An Online Approach for Entanglement Verification Using Classical Shadows

TL;DR

This paper introduces an online classical shadow-based method for entanglement verification that operates concurrently with quantum experiments, reducing sample requirements and improving efficiency.

Contribution

It develops unbiased online estimators for mixed state entanglement verification using PT-moments, optimizing memory and computational trade-offs.

Findings

Online estimators certify entanglement reliably.

Requires fewer samples than existing methods.

Enables real-time entanglement detection during experiments.

Abstract

Quantum measurements are slow, while classical processors are fast, yet existing hybrid protocols never exploit this asymmetry. In this work, we propose an alternative formulation of classical estimators as online algorithms that are updated incrementally upon obtaining a new sample. Classical shadows are the natural fit for this approach: designed around the principle of measuring first and asking questions later, each snapshot is a self-contained classical description that can be processed immediately and independently. As a first demonstration, we focus on mixed state entanglement verification via PT-moments, moments of the partially transposed density matrix that provide experimentally accessible sufficient conditions for entanglement. We construct two unbiased online estimators that together characterize the fundamental challenge between memory footprint and per-shot computational cost: one scales to large systems at low moment order, the other handles high moment orders at the expense of memory exponential in system size. The online estimator certifies entanglement reliably and, by exploiting all $\binom{T}{m}$ combinations of snapshots, requires fewer samples than state-of-the-art baselines, turning entanglement detection from a purely offline diagnostic into a protocol that runs concurrently with the experiment.