Translating Bell Non-Locality to Prepare-and-Measure Scenarios under Dimensional Constraints

Understanding the connections between different quantum information protocols has been proven fruitful for both theoretical insights and experimental applications. In this work, we explore the relationship between non-local and prepare-and-measure scenarios, proposing a systematic way to translate bipartite Bell inequalities into dimensionally-bounded prepare-and-measure tasks. We identify sufficient conditions under which the translation preserves the quantum bound and self-testing properties, enabling a wide range of certification protocols originally developed for the non-local setting to be adapted to the sequential framework of prepare-and-measure with a dimensional bound. While the dimensionality bound is not device-independent, it still is a practical and experimentally reasonable assumption in many cases of interest. In some instances, we find new experimentally-friendly certification protocols. In others, we demonstrate equivalences with already known prepare-and-measure protocols, where self-testing results were previously established using alternative mathematical methods. Our results unify different quantum correlation frameworks, and contribute to the ongoing research effort of studying the interplay between parallel and sequential protocols.

💡 Research Summary

This paper investigates a systematic method for translating bipartite Bell‑inequality tests, which are the cornerstone of device‑independent (DI) quantum information protocols, into prepare‑and‑measure (PM) tasks that operate under a bounded‑dimension assumption. The authors start by reviewing the standard Bell scenario: two spatially separated parties, Alice and Bob, share an arbitrary quantum state and perform local measurements defined by POVMs. Correlations are captured by conditional probabilities p(a,b|x,y) and quantified by a Bell functional I_NL = Σ k_{abxy} p(a,b|x,y). The quantum bound b_NL^q is obtained by maximizing the associated Bell operator over all states and measurements, while the local bound b_NL^L characterises classical hidden‑variable models.

Self‑testing is introduced as the situation where the quantum bound is achieved uniquely (up to local isometries and complex conjugation) by a specific state‑measurement pair. A powerful tool for proving self‑testing is the sum‑of‑squares (SOS) decomposition of the shifted Bell operator: b_q·𝟙 – B_NL = Σ_m f_m† f_m, where each f_m is a non‑commutative polynomial in the measurement observables. When the optimal strategy saturates the bound, all f_m annihilate the shared state, providing algebraic constraints that can be turned into robust self‑testing statements.

The PM scenario considered here involves Alice receiving a classical input x and, optionally, an auxiliary input a (or a combined input pair (x,a)). She prepares a quantum system ρ_{x,a} of fixed dimension d and sends it to Bob through a quantum channel. Bob then receives his own input y and performs a measurement N_{b|y}, producing outcome b with probability P(b|x,a,y)=Tr