Quantum advantage in zero-error function computation with side information

• URL: http://arxiv.org/abs/2402.01549v4
• Date: Wed, 19 Mar 2025 17:16:54 GMT
• Title: Quantum advantage in zero-error function computation with side information
• Authors: Ruoyu Meng, Aditya Ramamoorthy,
• Abstract summary: We consider the problem of zero-error function computation with side information.<n>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.
• Score: 10.0060346233449
• License: http://creativecommons.org/licenses/by/4.0/
• 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 K\"orner'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.

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