Kochen-Specker Sets and the Rank-1 Quantum Chromatic Number

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 $\chi_q^{(1)}(G)$, which upper bounds the quantum chromatic number, but is defined under stronger constraints. We study its relation with the chromatic number $\chi(G)$ and the minimum dimension of orthogonal representations $\xi(G)$. It is known that $\xi(G) \leq \chi_q^{(1)}(G) \leq \chi(G)$. We answer three open questions about these relations: we give a necessary and sufficient condition to have $\xi(G) = \chi_q^{(1)}(G)$, we exhibit a class of graphs such that $\xi(G) < \chi_q^{(1)}(G)$, and we give a necessary and sufficient condition to have $\chi_q^{(1)}(G) < \chi(G)$. Our main tools are Kochen-Specker sets, collections of vectors with a traditionally important role in the study of noncontextuality 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, Hasegawa, Kikuchi, and Sasaki on the quantum chromatic number, we give a family of Kochen-Specker sets of growing dimension.Comment: 12 page

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

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

Better Non-Local Games from Hidden Matching

We construct a non-locality game that can be won with certainty by a quantum strategy using log n shared EPR-pairs, while any classical strategy has winning probability at most 1/2+O(log n/sqrt{n}). This improves upon a recent result of Junge et al. in a number of ways.Comment: 11 pages, late

Graph-theoretical Bounds on the Entangled Value of Non-local Games

We introduce a novel technique to give bounds to the entangled value of non-local games. The technique is based on a class of graphs used by Cabello, Severini and Winter in 2010. The upper bound uses the famous Lov\'asz theta number and is efficiently computable; the lower one is based on the quantum independence number, which is a quantity used in the study of entanglement-assisted channel capacities and graph homomorphism games.Comment: 10 pages, submission to the 9th Conference on the Theory of Quantum Computation, Communication, and Cryptography (TQC 2014

Parallel Repetition of Free Entangled Games: Simplification and Improvements

In a two-player game, two cooperating but non communicating players, Alice and Bob, receive inputs taken from a probability distribution. Each of them produces an output and they win the game if they satisfy some predicate on their inputs/outputs. The entangled value $\omega^*(G)$ of a game $G$ is the maximum probability that Alice and Bob can win the game if they are allowed to share an entangled state prior to receiving their inputs. The $n$-fold parallel repetition $G^n$ of $G$ consists of $n$ instances of $G$ where Alice and Bob receive all the inputs at the same time and must produce all the outputs at the same time. They win $G^n$ if they win each instance of $G$. Recently, there has been a series of works showing parallel repetition with exponential decay for projection games [DSV13], games on the uniform distribution [CS14] and for free games, i.e. games on a product distribution [JPY13]. This article is meant to be a follow up of [CS14], where we improve and simplify several parts of our previous paper. Our main result is that for any free game $G$ with value $\omega^*(G)=1-\varepsilon$, we have $\omega^*(G^n) \le (1 - \varepsilon^2)^{\Omega(\frac{n}{\log(l)})}$ where $l$ is the size of the output set of the game. This result improves on both the results in [JPY13] and [CS14]. The framework we use can also be extended to free projection games. We show that for a free projection game $G$ with value $\omega^*(G)=1-\varepsilon$, we have $\omega^*(G^n) \le (1 - \varepsilon)^{\Omega(n)}$.Comment: 17 pages, this paper is a follow up and supersedes our previous paper 'Parallel Repetition of Entangled Games with Exponential Decay via the Superposed Information Cost' [CS14, arXiv:1310.7787] v2: updated GS affiliatio

Belief-Invariant and Quantum Equilibria in Games of Incomplete Information

Drawing on ideas from game theory and quantum physics, we investigate nonlocal correlations from the point of view of equilibria in games of incomplete information. These equilibria can be classified in decreasing power as general communication equilibria, belief-invariant equilibria and correlated equilibria, all of which contain the familiar Nash equilibria. The notion of belief-invariant equilibrium has appeared in game theory before, in the 1990s. However, the class of non-signalling correlations associated to belief-invariance arose naturally already in the 1980s in the foundations of quantum mechanics. Here, we explain and unify these two origins of the idea and study the above classes of equilibria, and furthermore quantum correlated equilibria, using tools from quantum information but the language of game theory. We present a general framework of belief-invariant communication equilibria, which contains (quantum) correlated equilibria as special cases. It also contains the theory of Bell inequalities, a question of intense interest in quantum mechanics, and quantum games where players have conflicting interests, a recent topic in physics. We then use our framework to show new results related to social welfare. Namely, we exhibit a game where belief-invariance is socially better than correlated equilibria, and one where all non-belief-invariant equilibria are socially suboptimal. Then, we show that in some cases optimal social welfare is achieved by quantum correlations, which do not need an informed mediator to be implemented. Furthermore, we illustrate potential practical applications: for instance, situations where competing companies can correlate without exposing their trade secrets, or where privacy-preserving advice reduces congestion in a network. Along the way, we highlight open questions on the interplay between quantum information, cryptography, and game theory

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.