A Complete Set of Invariants for LU-Equivalence of Density Operators

TL;DR

This paper establishes a finite, complete set of polynomial invariants that determine when two mixed quantum states are equivalent under local unitary transformations, using algebraic geometry and invariant theory.

Contribution

It provides a complete set of invariants for LU-equivalence of density operators and classifies SLOCC invariants with degree bounds, advancing quantum state classification methods.

Findings

Two density operators are LU-equivalent iff they agree on polynomial invariants.

A finite set of polynomial invariants fully characterizes LU-equivalence.

Degree bounds for invariants are established for n-qubit pure states.

Abstract

We show that two density operators of mixed quantum states are in the same local unitary orbit if and only if they agree on polynomial invariants in a certain Noetherian ring for which degree bounds are known in the literature. This implicitly gives a finite complete set of invariants for local unitary equivalence. This is done by showing that local unitary equivalence of density operators is equivalent to local ${\rm GL}$ equivalence and then using techniques from algebraic geometry and geometric invariant theory. We also classify the SLOCC polynomial invariants and give a degree bound for generators of the invariant ring in the case of $n$-qubit pure states. Of course it is well known that polynomial invariants are not a complete set of invariants for SLOCC.