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Physicae Proceedings

Generation of bipartite bound entanglement

Frank E. S. Steinhoff

Published Mar 29, 2012• 0 citations
Publisher

Abstract

We introduce a class of entangled bipartite states that are positive under partial transposition operation,i.e., PPT bound entangled states. These states are based on realistical preparation procedures in optical systems, being thus a feasible option to generate and observe bipartite bound entanglement in high precision experiments. The method employed to detect entanglement has the advantages of being simple, analitical and independent of the range criterion. We also link PPT property of one-mode states that pass a beam-splitter to Hankel operator theory and related moment problems.

DOI
10.5196/physicae.proceedings.xiyrm.22

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We introduce a class of entangled bipartite states that are positive under partial transposition operation,i.e., PPT bound entangled states. These states are based on realistical preparation procedures in optical systems, being thus a feasible option to generate and observe bipartite bound entanglement in high precision experiments. The method employed to detect entanglement has the advantages of being simple, analitical and independent of the range criterion. We also link PPT property of one-mode states that pass a beam-splitter to Hankel operator theory and related moment problems.
We introduce a class of bipartite entangled continuous variable states that are positive under a partial transposition (PPT) operation, i.e., PPT bound entangled. These states are based on realistic preparation procedures in optical systems, being thus a feasible option to generate and observe genuinely bipartite bound entanglement in high precision experiments. One fundamental step in our scheme is to perform a non-Gaussian operation over a single-mode Gaussian state. This degaussification procedure is achieved through a modified single-photon addition, which is a procedure that is currently being investigated in diverse optical setups. Although dependent on a single-photon detection in an idler channel, the preparation can be made unconditional after a calibration of the apparatus. The detection and proof of bound entanglement is made by means of the range criterion, theory of Hankel operators and Gerschgorin Disk's perturbation theorems.
We develop a new method for entanglement detection in bipartite quantum states by using the violation of the rank-1-generated property of matrices. The positive-semidefinite matrices form a convex cone that has extremal elements of rank 1. But, convex conic subsets resulting from the presence of linear constraints allow extremal elements of rank >= 2. The problem of deciding when a matrix is rank-1 generated, i.e, a sum of rank-1 positive-semidefinite (PSD) matrices, has been studied extensively in optimization theory. This rank-1 generated property acts as an entanglement criterion, and we use this property to find novel classes of PPT (Positive under partial transposition) entangled states. We do this by mapping some faces of PPT density matrices to convex cones that are not rank-1 generated. We show that all separable states map to a rank-1 generated matrix. In general, the same is not true for the corresponding mapping of PPT entangled states. We also extend this approach to construct PPT entangled edge states. Finally, we provide witnesses that detect violations of the rank-1 generated property.
We construct a class of quantum bipartite $d\ensuremath{\bigotimes}d$ states which are positive under partial transposition (PPT states). This class is invariant under the maximal commutative subgroup of U(d)\mathrm{U}(d)U(d) and contains as special cases many well-known examples of PPT states. States from our class provide criteria for testing the indecomposability of positive maps. Such maps are crucial for constructing entanglement witnesses.
Moment-based relaxations of the positive partial transpose (PPT) criterion have been recently introduced, as a hierarchy of entanglement criteria involving only experimentally accessible quantities of a given bipartite state. The goal of this work is to study their typical detection performance on high-dimensional bipartite systems. Concretely, we investigate whether random bipartite mixed states on Cd⊗Cd\mathbb C^d\otimes\mathbb C^dCd⊗Cd, obtained as the marginal over an environment Cs\mathbb C^sCs of a uniformly distributed pure state, generically satisfy or violate them. For each fixed level m∈Nm\in\mathbb Nm∈N in this hierarchy of moment-based PPT criteria, we are able to identify a threshold environment dimension s=λmd2s=λ_md^2s=λm​d2 at which the behavior of the associated random state switches from violating to satisfying it, with probability going to 111 as ddd grows. The proof combines combinatorics of permutations techniques to estimate the average value of moments of partially transposed random states and concentration of measure arguments to bound the probability of deviating from such average, when the underlying local dimension ddd is large. We additionally need tools from the theory of Hankel determinant evaluation via orthogonal polynomials.
The positivity of the partial transpose is in general only a necessary condition for separability. There exist quantum states that are not separable, but nevertheless are positive under the partial transpose. States of this type are known as bound entangled states, meaning that these states are entangled but they do not allow distillation of pure entanglement by means of local operations and classical communication. We present a parametrization of a class of $2\ifmmode\times\else\texttimes\fi{}2$ bound entangled Gaussian states for bipartite continuous-variable quantum systems with two modes on each side. We propose an experimental protocol for preparing a particular bound entangled state in quantum optics. We then discuss the robustness properties of this protocol with respect to the occupation number of thermal inputs and the degrees of squeezing.