The uncertainty geometry of finite-dimensional position and momentum

TL;DR

This paper characterizes the full covariance matrix geometry of finite-dimensional position and momentum observables, revealing new insights into quantum state space, uncertainty, and applications in quantum metrology and entanglement detection.

Contribution

It introduces a systematic framework combining analytic, convex-geometric, and semidefinite programming methods to describe the admissible covariance matrices in finite-dimensional quantum systems.

Findings

Describes the admissible covariance matrix region using unitary invariants.

Identifies extremal states and generalizes minimum-uncertainty states.

Provides bounds for quantum parameter estimation and entanglement criteria.

Abstract

Uncertainty relations are usually stated as bounds on selected combinations of variances, but the full covariance matrix contains substantially richer information about the geometry of quantum state space and about the operational capabilities of quantum systems. Here we characterize the covariance matrices attainable by a finite-dimensional canonical pair of observables related by the discrete Fourier transform, the natural analogue of position and momentum in a finite Hilbert space. We combine analytic arguments with convex-geometric and semidefinite-programming methods based on joint numerical ranges to describe the admissible region through unitary invariants, in particular the trace and determinant of the covariance matrix. This provides a systematic way to identify extremal states, generalizing the notion of minimum-uncertainty states, and to quantify how the discrete uncertainty geometry approaches its continuous counterpart with increasing dimension. We further show that the resulting covariance-matrix characterization has direct consequences for applications: it yields accuracy bounds for multi parameter estimation protocols and separability criteria for finite-dimensional bipartite systems, including discrete analogues of continuous-variable EPR-type witnesses. Our results establish a systematic and versatile platform for connecting uncertainty relations, convex quantum geometry, metrology, and entanglement detection in finite-dimensional systems.