Exploring Imaginary Coordinates: Disparity in the Shape of Quantum State Space in Even and Odd Dimensions

TL;DR

This paper characterizes the constraints on quantum state coordinates, revealing a surprising difference in the shape of the state space between even and odd dimensions using a Bloch ball-type model.

Contribution

It provides a complete set of inequalities for real and imaginary coordinates of quantum states and uncovers a qualitative difference in state space boundaries across dimensions.

Findings

Complete characterization of constraints in quantum state coordinates.

Discovery of a qualitative difference in state space boundaries between even and odd dimensions.

Development of a three-dimensional Bloch ball-type model for quantum state space.

Abstract

The state of a finite-dimensional quantum system is described by a density matrix that can be decomposed into a real diagonal, a real off-diagonal and and an imaginary off-diagonal part. The latter plays a peculiar role. While it is intuitively clear that some of the imaginary coordinates cannot have the same extension as their real counterparts the precise relation is not obvious. We give a complete characterization of the constraints in terms of tight inequalities for real and imaginary Bloch-type coordinates. Our description entails a three-dimensional Bloch ball-type model for the state space. We uncover a surprising qualitative difference for the state-space boundaries in even and odd dimensions.