Local Unitary Invariants of Generic Multi-qubit States

TL;DR

This paper introduces a comprehensive set of polynomial local unitary invariants for generic multi-qubit states, enabling precise determination of local unitary equivalence, with explicit counts for two- and three-qubit systems.

Contribution

It provides a complete characterization of local unitary invariants for multi-qubit states, including explicit counts and comparison with existing invariants.

Findings

At most 12 invariants for two-qubit states

At most 90 invariants for three-qubit states

Comparison with Makhlin's invariants for two-qubit systems

Abstract

We present a complete set of local unitary invariants for generic multi-qubit systems which gives necessary and sufficient conditions for two states being local unitary equivalent. These invariants are canonical polynomial functions in terms of the generalized Bloch representation of the quantum states. In particular, we prove that there are at most 12 polynomial local unitary invariants for two-qubit states and at most 90 polynomials for three-qubit states. Comparison with Makhlin's 18 local unitary invariants is given for two-quibit systems.