Properties of operator systems, corresponding to channels
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Properties of operator systems, corresponding to channels V. I. Yashin1 Received: 1 December 2019 / Accepted: 29 April 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract It was shown in Duan (Super-activation of zero-error capacity of noisy quantum channels, 2009. arXiv:0906.2527), that in finite-dimensional Hilbert spaces each operator system corresponds to some channel, for which this operator system will be an operator graph. This work is devoted to finding necessary and sufficient conditions for this property to hold in infinite-dimensional case. Keywords Non-commutative graphs · Operator systems · Quantum error correction · Quantum channels
1 Introduction The theory of zero-error communication via classical channels was first introduced in C.Shannon’s renown paper [16]. He considered the problem of constructing zeroerror code as a problem of finding maximal independent subset in a confusability graph of a channel, and defined zero-error capacity of a channel as a regularisation of an independence number of a graph. The notion of zero-error capacity for quantum channels was first introduced in [14]. The authors of [11] proposed studying zeroerror communication for quantum channels as a “noncommutative” graph-theoretic problem. They introduced a noncommutative confusability graph for a noisy quantum channel and considered the problem of finding zero-error capacity for this graph. To find a correcting code for a quantum channel means to find a quantum anticlique for this channel [1–5,20]. This gave a perspective for creation of noncommutative graph theory as a theory of operator systems [7,15], which are studied as a part of functional analysis [9]. Some notions of classical graph theory were generalized to
The work is supported by Russian Science Foundation under the Grant No. 19-11-00086.
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V. I. Yashin [email protected] Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow, Russia 119991 0123456789().: V,-vol
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the quantum case, such as Lovász number [11], Ramsey theory [20], Turán problem [21] and chromatic number [6]. Mostly, operator graphs were studied over finite-dimensional Hilbert spaces. Generalising the theory to infinite dimensions means to come across a number of functional-analytic complexities, because the structure of quantum channels and operator systems in infinite-dimensional case is rather complicated. In [8,10] it was shown, that in finite-dimensional Hilbert spaces each operator system is a noncommutative confusability graph for some channel. In this work we consider the generalisation of this property for infinite-dimensional separable Hilbert spaces. Differently speaking, we investigate the images of unital completely positive maps in details. In Sect. 2 we remind basic notions of quantum communication theory. Section 3 contains known results about finite-dimensional quantum confusability graphs, which will be used in Sect. 4. In Sect. 4 we find the necessary and sufficient cond
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