Quantum advantage in zero-error function computation with side information
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866908729052495872 |
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| author | Meng, Ruoyu Ramamoorthy, Aditya |
| author_facet | Meng, Ruoyu Ramamoorthy, Aditya |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_01549 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum advantage in zero-error function computation with side information Meng, Ruoyu Ramamoorthy, Aditya Information Theory Combinatorics Quantum Physics 05 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. |
| title | Quantum advantage in zero-error function computation with side information |
| topic | Information Theory Combinatorics Quantum Physics 05 |
| url | https://arxiv.org/abs/2402.01549 |