Communication Complexity of Graph Isomorphism, Coloring, and Distance Games
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910500491624448 |
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| author | Botteron, Pierre Weber, Moritz |
| author_facet | Botteron, Pierre Weber, Moritz |
| contents | In quantum information, nonlocal games are particularly useful for differentiating classical, quantum, and non-signalling correlations. An example of differentiation is given by the principle of no-collapse of communication complexity, which is often interpreted as necessary for a feasible physical theory. It is satisfied by quantum correlations but violated by some non-signalling ones.
In this work, we investigate this principle in the context of three nonlocal games related to graph theory, starting from the well-known graph isomorphism and graph coloring games, and introducing a new game, the vertex distance game, with a parameter $D\in\mathbb N$, that generalizes the former two to some extent. For these three games, we prove that perfect non-signalling strategies collapse communication complexity under favorable conditions. We also define a refinement of fractional isomorphism of graphs, namely D-fractional isomorphisms, and we show that this characterizes perfect non-signalling strategies for the vertex distance game. Surprisingly, we observe that non-signalling strategies provide a finer distinction for the new game compared to classical and quantum strategies since the parameter D is visible only in the non-signalling setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_02199 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Communication Complexity of Graph Isomorphism, Coloring, and Distance Games Botteron, Pierre Weber, Moritz Quantum Physics Mathematical Physics Combinatorics In quantum information, nonlocal games are particularly useful for differentiating classical, quantum, and non-signalling correlations. An example of differentiation is given by the principle of no-collapse of communication complexity, which is often interpreted as necessary for a feasible physical theory. It is satisfied by quantum correlations but violated by some non-signalling ones. In this work, we investigate this principle in the context of three nonlocal games related to graph theory, starting from the well-known graph isomorphism and graph coloring games, and introducing a new game, the vertex distance game, with a parameter $D\in\mathbb N$, that generalizes the former two to some extent. For these three games, we prove that perfect non-signalling strategies collapse communication complexity under favorable conditions. We also define a refinement of fractional isomorphism of graphs, namely D-fractional isomorphisms, and we show that this characterizes perfect non-signalling strategies for the vertex distance game. Surprisingly, we observe that non-signalling strategies provide a finer distinction for the new game compared to classical and quantum strategies since the parameter D is visible only in the non-signalling setting. |
| title | Communication Complexity of Graph Isomorphism, Coloring, and Distance Games |
| topic | Quantum Physics Mathematical Physics Combinatorics |
| url | https://arxiv.org/abs/2406.02199 |