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 Lovász 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 of 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.

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 multisender case. Furthermore, we show that entanglement allows for a peculiar amplification of information which cannot happen classically, for a fixed number of uses of the channels.

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.

The quantum chromatic number of a graph G is sandwiched between its chromatic number and its clique number, which are well-known NP-hard quantities. We restrict our attention to the rank-1 quantum chromatic number χq(1)(G), which upper bounds the quantum chromatic number, but is defined under stronger constraints. We study its relation with the chromatic number χ(G) and the minimum dimension of orthogonal representations ξ(G). It is known that ξ(G) ≤ χq(1)(G) ≤ χ(G). We answer three open questions about these relations: we give a necessary and sufficient condition to have ξ(G) = χq(1)(G), we exhibit a class of graphs such that ξ(G) ≤ χq(1)(G), and we give a necessary and sufficient condition to have χq(1)(G) ≤ χ(G). Our main tools are Kochen-Specker sets, collections of vectors with a traditionally important role in the study of contextuality of physical theories and, more recently, in the quantification of quantum zero-error capacities. Finally, as a corollary of our results and a result by Avis et al on the quantum chromatic number, we give a family of Kochen-Specker sets of growing dimension.

Bell inequality violations correspond to behavior of entangled quantum systems that cannot be simulated classically. We give two new two-player games with Bell inequality violations that are stronger, fully explicit, and arguably simpler than earlier work.The first game is based on the Hidden Matching problem of quantum communication complexity, introduced by Bar-Yossef, Jayram, and Kerenidis. This game can be won with probability 1 by a quantum strategy using a maximally entangled state with local dimension n (e.g., log n EPR-pairs), while we show that the winning probability of any classical strategy differs from 1/2 by at most O(log n/√n).The second game is based on the integrality gap for Unique Games by Khot and Vishnoi and the quantum rounding procedure of Kempe, Regev, and Toner. Here n-dimensional entanglement allows to win the game with probability 1/(log n)2, while the best winning probability without entanglement is 1/n. This near-linear ratio ("Bell inequality violation'') is near-optimal, both in terms of the local dimension of the entangled state, and in terms of the number of possible outputs of the two players.