Bounds on Multipartite Nonlocality via Reduction to Biased Nonlocality

TL;DR

The paper tackles the problem of bounding genuine multipartite nonlocality in a $THRESHOLD$-function setting by introducing the LOCCG model and proving a reduction to biased bipartite nonlocal games. This reduction enables precise quantum-classical comparisons for various threshold families, showing a quantum advantage for $THRESHOLD(n-1,1)$ and, in contrast, no advantage for general $AND(n-k,k)$ with $k>1$, while revealing a nuanced, input-biased behavior for $MAJORITY(n-k,k)$ with finite-n advantages that fade asymptotically for fixed $k$ but may persist near symmetric groupings. The authors validate optimality using semidefinite programming (Tsirelson-type SDP) and provide explicit protocols and dual solutions, linking multipartite LOCCG bounds to well-studied biased CHSH games. Overall, the work offers a principled bridge from multipartite to bipartite nonlocality principles and suggests broader applicability to non-XOR and nonlocal-box scenarios, with implications for multipartite information theories.

Abstract

Multipartite information principles are needed to understand nonlocal quantum correlations. Towards that end, we provide optimal bounds on genuine multipartite nonlocality for classes of THRESHOLD games using the LOCCG (Local Operations with Grouping) model. Our proof develops a reduction between multipartite nonlocal and biased bipartite nonlocal games. Generalizing this reduction to a larger class of games may build a bridge from multipartite to bipartite principles.

Figures (9)

• Figure 1: Three player Bell scenario where players $1$ and $2$ are grouped together via communication while players $2$ and $3$ share a different bipartite resource.
• Figure 2: Different groupings of the LOCCG model using communication and shared entanglement as resource. On the left we have grouping $(n-1,1)$ with communication allowed between $n-1$ players in the set $I \cup I'$ and shared entanglement across the $n-1$ players in the set $I \cup J$. Note that both the sets $I'$ and $J$ in this case contain a single player. Figure on the right shows general grouping $(n-k,k)$, with communication allowed only between $n-k$ players in the set $I \cup I'$ and $k$ players in the set $J \cup J'$. At the same time, $n-k$ players in the set $I \cup J$ share an entangled state $\ket{\psi}$, while $k$ players in the set $I' \cup J'$ share an entangled state $\ket{\phi}$. The idea being that even though groups can overlap, no resource is shared across a group of size larger than $n-k$. For simplicity we have omitted in the visualization entanglement that may be shared within each group $A$ and $B$.
• Figure 3: Visual summary of the games in our results. Quantum advantage exists for $THRESHOLD(n-1,1)$, while there is no advantage for $AND(n-k,k)$. We show that quantum advantage exists for choices of $MAJORITY(n-k,k)$, but it is open to determine how the advantage behaves as we scale $k$ in terms of $n$.
• Figure 4: The LOCCG model is shown on the left and the reduced biased game is on the right.
• Figure 5: Input mapping of LOCCG $(2,1)$ group $A$'s input $x$ to a biased CHSH game with $n=t=3$.

Theorems & Definitions (8)

• Definition 1
• Theorem 1
• Corollary 1
• Theorem 2
• Lemma 1
• Lemma 2
• Conjecture 1
• Corollary 2