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arXiv:1006.1317 · Physical Review A

Average entanglement for Markovian quantum trajectories

Sylvain Vogelsberger, Dominique Spehner

Published Nov 23, 2010• 24 citations
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Abstract

We study the evolution of the entanglement of noninteracting qubits coupled to reservoirs under monitoring of the reservoirs by means of continuous measurements. We calculate the average of the concurrence of the qubits wave function over all quantum trajectories. For two qubits coupled to independent baths subjected to local measurements, this average decays exponentially with a rate depending on the measurement scheme only. This contrasts with the known disappearance of entanglement after a finite time for the density matrix in the absence of measurements. For two qubits coupled to a common bath, the mean concurrence can vanish at discrete times. Our analysis applies to arbitrary quantum jump or quantum state diffusion dynamics in the Markov limit. We discuss the best measurement schemes to protect entanglement in specific examples.

DOI
10.1103/physreva.82.052327

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We study the evolution of the entanglement of noninteracting qubits coupled to reservoirs under monitoring of the reservoirs by means of continuous measurements. We calculate the average of the concurrence of the qubits wave function over all quantum trajectories. For two qubits coupled to independent baths subjected to local measurements, this average decays exponentially with a rate depending on the measurement scheme only. This contrasts with the known disappearance of entanglement after a finite time for the density matrix in the absence of measurements. For two qubits coupled to a common bath, the mean concurrence can vanish at discrete times. Our analysis applies to arbitrary quantum jump or quantum state diffusion dynamics in the Markov limit. We discuss the best measurement schemes to protect entanglement in specific examples.
Entanglement is a key resource in quantum information. It can be destroyed or sometimes created by interactions with a reservoir. In recent years, much attention has been devoted to the phenomena of entanglement sudden death and sudden birth, i.e., the sudden disappearance or revival of entanglement at finite times resulting from a coupling of the quantum system to its environment [1, 2, 3]. We investigate the evolution of the entanglement of noninteracting qubits coupled to reservoirs under monitoring of the reservoirs by means of continuous measurements. Because of these measurements, the qubits remain at all times in a pure state, which evolves randomly. To each measurement result (or "realization") corresponds a quantum trajectory in the Hilbert space of the qubits. We show that for two qubits coupled to independent baths subjected to local measurements, the average of the qubits' concurrence over all quantum trajectories is either constant or decays exponentially. The corresponding decay rate depends on the measurement scheme only. This result contrasts with the entanglement sudden death phenomenon exhibited by the qubits' density matrix in the absence of measurements. Our analysis applies to arbitrary quantum jump dynamics (photon counting) as well as to quantum state diffusion (homodyne or heterodyne detections) in the Markov limit. We discuss the best measurement schemes to protect the entanglement of the qubits. We also analyze the case of two qubits coupled to a common bath. Then, the average concurrence can vanish at discrete times and may coincide with the concurrence of the density matrix. The results explained in this article have been presented during the "Fifth International Workshop DICE2010" by the first author and have been the subject of a prior publication [4].
We present a comprehensive implementation of the quantum trajectory theory for the description of the entanglement dynamics in a Markovian open quantum system made of two qubits. We introduce the average concurrence to characterize the entanglement in the system and derive a deterministic evolution equation for it that depends on the ways in which information is read from the environment. This buildt-in flexibility of the method is used to address two actual issues in quantum information: entanglement protection and entanglement estimation. We identify general physical situations in which an entanglement protection protocol based on local monitoring of the environment can be implemented. Additionally, we methodically find unravelings of the system dynamics providing analytical tight bounds for the unmonitored entanglement in the system at all times. We conclude by showing the independence of the method from the choice of entanglement measure.
We analyze the dynamics of the entanglement in two independent non-Markovian channels. In particular, we focus on the entanglement dynamics as a function of the initial states and the channel parameters like the temperature and the ratio rrr between ω0ω_0ω0​ the characteristic frequency of the quantum system of interest, and ωcω_cωc​ the cut-off frequency of Ohmic reservoir. We give a stationary analysis of the concurrence and find that the dynamic of non-markovian entanglement concurrence Cρ(t)\mathcal{C}_ρ(t)Cρ​(t) at temperature kBT=0k_BT=0kB​T=0 is different from the kBT>0k_BT>0kB​T>0 case. We find that "entanglement sudden death" (ESD) depends on the initial state when kBT=0k_BT=0kB​T=0, otherwise the concurrence always disappear at finite time when kBT>0k_BT>0kB​T>0, which means that ESD must happen. The main result of this paper is that the non-Markovian entanglement dynamic is fundamentally different from the Markovian one. In the Markovian channel, entanglement decays exponentially and vanishes only asymptotically, but in the non-Markovian channel the concurrence Cρ(t)\mathcal{C}_ρ(t)Cρ​(t) oscillates, especially in the high temperature case. Then an open-loop controller adjusted by the temperature is proposed to control the entanglement and prolong the ESD time.
We analyze the dynamics of quantum correlations between two qubits coupled to a linear chain of oscillators. The chain mediates interactions between the qubits and acts as a non-Markovian reservoir. The model is amenable to an analytical solution when the initial state of the chain is Gaussian. We study the dynamics of the qubits' concurrence starting from a separable state and assuming that the chain spectrum is gapped and the chain is initially in a thermal state. We identify three relevant regimes that depend on the strength of the qubit-chain coupling in relation to the spectral gap. These are (i) the weak-coupling regime, where the qubits are entangled at the asymptotics, (ii) the strong-coupling regime, where the concurrence can exhibit collapses followed by revivals with exponentially attenuated amplitude, and (iii) the thermal damping regime, where the concurrence rapidly vanishes due to the chain's thermal excitations. In all cases, if entanglement is generated, this occurs after a finite time has elapsed. This time scale depends exponentially on the qubits' distance and is determined by the spectral properties of the chain. Entanglement irreversible decay, on the other hand, is due to the dissipative effect induced by the coupling with the chain and is controlled by the coupling strength between the chain and qubits. This study unravels the basic mechanisms leading to entanglement in a non-Markovian bath and allows one to identify the key resources for realizing quantum coherent dynamics of open systems.
The fully quantized model of double qubits coupled to a common bath is solved using the quantum state diffusion (QSD) approach in the non-Markovian regime. We have established the explicit time-local non-Markovian QSD equations for the two-qubit dissipative and dephasing models. Diffusive quantum trajectories are applied to the entanglement estimation of two-qubit systems in a non-Markovian regime. In both cases, non-Markovian features of entanglement evolution are revealed through quantum diffusive unravellings in the system state space.
The fully quantized model of double qubits coupled to a common bath is solved using the quantum state diffusion (QSD) approach in the non-Markovian regime. We have established the explicit time-local non-Markovian QSD equations for the two-qubit dissipative and dephasing models. Diffusive quantum trajectories are applied to the entanglement estimation of two-qubit systems in a non-Markovian regime. In both cases, non-Markovian features of entanglement evolution are revealed through quantum diffusive unravellings in the system state space.