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

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