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arXiv:2103.14091 · Illinois Journal of Mathematics

Information theoretic parameters of noncommutative graphs and convex corners

Gareth Boreland, Ivan G. Todorov, Andreas Winter

Published Mar 22, 2022• 2 citations
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Abstract

We establish a second anti-blocker theorem for noncommutative convex corners, show that the anti-blocking operation is continuous on bounded sets of convex corners, and define optimization parameters for a given convex corner that generalize well-known graph theoretic quantities. We define the entropy of a state with respect to a convex corner, characterize its maximum value in terms of a generalized fractional chromatic number and establish entropy splitting results that demonstrate the entropic complementarity between a convex corner and its anti-blocker. We identify two extremal tensor products of convex corners and examine the behavior of the introduced parameters with respect to tensoring. Specializing to noncommutative graphs, we obtain quantum versions of the fractional chromatic number and the clique covering number, as well as a notion of noncommutative graph entropy of a state, which we show to be continuous with respect to the state and the graph. We define the Witsenhausen rate of a noncommutative graph and compute the values of our parameters in some specific cases.

DOI
10.1215/00192082-9799163

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This thesis explores connections between information theory and graph theory. We prove a number of seemingly new results on graph entropy, a classical information theoretic quantity introduced by Korner. Our work includes the determination of the graph entropy of the odd cycles and their complements under certain probability distributions. We develop a theory of convex corners in finite dimensional spaces of matrices appropriate for applications in quantum information, and discuss the concept of entropy over a convex corner. We recall the definition of a non-commutative graph from the work of Duan, Severini and Winter, and with a given non-commutative graph we associate a number of convex corners, before proving a "quantum sandwich theorem". We define several new parameters for non-commutative graphs, and show them to be generalisations of the corresponding graph parameters. This includes two quantum versions of the Lovasz number, one of which is seen to be an upper bound on the Shannon capacity of an associated quantum channel. Finally we return to examine graph entropy in the case of a non-i.i.d. classical source, and attempt to generalise the Kolmogorov-Sinai entropy of a dynamical system to this setting.
Quantum graph theory, also known as non-commutative graph theory, is an operator space generalization of graph theory. The independence number, and Lovasz theta function were generalized to this setting by Duan, Severini, and Winter [DSW13] and two different version of the chromatic number were introduced by Stahlke [Sta16] and Paulsen [HPP16]. We introduce two new generalizations of the chromatic number to non-commutative graphs and provide an upper bound on the parameter of Stahlke. We provide a generalization of the graph complement and show the chromatic number of the orthogonal complement of a non-commutative graph is bounded below by its theta number. We also provide a generalization of both Sabidussi's Theorem and Hedetniemi's conjecture to non-commutative graphs. The study of non-local games considers scenarios in which separated players collaborate to provide satisfying responses to questions given by a referee. The condition of separating players makes non-local games an excellent setting to gain insight into quantum phenomena such as entanglement and non-locality. Non-local games can also provide protocols known as self-tests. Self-testing allows an experimenter to interact classically with a black box quantum system and certify that a specific entangled state was present and a specific set of measurements were performed. The most studied self-test is the CHSH game which certifies the presence of a single EPR entangled state and the use of anti-commuting Pauli measurements. We introduce an algebraic generalization of CHSH and obtain a self-test for non-Pauli operators resolving an open question posed by Coladangelo and Stark (QIP 2017). Our games also provide a self-test for states other than the maximally entangled state, and hence resolves the open question posed by Cleve and Mittal (ICALP 2012). This thesis is primarily based on two collaborative works written by the author and several coauthors [CMMN20], [KM19].
The zero-error capacity of a classical channel is a parameter of its confusability graph, and is equal to the minimum of the values of graph parameters that are additive under the disjoint union, multiplicative under the strong product, monotone under homomorphisms between the complements, and normalized. We show that any such function either has uncountably many extensions to noncommutative graphs with similar properties, or no such extensions at all. More precisely, we find that every extension has an exponent that characterizes its values on the confusability graphs of identity quantum channels, and the set of admissible exponents is either an unbounded subinterval of [1,∞)[1,\infty)[1,∞) or empty. In particular, the set of admissible exponents for the Lovász number, the projective rank, and the fractional Haemers bound over the complex numbers are maximal, while the fractional clique cover number does not have any extensions.
We continue the study of the quantum channel version of Shannon's zero-error capacity problem. We generalize the celebrated Haemers bound to noncommutative graphs (obtained from quantum channels). We prove basic properties of this bound, such as additivity under the direct sum and submultiplicativity under the tensor product. The Haemers bound upper bounds the Shannon capacity of noncommutative graphs, and we show that it can outperform other known upper bounds, including noncommutative analogues of the Lovász theta function (Duan-Severini-Winter, IEEE Trans. Inform. Theory, 2013 and Boreland-Todorov-Winter, arXiv preprint, 2019).
We study quantum entanglement and some of its applications in graph theory and zero-error information theory. In Chapter 1 we introduce entanglement and other fundamental concepts of quantum theory. In Chapter 2 we address the question of how much quantum correlations generated by entanglement can deviate from classical predictions. We focus on non-local games: experiments in which two players are separated and forbidden to communicate, and have to collaborate to accomplish a task. We give two games exhibiting quantum-classical separations close to optimal. Remarkably, our results in theoretical physics are inspired by theoretical computer science. Chapter 3 is dedicated to the study of quantum graph parameters. Well-known quantities such as the chromatic number and the independence number of a graph can be interpreted as parameters for non-local games. A definition of quantum graph parameters follows from this fact. We contribute to the field in a number of ways. Among other results, we find a surprising characterization of the quantum chromatic number that relates to the Kochen-Specker theorem, a result in the foundations of quantum mechanics. In Chapter 4, we move to zero-error information theory. We study the zero error capacity of a classical noisy channel when the sender and the receiver can use quantum entanglement. We initiate the study of the source problem and source-channel problem with entanglement and we find channels and sources that exhibit a strong divergence in quantum and classical behaviours. To do that, we use results in combinatorics, linear algebra, optimization and number theory.