Improving zero-error classical communication with entanglement

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 behaviour constitutes the field of classical zero-error information theory, the quantum generalisation of which has started to develop recently. 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 with no chance of error. In particular, we show how to construct such a channel based on any proof of the Bell-Kochen-Specker theorem. This is a new example of the use of quantum effects to improve the performance of a classical task. We investigate the connection between this phenomenon and that of ``pseudo-telepathy'' games. The use of generalised non-signalling correlations to assist in this task is also considered. In this case, a particularly elegant theory results and, remarkably, it is sometimes possible to transmit information with zero-error using a channel with no unassisted zero-error capacity.Comment: 6 pages, 2 figures. Version 2 is the same as the journal version plus figure 1 and the non-signalling box exampl

Multi-party zero-error classical channel coding with entanglement

We study the effects of quantum entanglement on the performance of two classical zero-error communication tasks among multiple parties. Both tasks are generalizations of the two-party zero-error channel-coding problem, where a sender and a receiver want to perfectly communicate messages through a one-way classical noisy channel. If the two parties are allowed to share entanglement, there are several positive results that show the existence of channels for which they can communicate strictly more than what they could do with classical resources. In the first task, one sender wants to communicate a common message to multiple receivers. We show that if the number of receivers is greater than a certain threshold then entanglement does not allow for an improvement in the communication for any finite number of uses of the channel. On the other hand, when the number of receivers is fixed, we exhibit a class of channels for which entanglement gives an advantage. The second problem we consider features multiple collaborating senders and one receiver. Classically, cooperation among the senders might allow them to communicate on average more messages than the sum of their individual possibilities. We show that whenever a channel allows single-sender entanglement-assisted advantage, then the gain extends also to the multi-sender case. Furthermore, we show that entanglement allows for a peculiar amplification of information which cannot happen classically, for a fixed number of uses of a channel with multiple senders.Comment: Some proofs have been modifie

Entanglement-assisted zero-error source-channel coding

We study the use of quantum entanglement in the zero-error source-channel coding problem. Here, Alice and Bob are connected by a noisy classical one-way channel, and are given correlated inputs from a random source. Their goal is for Bob to learn Alice's input while using the channel as little as possible. In the zero-error regime, the optimal rates of source codes and channel codes are given by graph parameters known as the Witsenhausen rate and Shannon capacity, respectively. The Lov\'asz theta number, a graph parameter defined by a semidefinite program, gives the best efficiently-computable upper bound on the Shannon capacity and it also upper bounds its entanglement-assisted counterpart. At the same time it was recently shown that the Shannon capacity can be increased if Alice and Bob may use entanglement. Here we partially extend these results to the source-coding problem and to the more general source-channel coding problem. We prove a lower bound on the rate of entanglement-assisted source-codes in terms Szegedy's number (a strengthening of the theta number). This result implies that the theta number lower bounds the entangled variant of the Witsenhausen rate. We also show that entanglement can allow for an unbounded improvement of the asymptotic rate of both classical source codes and classical source-channel codes. Our separation results use low-degree polynomials due to Barrington, Beigel and Rudich, Hadamard matrices due to Xia and Liu and a new application of remote state preparation.Comment: Title has been changed. Previous title was 'Zero-error source-channel coding with entanglement'. Corrected an error in Lemma 1.

Improved Quantum Communication Complexity Bounds for Disjointness and Equality

We prove new bounds on the quantum communication complexity of the disjointness and equality problems. For the case of exact and non-deterministic protocols we show that these complexities are all equal to n+1, the previous best lower bound being n/2. We show this by improving a general bound for non-deterministic protocols of de Wolf. We also give an O(sqrt{n}c^{log^* n})-qubit bounded-error protocol for disjointness, modifying and improving the earlier O(sqrt{n}log n) protocol of Buhrman, Cleve, and Wigderson, and prove an Omega(sqrt{n}) lower bound for a large class of protocols that includes the BCW-protocol as well as our new protocol.Comment: 11 pages LaTe

Quantum information and physics: Some future directions

I consider some promising future directions for quantum information theory that could influence the development of 21st century physics. Advances in the theory of the distinguishability of superoperators may lead to new strategies for improving the precision of quantum-limited measurements. A better grasp of the properties of multi-partite quantum entanglement may lead to deeper understanding of strongly-coupled dynamics in quantum many-body systems, quantum field theory, and quantum gravity

A Generalization of Kochen-Specker Sets Relates Quantum Coloring to Entanglement-Assisted Channel Capacity

We introduce two generalizations of Kochen-Specker (KS) sets: projective KS sets and generalized KS sets. We then use projective KS sets to characterize all graphs for which the chromatic number is strictly larger than the quantum chromatic number. Here, the quantum chromatic number is defined via a nonlocal game based on graph coloring. We further show that from any graph with separation between these two quantities, one can construct a classical channel for which entanglement assistance increases the one-shot zero-error capacity. As an example, we exhibit a new family of classical channels with an exponential increase.Comment: 16 page

Entanglement can increase asymptotic rates of zero-error classical communication over classical channels

It is known that the number of different classical messages which can be communicated with a single use of a classical channel with zero probability of decoding error can sometimes be increased by using entanglement shared between sender and receiver. It has been an open question to determine whether entanglement can ever increase the zero-error communication rates achievable in the limit of many channel uses. In this paper we show, by explicit examples, that entanglement can indeed increase asymptotic zero-error capacity, even to the extent that it is equal to the normal capacity of the channel. Interestingly, our examples are based on the exceptional simple root systems E7 and E8.Comment: 14 pages, 2 figur