Geometrically Underpinned Maximally Entangled States Bases

TL;DR

This paper uses finite geometry to construct maximally entangled states in prime-dimensional Hilbert spaces, providing a geometric framework that aids in understanding and solving quantum measurement problems like the Mean King Problem.

Contribution

It introduces a geometric approach using dual affine plane geometry to systematically construct maximally entangled bases and explores their application to quantum measurement challenges.

Findings

Maximally entangled states form an orthonormal basis derived from geometric lines.

Mutually unbiased bases are linked to geometric points, facilitating state construction.

The approach offers a transparent solution to the Mean King Problem.

Abstract

Finite geometry is used to underpin finite, two d-dimensional particles Hilbert space, d=prime 6= 2. A central role is allotted to states with mutual unbiased bases (MUB) labeling. Dual affine plane geometry (DAPG) points underpin single particle, MUB labeled, product states. The DAPG lines are shown to underpin maximally entangled states which form an orthonormal basis spanning the space. The relevance of mutually unbiased collective coordinates bases (MUCB) for dealing with maximally entangled states is discussed and shown to provide an economic alternative mode of study. These maximally entangled, geometrically reasoned states, provide the resource to a transparent solution to what may be termed tracking of the Mean King Problem (MKP): here Alice prepares a state measured by King along some orientation which Alice succeed in identifying with a subsequent measurement. Brief expositions of the topics considered: MUB, DAPG, MUCB and the MKP are included, rendering the paper self contained.