Geometrical self-testing of partially entangled two-qubit states

TL;DR

This paper introduces a geometric method for self-testing partially entangled two-qubit states, enabling the verification of quantum states through measurement correlations and extending to arbitrary measurement scenarios.

Contribution

It presents a novel geometric proof for self-testing partially entangled states and proposes protocols that can be scaled to many measurements.

Findings

A geometric condition uniquely determines the state vectors.

Self-testing is achievable with an additional measurement and a unitary observable.

The approach identifies extremal points of nonlocal correlations through geometry.

Abstract

Quantum nonlocality has recently been intensively studied in connection to device-independent quantum information processing, where the extremal points of the set of quantum correlations play a crucial role through self-testing. In most protocols, the proofs for self-testing rely on the maximal violation of the Bell inequalities, but there is another known proof based on the geometry of state vectors to self-test a maximally entangled state. We present a geometrical proof in the case of partially entangled states. We show that, when a set of correlators in the simplest Bell scenario satisfies a condition, the geometry of the state vectors is uniquely determined. The realization becomes self-testable when another unitary observable exists on the geometry. Applying this proven fact, we propose self-testing protocols by intentionally adding one more measurement. This geometrical scheme for self-testing is superior in that, by using this as a building block and repeatedly adding measurements, a realization with an arbitrary number of measurements can be self-tested. Besides the application, we also attempt to describe nonlocal correlations by guessing probabilities of distant measurement outcomes. In this description, the quantum set is also convex, and a large class of extremal points is identified by the uniqueness of the geometry.