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arXiv:1610.08839 · Physical Review A

Rényi formulation of entanglement criteria for continuous variables

Alexey E. Rastegin

Published Apr 21, 2017• 8 citations
arXivHTMLPDF

Abstract

Entanglement criteria for an $n$-partite quantum system with continuous variables are formulated in terms of R\'enyi entropies. R\'enyi entropies are widely used as a good information measure due to many nice properties. Derived entanglement criteria are based on several mathematical results such as the Hausdorff-Young inequality, Young's inequality for convolution and its converse. From the historical viewpoint, the formulations of these results with sharp constants were obtained comparatively recently. Using the position and momentum observables of subsystems, one can build two total-system measurements with the following property. For product states, the final density in each global measurement appears as a convolution of $n$ local densities. Hence, restrictions in terms of two R\'enyi entropies with constrained entropic indices are formulated for $n$-separable states of an $n$-partite quantum system with continuous variables. Experimental results are typically sampled into bins between prescribed discrete points. For these aims, we give appropriate reformulations of the derived entanglement criteria.

DOI
10.1103/physreva.95.042334

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Entanglement criteria for an nnn-partite quantum system with continuous variables are formulated in terms of R'enyi entropies. R'enyi entropies are widely used as a good information measure due to many nice properties. Derived entanglement criteria are based on several mathematical results such as the Hausdorff-Young inequality, Young's inequality for convolution and its converse. From the historical viewpoint, the formulations of these results with sharp constants were obtained comparatively recently. Using the position and momentum observables of subsystems, one can build two total-system measurements with the following property. For product states, the final density in each global measurement appears as a convolution of nnn local densities. Hence, restrictions in terms of two R'enyi entropies with constrained entropic indices are formulated for nnn-separable states of an nnn-partite quantum system with continuous variables. Experimental results are typically sampled into bins between prescribed discrete points. For these aims, we give appropriate reformulations of the derived entanglement criteria.
We derive a family of entanglement criteria for continuous-variable systems based on the R\'enyi entropy of complementary distributions. We show that these entanglement witnesses can be more sensitive than those based on second-order moments, as well as previous tests involving the Shannon entropy [Phys. Rev. Lett. 103, 160505 (2009)]. We extend our results to include the case of discrete sampling. We provide several numerical results which show that our criteria can be used to identify entanglement in a number of experimentally relevant quantum states.
Abstract. Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or in general by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two–mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p−entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixedstate entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions. 1.
Quantum entanglement of pure states of a bipartite system is defined as the amount of local or marginal (i.e. referring to the subsystems) entropy. For mixed states this identification vanishes, since the global loss of information about the state makes it impossible to distinguish between quantum and classical correlations. Here we show how the joint knowledge of the global and marginal degrees of information of a quantum state, quantified by the purities or, in general, by information entropies, provides an accurate characterization of its entanglement. In particular, for Gaussian states of continuous variable systems, we classify the entanglement of two-mode states according to their degree of total and partial mixedness, comparing the different roles played by the purity and the generalized p-entropies in quantifying the mixedness and bounding the entanglement. We prove the existence of strict upper and lower bounds on the entanglement and the existence of extremally (maximally and minimally) entangled states at fixed global and marginal degrees of information. This results allow for a powerful, operative method to measure mixed-state entanglement without the full tomographic reconstruction of the state. Finally, we briefly discuss the ongoing extension of our analysis to the quantification of multipartite entanglement in highly symmetric Gaussian states of arbitrary 1 × N-mode partitions.
A general procedure to construct criteria for identifying genuine multipartite continuous-variable entanglement is presented. It relies on the definition of adequate global operators describing the multipartite system, the positive partial transpose criterion of separability, and quantum-mechanical uncertainty relations. As a consequence, each criterion encountered consists of a single inequality that is nicely computable and experimentally feasible. Violation of the inequality is a sufficient condition for genuine multipartite entanglement. Additionally, we show that the previous work of van Loock and Furusawa [P. van Loock and A. Furusawa, Phys. Rev. A 67, 052315 (2003)] is a special case of our result.