Einstein-Podolsky-Rosen uncertainty limits for bipartite multimode states

TL;DR

This paper explores the limits of Einstein-Podolsky-Rosen (EPR) uncertainty for bipartite multimode quantum states, providing a unified framework based on Heisenberg uncertainty relations to assess entanglement and steering.

Contribution

It introduces a unitary quantum-mechanical approach to quantify entanglement and EPR steering in continuous-variable multimode states using uncertainty relations.

Findings

Derived bounds on EPR uncertainty sums for bipartite multimode states.

Established necessary conditions for separability and EPR unsteerability.

Confirmed that Gaussian states satisfy known criteria for entanglement and steering.

Abstract

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 $(N\, \text{vs} \,1)$-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 ones. The analysis of the minimal properly normalized sums of these variances yields necessary conditions of separability and EPR unsteerability of $(N\, \text{vs} \,1)$-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.