On inner geometry of noncommutative operator graphs
Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, 119991, Moscow, Russia
Accepted: 20 October 2020
Published online: 29 October 2020
Operator systems (noncommutative operator graphs in other terminology) play a major role in the theory of quantum error correcting codes. Any operator graph is associated with a number of quantum channels. The possibility to transmit quantum information through a quantum channel with zero error is determined by the geometrical properties of the corresponding graph. Noncommutative operator graphs are known to be generated by positive operator-valued measures (POVMs). In turn, many principal POVMs consist of multiple of projections. We construct the model in which the graph is a linear envelope of two projection-valued resolutions of identities in a Hilbert space. Conditions for the existence of quantum anticliques (error-correcting codes) for the graph are investigated. The connection with Shirokov’s example of quantum superactivation (Shirokov in Probl Inform Transm 51(2):87–102, 2015; Shirokov and Shulman in Commun Math Phys 335:1159, 2015) is revealed.
© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2020