Uncertainty relations for the support of quantum states

TL;DR

This paper extends uncertainty relations for quantum state support sizes from two bases to complete sets of mutually unbiased bases in prime dimensions, providing new bounds and identifying states that achieve them.

Contribution

It generalizes Tao's uncertainty relation to multiple bases in prime dimensions and constructs sharp bounds for small dimensions, advancing understanding of quantum support limitations.

Findings

Bound is sharp for dimension three.

Numerical results suggest bounds cannot be saturated generally.

Constructed sharp bounds for dimensions 2 to 7.

Abstract

Given a narrow signal over the real line, there is a limit to the localisation of its Fourier transform. In spaces of prime dimensions, Tao derived a sharp state-independent uncertainty relation which holds for the support sizes of a pure qudit state in two bases related by a discrete Fourier transform. We generalise Tao's uncertainty relation to complete sets of mutually unbiased bases in spaces of prime dimensions. The bound we obtain appears to be sharp for dimension three only. Analytic and numerical results for prime dimensions up to nineteen suggest that the bound cannot be saturated in general. For prime dimensions two to seven we construct sharp bounds on the support sizes in $(d+1)$ mutually unbiased bases and identify some of the states achieving them.