Neural quantum states for entanglement depth certification from randomized Pauli measurements

TL;DR

This paper introduces a neural quantum state approach for certifying entanglement depth from randomized Pauli measurements, avoiding full state tomography and enabling scalable, direct entanglement verification.

Contribution

It develops a likelihood-based model selection method using neural quantum states with entanglement constraints, allowing certification directly from measurement data.

Findings

Successfully validated on simulated six- and ten-qubit states

Robustness demonstrated under local noise conditions

Provides interpretability diagnostics for entanglement patterns

Abstract

Entanglement depth quantifies how many qubits share genuine multipartite entanglement, but certification typically relies on tailored witnesses or full tomography, both of which scale poorly with system size. We recast entanglement-depth and non-$k$-separability certification as likelihood-based model selection among neural quantum states whose architecture enforces a chosen entanglement constraint. A hierarchy of separable neural quantum states is trained on finite-shot local Pauli outcomes and compared against an unconstrained reference model trained on the same data. When all constrained models are statistically disfavored, the data certify entanglement beyond the imposed limit directly from measurement statistics, without reconstructing the density matrix. We validate the method on simulated six- and ten-qubit datasets targeting GHZ, Dicke, and Bell-pair states, and demonstrate robustness for mixed states under local noise. Finally, we discuss lightweight interpretability diagnostics derived from trained parameters that expose coarse entanglement patterns and qubit groupings directly from bitstring statistics.