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

TL;DR

This work develops a systematic method to translate Bell nonlocality, a hallmark of parallel quantum correlations, into dimension-bounded prepare-and-measure (PM) scenarios, enabling the transfer of self-testing and certification techniques from device-independent to semi-device-independent settings. Using the swap trick and SOS decompositions, the authors show that for qubits the PM quantum bound $b_q^{PM}$ matches the Bell bound $b_q^{NL}$ under maximally entangled inputs, and extend these insights to qudits with conjectured generalization. They illustrate the approach across several Bell families, including CHSH-type, generalized CHSH, and the Elegant Bell inequality, and demonstrate that many PM games are equivalent to known QRACs, unifying diverse certification protocols. The results yield practical PM certifiable tasks, e.g., self-testing of qubit measurements and symmetric network configurations, while offering a pathway to relax entanglement requirements in network-device-independent protocols through dimension-bounded channels. Overall, the framework links parallel and sequential quantum correlations, broadening the toolbox for semi-DI quantum information processing with concrete experimental relevance.

Abstract

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.

Figures (7)

• Figure 1: Two standard bipartite correlations structures: the Bell on the left and the prepare-and-measure scenario on the right side.
• Figure 2: On the left, the standard non-local scenario, maximised by a maximally entangled state of dimension $d$. On the right, its translated prepare-and-measure version, where Alice is now receiving two inputs and communicates with Bob through a dimensionally bounded quantum channel.
• Figure 3: Set of states for the prepare-and-measure tilted CHSH inequality, represented on a disc of the Bloch sphere. On the left, the set corresponding to the states steerred from Alice to Bob, in the maximally winning configuration of the non-local game. On the right, an allowed set of states in the prepare-and-measure game, that scores a higher value. Different thicknesses of the lines represent different sampling probabilities.
• Figure 4: Our prepare-and-measure CHSH on the left and the QRAC $2\to1$ on the right. All labels are binary strings. The structure of the games is identical, but the winning conditions (below the figures) are not equivalent.
• Figure 5: The bilocal network involves two mutually independent sources and three parties. The fact that the sources are mutually independent affects the form of achievable global classical and quantum correlations.

Theorems & Definitions (14)

• Definition 1
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• Definition 9
• Lemma 1