On construction of anticliques for non-commutative operator graphs

In this paper anticliques for non-commutative operator graphs generated by the generalized Pauli matrices are constructed. It is shown that application of entangled states for the construction of code space K allows one to substantially increase the dimension of a non-commutative operator graph for which the projection on K is an anticlique.Comment: 11 pages, typos are correcte

Zero-error channel capacity and simulation assisted by non-local correlations

Shannon's theory of zero-error communication is re-examined in the broader setting of using one classical channel to simulate another exactly, and in the presence of various resources that are all classes of non-signalling correlations: Shared randomness, shared entanglement and arbitrary non-signalling correlations. Specifically, when the channel being simulated is noiseless, this reduces to the zero-error capacity of the channel, assisted by the various classes of non-signalling correlations. When the resource channel is noiseless, it results in the "reverse" problem of simulating a noisy channel exactly by a noiseless one, assisted by correlations. In both cases, 'one-shot' separations between the power of the different assisting correlations are exhibited. The most striking result of this kind is that entanglement can assist in zero-error communication, in stark contrast to the standard setting of communicaton with asymptotically vanishing error in which entanglement does not help at all. In the asymptotic case, shared randomness is shown to be just as powerful as arbitrary non-signalling correlations for noisy channel simulation, which is not true for the asymptotic zero-error capacities. For assistance by arbitrary non-signalling correlations, linear programming formulas for capacity and simulation are derived, the former being equal (for channels with non-zero unassisted capacity) to the feedback-assisted zero-error capacity originally derived by Shannon to upper bound the unassisted zero-error capacity. Finally, a kind of reversibility between non-signalling-assisted capacity and simulation is observed, mirroring the famous "reverse Shannon theorem".Comment: 18 pages, 1 figure. Small changes to text in v2. Removed an unnecessarily strong requirement in the premise of Theorem 1

Maximum privacy without coherence, zero-error

We study the possible difference between the quantum and the private capacities of a quantum channel in the zero-error setting. For a family of channels introduced by Leung et al. [Phys. Rev. Lett. 113, 030512 (2014)], we demonstrate an extreme difference: the zero-error quantum capacity is zero, whereas the zero-error private capacity is maximum given the quantum output dimension

Covariant quantum combinatorics with applications to zero-error communication

We develop the theory of quantum (a.k.a. noncommutative) relations and quantum (a.k.a. noncommutative) graphs in the finite-dimensional covariant setting, where all systems (finite-dimensional $C^*$-algebras) carry an action of a compact quantum group $G$, and all channels (completely positive maps preserving the canonical $G$-invariant state) are covariant with respect to the $G$-actions. We motivate our definitions by applications to zero-error quantum communication theory with a symmetry constraint. Some key results are the following: 1) We give a necessary and sufficient condition for a covariant quantum relation to be the underlying relation of a covariant channel. 2) We show that every quantum confusability graph with a $G$-action (which we call a quantum $G$-graph) arises as the confusability graph of a covariant channel. 3) We show that a covariant channel is reversible precisely when its confusability $G$-graph is discrete. 4) When $G$ is quasitriangular (this includes all compact groups), we show that covariant zero-error source-channel coding schemes are classified by covariant homomorphisms between confusability $G$-graphs.Comment: 38 pages, many diagram