Distinguishing Graph States by the Properties of Their Marginals

TL;DR

This paper introduces new methods based on marginal properties of graph states to distinguish their entanglement classes under local unitaries, extending the known bounds for the LU-LC conjecture up to 10 qubits.

Contribution

It develops LU-invariants from marginals and generalized tools for testing local Clifford equivalence, improving the classification of graph states for larger qubit systems.

Findings

LU-invariants uniquely identify graph state classes up to 8 qubits

Methods extend to larger graphs via graph condensation techniques

LU-LC conjecture holds for up to 10 qubits with qubit permutations

Abstract

Graph states are a class of multi-partite entangled quantum states that are ubiquitous in quantum information. We study equivalence relations between graph states under local unitaries (LU) to obtain distinguishing methods both in local and in networked settings. Based on the marginal structure of graph states, we introduce a family of easy-to-compute LU-invariants. We show that these invariants uniquely identify the entanglement classes of every graph state up to 8 qubits and discuss their reliability for larger numbers of qubits. To handle larger graphs, we generalize tools to test for local Clifford (LC) equivalence of graph states that work by condensing large graphs into smaller graphs. In turn, we show that statements on the equivalence of these smaller graphs (which are easier to compute) can be used to infer statements on the equivalence of the original, larger graphs. We analyze LU-equivalence in two key settings - with and without allowing for the permutation of qubits. We identify entanglement classes, whose marginal structure does not allow us to distinguish them. As a result, we increase the bound on the number of qubits where the LU-LC conjecture holds from 8 to 10 qubits in the setting where qubit permutations are allowed.