Construction of Equientangled Bases in Arbitrary Dimensions via Quadratic Gauss Sums and Graph States

TL;DR

This paper constructs families of equientangled bases in any dimension, interpolating between product and maximally entangled states, using quadratic Gauss sums and graph states, thus solving an open problem for arbitrary dimensions.

Contribution

It provides two explicit constructions of equientangled bases in arbitrary dimensions, one based on quadratic Gauss sums and the other on graph states, extending previous results.

Findings

Existence of equientangled bases in all dimensions.

Explicit constructions using quadratic Gauss sums.

Generalization to multipartite bases.

Abstract

Recently [Karimipour and Memarzadeh, Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space each of dimension $D$ with the following properties: (i) The family continuously interpolates between the product basis and the maximally entangled basis as some parameter $t$ is varied, and (ii) for a fixed $t$, all basis states have the same amount of entanglement. The authors derived a necessary condition and provided explicit solutions for $D \leq 5$ but the existence of a solution for arbitrary dimensions remained an open problem. We prove that such families exist in arbitrary dimensions by providing two simple solutions, one employing the properties of quadratic Gauss sums and the other using graph states. The latter can be generalized to multipartite equientangled bases with more than two parties.