Non-commutative Graphs Research Articles

Chromatic numbers, Sabidussi's Theorem and Hedetniemi's conjecture for non-commutative graphs

Non-commutative graph theory is an operator space generalization of graph theory. Well known graph parameters such as the independence number and Lovász theta function were first generalized to this setting by Duan, Severini, and Winter [1]. In [10], Stahlke introduces a generalization of the chromatic number of a graph and obtains a Lovász sandwich inequality for certain operator spaces.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.

Algebraic Methods of the Study of Quantum Information Transfer Channels

Kraus representation of quantum information transfer channels is widely used in practice. We present examples of Kraus decompositions for channels that possess the covariance property with respect to the maximal commutative group of unitary operators. We show that in some problems (for example, the problem on the estimate of the minimal output entropy of the channel), the choice of a Kraus representation with nonminimal number of Kraus operators is relevant. We also present certain algebraic properties of noncommutative operator graphs generated by Kraus operators for the case of quantum channels that demonstrate the superactivation phenomenon.

On Non-Commutative Operator Graphs Generated by Reducible Unitary Representation of the Heisenberg-Weyl Group

We consider a reducible unitary representation of Heisenberg-Weyl group in a tensor product of two Hilbert spaces. A non-commutative operator graph generated by this representation is introduced. It is shown that spectral projections of unitaries in the representation are anticliques (quantum error-correcting codes) for this graph. The obtained codes are appeared to be linear envelopes of entangled vectors.

On non-commutative operator graphs generated by covariant resolutions of identity

We study non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary representations of a compact group. Our main goal is searching for orthogonal projections which are anticliques (error-correcting codes) for such graphs. A special attention is paid to the covariance with respect to unitary representations of the circle group. We determine a tensor product structure in the space of representation under which the obtained anticliques are generated by entangled vectors.

Complexity and Capacity Bounds for Quantum Channels

We generalise some well-known graph parameters to operator systems by considering their underlying quantum channels. In particular, we introduce the quantum complexity as the dimension of the smallest co-domain Hilbert space a quantum channel requires to realise a given operator system as its non-commutative confusability graph. We describe quantum complexity as a generalised minimum semidefinite rank and, in the case of a graph operator system, as a quantum intersection number. The quantum complexity and a closely related quantum version of orthogonal rank turn out to be upper bounds for the Shannon zero-error capacity of a quantum channel, and we construct examples for which these bounds beat the best previously known general upper bound for the capacity of quantum channels, given by the quantum Lov\'asz theta number.

On Construction of Anticliques for Noncommutative Operator Graphs

In this paper, we construct anticliques for noncommutative operator graphs generated by generalized Pauli matrices. It is shown that application of entangled states for the construction of the code space K allows one to substantially increase the dimension of a noncommutative operator graph for which the projection on K is an anticlique.

On General Properties of Non-Commutative Operator Graphs

In this paper we study the general properties of non-commutative operator graphs. The problem of the existence of quantum anticliques is considered. The covariant property for the resolution of the identity which generates the graph is investigated.

On Zero-Error Communication via Quantum Channels in the Presence of Noiseless Feedback

We initiate the study of zero-error communication via quantum channels when the receiver and sender have at their disposal a noiseless feedback channel of unlimited quantum capacity, generalizing Shannon's zero-error communication theory with instantaneous feedback. We first show that this capacity is a function only of the linear span of Choi-Kraus operators of the channel, which generalizes the bipartite equivocation graph of a classical channel, and which we dub "non-commutative bipartite graph". Then we go on to show that the feedback-assisted capacity is non-zero (with constant activating noiseless communication) if and only if the non-commutative bipartite graph is non-trivial, and give a number of equivalent characterizations. This result involves a far-reaching extension of the "conclusive exclusion" of quantum states [Pusey/Barrett/Rudolph, Nature Phys. 8:475-478]. We then present an upper bound on the feedback-assisted zero-error capacity, motivated by a conjecture originally made by Shannon and proved later by Ahlswede. We demonstrate this bound to have many good properties, including being additive and given by a minimax formula. We also prove that this quantity is the entanglement-assisted capacity against an adversarially chosen channel from the set of all channels with the same Choi-Kraus span, which can also be interpreted as the feedback-assisted unambiguous capacity. The proof relies on a generalization of the "Postselection Lemma" [Christandl/Koenig/Renner, PRL 102:020504] that allows to reflect additional constraints, and which we believe to be of independent interest. We illustrate our ideas with a number of examples, including classical-quantum channels and Weyl diagonal channels, and close with an extensive discussion of open questions.

