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arXiv:1304.1592 · Physical Review A

Bipartite bound entanglement in continuous variables through degaussification

Frank E. S. Steinhoff, M. C. de Oliveira, Jan Sperling, W. Vogel

Published Mar 11, 2014• 7 citations
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Abstract

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.

DOI
10.1103/physreva.89.032313

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The entanglement of Gaussian continuous-variable (CV) states is fully determined by the state's second moments. In contrast, some entangled non-Gaussian states evade every second-moment criterion, and non-Gaussian entanglement detection remains an experimental challenge. The p3p_3p3​-PPT criterion detects entanglement using moments of the partial transpose of the density matrix. This criterion was recently extended to CV systems using photon-number-resolving detectors and multi-copy interferometry; here we introduce a single-copy homodyne protocol that detects bipartite CV entanglement via the same criterion. Unbiased U-statistic estimators for the partial-transpose moments p2p_2p2​ and p3p_3p3​ are constructed directly from randomized homodyne data and used to evaluate the p3p_3p3​-PPT entanglement witnesses: a linear one for detection, and a quadratic one whose violation yields a dimension-free lower bound on the entanglement negativity. The protocol estimates p2p_2p2​ and p3p_3p3​ up to additive error ε\varepsilonε at Fock cutoff NNN from O((N+1)14/3/ε2)O((N+1)^{14/3}/\varepsilon^2)O((N+1)14/3/ε2) measurements at fixed confidence. We demonstrate the protocol on six families of Gaussian and non-Gaussian states, reaching 95%95\%95% empirical one-sided detection probability from ∼103\sim 10^3∼103 to 10410^4104 homodyne measurements for states with nˉ≈2\bar{n} \approx 2nˉ≈2, placing non-Gaussian entanglement detection within reach of current homodyne experiments.
Photon subtraction and addition are experimental means of generating non-Gaussian states from Gaussian states. Coherent subtraction or addition is a combination of photon subtractions or additions. The resultant states are quite general non-Gaussian states. The states can be photon number entangled states with arbitrary coefficients. We derive the entanglement conditions for several classes of coherent subtraction or coherent addition bipartite continuous variable states. One of the entanglement conditions is necessary and sufficient.
We use quantum entanglement witnesses derived from Gaussian operators to study the separable criteria of continuous variable states. We transform the validity of a Gaussian witness to a Bosonic Gaussian channel problem. It follows that the maximal means of two-mode and some four-mode Gaussian operators over product pure states are achieved by vacuum (or coherent states and squeezed states) according to the properties of Bosonic Gaussian channels. Then we have necessary and sufficient criteria of separability not only for Gaussian quantum states, but also for non-Gaussian states prepared by photon adding to or/and subtracting from Gaussian states. The criterion can be further explicitly expressed with covariance matrix of the Gaussian state or covariance matrix of Gaussian kernel of the non-Gaussian state. This opens a way for precise detection of non-Gaussian entanglement.