An inherently infinite-dimensional quantum correlation

Coladangelo, Andrea; Stark, Jalex

doi:10.1038/s41467-020-17077-9

Cited by 17 publications

(13 citation statements)

References 22 publications

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“…However, these self-test results are presented in terms of violations of Bell inequalities, unlike the CHSH game which arises from a non-local game (with binary payoff). Although we note that it may be possible to realize the tilted CHSH protocol as a game [Col19], this would still only resolve the above question for entangled states of local dimension two. Our games also resolve in the negative the question "Can every LCS game be played optimally using the maximally entangled state?"…”

Section: Introductionmentioning

confidence: 94%

A generalization of CHSH and the algebraic structure of optimal strategies

Cui,

Mehta,

Mousavi

et al. 2019

Preprint

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Self-testing has been a rich area of study in quantum information theory. It allows an experimenter to interact classically with a black box quantum system and to test that a specific entangled state was present and a specific set of measurements were performed. Recently, selftesting has been central to high-profile results in complexity theory as seen in the work on entangled games PCP of Natarajan and Vidick (FOCS 2018), iterated compression by Fitzsimons et al. (STOC 2019), and NEEXP in MIP* due to Natarajan and Wright (FOCS 2019). The most studied self-test is the CHSH game which features a bipartite system with two isolated devices. This game certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. Most of the self-testing literature has focused on extending these results to self-test for tensor products of EPR states and tensor products of Pauli measurements.In this work, we introduce an algebraic generalization of CHSH by viewing it as a linear constraint system (LCS) game, exhibiting self-testing properties that are qualitatively different. These provide the first example of non-local games that self-test non-Pauli operators resolving an open questions posed by Coladangelo and Stark (QIP 2017). Our games also provide a selftest for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal (ICALP 2012). Additionally, our games have 1 bit question and log n bit answer lengths making them suitable candidates for complexity theoretic application. This work is the first step towards a general theory of self-testing arbitrary groups. In order to obtain our results, we exploit connections between sum of squares proofs, non-commutative ring theory, and the Gowers-Hatami theorem from approximate representation theory. A crucial part of our analysis is to introduce a sum of squares framework that generalizes the solution group of Cleve, Liu, and Slofstra (Journal of Mathematical Physics 2017) to the non-pseudo-telepathic regime. Finally, we give the first example of a game that is not a self-test. Our results suggest a richer landscape of self-testing phenomena than previously considered. * It is an interesting question to see whether this observation is always true.

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Section: Introductionmentioning

confidence: 94%

A generalization of CHSH and the algebraic structure of optimal strategies

Cui,

Mehta,

Mousavi

et al. 2019

Preprint

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“…It seems that this highly important for our work question cannot be answered unambiguously at present, although there are some indications that the problem can be experimentally approached. In particular, we mention a recent work 20 , where the authors use a concept of the so-called dimension witness 21 to design an experiment, where dimensionality of a Hilbert space is decidable. Namely, to obtain the lower bound for Hilbert spaces describing measured physical systems, one settles a correlation based on a particular variant of Clauser-Horne-Shimony-Holt (CHSH) game.…”

Section: Based On This We Show Thatmentioning

confidence: 99%

“…of obtaining λ i is the specialisation of the previous general formula (20) to the discrete case, since the projection measure λ A (A → λ i ) = (projection on the 1-dimensional eigenspace spanned by e i ) = P i , so that…”

Section: The Born Rule and Solovay Generic Randomness -The Finite Dimensional Casementioning

confidence: 99%

Random World and Quantum Mechanics

Król

Bielas

Asselmeyer-Maluga

2021

Preprint

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Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω , to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

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“…C qs = C qa , where the latter is the closure. For precise definitions of these sets see [CS18]. We use superscripts to denote question and answer set sizes.…”

Section: Proposition 3 (Completeness) Let {S D } D∈n Be the Family Omentioning

confidence: 99%

A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations

Coladangelo

2020

Quantum

Self Cite

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We describe a two-player non-local game, with a fixed small number of questions and answers, such that an ϵ-close to optimal strategy requires an entangled state of dimension 2Ω(ϵ−1/8). Our non-local game is inspired by the three-player non-local game of Ji, Leung and Vidick \cite{ji2018three}. It reduces the number of players from three to two, as well as the question and answer set sizes. Moreover, it provides an (arguably) elementary proof of the non-closure of the set of quantum correlations, based on embezzlement and self-testing. In contrast, previous proofs \cite{slofstra2019set, dykema2017non, musat2018non} involved representation theoretic machinery for finitely-presented groups and C∗-algebras.

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An inherently infinite-dimensional quantum correlation

Cited by 17 publications

References 22 publications

A generalization of CHSH and the algebraic structure of optimal strategies

A generalization of CHSH and the algebraic structure of optimal strategies

Random World and Quantum Mechanics

A two-player dimension witness based on embezzlement, and an elementary proof of the non-closure of the set of quantum correlations

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