# Purity-bounded uncertainty relations in multidimensional space -- generalized purity

**Authors:** Mikalai Karelin  
**arXiv:** [quant-ph/0411067](https://arxiv.org/abs/quant-ph/0411067)  
**Date:** 2004-11-09  
**Summary page:** https://www.summarizepaper.com/fr/arxiv-id/quant-ph--0411067v5/

## Abstract

Uncertainty relations for mixed quantum states (precisely, purity-bounded position-momentum relations, developed by Bastiaans and then by Man'ko and Dodonov) are studied in general multi-dimensional case. An expression for family of mixed states at the lower bound of uncertainty relation is obtained. It is shown, that in case of entropy-bounded uncertainty relations, lower-bound state is thermal, and a transition from one-dimensional problem to multi-dimensional one is trivial. Results of numerical calculation of the relation lower bound for different types of generalized purity are presented. Analytical expressions for general purity-bounded relations for highly mixed states are obtained.