Multipartite minimum uncertainty products

TL;DR

This paper investigates the fundamental limits of the multipartite uncertainty product for quantum states, demonstrating that entangled states can surpass bounds set by separable states and identifying a measurable gap.

Contribution

It extends previous bounds by considering pure states with spherically symmetric wave functions and constructs entangled states that approach the minimal uncertainty product.

Findings

Entangled states can beat the lower bound for separable states.

The infimum of the uncertainty product is not attained by any state.

A measurable gap exists between separable and entangled states.

Abstract

In our previous work we have found a lower bound for the multipartite uncertainty product of the position and momentum observables over all separable states. In this work we are trying to minimize this uncertainty product over a broader class of states to find the fundamental limits imposed by nature on the observable quantites. We show that it is necessary to consider pure states only and find the infimum of the uncertainty product over a special class of pure states (states with spherically symmetric wave functions). It is shown that this infimum is not attained. We also explicitly construct a parametrized family of states that approaches the infimum by varying the parameter. Since the constructed states beat the lower bound for separable states, they are entangled. We thus show that there is a gap that separates the values of a simple measurable quantity for separable states from entangled ones and we also try to find the size of this gap.