Non-commutative Graphs Research Articles

Szeged and Padmakar-Ivan Energies of Non-Commuting Graph for Dihedral Groups

The graph can represent the molecule structure and the -electron energy derives the concept of graph energy. The graph also can be related to the groups or rings as its vertex set. The non-commutative graph is a type of graph whose construction is determined by the structure of a group. This paper focuses on the energy of the non-commuting graph for dihedral groups using the Szeged and Padmakar-Ivan matrices. Both matrices are constructed based on the distance between two vertices in the graph. The eigenvalues ​​of these matrices lead to the formulation of the graph's energy values​. Interestingly, the energies obtained are equal to twice the spectral radius and are hyperenergetic.

Quantum No-signalling Correlations and Non-local Games

We introduce and examine three subclasses of the family of quantum no-signalling (QNS) correlations introduced by Duan and Winter: quantum commuting, quantum and local. We formalise the notion of a universal TRO of a block operator isometry, define an operator system, universal for stochastic operator matrices, and realise it as a quotient of a matrix algebra. We describe the classes of QNS correlations in terms of states on the tensor products of two copies of the universal operator system and specialise the correlation classes and their representations to classical-to-quantum correlations. We study various quantum versions of synchronous no-signalling correlations and show that they possess invariance properties for suitable sets of states. We introduce quantum non-local games as a generalisation of non-local games. We define the operation of quantum game composition and show that the perfect strategies belonging to a certain class are closed under channel composition. We specialise to the case of graph colourings, where we exhibit quantum versions of the orthogonal rank of a graph as the optimal output dimension for which perfect classical-to-quantum strategies of the graph colouring game exist, as well as to non-commutative graph homomorphisms, where we identify quantum versions of non-commutative graph homomorphisms introduced by Stahlke.

A note on the stabilizer formalism via noncommutative graphs

A note on the stabilizer formalism via noncommutative graphs

The quantum-to-classical graph homomorphism game

Motivated by non-local games and quantum coloring problems, we introduce a graph homomorphism game between quantum graphs and classical graphs. This game is naturally cast as a “quantum–classical game,” that is, a non-local game of two players involving quantum questions and classical answers. This game generalizes the graph homomorphism game between classical graphs. We show that winning strategies in the various quantum models for the game is an analog of the notion of non-commutative graph homomorphisms due to Stahlke [IEEE Trans. Inf. Theory 62(1), 554–577 (2016)]. Moreover, we present a game algebra in this context that generalizes the game algebra for graph homomorphisms given by Helton et al. [New York J. Math. 25, 328–361 (2019)]. We also demonstrate explicit quantum colorings of all quantum complete graphs, yielding the surprising fact that the algebra of the four coloring game for a quantum graph is always non-trivial, extending a result of Helton et al. [New York J. Math. 25, 328–361 (2019)].

Information theoretic parameters of noncommutative graphs and convex corners

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.

On the Construction of a Quantum Channel Corresponding to Non-commutative Graph for a Qubit Interacting with Quantum Oscillator

On the Construction of a Quantum Channel Corresponding to Non-commutative Graph for a Qubit Interacting with Quantum Oscillator

On Noncommutative Operator Graphs Generated by Resolutions of Identity

On Noncommutative Operator Graphs Generated by Resolutions of Identity

On Noncommutative Operator Graphs Generated by Resolutions of Identity

Некоммутативные операторные графы составляют важную часть квантовой теории исправления ошибок. В работе дается краткий обзор последних результатов, посвященных графам, порожденным разложениями единицы, для которых существует квантовый код, исправляющий ошибки. Обсуждаются примеры таких графов, затрагивается задача об описании квантового шума в рамках данной теории.

Noncommutative graphs based on finite-infinite system couplings: Quantum error correction for a qubit coupled to a coherent field

Quantum error correction plays a key role for quantum information transmission and quantum computing. In this work, we develop and apply the theory of non-commutative operator graphs to study error correction in the case of a finite-dimensional quantum system coupled to an infinite dimensional system. We consider as an explicit example a qubit coupled via the Jaynes-Cummings Hamiltonian with a bosonic coherent field. We extend the theory of non-commutative graphs to this situation and construct, using the Gazeau-Klauder coherent states, the corresponding non-commutative graph. As the result, we find the quantum anticlique, which is the projector on the error correcting subspace, and analyze it as a function of the frequencies of the qubit and the bosonic field. The general treatment is also applied to the analysis of the error correcting subspace for certain experimental values of the parameters of the Jaynes-Cummings Hamiltonian. The proposed scheme can be applied to any system that possess the same decomposition of spectrum of the Hamiltonian into a direct sum as in JC model, where eigenenergies in the two direct summands form strictly increasing sequences.

Quantum Graphs as Quantum Relations

The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovász number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.

Quantum Asymptotic Spectra of Graphs and Non-Commutative Graphs, and Quantum Shannon Capacities

