Communication complexity bounds from information causality

TL;DR

This work extends the information causality principle to a distributed, one-way, entanglement-assisted communication setting, deriving a universal, theory-independent lower bound on $C^*_ abla(f)$ expressed via a sum of conditional mutual informations. The bound reproduces known asymptotically tight lower bounds for INDEX$_n$, IP$_n$, and DISJ$_n$, and yields linear bounds for EQ$_n$ in the deterministic case while remaining trivial for $\varepsilon>0$; it also gives $\Omega(n-2k)$ bounds for $k$-INT$_n$. By linking information-theoretic constraints to the strength of quantum correlations in Bell tests, the results unify and strengthen the original information causality framework, offering a tool to bound quantum advantages and guide extensions to two-way communication and function-classifications. The framework thus provides a principled route to assess fundamental limits on quantum technologies from mutual-information axioms, with potential applications to nonlocality, PR-box analyses, and beyond.

Abstract

Communication complexity, which quantifies the minimum communication required for distributed computation, offers a natural setting for investigating the capabilities and limitations of quantum mechanics in information processing. We introduce an information-theoretic approach to study one-way communication complexity based solely on the axioms of mutual information. Within this framework, we derive an extended statement of the information causality principle, which recovers known lower bounds on the communication complexities for a range of functions in a simplified manner and leads to new results. We further prove that the extended information causality principle is at least as strong as the principle of non-trivial communication complexity in bounding the strength of quantum correlations attainable in Bell experiments. Our study establishes a new route for exploring the fundamental limits of quantum technologies from an information-theoretic viewpoint.

Figures (2)

• Figure 1: Considered scenario: at each round, Alice and Bob receive random inputs $x$ and $y$, taking values in $X$ and $Y$, respectively. Alice sends $m$ bits of data to Bob, who then produces a guess $g$ of the value of $f(x,y)$. Alice and Bob have access to unlimited shared entanglement.
• Figure 2: a) The process of classification of Boolean functions with equivalent form of the lower bound in \ref{['th:ULB']} for $X=Y=\Set{0,1}^2$. The root $0$ denotes the case of $f_{00}(x_0,x_1)=0$, and each node represents a choice for $f_{y_0y_1}$, with the leafs corresponding to the choice of $f_{11}$. The blue arrows represent twelve unique branches, which are then further reduced to eight non-equivalent extended IC statements. b) Relations between the eight equivalence classes in \ref{['eq:8_classes']} with respect to the constraints that they impose on the set of quantum correlations. An arrow between two classes represents that one class of functions implies the other.

Theorems & Definitions (14)

• Theorem 1
• Example 1
• Example 2
• Example 3
• Example 4
• Example 5
• Theorem 2
• proof
• Proposition 3
• proof