Maximally Entangled States

TL;DR

This paper characterizes maximally entangled states of two d-dimensional particles as product states in collective coordinates, introduces a phase space analogy, and explores their geometric and algebraic properties.

Contribution

It provides a novel geometric and algebraic framework for understanding maximally entangled states using phase space and finite geometry concepts.

Findings

Maximally entangled states form a basis analogous to a phase space array.

MES can be expressed as products of collective coordinate states.

Finite geometry offers insights into the structure of MES and their relation to mutually unbiased bases.

Abstract

Every Maximally Entangled State (MES) of two d-dimensional particles is shown to be a product state of suitably chosen collective coordinates. The state may be viewed as defining a "point" in a "phase space" like d^2 array representing d^2 orthonormal Maximally Entangled States basis for the Hilbert space. A finite geometry view of MES is presented and its relation with the afore mentioned "phase space" is outlined: "straight lines" in the space depict product of single particle mutually unbiased basis (MUB) states, inverting thereby Schmidt's diagonalization scheme in giving a product single particle states as a d-terms sum of maximally entangled states. To assure self sufficiency the essential mathematical results are summarized in the appendices.