General framework for verifying pure quantum states in the adversarial scenario

TL;DR

This paper develops a systematic framework for verifying pure quantum states in adversarial scenarios, providing analytical formulas and practical protocols that require resource costs comparable to nonadversarial cases.

Contribution

It introduces a general method and recipe for verifying pure quantum states in adversarial settings, bridging the gap with nonadversarial verification methods.

Findings

Analytical formula for minimal tests in homogeneous strategies.

Resource overhead is at most three times compared to nonadversarial verification.

Applicable to various quantum states using only local projective measurements.

Abstract

Bipartite and multipartite entangled states are of central interest in quantum information processing and foundational studies. Efficient verification of these states, especially in the adversarial scenario, is a key to various applications, including quantum computation, quantum simulation, and quantum networks. However, little is known about this topic in the adversarial scenario. Here we initiate a systematic study of pure-state verification in the adversarial scenario. In particular, we introduce a general method for determining the minimal number of tests required by a given strategy to achieve a given precision. In the case of homogeneous strategies, we can even derive an analytical formula. Furthermore, we propose a general recipe to verifying pure quantum states in the adversarial scenario by virtue of protocols for the nonadversarial scenario. Thanks to this recipe, the resource cost for verifying an arbitrary pure state in the adversarial scenario is comparable to the counterpart for the nonadversarial scenario, and the overhead is at most three times for high-precision verification. Our recipe can readily be applied to efficiently verify bipartite pure states, stabilizer states, hypergraph states, weighted graph states, and Dicke states in the adversarial scenario, even if only local projective measurements are accessible. This paper is an extended version of the companion paper Zhu and Hayashi, Phys. Rev. Lett. 123, 260504 (2019).