Rényi formulation of entanglement criteria for continuous variables

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.