Quantum advantage in zero-error function computation with side information

TL;DR

The paper investigates zero-error function computation with side information for classical and quantum encoders, centering on the f-confusion graph G and its m-fold version G^{(m)}. It establishes that G^{(m)} is always sandwiched between the strong product G^{⊗ m} and the OR product G^{∨ m}, and provides necessary and sufficient conditions for equality with either extreme based on C1/C2 explanations. It then develops rate characterizations for fixed-length classical coding (via χ(G^{(m)})), variable-length coding (via chromatic entropy and Körner entropy), and quantum coding (via the orthogonal rank ξ(ar{G^{(m)}})), including single-letter results for special graph classes like perfect graphs. The work presents a spectrum of examples showing diverse quantum-classical rate behaviors across the single-instance and asymptotic m regimes, including cases with quantum advantage, VLC advantage, or equivalence, and highlights the nuanced impact of the underlying graph structure on achievable rates. Overall, the results illuminate when quantum methods yield genuine advantages in zero-error function computation with side information and map rich rate phenomena to graph-entropy frameworks, with potential implications for quantum communication and distributed computation tasks.

Abstract

We consider the problem of zero-error function computation with side information. Alice and Bob have correlated sources $X,Y$ with joint p.m.f. $p_{XY}(\cdot, \cdot)$. Bob wants to calculate $f(X,Y)$ with zero error. Alice encodes $m$-length blocks $(m \geq 1)$ of her observations to Bob over error-free channels, which can be classical or quantum. We consider two classical settings. (i) Alice communicates via a fixed length code (FLC), and (ii) Alice communicates via a variable length code (VLC). In the FLC scenario, the minimum communication rate depends on the asymptotic growth of the chromatic number of an appropriately defined $m$-instance ``confusion graph'' $G^{(m)}$. In the VLC scenario, the corresponding rate is characterized by the asymptotics of the chromatic entropy of $G^{(m)}$. %and has single-letter characterization in terms of Körner's graph entropy if $G^{(m)}$ is $m$-times graph OR product. In the quantum setting, we only consider fixed length codes; the corresponding rate depends on the asymptotic growth of the orthogonal rank of the complement of $G^{(m)}$. The behavior of the communication rates depends critically on $G^{(m)}$, which is shown to be sandwiched between $G^{\boxtimes m}$ ($m$-times strong product) and $G^{\lor m}$ ($m$-times OR product) respectively. Our work presents necessary and sufficient conditions on the function $f(\cdot, \cdot)$ and joint p.m.f. $p_{XY}(\cdot,\cdot)$ such that $G^{(m)}$ equals either $G^{\boxtimes m}$ or $G^{\lor m}$. Our work explores the multitude of possible behaviors of the quantum and classical (FLC/VLC) rates in the single-instance case and the asymptotic (in $m$) case for several classes of confusion graphs.

Figures (11)

• Figure 1: Function computation with side information
• Figure 2: The figure shows the $f$-confusion graph $G$ of $f$ with p.m.f. defined in (\ref{['eq:corr_prop_eq']}). A coloring of $G$ can be performed with three colors. This is shown in the figure with red, black and green. When communicating over one instance, the classical rate can therefore be $\log_2 3$. It can be shown that the graph over two instances $G^{(2)}$ can be colored with five colors. This leads to a per-computation classical rate of $\frac{1}{2} \log_2 5$. This is in fact optimal Witsenhausen_76lovasz1979shannon.
• Figure 3: Strong product and OR product of $G$ with itself where $G$ is a path of two edges.
• Figure 4: $G_{13}$ with a $4$-coloring.
• Figure 5: $G_{13}$ with each undirected edge viewed as two opposite directed edges.

Theorems & Definitions (93)

• Definition 1
• Example 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Remark 1
• Definition 7
• Remark 2