Geometry of the set of mixed quantum states: An apophatic approach

TL;DR

This paper explores the geometric structure of the set of quantum states, especially for higher dimensions, using an apophatic approach and analyzing cross-sections and projections to understand its complex properties.

Contribution

It introduces an apophatic framework to study the geometry of quantum states and examines their cross-sections and projections, revealing new insights into their structure for dimensions N ≥ 3.

Findings

The set of quantum states has a richer structure for N ≥ 3 compared to the Bloch ball.

Cross-sections and projections of the set are dual to each other.

Certain properties of the set depend on the dimension N.

Abstract

The set of quantum states consists of density matrices of order $N$, which are hermitian, positive and normalized by the trace condition. We analyze the structure of this set in the framework of the Euclidean geometry naturally arising in the space of hermitian matrices. For N=2 this set is the Bloch ball, embedded in $\mathbb R^3$. For $N \geq 3$ this set of dimensionality $N^2-1$ has a much richer structure. We study its properties and at first advocate an apophatic approach, which concentrates on characteristics not possessed by this set. We also apply more constructive techniques and analyze two dimensional cross-sections and projections of the set of quantum states. They are dual to each other. At the end we make some remarks on certain dimension dependent properties.