Giannicola Scarpa

New Separations in Zero-Error Channel Capacity Through Projective Kochen–Specker Sets and Quantum Coloring

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.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.

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 χ_q⁽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) ≤ χ_q⁽¹⁾(G) ≤ χ(G). We answer three open questions about these relations: we give a necessary and sufficient condition to have ξ(G) = χ_q⁽¹⁾(G), we exhibit a class of graphs such that ξ(G) ≤ χ_q⁽¹⁾(G), and we give a necessary and sufficient condition to have χ_q⁽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.

Parallel repetition of entangled games with exponential decay via the superposed information cost

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 ω *(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.

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 Alices 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 Szegedys 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.

Near-Optimal and Explicit Bell Inequality Violations

Entangled quantum systems can exhibit correlations that cannot be simulated classically. For historical reasons such correlations are called “Bell inequality violations.” 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 strategy using a maximally entangled state with local dimension n (e. g., logn EPR-pairs), while we show that the winning probability of any classical strategy differs from 1/2 by at most O((logn)/ √ n). ∗An earlier version of this paper appeared in the Proceedings of the 26th IEEE Conference on Computational Complexity, pages 157–166, 2011. †Supported by a Vici grant from the Netherlands Organisation for Scientific Research (NWO), and by the European Commission under the project QCS (Grant No. 255961). ‡Supported by the Israel Science Foundation, by the Wolfson Family Charitable Trust, and by a European Research Council (ERC) Starting Grant. Part of the work done while a DIGITEO visitor in LRI, Orsay. §Supported by a Vidi grant from the Netherlands Organisation for Scientific Research (NWO), and by the European Commission under the project QCS (Grant No. 255961). ¶Supported by a Vidi grant from the Netherlands Organisation for Scientific Research (NWO), and by the European Commission under the project QCS (Grant No. 255961). ACM Classification: J.2 AMS Classification: 81P68

Exclusivity structures and graph representatives of local complementation orbits

We describe a construction that maps any connected graph G on three or more vertices into a larger graph, H(G), whose independence number is strictly smaller than its Lovasz number which is equal to its fractional packing number. The vertices of H(G) represent all possible events consistent with the stabilizer group of the graph state associated with G, and exclusive events are adjacent. Mathematically, the graph H(G) corresponds to the orbit of G under local complementation. Physically, the construction translates into graph-theoretic terms the connection between a graph state and a Bell inequality maximally violated by quantum mechanics. In the context of zero-error information theory, the construction suggests a protocol achieving the maximum rate of entanglement-assisted capacity, a quantum mechanical analogue of the Shannon capacity, for each H(G). The violation of the Bell inequality is expressed by the one-shot version of this capacity being strictly larger than the independence number. Finally, gi...

Multiparty 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 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.

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 Lovasz 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.