which can fulfill the dynamical invariant condition is a dynamical invariant, we construct a type of special dynamical invariant of which a part of the eigenvalues is still constant. The dynamical invariants in the subspace spanned by these eigenstates thus evolve unitarily.
We find dynamical invariants for open quantum systems described by the non-Markovian quantum-state-diffusion (QSD) equation. In stark contrast to closed systems where the dynamical invariant can be identical to the system density operator, these dynamical invariants no longer share the equation of motion for the density operator. Moreover, the invariants obtained with a biorthonormal basis can be used to render an exact solution to the QSD equation and the corresponding non-Markovian dynamics without using master equations or numerical simulations. Significantly we show that we can apply these dynamical invariants to reverse engineering a Hamiltonian that is capable of driving the system to the target state, providing a different way to design control strategy for open quantum systems.
A brief review of two recent topics on the Lindblad equation is presented. One is concerned with time evolution of the quantum Renyi entropy under the equation. The lower bound of the entropy rate is derived in a compact form. The other is about the concept of weak invariants, which generalize the Lewis-Riesenfeld invariant and are defined in such a way that they are not constant but their expectation values remain invariant in time. The Lindbladian operator describing the time-dependent damped quantum oscillator is identified, and the corresponding weak invariant is explicitly constructed as an example.
The unitary operator corresponding to the classical canonical transformation that connects a general closed system to an open system under adiabatic conditions is found. The quantum invariant operator of the adiabatic open system is derived from the unitary transformation of the quantum Hamiltonian of the closed system. On the basis of these results, we investigate the evolution of the general quantum adiabatic system and construct a revised adiabatic theorem. The adiabatic theorem developed here exactly reduces to the well-known Berry adiabatic theorem when the control parameter
We characterize the dynamical behavior of continuous-time, Markovian quantum systems with respect to a subsystem of interest. Markovian dynamics accurately describes a wide class of open quantum systems of relevance to quantum information processing, subsystem encodings offering a general pathway for faithfully representing quantum information. We provide explicit linear-algebraic characterizations of the notion of invariant and noiseless subsystem for Markovian master equations, under different robustness assumptions for model parameter and initial state variations. The stronger concept of an attractive quantum subsystem is introduced, and sufficient existence conditions are identified based on Lyapunov’s stability techniques. As a main control application, we address the potential of output-feedback Markovian control strategies for quantum pure state-stabilization and noiseless-subspace generation. In particular,
Weak invariants are time-dependent observables with conserved expectation values. Their fluctuations, however, do not remain constant in time. On the assumption that time evolution of the state of an open quantum system is given in terms of a completely positive map, the fluctuations monotonically grow even if the map is not unital, in contrast to the fact that monotonic increases of both the von Neumann entropy and Rényi entropy require the map to be unital. In this way, the weak invariants describe temporal asymmetry in a manner different from the entropies. A formula is presented for time evolution of the covariance matrix associated with the weak invariants in cases where the system density matrix obeys the Gorini-Kossakowski-Lindblad-Sudarshan equation.
We revise fundamental concepts in the dynamics of open quantum systems in the light of modern developments in the field. Our aim is to present a unified approach to the quantum evolution of open systems that incorporates the concepts and methods traditionally employed by different communities. We present in some detail the mathematical structure and the general properties of the dynamical maps underlying open system dynamics. We also discuss the microscopic derivation of dynamical equations, including both Markovian and non-Markovian evolutions.