Geometry of the set of synchronous quantum correlations

Russell, Travis

doi:10.1063/1.5115010

Cited by 4 publications

(3 citation statements)

References 23 publications

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“…However, no similar dimension reduction argument is known when the number of measurements or number of measurement outcomes is greater than 2. A result of Russell describes another linear slice, the synchronous correlations, in C q (3, 3, 2, 2), but again this description does not extend to other numbers of measurements and outcomes [Rus20].…”

Section: Introductionmentioning

confidence: 96%

The membership problem for constant-sized quantum correlations is undecidable

Fu¹,

Miller²,

Slofstra³

2021

Preprint

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When two spatially separated parties make measurements on an unknown entangled quantum state, what correlations can they achieve? How difficult is it to determine whether a given correlation is a quantum correlation? These questions are central to problems in quantum communication and computation. Previous work has shown that the general membership problem for quantum correlations is computationally undecidable. In the current work we show something stronger: there is a family of constant-sized correlations -that is, correlations for which the number of measurements and number of measurement outcomes are fixed -such that solving the quantum membership problem for this family is computationally impossible. Thus, the undecidability that arises in understanding Bell experiments is not dependent on varying the number of measurements in the experiment. This places strong constraints on the types of descriptions that can be given for quantum correlation sets. Our proof is based on a combination of techniques from quantum self-testing and from undecidability results of the third author for linear system nonlocal games.

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Section: Introductionmentioning

confidence: 96%

The membership problem for constant-sized quantum correlations is undecidable

Fu¹,

Miller²,

Slofstra³

2021

Preprint

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“…In another paper, Dykema-Paulsen-Parakash [6] provided a negative answer to the strong Tsirelson problem (whether or not C q (n, m) = C qc (n, m)) in the setting of synchronous correlation sets with n = 5 and m = 2. In [16] the author determined the geometry of the synchronous quantum correlation set in the setting of n = 3 and m = 2 but left the solution to Tsirelson's problems in this setting open. In this paper we complete the study of the n = 3, m = 2 case by showing that the strong Tsirelson problem has an affirmative answer in this setting.…”

Section: Introductionmentioning

confidence: 99%

Two-outcome synchronous correlation sets and Connes' embedding problem

Russell¹

2019

Preprint

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We show that Connes' embedding problem is equivalent to the weak Tsirelson problem in the setting of two-outcome synchronous correlation sets. We further show that the extreme points of two-outcome synchronous correlation sets can be realized using a certain class of universal C*-algebras. We examine these algebras in the three-experiment case and verify that the strong and weak Tsirelson problems have affirmative answers in that setting.

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“…Proof. By [12], the set C s q in our (three input, two output) case is closed. Thus, the supremum (11) used to compute ω q (G, π) is achieved.…”

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confidence: 96%

Products of synchronous games

Mančinska,

Paulsen,

Todorov

et al. 2021

Preprint

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We show that the *-algebra of the product of two synchronous games is the tensor product of the corresponding *-algebras. We prove that the product game has a perfect C*-strategy if and only if each of the individual games does, and that in this case the C*-algebra of the product game is *-isomorphic to the maximal C*-tensor product of the individual C*-algebras. We provide examples of synchronous games whose synchronous values are strictly supermultiplicative.

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Geometry of the set of synchronous quantum correlations

Cited by 4 publications

References 23 publications

The membership problem for constant-sized quantum correlations is undecidable

The membership problem for constant-sized quantum correlations is undecidable

Two-outcome synchronous correlation sets and Connes' embedding problem

Products of synchronous games

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