Extending the Known Region of Nonlocal Boxes that Collapse Communication Complexity

TL;DR

The paper investigates non-signalling boxes (NS) and the phenomenon of communication complexity (CC) collapse, seeking conditions under which CC becomes trivial for any computable function $f$. It extends the BBLMTU protocol by introducing local uniformity and a hierarchy of protocols $\mathcal{P}_k$, enabling distributed computation of $f$ with increasing bias using multiple copies of a box and shared randomness. A key analytic contribution is the bias map $\mu_{k+1}=F(\mu_k)$ with $F(\mu)=\frac{\mu}{16}(A+B-\mu^2(A-B))$, where $A=(\eta_{00}+\eta_{01}+\eta_{10}+\eta_{11})^2$ and $B=2\eta_{00}^2+4\eta_{01}\eta_{10}+2\eta_{11}^2$, derived from the box biases $\eta_{xy}$. The condition $A+B>16$ guarantees three fixed points $\{0,\pm\mu_*\}$ with $0$ repulsive, ensuring convergence to a positive fixed point and proving the existence of a universal $p>1/2$ such that CC collapses for any $f$. Applying the result to two NS-slices, Case 1 (PR–PR′–I) and Case 2 (PR–SR–I), yields explicit analytic boundary inequalities in CHSH coordinates that extend previously known collapsing regions, though such boxes are deemed unlikely to occur in Nature.

Abstract

Non-signalling boxes (NS) are theoretical resources defined by the principle of no-faster-than-light communication. They generalize quantum correlations, and some of them are known to collapse communication complexity (CC). However, this collapse is strongly believed to be unachievable in Nature, so its study provides intuition on which theories are unrealistic. In the present letter, we find a better sufficient condition for a nonlocal box to collapse CC, thus extending the known collapsing region. In some slices of NS, we show this condition coincides with an area outside of an ellipse.

Figures (4)

• Figure 1: Communication complexity game. (Colors online.) Lowercase letters $a,a'\!,b,x,y$ are bits, and capital letters are strings: $X\in\{0,1\}^n$, $Y\in\{0,1\}^m$ and $B\in\{0,1\}^k$. Let $f:\{0,1\}^n\times\{0,1\}^m\to\{0,1\}$ be known. Once the game starts, Alice and Bob are spacelike separated and the referee sends them the respective strings $X$ and $Y$. The goal is that Alice answers a bit $a'$ such that $a'=f(X,Y)$. To achieve it, Bob is allowed to send some communication bits to Alice, but these bits are costly so he wants to send as few as possible. They may also use as many copies as they want of an NLB.
• Figure 2: Historical overview of collapsing boxes, drawn in the slice of $\mathcal{N\!S}$ passing through $\mathtt{PR}$ and $\mathtt{SR}$ and $\mathtt{I}$. (Colors online.) In red and purple are represented respectively the non-collapsing and the collapsing boxes. In blue is drawn the region of boxes for which we do not know yet if they collapse communication complexity. See BCMW10BM06FWW09Mor16SWH20NSSRRB22PRLEWC22bPopescu14Karvonen21NPA08 for impossibility results and others.
• Figure 3: Distributively compute the products $r_1s_1$ and $r_2s_2$ with probability bias $\eta_{r_1, s_1}$ and $\eta_{r_2, s_2}$ respectively.
• Figure 4: Two slices of $\mathcal{N\!S}$ (colors online). In purple is drawn the prior (analytically) known collapsing region. We extend it using Theorem \ref{['thm: sufficient condition']}: the black area is the new analytic collapsing region. The red area corresponds to the area of non-collapsing boxes. The blue area is the gap to be filled in red or purple (open problem). Drawings (a) and (b) represent the slices of $\mathcal{N\!S}$ passing through resp. $\{\mathtt{PR}, \mathtt{PR'},\mathtt{I}\}$ (case 1, finding interest in Branciard11) and $\{\mathtt{PR},\mathtt{SR},\mathtt{I}\}$ (case 2, finding interest in BS09).

Theorems & Definitions (3)

• Theorem 1: Sufficient Condition
• proof
• Remark