Given one or more uses of a classical channel, only a certain number of messages can be transmitted with zero probability of error. The study of this number and its asymptotic behavior constitutes the field of classical zero-error information theory. We show that, given a single use of certain classical channels, entangled states of a system shared by the sender and receiver can be used to increase the number of (classical) messages which can be sent without error. In particular, we show how to construct such a channel based on any proof of the Kochen-Specker theorem. We investigate the connection to pseudotelepathy games. The use of generalized nonsignaling correlations to assist in this task is also considered. In this case, an elegant theory results and, remarkably, it is sometimes possible to transmit information with zero error using a channel with no unassisted zero-error capacity.

The starting point of the present investigation is the well-known result by Helstrom, identifying the best achievable bias in distinguishing two quantum states under all measurements as (half) the trace norm of their difference. We turn this around, noticing that every sufficiently rich set M of measurements on a fixed quantum system defines a statistical norm · M on the states of that system, via the optimal bias achievable when restricted to M. These norms are all upper bounded by the usual trace norm, and in finite dimensional Hilbert spaces they are all equivalent to the trace norm in the sense that there exist "constants of domination" λ and µ, such thatwhich are optimal in the sense that there exist states such that the bounds are tight. In other words, if we rate the performance of a set of measurements in distinguishing a given pair of states (of equal prior probability) as the ratio of the largest bias that can be obtained by such measurements to the best bias achievable when allowing all measurements, then λ and µ determine the worst and best case performance respectively for any pair of states. Here we set ourselves the task of computing, or at least bounding such constants for various sets of measurements M.Specifically, we look at the case that M consists only of a single measurement, namely the uniformly random POVM, 2-designs, and 4-designs where we find asymptotically tight bounds for λ and µ. Furthermore, we analyse the multipartite setting where the set of measurements consists of all POVMs implementable by local operations and classical communication (among other, related classes).In the case of two parties, we show that the lower domination constant λ is the same as that of a tensor product of local uniformly random POVMs up to a constant. This answers in the affirmative an open question about the (near-)optimality of bipartite data hiding: The bias that can be achieved by LOCC in discriminating two orthogonal states of a d × d bipartite system is Ω(1/d), which is known to be tight. Finally, we use our analysis to derive certainty relations (in the sense of Sanchez-Ruiz) for any such measurements and to lower bound the locally accessible information for bipartite systems. * Electronic address: william.matthews@bris.ac.uk † Electronic address: wehner@caltech.edu ‡ Electronic address: a.j.winter@bris.ac.uk Proof . For a given ξ = ρ − σ and POVM (M k ) n k=1 ∈ M, the bias achieved is simply n k=1 | Tr (M k ξ) |. The same bias is achieved by the 2-outcome POVM (M + , M − ) ∈ M 2 , where M + = k∈P M k , P = {k ∈ [n] : Tr(M k ξ) ≥ 0}, M − = k∈N M k = 1 1 − M + , N = {k ∈ [n] : Tr(M k ξ) < 0}and clearly no other grouping of the elements of this POVM can result in a larger bias.Note that M has a non-empty interior (and then contains the origin in its interior) if and only if the collection M is informationally complete, which the case if and only if M 2 is informationally complete. Mathematically the information-completeness is expressed by M, spanning the whole operator space. Furthermore, note that ...

We derive upper bounds on the rate of transmission of classical information over quantum channels by block codes with a given blocklength and error probability, for both entanglement-assisted and unassisted codes, in terms of a unifying framework of quantum hypothesis testing with restricted measurements. Our bounds do not depend on any special property of the channel (such as memorylessness) and generalize both a classical converse of Polyanskiy, Poor, and Verdú as well as a quantum converse of Renner and Wang, and have a number of desirable properties. In particular, our bound on entanglement-assisted codes is a semidefinite program and for memoryless channels, its large blocklength limit is the well-known formula for entanglement-assisted capacity due to Bennett, Shor, Smolin, and Thapliyal.Index Terms-Channel coding, quantum channels, block codes, finite blocklength, quantum entanglement.

We derive one-shot upper bounds for quantum noisy channel codes. We do so by regarding a channel code as a bipartite operation with an encoder belonging to the sender and a decoder belonging to the receiver, and imposing constraints on the bipartite operation. We investigate the power of codes whose bipartite operation is non-signalling from Alice to Bob, positive-partial transpose (PPT) preserving, or both, and derive a simple semidefinite program for the achievable entanglement fidelity. Using the semidefinite program, we show that the non-signalling assisted quantum capacity for memoryless channels is equal to the entanglement-assisted capacity. We also relate our PPT-preserving codes and the PPT-preserving entanglement distillation protocols studied by Rains. Applying these results to a concrete example, the 3-dimensional Werner-Holevo channel, we find that codes that are non-signalling and PPT-preserving can be strictly less powerful than codes satisfying either one of the constraints, and therefore provide a tighter bound for unassisted codes. Furthermore, PPT-preserving non-signalling codes can send one qubit perfectly over two uses of the channel, which has no quantum capacity. We discuss whether this can be interpreted as a form of superactivation of quantum capacity.