We describe the main classes of non-signalling bipartite correlations in terms of states on operator system tensor products. This leads to the introduction of another new class of games, called reflexive games, which are characterised as the hardest non-local games that can be won using a given set of strategies. We provide a characterisation of their perfect strategies in terms of operator system quotients. We introduce a new class of non-local games, called imitation games, in which the players display linked behaviour, and which contain as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and unique games. We associate a C*-algebra C*(G)\$C\^{*}($\backslash$mathcal {G})\$ to any imitation game G\$$\backslash$mathcal {G}\$, and show that the existence of perfect quantum commuting (resp. quantum, local) strategies of G\$$\backslash$mathcal {G}\$ can be characterised in terms of properties of this C*-algebra. We single out a subclass of imitation games, which we call mirror games, and provide a characterisation of their quantum commuting strategies that has an algebraic flavour, showing in addition that their approximately quantum perfect strategies arise from amenable traces on the encoding C*-algebra.

}, isbn = {1572-9656}, doi = {10.1007/s11040-020-9331-7}, url = {https://doi.org/10.1007/s11040-020-9331-7}, author = {Lupini, M. and Man{\v c}inska, L. and Paulsen, V. I. and Roberson, D. E. and Scarpa, G. and Severini, S. and Todorov, I. G. and Winter, A.} } @article {duan_zero-error_2016, title = {On {Zero}-{Error} {Communication} via {Quantum} {Channels} in the {Presence} of {Noiseless} {Feedback}}, journal = {IEEE Transactions on Information Theory}, volume = {62}, number = {9}, year = {2016}, month = {sep}, pages = {5260{\textendash}5277}, abstract = {We initiate the study of zero-error communication via quantum channels when the receiver and the sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon{\textquoteright}s zero-error communication theory with instantaneous feedback. We first show that this capacity is only a function of the linear span of Choi-Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub non-commutative bipartite graph. Then, we go on to show that the feedback-assisted capacity is non-zero (allowing for a constant amount of activating noiseless communication) if and only if the non-commutative bipartite graph is non-trivial, and give a number of equivalent characterizations. This result involves a far-reaching extension of the conclusive exclusion of quantum states. We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede. We demonstrate that this bound to have many good properties, including being additive and given by a minimax formula. We also prove a coding theorem showing that this quantity is the entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Choi-Kraus span, which can also be interpreted as the feedback-assisted unambiguous capacity. The proof relies on a generalization of the Postselection Lemma (de Finetti reduction) that allows to reflect additional constraints, and which we believe to be of independent interest. This capacity is a relaxation of the feedback-assisted zero-error capacity; however, we have to leave open the question of whether they coincide in general. We illustrate our ideas with a number of examples, including classical-quantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions.}, keywords = {bipartite equivocation graph, Bipartite graph, Capacity planning, Choi-Kraus operators, coding theorem, commutative bipartite graph, Electronic mail, encoding, entanglement-assisted capacity, feedback, feedback-assisted capacity, feedback-assisted zero-error capacity, information theory, noiseless feedback channel, quantum channels, quantum communication, quantum entanglement, Quantum information, Receivers, Shannon{\textquoteright}s zero-error communication, telecommunication channels, unlimited quantum capacity, zero-error capacity}, issn = {0018-9448}, doi = {10.1109/TIT.2016.2562580}, author = {Duan, R. and Severini, S. and Winter, A.} }