Optimal Dynamical Characterization of Entanglement

Abstract

We show that the time evolution of entanglement under incoherent environment coupling can be faithfully recovered by monitoring the system according to a suitable measurement scheme.PACS numbers: 03.67.Mn,03.65.Yz,42.50.Lc Quantum information processing requires the ability to produce entangled states and coherently perform operations on them. Under realistic laboratory conditions, however, entanglement is degraded through uncontrolled coupling to the environment. It is of crucial practical importance to quantify this degradation process [1][2][3], though also extremely difficult in general, due to the intricate mathematical notions upon which our understanding of entanglement relies [4][5][6]. Up to now, no general observable is known which would complement such essentially formal concepts with a specific experimental measurement setup.In the present Letter, we come up with a dynamical characterization of entanglement, through the continuous observation of a quantum system which evolves under incoherent coupling to an environment. We show that, at least for small, yet experimentally relevant systems, there is an optimal measurement strategy to monitor the entanglement of the time evolved, mixed system state. Mixed state entanglement is then given as the average entanglement of the pure states generated by single realisations of the optimal measurement-induced, stochastic time evolution.Consider a bipartite quantum system composed of subsystems A and B, interacting with its environment. Due to this coupling, an initially pure state |Ψ 0 of the composite system will evolve into a mixed state ρ(t), in a way governed by the master equatioṅwhere the Hamiltonian H generates the unitary system dynamics. The superoperators L k describe the effects of the environment on the system, and, for a Markovian bath, have the standard form [7]where the operators J k depend on the specific physical situation under study.To extract the time evolution of entanglement under this incoherent dynamics, one solution is to evaluate a given entanglement measure M (ρ) for the solution ρ(t), at all times t. One starts from one of the known pure state measures M (Ψ) [5,6,8], together with a pure state decomposition of ρ,where the p i are the positive, normalized weights of each pure state |Ψ i . The most naive generalization for a mixed state would then be to consider the averagewhich, however, is not suitable, since the decomposition (3) is not unique: M would thus give rise to different values of entanglement for different valid decompositions of ρ [9], inconsistently with the general requirements for a bona fide entanglement measure [5,6]. The proper definition of M (ρ) therefore is the infimum of all possible averages M [10], but holds two main drawbacks: (i) it turns into a hard numerical problem for higher dimensional or multipartite systems, and, (ii) even for bipartite qubits, where analytical solutions for some measures M (ρ) are known [8], there is no obvious interpretation of this optimal decomposition, in physical terms. Our a...

Vacuum as a less hostile environment to entanglement

“…This clearly shows that for pure states of the form (17) with |αδ − βγ| > |δ| 2 the entanglement is, under the vacuum decoherence, never completely lost [27]. There is an alternative way to arrive at this conclusion, using the result in Section III.…”

Physically realizable entanglement by local continuous measurements

“…The fact that one can reconstruct the state allows one to recover the average value of any observable in time by averaging over the trajectories. However, when it comes to obtaining the entanglement of the system, the restrictions imposed by the unraveled time evolution may be too strong [4].In this paper, we investigate equivalent local measurement strategies over the independent reservoirs that cause two entangled qubits to lose coherence and entanglement in time. The restriction on the ensembles that can be continuously generated in time becomes clear when we compare the minimum and maximum average entanglement [5] resulting from locally realizable unravellings to the extremes over all possible decompositions of a given density matrix, respectively the entanglement of formation E F [6] and the entanglement of assistance E A [7].…”

“…In this case, classical communication is required given that a single decay completely destroys the entanglement of the system. Also note that, in this case, it is particularly important to use E F as a lower bound for the average entanglement since the Wootters concurrence, for example, still gives the same average values for all unravellings, as shown in [4,15].…”

“…The fact that one can reconstruct the state allows one to recover the average value of any observable in time by averaging over the trajectories. However, when it comes to obtaining the entanglement of the system, the restrictions imposed by the unraveled time evolution may be too strong [4].…”

“…When restricted to U = U A ⊗ U B this decomposition always decreases the average entanglement as should be expected given that the added noise blurs the outcomes of the measurements on the reservoirs. This fact is used in [4] to minimize the average concurrence.…”