Bargmann invariants and local unitary equivalence

Abstract

In this paper, we study the local unitary equivalence of quantum states on ${\mathbb{C}}^n\otimes {\mathbb{C}}^n$, which is an important notion in quantum information theory. For the case $n=2$, the local unitary orbits of two-qubit states are completely determined by their local unitary Bargmann invariants. We show that the local unitary Bargmann invariants do not form a complete set of invariants of local unitary orbits for $n⩾3$, which negatively answers a problem proposed by L. Zhang and B. Xie.