Optimal Verification of the Bell State and Greenberger-Horne-Zeilinger States in Untrusted Quantum Networks

TL;DR

This paper introduces a geometric approach to verify Bell and Greenberger-Horne-Zeilinger states in untrusted quantum networks using local measurements, achieving optimal efficiency and entanglement detection.

Contribution

It presents a novel geometric method for optimal verification protocols for Bell and GHZ states in untrusted networks, requiring only local measurements.

Findings

Protocols are tied to probability distributions on the Bloch sphere.

Verification protocols are nearly as efficient as standard quantum state verification.

Constructed optimal protocols for GHZ state verification and entanglement detection.

Abstract

Bipartite and multipartite entangled states are basic ingredients for constructing quantum networks and their accurate verification is crucial to the functioning of the networks, especially for untrusted networks. Here we propose a simple approach for verifying the Bell state in an untrusted quantum network in which one party is not honest. Only local projective measurements are required for the honest party. It turns out each verification protocol is tied to a probability distribution on the Bloch sphere and its performance has an intuitive geometric meaning. This geometric picture enables us to construct the optimal and simplest verification protocols, which are also very useful to detecting entanglement in the untrusted network. Moreover, we show that our verification protocols can achieve almost the same sample efficiencies as protocols tailored to standard quantum state verification. Furthermore, we establish an intimate connection between the verification of Greenberger-Horne-Zeilinger states and the verification of the Bell state. By virtue of this connection we construct the optimal protocol for verifying Greenberger-Horne-Zeilinger states and for detecting genuine multipartite entanglement.