Entanglement in general two-mode continuous-variable states: Local approach and mapping to a two-qubit system

Abstract

We present a new approach to the analysis of entanglement in smooth bipartite continuousvariable states. One or both parties perform projective filterings via preliminary measurements to determine whether the system is located in some region of space; we study the entanglement remaining after filtering. For small regions, a two-mode system can be approximated by a pair of qubits and its entanglement fully characterized, even for mixed states. Our approach may be extended to any smooth bipartite pure state or two-mode mixed state, leading to natural definitions of concurrence and negativity densities. For Gaussian states both these quantities are constant throughout configuration space.There has been growing interest in the quantification of entanglement in quantum systems. However, many systems of interest have continuous, rather than discrete, degrees of freedom [1]. The general characterization of entanglement in such continous-variable systems is a very difficult problem; most of the known results are for Gaussian states, where the logarithmic negativity [2] can be calculated for arbitrary bipartite divisions [3]. The entanglement of formation [4] is known exactly only for symmetric bipartite Gaussian states [5], and further measures of entanglement in multipartite Gaussian states are a topic of active current research [1,6]. For non-Gaussian states there is some progress in finding criteria for entanglement [7], but much less in quantifying it.In this letter we present a new approach to this problem, allowing entanglement to be quantified locally in arbitrary (including non-Gaussian) continuous-variable states. We concentrate on entanglement localized near particular regions in configuration space, which we analyse via a thought experiment in which the entangled state is first measured to localise it. This corresponds to a particular type of projective filtering, used to identify the distribution of entanglement in a state which has a preexisting bipartite structure. This contrasts with previous studies [8,9,10] of the entanglement of a finite region of space with the rest of the system. We show that in the limit where the sizes of the measured regions become very small the description of each mode in the system becomes isomorphic to a single qubit. In two-mode systems simple expressions result for entanglement monotones (concurrence and negativity), yielding natural definitions for corresponding densities in configuration space.The filtering process. Let Alice and Bob share a state of two distinguishable one-dimensional particles (or two modes). Alice can measure only the position of her particle or mode (coordinate q A ), Bob the position of his (coordinate q B ). They filter their state by determining whether or not the particles are found in particular regions of configuration space, and discard instances in which they are not. We refer to the resulting subensemble as the 'discarding ensemble'. For example, if Alice measures whether her coordinate is in a sub-region a, the corresponding projector iŝwhere...