Numerical detection of Gaussian entanglement and its application to the identification of bound entangled Gaussian state
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Numerical detection of Gaussian entanglement and its application to the identification of bound entangled Gaussian states Shan Ma1,2 · Shibei Xue3,4 · Yu Guo5 · Chuan-Cun Shu6 Received: 5 February 2020 / Accepted: 10 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We present a numerical method for solving the separability problem of Gaussian quantum states in continuous-variable quantum systems. We show that the separability problem can be cast as an equivalent problem of determining the feasibility of a set of linear matrix inequalities. Thus, it can be efficiently solved using existent numerical solvers. We apply this method to the identification of bound entangled Gaussian states. We show that the proposed method can be used to identify bound entangled Gaussian states that could be simple enough to be producible in quantum optics. Keywords Entanglement · Separability · Gaussian states · Bound entanglement · Continuous variable
1 Introduction Quantum entanglement plays a central role in quantum information technologies, e.g., in quantum computation, quantum communication, and quantum metrology [1–7]. In
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Chuan-Cun Shu [email protected]
1
School of Automation, Central South University, Changsha 410083, China
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Peng Cheng Laboratory, Shenzhen 518000, China
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Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China
4
Key Laboratory of System Control and Information Processing, Ministry of Education of China, Shanghai 200240, China
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Hunan Provincial Key Laboratory of Flexible Electronic Materials Genome Engineering, School of Physics and Electronic Science, Changsha University of Science and Technology, Changsha 410114, China
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Hunan Key Laboratory of Super-Microstructure and Ultrafast Process, School of Physics and Electronics, Central South University, Changsha 410083, China 0123456789().: V,-vol
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recent years, a great deal of research effort has been put into the analysis of the entanglement properties of multiparticle systems [8–17]. While most of the effort has been devoted to systems with finite-dimensional Hilbert spaces, in particular discretevariable qubit states, recently there has been considerable interest in the continuousvariable (CV) case [18–26]. Gaussian states, as a particularly useful class of CV states, are commonly produced in quantum optics laboratories. Given a Gaussian state of a bipartite CV system, the most fundamental problem in CV quantum information theory is to determine whether the state is entangled or not with respect to the splitting. Consider two CV quantum systems A with m modes and B with n modes having infinite-dimensional Hilbert spaces H A and H B , respectively. The global bipartite system A + B with m + n modes has a Hilbert space H = H A ⊗ H B . By definition, a quantum state ρˆ of the global bipartite system A + B is said to be separable if it can be written as a convex sum of pure product states, namely, p j ρˆ jA ⊗ ρˆ Bj , (1) ρˆ =
j
where p j ≥ 0
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