Rounding near-optimal quantum strategies for nonlocal games to strategies using maximally entangled states
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866918143485542400 |
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| author | Paddock, Connor |
| author_facet | Paddock, Connor |
| contents | We establish approximate rigidity results for several well-known families of nonlocal games. In particular, we show that near-perfect quantum strategies for boolean constraint system (BCS) games are approximate representations of the corresponding BCS algebra. Likewise, for the class of XOR nonlocal games, we show that near-optimal quantum strategies are approximate representations of the corresponding $*$-algebra associated with optimal quantum values for the game. In both cases, the approximate representations are with respect to the normalized Hilbert-Schmidt norm and independent of the Hilbert space or quantum state employed in the strategy. We also show that approximate representation of the BCS (resp.~XOR-algebra) yields measurement operators for near-perfect (resp.~near-optimal) quantum strategies where the players employ a maximally entangled state in the game. As a corollary, every near-perfect BCS (resp.~near-optimal XOR) quantum strategy is close to a near-perfect (resp.~near-optimal) quantum strategy using a maximally entangled state. Lastly, we establish that every synchronous algebra is $*$-isomorphic to a certain BCS algebra called the SynchBCS algebra. This allows us to apply our BCS rigidity results to the class of synchronous games as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_02525 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Rounding near-optimal quantum strategies for nonlocal games to strategies using maximally entangled states Paddock, Connor Quantum Physics We establish approximate rigidity results for several well-known families of nonlocal games. In particular, we show that near-perfect quantum strategies for boolean constraint system (BCS) games are approximate representations of the corresponding BCS algebra. Likewise, for the class of XOR nonlocal games, we show that near-optimal quantum strategies are approximate representations of the corresponding $*$-algebra associated with optimal quantum values for the game. In both cases, the approximate representations are with respect to the normalized Hilbert-Schmidt norm and independent of the Hilbert space or quantum state employed in the strategy. We also show that approximate representation of the BCS (resp.~XOR-algebra) yields measurement operators for near-perfect (resp.~near-optimal) quantum strategies where the players employ a maximally entangled state in the game. As a corollary, every near-perfect BCS (resp.~near-optimal XOR) quantum strategy is close to a near-perfect (resp.~near-optimal) quantum strategy using a maximally entangled state. Lastly, we establish that every synchronous algebra is $*$-isomorphic to a certain BCS algebra called the SynchBCS algebra. This allows us to apply our BCS rigidity results to the class of synchronous games as well. |
| title | Rounding near-optimal quantum strategies for nonlocal games to strategies using maximally entangled states |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2203.02525 |