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

2016/12/26 by Hoshang Heydari, Heydari, Hoshang
Mathematics · #Advanced Algebra and Geometry #Advanced Operator Algebra Research #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Quantum Physics (quant-ph)

paper · pdf · doi:10.48550/arxiv.1612.08428

openalex publication_date 2016/12/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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