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Title: Multiple operator integrals, Haagerup and Haagerup-like tensor products, and operator ideals: MULTIPLE OPERATOR INTEGRALS, HAAGERUP AND HAAGERUP-LIKE TENSOR
NSF-PAR ID:
10035302
Author(s) / Creator(s):
 ;  
Publisher / Repository:
DOI PREFIX: 10.1112
Date Published:
Journal Name:
Bulletin of the London Mathematical Society
Volume:
49
Issue:
3
ISSN:
0024-6093
Page Range / eLocation ID:
463 to 479
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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  3. Abstract

    We consider 3XOR games with perfect commuting operator strategies. Given any 3XOR game, we show existence of a perfect commuting operator strategy for the game can be decided in polynomial time. Previously this problem was not known to be decidable. Our proof leads to a construction, showing a 3XOR game has a perfect commuting operator strategy iff it has a perfect tensor product strategy using a 3 qubit (8 dimensional) GHZ state. This shows that for perfect 3XOR games the advantage of a quantum strategy over a classical strategy (defined by the quantum-classical bias ratio) is bounded. This is in contrast to the general 3XOR case where the optimal quantum strategies can require high dimensional states and there is no bound on the quantum advantage. To prove these results, we first show equivalence between deciding the value of an XOR game and solving an instance of the subgroup membership problem on a class of right angled Coxeter groups. We then show, in a proof that consumes most of this paper, that the instances of this problem corresponding to 3XOR games can be solved in polynomial time.

     
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