Detecting Gaussian entanglement via local quantities

¹ School of Mathematics and Physics, University of Science and Technology Beijing - Beijing 100083, China
² Academy of Mathematics and Systems Science, Chinese Academy of Sciences - Beijing 100190, China
³ School of Mathematical Sciences, University of Chinese Academy of Sciences - Beijing 100049, China
⁴ Beijing Academy of Quantum Information Sciences - Beijing 100193, China
⁵ State Key Laboratory of Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, Peking University - Beijing 100871, China

^(a) zhangyue@baqis.ac.cn (corresponding author)

Received: 24 March 2021
Accepted: 7 June 2021

Abstract

Entanglement detection and quantification usually require the knowledge of correlations and thus is a global issue. For both theoretical and practical reasons, it is desirable to characterize entanglement via local quantities. Of course, this can only be achieved in special setups with some underlying structure encoding the correlations. For the often encountered bilinear interactions such as beamsplitter experiments, we propose an operational and friendly entanglement criterion for the two-mode Gaussian output states. Specifically, we show that in order to detect and evaluate the entanglement of the output states of a bilinear interaction involving uncorrelated Gaussian input states, we do not need to know the joint covariance matrix of the whole two-mode output states. Instead, we only need the respective covariances of the reduced states of the two input modes and that of the two output modes, which are all local quantities pertaining to a single mode and can be easily measured locally without explicitly considering the correlations between the modes. This is achieved by exploiting the input-output characteristics of bilinear interactions. The result recasts the celebrated entanglement criteria of Simon (Phys. Rev. Lett., 84 (2000) 2726) and Duan et al. (Phys. Rev. Lett., 84 (2000) 2722) into a more convenient and simple form.