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Journal of Physics A Mathematical and General

Measures of entanglement based on decoherence

Ph. Blanchard, Lech Jakóbczyk, Robert Olkiewicz

Published Oct 8, 2001• 7 citations
Publisher

Abstract

New measures of entanglement - the time of disentanglement and the rate of decoherence - of states of a compound system are proposed. They are based on the dynamical properties of the system induced by the measurement-like interaction with an environment. In particular, it is shown that in the case of two qubits the time of disentanglement may serve as a practical measure of entanglement of mixed states.

DOI
10.1088/0305-4470/34/41/308

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New measures of entanglement - the time of disentanglement and the rate of decoherence - of states of a compound system are proposed. They are based on the dynamical properties of the system induced by the measurement-like interaction with an environment. In particular, it is shown that in the case of two qubits the time of disentanglement may serve as a practical measure of entanglement of mixed states.
We analyze the dynamics of entanglement between two qubits which interact through collective and local environments. Our approach is based on a resonance theory which assumes a small interaction between qubits and environments and which gives rigorous perturbation theory results, valid for all times. We obtain expressions for (i) characteristic time-scales for decoherence, relaxation, disentanglement, and for (ii) the evolution of observables, valid uniformly in time t≥0t\geq 0t≥0. We introduce a classification of decoherence times based on clustering of the reduced density matrix elements, persisting on all time-scales. We examine characteristic dynamical properties such as creation, death and revival of entanglement. We discuss possible applications of our results for superconducting quantum computation and quantum measurement technologies.
Entanglement in quantum systems is usually degraded by interaction with the environment. From time to time, some parties of a multipartite entangled system may become decoherent with other parties of the system due to the interference with the environment. In this situation, it is interesting to know how much information the residual system would keep on carrying. In this paper, as a starting point for any entangled system, we investigate the property of the 2 x N x M state with qubit being disentangled, which is characterized by the measurement of robustness.
Interaction with environment may lead to the transition of quantum system from pure state to the mixed one. In this case, the problem of definition of entanglement may arise. In particular, quantitative measure of entanglement concurrence is introduced differently for pure and mixed states. To solve this problem, alternative definition of concurrence of mixed states is introduced in present work. In particular, concurrence of mixed states is determined as an average of pure states concurrences. Averaging is carried out over the quantum ensemble, corresponding to all possible realization of random variable.
The decoherence of entanglement of two-qubit states is investigated by means of the non-Markovian quantum master equation. It is assumed that the decoherence is caused by environmental systems in non-stationary states, which evolve from initial states into equilibrium states. In this case, the non-Markovian nature of the environmental systems is of essential importance. It is shown that the disentanglement time after which there is no qubit entanglement is determined by the non-Markovian effect and the thermal effect of the environmental systems.
We show that the methods for quantification of system-environment entanglement that were recently developed for interactions that lead to pure decoherence of the system can be straightforwardly generalized to time-dependent Hamiltonians of the same type. This includes the if-and-only-if criteria of separability, as well as the entanglement measure applicable to qubit systems, and methods of detection of entanglement by operations and measurements performed solely on the system without accessing the environment. We use these methods to study the nature of the decoherence of a qubit-oscillator system. Qubit-oscillator entanglement is essential for developing bosonic quantum technology with quantum non-Gaussian states and its applications in quantum sensing and computing. The dominating bosonic platforms---trapped ions, electromechanics, and superconducting circuits---are based on the time-dependent gates that use such entanglement to achieve new quantum sensors and quantum error correction. The steplike time dependence of the Hamiltonian that is taken into account allows us to capture the complex interplay between the buildups of classical and quantum correlations, which could not be replicated in time-independent scenarios.
Quantum decoherence refers to the phenomenon when the interaction of a quantum system with its environment results in the degradation of quantum coherence. Decoherence is considered to be the most popular mechanism responsible for the emergence of classicality from quantum mechanics. The issue of formulating a measure of decoherence is addressed here. The approach taken here is that decoherence results from the entanglement of a quantum system with certain environment degrees of freedom, and quantifying this entanglement should yield the most natural measure of decoherence. A simple measure of decoherence is presented based on this notion, and it is examined for various example systems. The measure proves to be effective and is relatively straightforward to compute. In addition, a method has been proposed to measure decoherence in a Mach-Zehnder interferometer which may be useful in neutron interferometry.