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arXiv:1006.1494 · Physical Review A

Detection of bound entanglement in continuous-variable systems

Chengjie Zhang, Hyunchul Nha, Yong-Sheng Zhang, Guang-Can Guo

Published Sep 23, 2010• 8 citations
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Abstract

We present several entanglement conditions in order to detect bound entangled states in continuous-variable systems. Specifically, Werner and Wolf [Phys. Rev. Lett. 86, 3658 (2001)] and Horodecki and Lewenstein [Phys. Rev. Lett. 85, 2657 (2000)] have proposed examples of a bound entangled Gaussian state and a bound entangled non-Gaussian state, respectively, of which entanglement can be detected by using our entanglement conditions.

DOI
10.1103/physreva.82.032323

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We present several entanglement conditions in order to detect bound entangled states in continuous-variable systems. Specifically, Werner and Wolf [Phys. Rev. Lett. 86, 3658 (2001)] and Horodecki and Lewenstein [Phys. Rev. Lett. 85, 2657 (2000)] have proposed examples of a bound entangled Gaussian state and a bound entangled non-Gaussian state, respectively, of which entanglement can be detected by using our entanglement conditions.
We study the detection of continuous-variable entanglement, for which most of the existing methods designed so far require a full specification of the devices, and we present protocols for entanglement detection in a scenario where the measurement devices are completely uncharacterized. We first generalize, to the continuous variable regime, the seminal results by Buscemi [Phys. Rev. Lett. 108, 200401 (2012)PRLTAO0031-900710.1103/PhysRevLett.108.200401] and Branciard et al. [Phys. Rev. Lett. 110, 060405 (2013)PRLTAO0031-900710.1103/PhysRevLett.110.060405], showing that all entangled states can be detected in this scenario. Most importantly, we then describe a practical protocol that allows for the measurement-device-independent certification of entanglement of all two-mode entangled Gaussian states. This protocol is feasible with current technology as it makes use only of standard optical setups such as coherent states and homodyne measurements.
Currently available separability criteria for continuous-variable states are generally based on the covariance matrix of quadrature operators. The well-known separability criterion of Duan et al. [L. M. Duan et al., Phys. Rev. Lett. 84, 2722 (2000)] and Simon [R. Simon, Phys. Rev. Lett. 84, 2726 (2000)], for example, gives a necessary and sufficient condition for a two-mode Gaussian state to be separable, but leaves many entangled non-Gaussian states undetected. Here we introduce an improvement of this criterion that enables a stronger entanglement detection. The improved condition is based on the knowledge of an additional parameter, namely, the degree of Gaussianity, and exploits a connection with Gaussianity-bounded uncertainty relations [A. Mandilara and N. J. Cerf, Phys. Rev. A 86, 030102 (2012)]. We exhibit families of non-Gaussian entangled states whose entanglement remains undetected by the Duan-Simon criterion.
Entanglement detection and estimation are fundamental problems in quantum information science. Compared with discrete-variable states, for which lots of efficient entanglement detection criteria and lower bounds of entanglement measures have been proposed, the continuous-variable entanglement is much less understood. Here we shall present a family of entanglement witnesses based on continuous-variable local orthogonal observables (CVLOOs) to detect and estimate entanglement of Gaussian and non-Gaussian states, especially for bound entangled states. By choosing an optimal set of CVLOOs, our entanglement witness is equivalent to the realignment criterion and can be used to detect bound entanglement of a class of 2+2 mode Gaussian states. Via our entanglement witness, lower bounds of two typical entanglement measures for arbitrary two-mode continuous-variable states are provided.
The dimensionality of entanglement in high-dimensional quantum systems is quantified by the Schmidt number, whose large value signals potential advantages for quantum communication and quantum computation. A commonly used approach to practical Schmidt-number detection in finite-dimensional systems is based on the estimation of fidelities to pure states with large entanglement dimensionality or lower bounds to these fidelities. However, for mode-entangled continuous-variable systems, the corresponding measurements are typically difficult to realize. Here, we introduce and assess fidelity bounds for Schmidt-number detection based on practically more accessible measurements of the first and second statistical moments of arbitrary continuous-variable quantum states. We provide analytical formulas that allow evaluating these bounds and test them for families of Gaussian and non-Gaussian states, including two-mode squeezed thermal states and photon-added displaced two-mode squeezed vacuum states. We compare our bounds to other methods for the certification of high-dimensional entanglement based on the same measurements.
By using the dynamics described with the quantum Langevin equation and the inseparability criterion for continuous-variable systems [L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000).], we discuss a method to judge whether entanglement exists in the evolutions of bipartite systems with continuous variables in their baths. By using this method we investigate a nontrivial example, namely, we judge when the entanglement exists in the evolution of the two coupled anharmonic oscillators in their environments.
We examine the implications of several recently derived conditions [Hillery and Zubairy, Phys. Rev. Lett. 96, 050503 (2006)] for determining when a two-mode state is entangled. We first find examples of non-Gaussian states that satisfy these conditions. We then apply the entanglement conditions to the study of several linear devices, the beam splitter, the parametric amplifier, and the linear phase-insensitive amplifier. For the first two, we find conditions on the input states that guarantee that the output states are entangled. For the linear amplifier, we determine in the limit of high and no gain, when an entangled input leads to an entangled output. Finally, we show how application of two two-mode entanglement conditions to a three-mode state can serve as a test of genuine three-mode entanglement.