Certifying entanglement dimensionality by random Pauli sampling

TL;DR

This paper presents a scalable, measurement-based protocol using random Pauli sampling to efficiently certify the entanglement dimensionality of multi-qubit states, significantly reducing sample complexity compared to worst-case scenarios.

Contribution

The authors develop a novel Pauli-measurement algorithm that certifies Schmidt number with improved average-case sample complexity and introduces a proof framework leveraging local pseudorandom unitaries.

Findings

Achieves polynomial sample complexity in the number of qubits.

Transforms worst-case complexity into average-case with high probability.

Provides a scalable method for high-dimensional entanglement certification.

Abstract

We introduce a Pauli-measurement-based algorithm to certify the Schmidt number of $n$-qubit pure states. Our protocol achieves an average-case sample complexity of $\caO(\mathrm{poly}(n)\chi^2)$, a substantial improvement over the $\caO(2^n \chi)$ worst-case bound. By utilizing local pseudorandom unitaries, we ensure the worst case can be transformed into the average-case with high probability. This work establishes a scalable approach to high-dimensional entanglement certification and introduces a proof framework for random Pauli sampling.