We study quantum versions of the Shannon capacity of graphs and non-commutative graphs. We introduce the asymptotic spectrum of graphs with respect to quantum and entanglement-assisted homomorphisms, and we introduce the asymptotic spectrum of non-commutative graphs with respect to entanglement-assisted homomorphisms. We apply Strassen’s spectral theorem (J. Reine Angew. Math., 1988) in order to obtain dual characterizations of the corresponding Shannon capacities and asymptotic preorders in terms of their asymptotic spectra. This work extends the study of the asymptotic spectrum of graphs initiated by Zuiddam (Combinatorica, 2019) to the quantum domain. We then exhibit spectral points in the new quantum asymptotic spectra and discuss their relations with the asymptotic spectrum of graphs. In particular, we prove that the (fractional) real and complex Haemers bounds upper bound the quantum Shannon capacity, which is defined as the regularization of the quantum independence number (Mancinska and Roberson, J. Combin. Theory Ser. B, 2016), and that the fractional real and complex Haemers bounds are elements in the quantum asymptotic spectrum of graphs. This is in contrast to the Haemers bounds defined over certain finite fields, which can be strictly smaller than the quantum Shannon capacity. Moreover, since the Haemers bound can be strictly smaller than the Lovasz theta function (Haemers, IEEE Trans. Inf. Theory, 1979), we find that the quantum Shannon capacity and the Lovasz theta function do not coincide. As a consequence, two well-known conjectures in quantum information theory, namely: 1) the entanglement-assisted zero-error capacity of a classical channel is equal to the Lovasz theta function and 2) maximally entangled states and projective measurements are sufficient to achieve the entanglement-assisted zero-error capacity, cannot both be true.

On Errors Generated by Unitary Dynamics of Bipartite Quantum Systems

Given a quantum channel it is possible to define the non-commutative operator graph whose properties determine a possibility of error-free transmission of information via this channel. The corresponding graph has a straight definition through Kraus operators determining quantum errors. We are discussing the opposite problem of a proper definition of errors that some graph corresponds to. Taking into account that any graph is generated by some POVM we give a solution to such a problem by means of the Naimark dilatation theorem. Using our approach we construct errors corresponding to the graphs generated by unitary dynamics of bipartite quantum systems. The cases of POVMs on the circle group ${\mathbb Z}_n$ and the additive group $\mathbb R$ are discussed. As an example we construct the graph corresponding to the errors generated by dynamics of two mode quantum oscillator.

On inner geometry of noncommutative operator graphs

Operator systems (noncommutative operator graphs in other terminology) play a major role in the theory of quantum error correcting codes. Any operator graph is associated with a number of quantum channels. The possibility to transmit quantum information through a quantum channel with zero error is determined by the geometrical properties of the corresponding graph. Noncommutative operator graphs are known to be generated by positive operator-valued measures (POVMs). In turn, many principal POVMs consist of multiple of projections. We construct the model in which the graph is a linear envelope of two projection-valued resolutions of identities in a Hilbert space. Conditions for the existence of quantum anticliques (error-correcting codes) for the graph are investigated. The connection with Shirokov’s example of quantum superactivation (Shirokov in Probl Inform Transm 51(2):87–102, 2015; Shirokov and Shulman in Commun Math Phys 335:1159, 2015) is revealed.

Sandwich theorems and capacity bounds for non-commutative graphs

We define non-commutative versions of the vertex packing polytope, the theta convex body and the fractional vertex packing polytope of a graph, and establish a quantum version of the Sandwich Theorem of Grötschel, Lovász and Schrijver (1986) [7]. We define new non-commutative versions of the Lovász number of a graph which lead to an upper bound of the zero-error capacity of the corresponding quantum channel that can be genuinely better than the one established by Duan, Severini and Winter (2013) [5]. We define non-commutative counterparts of widely used classical graph parameters and establish their interrelation.

An infinite quantum Ramsey theorem

We prove an infinite Ramsey theorem for noncommutative graphs realized as unital self-adjoint subspaces of linear operators acting on an infinite dimensional Hilbert space. Specifically, we prove that if V is such a subspace, then provided there is no obvious obstruction, there is an infinite rank projection P with the property that the compression PVP is either maximal or minimal in a certain natural sense.

The rainbow $k$--connectivity of the non-commutative graph of a finite group

The non-commuting graph Γ( G ) of a non-abelian group G is defined as follows. The vertex set V (Γ( G )) of ℾ( G ) is G Z ( G ) where Z ( G ) denotes the center of G and two vertices x and y are adjacent if and only if xy ≠ yx . We prove that the rainbow k -connectivity of Γ( G ) is equal to ⌈k/2⌉ + 2, for 3 ≤ k ≤ | Z ( G )|.

Non-Commutative Graphs in the Fock Space over One-Particle Hilbert Space

In the present paper we continue our study of non-commutative operator graphs in infinite-dimensional spaces. We consider examples of the non-commutative operator graphs generated by resolutions of identity corresponding to the Heisenberg–Weyl group of operators acting on the Fock space over one-particle state space. The problem of quantum error correction for such graphs is discussed.

The Haemers Bound of Noncommutative Graphs

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 Lovasz theta function (Duan–Severini–Winter, IEEE Trans. Inform. Theory , 2013 and Boreland–Todorov–Winter, arXiv preprint , 2019).

Non-commutative graphs and quantum error correction for a two-mode quantum oscillator

An important topic in quantum information is the theory of error correction codes. Practical situations often involve quantum systems with states in an infinite-dimensional Hilbert space, for example, coherent states. Motivated by these practical needs, we develop the theory of non-commutative graphs, which is a tool to analyze error correction codes, to infinite-dimensional Hilbert spaces. As an explicit example, a family of non-commutative graphs associated with the Schrodinger equation describing the dynamics of a two-mode quantum oscillator is constructed and maximal quantum anticliques for these graphs are found.

On Linear Structure of Non-commutative Operator Graphs

We continue the study of non-commutative operator graphs generated by resolutions of identity covariant with respect to unitary actions of the circle group and the Heisenber-Weyl group as well. It is shown that the graphs generated by the circle group has the system of unitary generators fulfilling permutations of basis vectors. For the graph generated by the Heisenberg-Weyl group the explicit formula for a dimension is given. Thus, we found a new description of the linear structure for the operator graphs introduced in our previous works.