Entanglement and squeezing in continuous-variable systems

Manuel Gessner, Luca Pezzè, and Augusto Smerzi

QSTAR, INO-CNR and LENS, Largo Enrico Fermi 2, I-50125 Firenze, Italy

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We introduce a multi-mode squeezing coefficient to characterize entanglement in $N$-partite continuous-variable systems. The coefficient relates to the squeezing of collective observables in the $2N$-dimensional phase space and can be readily extracted from the covariance matrix. Simple extensions further permit to reveal entanglement within specific partitions of a multipartite system. Applications with nonlinear observables allow for the detection of non-Gaussian entanglement.

Quantum mechanical squeezing shrinks the variance of one observable at the expense of another, and can thereby reduce the quantum noise in a measurement. In collective spin systems, it is possible to quantify the degree of squeezing and relate it to entanglement using spin-squeezing coefficients. Similar tools have so far been unavailable for continuous-variable systems, due to the unbounded Hilbert space. In this article, we introduce a bosonic squeezing coefficient that identifies multimode continuous-variable squeezing. Using this coefficient, we can show that multimode squeezing indeed implies mode entanglement. The coefficient can be obtained directly from the covariance matrix, which is accessible in many experiments.

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