Irreducible multi-partite correlations as an order parameter for k-local nontrivial states

TL;DR

This paper introduces an order parameter based on irreducible multi-partite correlations to identify k-local nontrivial quantum states, providing a new way to distinguish complex quantum phases without relying on geometric locality.

Contribution

The paper proposes a novel order parameter that captures irreducible multi-partite correlations, enabling the detection of k-local nontrivial states beyond geometric locality.

Findings

Successfully applied to toric code and stabilizer states

Revealed connections with quantum error correction thresholds

Linked to classical percolation phenomena

Abstract

Geometrically nontrivial quantum states can be defined as states that cannot be prepared by a constant depth geometrically local unitary circuit starting from a product state. However, for topological phases, as well as a large class of quantum error correcting codes without an underlying geometric structure, the required circuit depth remains infinite even if we replace the condition of geometric locality with the weaker condition of k-locality. Motivated by this observation, we look for a non-geometric quantity that can capture k-local non-triviality of a given state, for example, we ask if it is possible to distinguish the ground state of the toric code from a trivial state without having access to the position label of the qubits. We observe that a fundamental property of k-local nontrivial states is the presence of irreducible many-partite correlations shared between an infinitely large number of randomly chosen parties, i.e. correlations that cannot be inferred by accessing only a finite number of parties. We introduce an order parameter designed to capture such correlations. We demonstrate the utility of our order parameter by applying it to a wide variety of examples: The toric code on a square lattice, random stabilizer states, quantum expander codes, and a particular holographic stabilizer state. We discuss general relations between this order parameter and the erasure thresholds of quantum error correcting codes as well as the classical bond percolation problem.