25 research outputs found
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
Graph Homomorphisms for Quantum Players
A homomorphism from a graph X to a graph Y is an adjacency preserving mapping f:V(X) -> V(Y). We consider a nonlocal game in which Alice and Bob are trying to convince a verifier with certainty that a graph X admits a homomorphism to Y. This is a generalization of the well-studied graph coloring game. Via systematic study of quantum homomorphisms we prove new results for graph coloring. Most importantly, we show that the Lovász theta number of the complement lower bounds the quantum chromatic number, which itself is not known to be computable. We also show that other quantum graph parameters, such as quantum independence number, can differ from their classical counterparts. Finally, we show that quantum homomorphisms closely relate to zero-error channel capacity. In particular, we use quantum homomorphisms to construct graphs for which entanglement-assistance increases their one-shot zero-error capacity.Published versio
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
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.
On a tracial version of Haemers bound
We extend upper bounds on the quantum independence number and the quantum
Shannon capacity of graphs to their counterparts in the commuting operator
model. We introduce a von Neumann algebraic generalization of the fractional
Haemers bound (over ) and prove that the generalization upper
bounds the commuting quantum independence number. We call our bound the tracial
Haemers bound, and we prove that it is multiplicative with respect to the
strong product. In particular, this makes it an upper bound on the Shannon
capacity. The tracial Haemers bound is incomparable with the Lov\'asz theta
function, another well-known upper bound on the Shannon capacity. We show that
separating the tracial and fractional Haemers bounds would refute Connes'
embedding conjecture.
Along the way, we prove that the tracial rank and tracial Haemers bound are
elements of the (commuting quantum) asymptotic spectrum of graphs (Zuiddam,
Combinatorica, 2019). We also show that the inertia bound (an upper bound on
the quantum independence number) upper bounds the commuting quantum
independence number.Comment: 39 pages, comments are welcom