Geometrical view of the Mean King Problem

TL;DR

This paper employs finite geometry, specifically dual affine plane geometry, to provide a novel, geometrical solution to the Mean King Problem using maximally entangled states and mutually unbiased bases in a prime-dimensional Hilbert space.

Contribution

It introduces a new geometrical framework using DAPG to understand and solve the Mean King Problem with maximally entangled states and MUBs in prime dimensions.

Findings

DAPG lines underpin maximally entangled states

New geometrical solution to the Mean King Problem

Self-contained exposition of MUB, DAPG, and MKP

Abstract

Finite geometry is used to underpin finite, $d^2$, dimensional Hilbert space accommodating two particles, d dimensional each. d=prime $\ne2$. Central role is allotted to states with mutual unbiased bases (MUB) labelling underpinned with points of finite dual affine plane geometry (DAPG). The DAPG lines are shown to underpin maximally entangled states which form an orthonormal basis spanning the space and provide a novel, geometrical view to a new solution of the Mean King Problem (MKP). Brief expositions to the topics considered: MUB, DAPG and the MKP are included rendering the paper self contained.