The one-shot zero-error classical capacity of a quantum channel is the amount of classical information that can be transmitted with zero probability of error by a single use. Then the one-shot zero-error classical capacity equals to the logarithmic value of the independence number of the noncommutative graph induced by the channel. Thus the additivity of the one-shot zero-error classical capacity of a quantum channel is equivalent to the multiplicativity of the independence number of the noncommutative graph. The independence number is not multiplicative in general, and it is not clearly understood when the multiplicativity occurs. In this work, we present sufficient conditions for multiplicativity of the independence number, and we give explicit examples of quantum channels. Furthermore, we consider a block form of noncommutative graphs, and provide conditions when the independence number is multiplicative.
The one-shot zero-error classical capacity of a quantum channel is the amount of classical information that can be transmitted with zero probability of error by a single use. Then the one-shot zero-error classical capacity equals to the logarithmic value of the independence number of the noncommutative graph induced by the channel. Thus the additivity of the one-shot zero-error classical capacity of a quantum channel is equivalent to the multiplicativity of the independence number of the noncommutative graph. The independence number is not multiplicative in general, and it is not clearly understood when the multiplicativity occurs. In this work, we present sufficient conditions for multiplicativity of the independence number, and we give explicit examples of quantum channels. Furthermore, we consider a block form of noncommutative graphs, and provide conditions when the independence number is multiplicative.
Recently the one-shot quantum no-signalling assisted zero-error classical capacity of a quantum channel has been formulated as a semidefinite programming (SDP) depending only on the Choi-Kraus operator space of the channel. In this paper, we study the \textit{activated quantum no-signalling assisted zero-error classical capacity} by first allowing the assistance from some noiseless forward communication channel and later paying back the cost of the helper. We show that the one-shot activated capacity can also be formulated as a SDP and derive a number of striking properties of this number. In particular, this number is additive under direct sum, and is always greater than or equal to the super-dense coding bound. An a remarkable consequence, we find that one bit noiseless classical communication is able to fully activate any classical-quantum channel to achieve its asymptotic capacity, or the semidefinite fractional packing number. We also discuss the condition under which a noisy channel can activate an activatable channel. Interestingly, a channel is able to activate itself if its one-shot capacity is greater than or equal to the Golden Ratio - (1+5)/2. We also show that the asymptotic activated capacity is still equal to the usual no-signalling assisted capacity. Finally, we show that general the asymptotic no-signalling assisted zero-error capacity does not equal to the semidefinite (fractional) packing number by an explicit construction.
Alice and Bob receive a bipartite state (possibly entangled) from some finite collection or from some subspace. Alice sends a message to Bob through a noisy quantum channel such that Bob may determine the initial state, with zero chance of error. This framework encompasses, for example, teleportation, dense coding, entanglement assisted quantum channel capacity, and one-way communication complexity of function evaluation. With classical sources and channels, this problem can be analyzed using graph homomorphisms. We show this quantum version can be analyzed using homomorphisms on non-commutative graphs (an operator space generalization of graphs). Previously the Lovász ϑ number has been generalized to non-commutative graphs; we show this to be a homomorphism monotone, thus providing bounds on quantum source-channel coding. We generalize the Schrijver and Szegedy numbers, and show these to be monotones as well. As an application, we construct a quantum channel whose entanglement assisted zero-error one-shot capacity can only be unlocked using a non-maximally entangled state. These homomorphisms allow definition of a chromatic number for non-commutative graphs. Many open questions are presented regarding the possibility of a more fully developed theory.
Duan and Winter studied the one-shot zero-error classical capacity of a quantum channel assisted by quantum non-signalling correlations, and formulated this problem as a semidefinite program depending only on the Kraus operator space of the channel. For the class of classical-quantum channels, they showed that the asymptotic zero-error classical capacity assisted by quantum non-signalling correlations, minimized over all classicalquantum channels with a confusability graph G, is exactly log ϑ(G), where ϑ(G) is the celebrated Lov´asz theta function. In this paper, we show that the one-shot capacity for a classical-quantum channel, induced from a circulant graph G defined by equal-sized cyclotomic cosets, is logbϑ(G)c, which further implies that its asymptotic capacity is log ϑ(G). This type of graphs include the cycle graphs of odd length, the Paley graphs of prime vertices, and the cubit residue graphs of prime vertices. Examples of other graphs are also discussed. This gives Lov´asz ϑ function another operational meaning in zero-error classical-quantum communication.
We study the one-shot zero-error classical capacity of a quantum channel assisted by quantum no-signalling correlations, and the reverse problem of exact simulation of a prescribed channel by a noiseless classical one. Quantum no-signalling correlations are viewed as two-input and two-output completely positive and trace preserving maps with linear constraints enforcing that the device cannot signal. Both problems lead to simple semidefinite programmes (SDPs) that depend only on the Choi-Kraus (operator) space of the channel. In particular, we show that the zero-error classical simulation cost is precisely the conditional min-entropy of the Choi-Jamiołkowski matrix of the given channel. The zero-error classical capacity is given by a similar-looking but different SDP; the asymptotic zero-error classical capacity is the regularization of this SDP, and in general, we do not know of any simple form. Interestingly, however, for the class of classical-quantum channels, we show that the asymptotic capacity is given by a much simpler SDP, which coincides with a semidefinite generalization of the fractional packing number suggested earlier by Aram Harrow. This finally results in an operational interpretation of the celebrated Lovász ϑ function of a graph as the zero-error classical capacity of the graph assisted by quantum no-signalling correlations, the first information theoretic interpretation of the Lovász number.
In this paper, we explicitly evaluate the one-shot quantum nonsignaling-assisted zero-error classical capacities M0QNS for qubit channels. In particular, we show that for nonunital qubit channels M0QNS=0, which implies that in the one-shot setting nonunital qubit channels cannot transmit any information with zero probability of error even when assisted by quantum nonsignaling correlations. Furthermore, we show that for qubit channels M0QNS equals the one-shot entanglement-assisted zero-error classical capacity. This means that for a single use of a qubit channel quantum nonsignaling correlations are not more powerful than shared entanglement.
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'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.