Geometric uncertainty relation, the symplectic area, and the J-holomorphic maps for mixed quantum states

TL;DR

This paper explores the connection between geometric uncertainty, symplectic area, and J-holomorphic maps in mixed quantum states, revealing that the covariance matrix determinant relates to harmonic energy of area-minimizing holomorphic maps.

Contribution

It establishes a novel relation between quantum covariance determinants, symplectic geometry, and J-holomorphic maps in the context of mixed quantum states.

Findings

Covariance matrix determinant equals the squared metric area.

Geometric uncertainty compares metric and symplectic areas.

Harmonic energy of holomorphic maps corresponds to covariance determinant.

Abstract

In this paper we will establish a relation between geometric uncertainty relation and the determinant of the quantum covariance matrix for mixed quantum states. We will show that determinant of the covariance matrix represents the squared metric area of a parallelogram. In this setting the geometric uncertainty relation compares a metric area to a symplectic area. Moreover, we will in details investigate relation between J-holomorphic maps and geometric uncertainty relation for mixed quantum states. We will argue that determinant of the quantum covariance matrix is equal to the harmonic energy of a holomorphic map that minimize the areas.