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arXiv:quant-ph/0507144 · New Journal of Physics

Inseparability inequalities for higher order moments for bipartite systems

G. S. Agarwal, Asoka Biswas

Published Oct 7, 2005• 146 citations
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Abstract

There are several examples of bipartite entangled states of continuous variables for which the existing criteria for entanglement using the inequalities involving the second-order moments are insufficient. We derive new inequalities involving higher order correlation, for testing entanglement in non-Gaussian states. In this context, we study an example of a non-Gaussian state, which is a bipartite entangled state of the form . Our results open up an avenue to search for new inequalities to test entanglement in non-Gaussian states.

DOI
10.1088/1367-2630/7/1/211

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There are several examples of bipartite entangled states of continuous variables for which the existing criteria for entanglement using the inequalities involving the second-order moments are insufficient. We derive new inequalities involving higher order correlation, for testing entanglement in non-Gaussian states. In this context, we study an example of a non-Gaussian state, which is a bipartite entangled state of the form . Our results open up an avenue to search for new inequalities to test entanglement in non-Gaussian states.
There are several examples of bipartite entangled states of continuous variables for which the existing criteria for entanglement using the inequalities involving the second order moments are insufficient. We derive new inequalities involving higher order correlation, for testing entanglement in non-Gaussian states. In this context we study an example of a non-Gaussian state, which is a bipartite entangled state of the form psi(xrma,xrmb)propto(alphaxrma+betaxrmb)e−(xrma2+xrmb2)/2\\psi(x_{\\rm a},x_{\\rm b})\\propto (\\alpha x_{\\rm a}+\\beta x_{\\rm b})e^{-(x_{\\rm a}^2+x_{\\rm b}^2)/2}psi(xrma​,xrmb​)propto(alphaxrma​+betaxrmb​)e−(xrma2​+xrmb2​)/2. Our results open up an avenue to search for new inequalities to test entanglement in non-Gaussian states.
We derive two types of sets of higher-order conditions for bipartite entanglement in terms of continuous variables. One corresponds to an extension of the well-known Duan inequalities from second to higher moments describing a kind of higher-order Einstein-Podolsky-Rosen (EPR) correlations. Only the second type, however, expressed by powers of the mode operators leads to tight conditions with a hierarchical structure. We start with a minimization problem for the single-partite case and, using the results obtained, establish relevant inequalities for higher-order moments satisfied by all bipartite separable states. We give an explicit example of a non-Gaussian state that exhibits fourth-order but no second-order EPR correlations. Similarly, a certain fourth-order condition cannot be violated by any Gaussian state and we present non-Gaussian states whose entanglement is detected by that condition. Violations of all our conditions are provided, so they can all be used as entanglement tests.
Entanglement is central to quantum physics, yet detecting and exploiting it in non-Gaussian systems remains a major challenge. In continuous variable platforms, standard inseparability criteria based on Gaussian statistics-such as the Duan-Simon criterion-fail when quantum correlations are encoded in higher moments of the field quadratures. Here we introduce a framework for detecting non-Gaussian entanglement using higher-order quadrature cumulants. In the Gaussian limit, the lowest order condition reduces to the Duan criterion, while higher-order violations reveal entanglement inaccessible to second-order methods. We experimentally demonstrate a tomography-free certification of inseparability in photon-subtracted squeezed states of light, where Gaussian witnesses fail despite the presence of entanglement. We further show that such higher-order inseparability enables enhanced quantum teleportation of Wigner negativity as compared to Gaussian protocols. These results identify higher-order cumulants as natural observables for non-Gaussian entanglement and open new routes to harnessing non-Gaussian resources in continuous variable quantum technologies.
The characterisation of quantum correlations, also known as entanglement, is central to quantum information theory. This chapter devotes to techniques and criteria based on second moments to detect the entanglement of Gaussian states. It also devotes to the description of possible ways to quantify such quantum correlations, bearing in mind that the determination of entanglement monotones of clear operational significance is still an open problem, even in the restricted arena of Gaussian states. The chapter addresses the issues of entanglement distillation and quantum non-locality tests, in whose contexts we shall confront other non-Gaussian operations and states, such as photon subtracted states. It introduces a general criteria based on higher order statistical moments, which will be put to use to detect the quantum correlations of an entangled cat-like state and addresses the issues of entanglement distillation and quantum non-locality tests.