R\'enyi formulation of entanglement criteria for continuous variables

TL;DR

This paper develops entanglement criteria for continuous-variable quantum systems using Rényi entropies, leveraging mathematical inequalities and convolution properties, with practical reformulations for experimental data analysis.

Contribution

It introduces novel entanglement criteria based on Rényi entropies for continuous variables, utilizing recent mathematical inequalities and convolution properties.

Findings

Derived entanglement criteria using Rényi entropies.

Criteria applicable to n-separable states in continuous-variable systems.

Reformulations suitable for experimental data sampling.

Abstract

Entanglement criteria for an $n$-partite quantum system with continuous variables are formulated in terms of R\'{e}nyi entropies. R\'{e}nyi 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\'{e}nyi 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.