Two-outcome synchronous correlation sets and Connes' embedding problem

Russell, Travis

doi:10.26421/qic20.5-6-1

Cited by 4 publications

(5 citation statements)

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“…On the other hand, if one only deals with correlations rather than non-local games, then the synchronous correlations naturally occur as a closed face of a larger bisynchronous correlation set. This is similar to how C s t (n, k) arises as a closed face of C s t (nk, 2) (see [23] and also [10]).…”

Section: Bisynchronous Gamessupporting

confidence: 64%

Synchronous games with $*$-isomorphic game algebras

Harris¹

2021

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We establish several strong equivalences of synchronous non-local games, in the sense that the corresponding game algebras are * -isomorphic. We first show that the game algebra of any synchronous game on n inputs and k outputs is * -isomorphic to the game algebra of an associated bisynchronous game on nk inputs and nk outputs. As a result, we show that there are bisynchronous games with equal question and answer sets, whose optimal strategies only exist in the quantum commuting model, and not in the quantum approximate model. Moreover, we exhibit a bisynchronous game with 20 questions and 20 answers that has a non-zero game algebra, but no winning commuting strategy, resolving a problem of V.I. Paulsen and M. Rahaman. We also exhibit a * -isomorphism between any synchronous game algebra with n questions and k > 3 answers and a synchronous game algebra with n(k − 2) questions and 3 answers.

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Section: Bisynchronous Gamessupporting

confidence: 64%

Synchronous games with $*$-isomorphic game algebras

Harris¹

2021

Preprint

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“…Letting ω povm (G, π) denote the max value of the game over all densities of the form τ (E x,a E y,b ) for POVMs {E x,a } in a finite dimensional von Neumann algebra with a trace τ , the last assertion of the proposition has main assumption which implies ω s q (G, π) ≤ ω povm (G, π) ≤ ω qc (G, π) = ω q (G, π); the second inequality following from Lemma 5.2 of [Ru20]. Thus our maximizing PVM strategy is a POVM maximizer which amounts to the demanding hypothesis of Proposition 7.9.…”

Section: Optimality Conditionsmentioning

confidence: 97%

“…Here, we consider optimizing trace functionals over the bigger set of POVM's, which might well produce a higher maximum. Note that if {E x,a } are only POVM's, then setting p(a, b|x, y) = τ (E x,a E y,b ), does define a density in C qc , see Lemma 5.2 of [Ru20]. But it will not necessarily be a synchronous density.…”

Section: Optimality Conditionsmentioning

confidence: 99%

“…Thus, the set of densities that can be obtained in this fashion is strictly larger than the synchronous densities, but it is also known to be smaller than the set of all densities in C qc . For more details on this set of densities see [Ru20]. We write ω povm (G, π) for the value of a game over the set of densities obtained as traces of POVM's.…”

Section: Optimality Conditionsmentioning

confidence: 99%

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Synchronous Values of Games

Helton¹,

Mousavi²,

Nezhadi³

et al. 2021

Preprint

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We study synchronous values of games, especially synchronous games. It is known that a synchronous game has a perfect strategy if and only if it has a perfect synchronous strategy. However, we give examples of synchronous games, in particular graph colouring games, with synchronous value that is strictly smaller than their ordinary value. Thus, the optimal strategy for a synchronous game need not be synchronous.We derive a formula for the synchronous value of an XOR game as an optimization problem over a spectrahedron involving a matrix related to the cost matrix.We give an example of a game such that the synchronous value of repeated products of the game is strictly increasing. We show that the synchronous quantum bias of the XOR of two XOR games is not multiplicative.Finally, we derive geometric and algebraic conditions that a set of projections that yields the synchronous value of a game must satisfy.

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“…quantum commuting correlations) with n questions and k = 2 answers. For k > 2, it was shown in [16] and [8] that a particular affine slice of the set D q (nk) (resp. D qc (nk)) is affinely isomorphic to the set of synchronous quantum correlations (resp.…”

Section: Preliminariesmentioning

confidence: 99%

A synchronous NPA hierarchy with applications

Russell¹

2021

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We present an adaptation of the NPA hierarchy to the setting of synchronous correlation matrices. Our adaptation improves upon the original NPA hierarchy by using smaller certificates and fewer constraints, although it can only be applied to certify synchronous correlations. We recover characterizations for the sets of synchronous quantum commuting and synchronous quantum correlations. For applications, we show that the existence of symmetric informationally complete positive operator-valued measures and maximal sets of mutually unbiased bases can be verified or invalidated with only two certificates of our adapted NPA hierarchy.

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Two-outcome synchronous correlation sets and Connes' embedding problem

Cited by 4 publications

References 0 publications

Synchronous games with $*$-isomorphic game algebras

Synchronous games with $*$-isomorphic game algebras

Synchronous Values of Games

A synchronous NPA hierarchy with applications

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