Maximal Entanglement, Collective Coordinates and Tracking the King

TL;DR

This paper explores the structure of maximally entangled states in prime-dimensional systems, their geometric representation, and their application to quantum communication protocols like the Mean King Problem and Tracking the King.

Contribution

It introduces a geometric framework for MES using finite geometry and demonstrates its utility in solving quantum measurement problems and developing new communication channels.

Findings

MES form a basis linked to finite geometry lines

The geometric representation aids in solving the Mean King Problem

Tracking the King is identified as a new quantum communication channel

Abstract

Maximal entangled states (MES) provide a basis to two d-dimensional particles Hilbert space, d=prime $\ne 2$. The MES forming this basis are product states in the collective, center of mass and relative, coordinates. These states are associated (underpinned) with lines of finite geometry whose constituent points are associated with product states carrying Mutual Unbiased Bases (MUB) labels. This representation is shown to be convenient for the study of the Mean King Problem and a variant thereof, termed Tracking the King which proves to be a novel quantum communication channel. The main topics, notions used are reviewed in an attempt to have the paper self contained.