Entanglement Verification, with or without tomography

TL;DR

This paper investigates the challenge of certifying multipartite entanglement efficiently without full state tomography, revealing fundamental limitations and proposing near-optimal local measurement schemes for pure states.

Contribution

It demonstrates that universal entanglement detection among all states cannot be achieved without full tomography and provides efficient methods for pure state entanglement detection using local measurements.

Findings

Universal entanglement detection requires full state tomography.

Almost all multipartite correlations need full tomography for certification.

Efficient local measurement schemes are possible for pure state entanglement detection.

Abstract

Multipartite entanglement has been widely regarded as key resources in distributed quantum computing, for instance, multi-party cryptography, measurement based quantum computing, quantum algorithms. It also plays a fundamental role in quantum phase transitions, even responsible for transport efficiency in biological systems. Certifying multipartite entanglement is generally a fundamental task. Since an $N$ qubit state is parameterized by $4^N-1$ real numbers, one is interested to design a measurement setup that reveals multipartite entanglement with as little effort as possible, at least without fully revealing the whole information of the state, the so called "tomography", which requires exponential energy. In this paper, we study this problem of certifying entanglement without tomography in the constrain that only single copy measurements can be applied. This task is formulate as a membership problem related to a dividing quantum state space, therefore, related to the geometric structure of state space. We show that universal entanglement detection among all states can never be accomplished without full state tomography. Moreover, we show that almost all multipartite correlation, include genuine entanglement detection, entanglement depth verification, requires full state tomography. However, universal entanglement detection among pure states can be much more efficient, even we only allow local measurements. Almost optimal local measurement scheme for detecting pure states entanglement is provided.