Sample Complexity for Embedded Multipartite Entanglement Witness via Pauli and Clifford Classical Shadows

TL;DR

This paper investigates the sample complexity of detecting multipartite entanglement in large qubit systems using classical shadows, revealing how measurement strategies shift from Pauli to Clifford based on the witness's globalness.

Contribution

It introduces variance bounds for classical shadow estimations of entanglement witnesses, demonstrating a transition in measurement efficiency from Pauli to Clifford strategies.

Findings

Distinct measurement scaling regimes identified for local vs. global witnesses.

Numerical simulations confirm the crossover from Pauli to Clifford efficiency.

Provides theoretical bounds for sample complexity in multipartite entanglement detection.

Abstract

Detecting multipartite entanglement in many qubit systems is measurement-intensive, motivating protocols that estimate only selected observables with provable efficiency. In this work we use the classical shadow protocol to study the sample complexity required to estimate a family of subsystem $n$-partite entanglement witness embedded in an larger $N$-qubit system. We derive ensemble dependent variance bounds that lead to qualitatively distinct scaling for the snapshots cost at fixed additive error $\epsilon$ with numerical simulations confirm these trends, exhibiting a clear crossover from Pauli favorable performance for local witness to Clifford favorable performance as the witness becomes more global.