Testing equivalence of pure quantum states and graph states under SLOCC

TL;DR

This paper presents a classical algorithm to determine SLOCC-equivalence of pure quantum states and graph states using stabilizer formalism, simplifying the process compared to direct methods.

Contribution

It derives necessary and sufficient conditions for SLOCC-equivalence and provides an efficient algorithm leveraging stabilizer states and local unitary transformations.

Findings

The algorithm efficiently detects non-equivalence to graph states.

All stabilizer states are shown to be equivalent to graph states via local unitaries.

The method simplifies SLOCC-equivalence testing for pure states.

Abstract

A set of necessary and sufficient conditions are derived for the equivalence of an arbitrary pure state and a graph state on n qubits under stochastic local operations and classical communication (SLOCC), using the stabilizer formalism. Because all stabilizer states are equivalent to a graph state by local unitary transformations, these conditions constitute a classical algorithm for the determination of SLOCC-equivalence of pure states and stabilizer states. This algorithm provides a distinct advantage over the direct solution of the SLOCC-equivalence condition for an unknown invertible local operator S, as it usually allows for easy detection of states that are not SLOCC-equivalent to graph states.