Orthogonal pairs and mutually unbiased bases

TL;DR

This paper explores the algebraic and geometric structures of orthogonal decompositions in Lie algebras and their application to mutually unbiased bases in quantum information, providing new existence results for these bases in six-dimensional space.

Contribution

It establishes the existence of a 4-dimensional family of mutually unbiased bases in C^6, solving a longstanding problem in quantum information theory.

Findings

Existence of a 4-dimensional family of orthogonal pairs in sl(6)

Existence of a 4-dimensional family of mutually unbiased bases in C^6

Resolution of a long-standing problem in quantum information theory

Abstract

The paper is devoted to representation theoretic and algebraic geometric aspects of the theory of orthogonal decompositions of Lie algebra sl(n) into Cartan subalgebras orthogonal with respect to Killing form and the relevant theory of mutually unbiased bases. We describe the main steps of our proof for the existence of 4-dimensional family of orthogonal pairs of Cartan subalgebras in sl(6). As an application, we prove existence of a real 4-dimensional family of pairs of mutually unbiased bases in C^6, thus solving the long standing problem in Quantum Information Theory.