# Optimal measurement strategies for fast entanglement detection

## TL;DR

This paper develops optimal measurement strategies for efficiently detecting entanglement in multi-qubit systems, significantly reducing the number of measurements needed compared to random approaches.

## Contribution

It introduces a novel method using truncated moment sequences to identify the most efficient measurement strategies for entanglement detection, especially for symmetric states.

## Key findings

- Diagonal moment matrix measurements are highly efficient.
- Number of measurements grows polynomially with qubits.
- Detection efficiency improves rapidly with system size.

## Abstract

With the advance of quantum information technology, the question of how to most efficiently test quantum circuits is becoming of increasing relevance. Here we introduce the statistics of lengths of measurement sequences that allows one to certify entanglement across a given bi-partition of a multi-qubit system over the possible sequence of measurements of random unknown states, and identify the best measurement strategies in the sense of the (on average) shortest measurement sequence of (multi-qubit) Pauli measurements. The approach is based on the algorithm of truncated moment sequences that allows one to deal naturally with incomplete information, i.e. information that does not fully specify the quantum state. We find that the set of measurements corresponding to diagonal matrix elements of the moment matrix of the state are particularly efficient. For symmetric states their number grows only like the third power of the number $N$ of qubits. Their efficiency grows rapidly with $N$, leaving already for $N=4$ less than a fraction $10^{-6}$ of randomly chosen entangled states undetected.