New bounds on classical and quantum one-way communication complexity

In this paper we provide new bounds on classical and quantum distributional communication complexity in the two-party, one-way model of communication. In the classical model, our bound extends the well known upper bound of Kremer, Nisan and Ron to include non-product distributions. We show that for a boolean function f:X x Y -> {0,1} and a non-product distribution mu on X x Y and epsilon in (0,1/2) constant: D_{epsilon}^{1, mu}(f)= O((I(X:Y)+1) vc(f)), where D_{epsilon}^{1, mu}(f) represents the one-way distributional communication complexity of f with error at most epsilon under mu; vc(f) represents the Vapnik-Chervonenkis dimension of f and I(X:Y) represents the mutual information, under mu, between the random inputs of the two parties. For a non-boolean function f:X x Y ->[k], we show a similar upper bound on D_{epsilon}^{1, mu}(f) in terms of k, I(X:Y) and the pseudo-dimension of f’ = f/k. In the quantum one-way model we provide a lower bound on the distributional communication complexity, under product distributions, of a function f, in terms the well studied complexity measure of f referred to as the rectangle bound or the corruption bound of f . We show for a non-boolean total function f : X x Y -> Z and a product distribution mu on XxY, Q_{epsilon^3/8}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)), where Q_{epsilon^3/8}^{1, mu}(f) represents the quantum one-way distributional communication complexity of f with error at most epsilon^3/8 under mu and rec_ epsilon^{1, mu}(f) represents the one-way rectangle bound of f with error at most epsilon under mu . Similarly for a non-boolean partial function f:XxY -> Z U {*} and a product distribution mu on X x Y, we show, Q_{epsilon^6/(2 x 15^4)}^{1, mu}(f) = Omega(rec_ epsilon^{1, mu}(f)).

💡 Research Summary

The paper studies two‑party one‑way communication, where Alice sends a single message to Bob, and investigates both classical and quantum distributional communication complexity under arbitrary input distributions. The authors first revisit the classical upper bound of Kremer, Nisan and Ron (KNR), which states that for any Boolean function f and any product distribution μ, the one‑way distributional complexity D₁,μ^ε(f) is O(VC(f)). They observe that this bound cannot hold for non‑product distributions because it would imply a universal O(VC(f)) bound on public‑coin one‑way complexity, which is false for functions such as Greater‑Than. To address this gap they introduce the mutual information I(X : Y) of the input pair (X,Y) under μ as a measure of correlation. Their main classical result (Theorem 2) shows that for any Boolean f, any (possibly correlated) distribution μ, and any constant error ε∈(0,½),

 D₁,μ^ε(f) ≤ κ·(I(X : Y)+1)·VC(f),

where κ is a universal constant. Consequently, by Yao’s principle, the public‑coin one‑way randomized complexity satisfies

 R₁, pub(f) = O((I(X : Y)+1)·VC(f))

where the mutual information is taken under a hard distribution for f. This refines the deterministic bound D₁(f)=O(VC(f)·log|Y|) because I(X : Y) ≤ log|Y| and can be much smaller (e.g., for Inner‑Product under the uniform product distribution I=0). The authors also compare with known relations such as R₁(f)=O(Q₁(f)·log|Y|) and note that their bound can be stronger when I is small.

For non‑Boolean functions f:X×Y→{1,…,k} they generalize VC‑dimension to the pseudo‑dimension of the normalized function f′=f/k. Theorem 3 gives an upper bound

 D₁,μ^{3ε}(f) ≤ κ·k·(log(1/ε)+log k)·(I(X : Y)+log k)·Pd_ε(f′),

where Pd_ε(f′) is the ε‑pseudo‑dimension. This captures the dependence on the output alphabet size k and the “real‑valued” complexity of f′.

Turning to the quantum setting, the authors focus on product distributions μ. They study the one‑way rectangle (corruption) bound rec₁,μ^ε(f), a well‑known lower‑bound technique for classical communication. Theorem 4 proves that if rec₁,μ^ε(f) > 2·log(1/ε), then the quantum one‑way distributional complexity satisfies

 Q₁,μ^{ε³/8}(f) ≥ ½·(1−2ε)·(S(ε/2)−S(ε/4))·(⌊rec₁,μ^ε(f)⌋−1) = Ω(rec₁,μ^ε(f)),

where S(p) is the binary entropy. For partial functions a similar bound holds with a different constant factor. This shows that, under product distributions, the rectangle bound also lower‑bounds quantum one‑way communication, extending the classical relationship to the quantum regime. The authors note that the product‑distribution assumption is essential: there exist total functions and non‑product μ for which Q₁,μ^ε(f) is exponentially smaller than rec₁,μ^ε(f) (as implicit in Gavinsky et al. 2007).

The paper also connects these technical results to cryptographic applications. By relating the rectangle bound of an extractor to the minimum min‑entropy required for extracting a uniform bit, they recover and generalize a result of König and Terhal (2008) showing that any Boolean extractor secure against quantum adversaries with bounded memory can be analyzed via its rectangle bound.

The structure of the paper is as follows: Section 2 reviews information‑theoretic preliminaries and formal definitions of one‑way communication models. Section 3 presents the classical upper bounds, proving Theorems 2 and 3 and discussing their implications. Section 4 establishes the quantum lower bounds (Theorem 4) and discusses why product distributions are necessary. Section 5 applies the quantum lower bound to the security analysis of randomness extractors against quantum adversaries. Section 6 concludes with open problems, such as extending the quantum lower bound to non‑product distributions or tightening the dependence on ε.

Overall, the work unifies several complexity measures—mutual information, VC/pseudo‑dimension, and rectangle bound—within the one‑way communication framework, yielding tighter classical upper bounds for correlated inputs and the first quantum lower bounds that directly tie quantum communication to the classical rectangle bound under product distributions. These contributions deepen our understanding of the interplay between information theory and communication complexity, and they have concrete implications for the design of communication‑efficient protocols and quantum‑secure randomness extractors.