# Minimum uncertainty states and completeness of non-negative quasi probability of finite-dimensional quantum systems

## TL;DR

This paper constructs minimum-uncertainty states and a non-negative quasi probability distribution for finite-dimensional quantum systems, analyzing their properties, limitations, and the conditions for completeness based on system dimension.

## Contribution

It introduces a new non-negative quasi probability distribution for finite-dimensional quantum systems and clarifies conditions for its completeness depending on system dimension.

## Key findings

- Minimum-uncertainty states saturate the uncertainty inequality.
- Constructed quasi probability distribution has optimal marginal properties.
- Completeness of the quasi probability depends on whether the dimension is odd or even.

## Abstract

We construct minimum-uncertainty states and a non-negative quasi probability distribution for quantum systems on a finite-dimensional space. We reexamine the theorem of Massar and Spindel for the uncertainty relationof the two unitary operators related by the discrete Fourier transformation. It is shown that some assumptions in their proof can be justified by the use of the Perron-Frobenius theorem. The minimum-uncertainty states are the ones that saturate this uncertainty inequality. The continuum limit is closely analyzed by introducing a scale factor in the limiting scheme. Using the minimum-uncertainty states, we construct a non-negative quasi probability distribution. Its marginal distributions are smeared out. However, we show that this quasi probability is optimal in the sense that there does not exist a non-negative quasi probability distribution with sharper marginal properties if the translational covariance in the phase space is assumed. Generally, it is desirable that the quasi probability is complete, i.e., it contains full information of the state. We show that the obtained quasi probability is indeed complete if the dimension of the state space is odd, whereas it is not if the dimension is even.