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Physical Review Letters

Nonlinear Entanglement and its Application to Generating Cat States

Yong Shen, Syed M. Assad, Nicolai B. Grosse, X. Y. Li, M. D. Reid, Ping Koy Lam

Published Mar 10, 2015• 41 citations
Publisher

Abstract

The Einstein-Podolsky-Rosen (EPR) paradox, which was formulated to argue for the incompleteness of quantum mechanics, has since metamorphosed into a resource for quantum information. The EPR entanglement describes the strength of linear correlations between two objects in terms of a pair of conjugate observables in relation to the Heisenberg uncertainty limit. We propose that entanglement can be extended to include nonlinear correlations. We examine two driven harmonic oscillators that are coupled via third-order nonlinearity can exhibit quadraticlike nonlinear entanglement which, after a projective measurement on one of the oscillators, collapses the other into a cat state of tunable size.

DOI
10.1103/physrevlett.114.100403

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The Einstein-Podolsky-Rosen (EPR) paradox, which was formulated to argue for the incompleteness of quantum mechanics, has since metamorphosed into a resource for quantum information. The EPR entanglement describes the strength of linear correlations between two objects in terms of a pair of conjugate observables in relation to the Heisenberg uncertainty limit. We propose that entanglement can be extended to include nonlinear correlations. We examine two driven harmonic oscillators that are coupled via third-order nonlinearity can exhibit quadraticlike nonlinear entanglement which, after a projective measurement on one of the oscillators, collapses the other into a cat state of tunable size.
We report on the generation of a continuous variable Einstein-Podolsky-Rosen (EPR) entanglement using an optical fiber interferometer. The Kerr nonlinearity in the fiber is exploited for the generation of two independent squeezed beams. These interfere at a beam splitter and EPR entanglement is obtained between the output beams. The correlation of the amplitude (phase) quadratures is measured to be 4.0+/-0.2 (4.0+/-0.4) dB below the quantum noise limit. The sum criterion for these squeezing variances 0.80+/-0.03<2 verifies the nonseparability of the state. The product of the inferred uncertainties for one beam (0.64+/-0.08) is well below the EPR limit of unity.
A generalization of the 1935 Einstein-Podolsky-Rosen (EPR) argument for measurements with continuous variable outcomes is presented to establish criteria for the demonstration of the EPR paradox, for situations where the correlation between spatially separated subsystems is not perfect. Two types of criteria for EPR correlations are determined. The first type are based on measurements of the variances of conditional probability distributions and are necessary to reflect directly the situation of the original EPR paradox. The second weaker set of EPR criteria are based on the proven failure of (Bell-type) local realistic theories which could be consistent with a local quantum description for each subsystem. The relationship with criteria sufficient to prove entanglement is established, to show that any demonstration of EPR correlations will also signify entanglement. It is also shown how a demonstration of entanglement between two spatially separated subsystems, if not able to interpreted as a violation of a Bell-type inequality, may be interpreted as a demonstration of the EPR correlations. In particular it is explained how the experimental observation of two-mode squeezing using spatially separated detectors will signify not only entanglement but EPR correlations defined in a general sense.
Certification and quantification of correlations for multipartite states of quantum systems appear to be a central task in quantum information theory. We give here a unitary quantum-mechanical perspective of both entanglement and Einstein-Podolsky-Rosen (EPR) steering of continuous-variable multimode states. This originates in the Heisenberg uncertainty relations for the canonical quadrature operators of the modes. Correlations of two-party (Nvs1)(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)(Nvs1)-mode states are examined by using the variances of a pair of suitable EPR-like observables. It turns out that the uncertainty sum of these nonlocal variables is bounded from below by local uncertainties and is strengthened differently for separable states and for each one-way unsteerable state. The analysis of the minimal properly normalized sums of these variances yields necessary conditions of separability and EPR unsteerability of (Nvs1)(N\phantom{\rule{0.16em}{0ex}}\text{vs}\phantom{\rule{0.16em}{0ex}}1)(Nvs1)-mode states in both possible ways of steering. When the states and the performed measurements are Gaussian, then these conditions are precisely the previously known criteria of separability and one-way unsteerability.