Can outcome communication explain Bell nonlocality?

TL;DR

This work investigates whether outcome communication alone can reproduce Bell-nonlocal correlations. It proves that for qubit–qudit states under projective measurements, an LHV+Out model is equivalent to an LHV model, highlighting a fundamental limitation of outcome-only communication in this broad setting. However, when measurement sets are restricted (e.g., Alice's measurements restricted to the upper Bloch-hemisphere), outcome communication can provide real advantages, producing separations from standard LHV models and revealing the pivotal role of antipodal measurements. The authors also present an explicit, near-exact LHV+Out construction for a nonlocal Werner state on a finite measurement set and, via convex-decomposition techniques, extend this to all projective measurements on a hemisphere, establishing a precise threshold of visibility and showing that nonlocality can persist even with outcome communication under restricted measurements. Overall, the paper clarifies when outcome communication helps or fails in classical simulations of quantum correlations and provides practical methods for constructing such models using polytope approximations and Frank–Wolfe type algorithms.

Abstract

A central aspect of quantum information is that correlations between spacelike separated observers sharing entangled states cannot be reproduced by local hidden variable (LHV) models, a phenomenon known as Bell nonlocality. If one wishes to explain such correlations by classical means, a natural possibility is to allow communication between the parties. In particular, LHV models augmented with two bits of classical communication can explain the correlations of any two-qubit state. Would this still hold if communication is restricted to measurement outcomes? While in certain scenarios with a finite number of inputs the answer is yes, we prove that if a model must reproduce all projective measurements, then for any qubit-qudit state the answer is no. In fact, a qubit-qudit under projective measurements admits an LHV model with outcome communication if and only if it already admits an LHV model without communication. On the other hand, we also show that when restricted sets of measurements are considered (for instance, when the qubit measurements are in the upper hemisphere of the Bloch ball), outcome communication does offer an advantage. This exemplifies that trivial properties in standard LHV scenarios, such as deterministic measurements and outcome-relabelling, play a crucial role in the outcome communication scenario.

Figures (2)

• Figure 1: Different causal structures for bipartite Bell scenarios. (a) Standard Bell scenario, where the parties are correlated solely through a shared hidden variable $\lambda$ and no communication occurs. (b) Bell scenario with general communication from Alice to Bob, where Alice’s message $f(x, a, \lambda)$ may depend on both her input and outcome and the hidden variable $\lambda$. (c) Bell scenario with outcome communication, where Alice’s message to Bob is restricted to her measurement outcome.
• Figure 2: Pictorial representation of the relations between the different sets of correlations: $\text{LHV}$ denotes the of local correlations; $\mathcal{Q}$ the set of quantum correlations; $\mathcal{NS}$ the set of nonsignalling correlations; and $\text{(LHV+Out)} \cap \mathcal{NS}$ the set of local correlations augmented with outcome communication, restricted to the nonsignalling space. Note that $\text{LHV} \subset \mathcal{Q}$ and $\text{LHV} \subset \text{(LHV+Out)} \cap \mathcal{NS}$. However, in general, $\mathcal{Q}$ and $\text{(LHV+Out)}$$\cap$$\mathcal{NS}$ are not subsets of one another.

Theorems & Definitions (11)

• Definition 1: LHV+Out model
• Definition 2: LHV+Out states
• Remark 1
• Theorem 1
• Theorem 1
• proof
• Theorem 2
• proof : Proof
• Proposition 1
• proof