Properties of operator systems, corresponding to channels

Yashin, V.I.

doi:10.1007/s11128-020-02693-7

Cited by 9 publications

(5 citation statements)

References 16 publications

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“…Удается показать, что любой операторный граф и в конечномерном, и в бесконечномерном случаях является замыканием линейной оболочки некоторого обобщенного разложения единицы [1,23]. Таким образом, вместо предположений о виде шума мы можем накладывать какие-то условия на представления единицы, порождающие операторный граф.…”

Section: квантовый шум и некоммутативные операторные графыunclassified

On Noncommutative Operator Graphs Generated by Resolutions of Identity

Amosov¹,

Мокеев²,

Mokeev

2021

Trudy Mat. Inst. Steklova

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Некоммутативные операторные графы составляют важную часть квантовой теории исправления ошибок. В работе дается краткий обзор последних результатов, посвященных графам, порожденным разложениями единицы, для которых существует квантовый код, исправляющий ошибки. Обсуждаются примеры таких графов, затрагивается задача об описании квантового шума в рамках данной теории.

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Section: квантовый шум и некоммутативные операторные графыunclassified

On Noncommutative Operator Graphs Generated by Resolutions of Identity

Amosov¹,

Мокеев²,

Mokeev

2021

Trudy Mat. Inst. Steklova

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“…Recently, there have been some interests in infinite dimensional operator systems in quantum information theory (see e.g. [5,19]). In this respects, it seems that an analogue of [4,Corollary 4.5] in the infinite dimensional case could be a useful tool.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, let us recall that a unital operator system T ⊆ L(H) is a graph system if it is closed under the weak- * -topology (see [16,19]). The following is a direct consequence of Theorem 3.17(a) and Corollary 3.9(b).…”

mentioning

confidence: 99%

Dual spaces of operator systems

Ng¹

2021

Preprint

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The aim of this article is to give an infinite dimensional analogue of a result of Choi and Effros concerning dual spaces of finite dimensional unital operator systems.An (not necessarily unital) operator system is a self-adjoint subspace of L(H), equipped with the induced matrix norm and the induced matrix cone. We say that an operator system T is dualizable if one can find an equivalent dual matrix norm on the dual space T * such that under this dual matrix norm and the canonical dual matrix cone, T * becomes a dual operator system.We show that an operator system T is dualizable if and only if the ordered normed space M∞(T ) sa satisfies a form of bounded decomposition property. In this case,is the largest dual matrix norm that is equivalent to and dominated by the original dual matrix norm on T * that turns it into a dual operator system, denoted by T d . It can be shown that T d is again dualizable. Furthermore, we will verify that that if S is either a C * -algebra or a unital operator system, then S is dualizable and the canonical weak- * -homeomorphism from the unital operator system S * * to the operator system (S d ) d is a completely isometric complete order isomorphism.Consequently, a nice duality framework for operator systems is obtained, which includes all C *algebras and all unital operator systems.

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“…So it could be meaningful to explore the problem of quantum error-correcting codes for graphs with parametrizations given in some separate way from mentioned error-correction formalism, in that description each conjecture on a graph also is the conjecture on a possible noise. It is shown [5][6][7], that any non-commutative operator graph is the closure of the linear envelope of some positive operator valued measure (POVM). So the problem of quantum error correction could be viewed in terms of POVMs.…”

Section: Introductionmentioning

confidence: 99%

“…Hence V is an operator system in the sense of [12]. Moreover any operator system is a non-commutative operator graph associated with some quantum channel [5][6][7]. If there is a projection P called a (quantum) anticlique [13] that has the property…”

Section: Introductionmentioning

confidence: 99%

On errors generated by unitary dynamics of bipartite quantum systems

Amosov,

Mokeev

2020

Preprint

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Given a quantum channel it is possible to define the noncommutative operator graph whose properties determine a possibility of errorfree transmission of information via this channel. The corresponding graph has a straight definition through Kraus operators determining quantum errors. We are discussing the opposite problem of a proper definition of errors that some graph corresponds to. Taking into account that any graph is generated by some POVM we give a solution to such a problem by means of the Naimark dilatation theorem. Using our approach we construct errors corresponding to the graphs generated by unitary dynamics of bipartite quantum systems. The cases of POVMs on the circle group Z n and the additive group R are discussed. As an example we construct the graph corresponding to the errors generated by dynamics of two mode quantum oscillator.

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Properties of operator systems, corresponding to channels

Cited by 9 publications

References 16 publications

On Noncommutative Operator Graphs Generated by Resolutions of Identity

On Noncommutative Operator Graphs Generated by Resolutions of Identity

Dual spaces of operator systems

On errors generated by unitary dynamics of bipartite quantum systems

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