Equientangled Bases in Arbitrary Dimensions and Quadratic Gauss Sums

TL;DR

This paper proves the existence of a continuous family of orthonormal bases in bipartite quantum systems of any dimension, where all states have equal entanglement, interpolating between product and maximally entangled bases, using quadratic Gauss sums.

Contribution

The authors demonstrate that such equientangled bases exist for all dimensions and provide an explicit parametrization using quadratic Gauss sums.

Findings

Existence of equientangled bases in all dimensions.

Explicit parametrization of these bases.

Illustrative examples of the bases' behavior.

Abstract

Recently [Karimipour and Memarzadeh, PhysRevA 73, 012329 (2006)] posed the problem of finding a continuous family of orthonormal bases in a bipartite space of two identical systems with the following properties: i) in each basis, all states have to be equally entangled, and ii) the family continuously interpolate between the product basis and the maximally entangled basis. The authors provided a necessary condition and simple examples for relatively small dimensions, but questioned the existence of a general solution for arbitrary dimensions. Employing the properties of quadratic Gauss sums, we prove that such a family of bases exists for all dimensions and provide an explicit simple parametrization. We illustrate the behaviour of our solution with particular examples.