On the Noncommutative Deformation of the Operator Graph Corresponding to the Klein Group

We study the noncommutative operator graph ℒ θ depending on a complex parameter θ recently introduced by M. E. Shirokov to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We define a noncommutative group G and an algebra A θ which is the quotient of ℂG by a special algebraic relation depending on θ such that the matrix representation ϕ of A θ results in the algebra ℳ θ generated by ℒ θ . In the case of θ = ±1, the representation ϕ degenerates into a faithful representation of ℂK 4 , where K 4 is the Klein group. Thus, ℒ θ can be regarded as a noncommutative deformation of the graph associated with the Klein group. Bibliography: 16 titles.

Structure of the algebra generated by a noncommutative operator graph which demonstrates the superactivation phenomenon for zero-error capacity

Structure of the algebra generated by a noncommutative operator graph which demonstrates the superactivation phenomenon for zero-error capacity

On the independence and domination numbers of replacement product graphs

This paper examines invariants of the replacement product of two graphs in terms of the properties of the component graphs. In particular, we present results on the independence number, the domination number, and the total domination number of these graphs. The replacement product is a noncommutative graph operation that has been widely applied in many areas. One of its advantages over other graph products is its ability to produce sparse graphs. The results in this paper give insight into how to construct large, sparse graphs with optimal independence or domination numbers.

Quantum Zero-Error Source-Channel Coding and Non-Commutative Graph Theory

Alice and Bob receive a bipartite state (possibly entangled) from some finite collection or from some subspace. Alice sends a message to Bob through a noisy quantum channel such that Bob may determine the initial state, with zero chance of error. This framework encompasses, for example, teleportation, dense coding, entanglement assisted quantum channel capacity, and one-way communication complexity of function evaluation. With classical sources and channels, this problem can be analyzed using graph homomorphisms. We show this quantum version can be analyzed using homomorphisms on non-commutative graphs (an operator space generalization of graphs). Previously the Lov\'{a}sz $\vartheta$ number has been generalized to non-commutative graphs; we show this to be a homomorphism monotone, thus providing bounds on quantum source-channel coding. We generalize the Schrijver and Szegedy numbers, and show these to be monotones as well. As an application we construct a quantum channel whose entanglement assisted zero-error one-shot capacity can only be unlocked by using a non-maximally entangled state. These homomorphisms allow definition of a chromatic number for non-commutative graphs. Many open questions are presented regarding the possibility of a more fully developed theory.

Zero-Error Communication via Quantum Channels, Noncommutative Graphs, and a Quantum Lovász Number

We study the quantum channel version of Shannon's zero-error capacity problem. Motivated by recent progress on this question, we propose to consider a certain subspace of operators (so-called operator systems) as the quantum generalization of the adjacency matrix, in terms of which the zero-error capacity of a quantum channel, as well as the quantum and entanglement-assisted zero-error capacities can be formulated, and for which we show some new basic properties. Most importantly, we define a quantum version of Lovász' famous ϑ function on general operator systems, as the norm-completion (or stabilization) of a “naive” generalization of ϑ. We go on to show that this function upper bounds the number of entanglement-assisted zero-error messages, that it is given by a semidefinite program, whose dual we write down explicitly, and that it is multiplicative with respect to the tensor product of operator systems (corresponding to the tensor product of channels). We explore various other properties of the new quantity, which reduces to Lovász' original ϑ in the classical case, give several applications, and propose to study the operator systems associated with channels as “noncommutative graphs,” using the language of Hilbert modules.

Spectra of non-commutative dynamical systems and graphs related to fractal groups

We study spectra of noncommutative dynamical systems, representations of fractal groups, and regular graphs. We explicitly compute these spectra for five examples of groups acting on rooted trees, and in three cases obtain totally disconnected sets.

Connes' Distance Function for Commutative and Noncommutative Graphs

By formulating the concept of a graph algebraically, i.e., as properties of thealgebra of functions over the set of vertices and the set of edges, we arrive at apurely algebraic concept of distance related to the one proposed by Connes formanifolds which easily extends also to the noncommutative case. The Diracoperator used in Connes' approach is replaced by a generalized difference operatorwhich can be defined on arbitrary graphs. We speculate on the question of howthis operator might be related to the concept of a Dirac operator on graphs.

Combinatorial and noncommutative list-processing and associated graphs

Recently the authors have proposed a list-processing approach to the modeling of algebraic quantum field theory methods in quantum mechanics in which the noncommutative algebra of quantum-mechanical operators is emulated by lists. The processing produces reordered sequences of elements of a ring with a unit commutator and generates dynamic structures which, for some initial arrangements, correspond to partially ordered graphs characterized by recurrence relations and combinatorial identities. Likewise, in another list-processing application to physical problems, a simulation of Feynman diagrams hinged on predominantly combinatorial aspects and demanded explicit generation of certain combinatorial objects. This motivated an investigation into the combinatorial nature of noncommutative list-processing and of recursive algorithms for explicit construction of combinatorial lists, which we now present. The emphasis is also placed on the consideration of associated graphs and the graph-theoretic origin of the appearance of recurrence relations in the reordering theorems of the noncommutative algebra.