A non-commutative unitary analogue of Kirchberg's conjecture

Harris, S.J.

doi:10.1512/iumj.2019.68.7598

Cited by 36 publications

(5 citation statements)

References 25 publications

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“…We show that any such quotient possesses the local lifting property [43]. This unifies a number of results in the literature, implying in particular [35,Theorem 4.9].…”

Section: Introductionsupporting

confidence: 85%

Quantum no-signalling correlations and non-local games

Todorov,

Turowska

2020

Preprint

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We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system, and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.

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“…We show that any such quotient possesses the local lifting property [43]. This unifies a number of results in the literature, implying in particular [35,Theorem 4.9].…”

Section: Introductionsupporting

confidence: 85%

Quantum no-signalling correlations and non-local games

Todorov,

Turowska

2020

Preprint

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show abstract

“…We show that any such quotient possesses the local lifting property [44]. This unifies a number of results in the literature, implying in particular [36,Theorem 4.9]. In Sect.…”

Section: Introductionsupporting

confidence: 86%

Quantum No-signalling Correlations and Non-local Games

Todorov,

Turowska

2024

Commun. Math. Phys.

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We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.

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“…Theorem A can be viewed as liftings of (1.1) to C * -algebras. It provides explicit connections between the two universal C * -algebras and relates to some recent results in [15]. Moreover, the analogous result between "reduced" algebras is also obtained.…”

Section: Introductionsupporting

confidence: 68%

“…Cleve, Liu and Paulsen showed in [7] that the protocol for embezzling entanglement corresponds to a state on the minimal tensor product B d ⊗ min B d that cannot be implemented as a vector state via tensor product representations. Their result is in parallel to Stolstra's result in the sense that the Brown algebra has analogues of Kirchberg's conjecture [15] and Tsirelson's problem [16]. Our idea is to apply the * -isomorphisms in Theorem A to translate the non-spatial correlation from…”

Section: Introductionmentioning

confidence: 85%

Quantum Teleportation and Super-Dense Coding in Operator Algebras

Gao

Harris

Junge

2019

International Mathematics Research Notices

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Let B d be the unital C * -algebra generated by the elements u jk , 0 ≤ i, j ≤ d − 1, satisfying the relations that [u j,k ] is a unitary operator, and let C * (F d 2 ) be the full group C * -algebra of free group of d 2 generators. Based on the idea of teleportation and super-dense coding in quantum information theory, we exhibit the two * -isomorphisms for certain actions of Z d . As an application, we show that for any d, m ≥ 2 with (d, m) = (2, 2), the matrix-valued generalization of the (tensor product) quantum correlation set of d inputs and m outputs is not closed.

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A non-commutative unitary analogue of Kirchberg's conjecture

Cited by 36 publications

References 25 publications

Quantum no-signalling correlations and non-local games

Quantum no-signalling correlations and non-local games

Quantum No-signalling Correlations and Non-local Games

Quantum Teleportation and Super-Dense Coding in Operator Algebras

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