Nonstrict inequality for Schmidt coefficients of three-qubit states

TL;DR

This paper derives nonstrict inequalities relating the Schmidt coefficients of three-qubit states, revealing constraints that differ from bipartite cases and providing bounds on these coefficients.

Contribution

It introduces new nonstrict inequalities for three-qubit Schmidt coefficients, highlighting restrictions absent in bipartite systems.

Findings

Largest Schmidt coefficient bounds other coefficients

Existence of an upper bound for the remaining coefficient

Constraints differ from bipartite Schmidt decompositions

Abstract

Generalized Schmidt decomposition of pure three-qubit states has four positive and one complex coefficients. In contrast to the bipartite case, they are not arbitrary and the largest Schmidt coefficient restricts severely other coefficients. We derive a nonstrict inequality between three-qubit Schmidt coefficients, where the largest coefficient defines the least upper bound for the three nondiagonal coefficients or, equivalently, the three nondiagonal coefficients together define the greatest lower bound for the largest coefficient. In addition, we show the existence of another inequality which should establish an upper bound for the remaining Schmidt coefficient.