Towards variance-matrix characterization of complementarity relations in a continuous variable system

TL;DR

This paper explores how complementarity relations in bipartite continuous variable quantum systems can be characterized using the variance matrix, extending concepts from discrete systems and highlighting the complexity when symmetry conditions are absent.

Contribution

It establishes a connection between complementarity relations and the variance matrix for symmetric two-mode states, pioneering an operational approach for continuous variable systems.

Findings

Complementarity relations relate to variance matrix elements in symmetric states.

Non-trivial connections arise when symmetry conditions are not met.

First step towards operational characterization of complementarity in continuous variables.

Abstract

We discuss complementarity relations in a bipartite continuous variable system. Building up from the work done on discrete d-dimensional systems, we prove that for symmetric two-mode states, quantum complementarity relations can be put in a simple relation with the elements of the variance matrix. When this condition is not satisfied, such a connection becomes non-trivial. Our investigation is the first step towards an operative characterization of the complementarity in a scenario that has not been investigated so far.