Connes implies Tsirelson: a simple proof

• URL: http://arxiv.org/abs/2209.07940v1
• Date: Fri, 16 Sep 2022 13:59:42 GMT
• Title: Connes implies Tsirelson: a simple proof
• Authors: Alexander Frei
• Abstract summary: We show that the Connes embedding problem implies the synchronous Tsirelson conjecture. We also give a different construction of Connes' algebra $mathcalRomega$ appearing in the Connes embedding problem.
• Score: 91.3755431537592
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: More precisely, we give a simple and very short proof of "the Connes embedding problem implies the synchronous Tsirelson conjecture" that relies on only two elementary ingredients: 1) the well-known description of synchronous correlations as traces on the algebra per player $C^*(\mathrm{player})=C^*(\mathrm{inputs}|\mathrm{outputs})$ and 2) an elementary lifting result by Kim, Paulsen and Schafhauser. Moreover, this bypasses every of the deep results by Kirchberg as well as any other implicit reformulation as the microstates conjecture and thelike. Meanwhile, we also give a different construction of Connes' algebra $\mathcal{R}^\omega$ appearing in the Connes embedding problem, which is more suitable for the purposes of quantum information theory and much easier to comprehend for the reader without any prior knowledge in operator algebras. Most importantly, however, we present this proof for the following reason: Since the recent refutation of the synchronous Tsirelson conjecture by MIP*=RE, there exists a nonlocal game which violates the synchronous Tsirelson conjecture, and by the proof of MIP*=RE even a synchronous such game. The approach however is based on contradiction with the undecidability of the Halting problem, and so remains implicit. As such the quest now has started to give an explicit example of a synchronous game violating the synchronous Tsirelson conjecture together with a direct argument for the failure, and the current article serves as a direct translation to the corresponding operator algebra and its tracial state violating the Connes embedding problem.

Related papers

• Topics in Non-local Games: Synchronous Algebras, Algebraic Graph Identities, and Quantum NP-hardness Reductions [0.0]
  We prove the equivalence between the hereditary and $C*$ models questioned in (Helton et al., New York J. Math.) We also extend the quantum-version NP-hardness reduction $texttt3-SAT* leq_p textttClique*$ by exhibiting another instance of such reduction.
  arXiv  Detail & Related papers  (2024-08-19T16:01:21Z)
• Nonlocality under Computational Assumptions [51.020610614131186]
  A set of correlations is said to be nonlocal if it cannot be reproduced by spacelike-separated parties sharing randomness and performing local operations. We show that there exist (efficient) local producing measurements that cannot be reproduced through randomness and quantum-time computation.
  arXiv  Detail & Related papers  (2023-03-03T16:53:30Z)
• Global Nash Equilibrium in Non-convex Multi-player Game: Theory and Algorithms [66.8634598612777]
  We show that Nash equilibrium (NE) is acceptable to all players in a multi-player game. We also show that no one can benefit unilaterally from the general theory step by step.
  arXiv  Detail & Related papers  (2023-01-19T11:36:50Z)
• The quantum commuting model (Ia): The CHSH game and other examples: Uniqueness of optimal states [91.3755431537592]
  We use the universal description of quantum commuting correlations as state space on the universal algebra for two player games. We find that the CHSH game leaves a single optimal state on this common algebra.
  arXiv  Detail & Related papers  (2022-10-07T17:38:31Z)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• Nonlocal Games, Compression Theorems, and the Arithmetical Hierarchy [0.12891210250935145]
  We investigate the connection between the complexity of nonlocal games and the arithmetical hierarchy. We show that exactly computing the quantum value is strictly harder than approximating it, and also strictly harder than computing the commuting operator value.
  arXiv  Detail & Related papers  (2021-10-09T21:53:02Z)
• The Connes Embedding Problem: A guided tour [0.0]
  The Connes Embedding Problem (CEP) is a problem in the theory of tracial von Neumann algebras. We outline two such proofs, one following the "traditional" route that goes via Kirchberg's QWEP problem in C*-algebra theory and Tsirelson's problem in quantum information theory.
  arXiv  Detail & Related papers  (2021-09-26T19:45:36Z)
• Counterexamples to the Low-Degree Conjecture [80.3668228845075]
  A conjecture of Hopkins posits that for certain high-dimensional hypothesis testing problems, no-time algorithm can outperform so-called "simple statistics" This conjecture formalizes the beliefs surrounding a line of recent work that seeks to understand statistical-versus-computational tradeoffs.
  arXiv  Detail & Related papers  (2020-04-17T21:08:11Z)
• MIP*=RE [9.42581332803306]
  We show that the class MIP* of languages can be decided by a classical verifier interacting with multiple quantum provers sharing entanglement. We provide a refutation of Connes' theory of von Neumann algebras.
  arXiv  Detail & Related papers  (2020-01-13T16:32:40Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.