A compositional approach to quantum functions

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.

Recently, the work on quantum automorphism groups of graphs has seen renewed progress, which we expand in this paper. Quantum symmetry is a richer notion of symmetry than the classical symmetries of a graph. In general, it is non-trivial to decide whether a given graph does have quantum symmetries or not. For vertex-transitive graphs, the quantum symmetries have already been determined in earlier work on up to 11 and 13 vertices. This paper fills the gap by determining for all vertex-transitive graphs on 12 vertices, whether they have quantum symmetries and for most of these graphs, we also give their quantum automorphism group explicitly.

Unitary transformations of fibre functors

We clarify the correspondence between two approaches to quantum graphs: via quantum adjacency matrices and via quantum relations. We show how the choice of a (possibly non-tracial) weight manifests itself on the quantum relation side and suggest an extension of the theory of quantum graphs to the infinite dimensional case. Then we use this framework to introduce quantum graphs associated to discrete quantum groups, leading to a new definition of a quantum Cayley graph.

Connectivity for quantum graphs

We study some spectral problems for quantum graphs with a potential. On the one hand we analyze the quantitative dependence of bound states of the Schrodingeroperator on the potential. On the other hand we generalize certain basic identities from the one-dimensional scattering theory to quantum graphs. The first paper is concerned with the study of the discrete negative spectra of quantum graphs. We use the method of trace identities (sum rules) to derive inequalities of Lieb-Thirring, Payne-Polya-Weinberger, and Yang types, among others. We show that the sharp constants of these inequalities and even their forms depend on the topology of the graph. Conditions are identified under which the sharp constants are the same as for the classical inequalities; in particular, this is true in the case of trees. We also provide some counterexamples where the classical form of the inequalities is false. The second paper deals with the scattering problem for the Schrodinger equation on the half-line with the Robin boundary condition at the origin. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to a representation for the perturbation determinant and to trace identities of Buslaev-Faddeev type. In the third paper we generalize results from the half-line case to the full graph case. More precisely, we consider the Schrodinger problem on a star shaped graph with n edges joined at a single vertex. A trace formula is derived for the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to representations for the perturbation determinant and the spectral shift function, and to an analog of Levinson's formula. Besides these three articles this thesis also contains some further results. The method of sum rules is applied to the modified Schrodinger operator with variable coefficients to obtain a Lieb-Thirring type inequality with optimal constant. Furthermore, Lieb-Thirring inequalities are studied for star shaped graphs by using variational arguments and the method of symmetry decomposition of the corresponding Hilbert space. In several cases this leads to optimal constants in the inequalities. Diese Arbeit beinhaltet Untersuchungen von Spektralproblemen fur Quantengraphen mit Potential. Einerseits analysieren wir die quantitative Abhangigkeit der Eigenwerte des Schrodingeroperators vom Potential. Andererseits verallgemeinern wir gewisse elementare Identitaten von der eindimensionalen Streutheorie auf Quantengraphen. Im ersten Artikel wird das diskrete negative Spektrum von Quantengraphen analysiert. Dabei verwenden wir die Methode der Summenformeln, um Ungleichungen vom Lieb-Thirring-, Payne-Polya-Weinberger- und Yang- Typ herzuleiten. Wir zeigen, dass die scharfen Konstanten fur diese Ungleichungen, und sogar ihre Gestalt, von der Topologie des Graphen abhangen. Es werden Bedingungen angegeben, unter welchen die scharfen Konstanten mit den scharfen Konstanten fur den klassischen Fall des Ganzraumes ubereinstimmen. Insbesondere ist dies der Fall fur metrische Baume. Desweiteren konstruieren wir Gegenbeispiele, fur welche die klassischen Konstanten falsch sind. Der zweite Artikel befasst sich mit dem Streuproblem fur den Schrodingeroperator auf der Halbachse mit Robin-Randbedingung. Wir leiten einen Ausdruck fur die Spur der Differenz von der gestorten und der ungestorten Resolvente her, welche eine Wronski-Determinante beinhaltet. Dies fuhrt zu einer Darstellung der Storungsdeterminante und zu Spurformeln vom Buslaev-Faddeev-Typ. Im dritten Artikel werden Resultate aus dem zweiten Artikel verallgemeinert auf sternformige Quantengraphen, die aus n Halbachsen bestehen, welche in einem einzigen Knotenpunkt verbunden sind. Wir beweisen eine Spurformel fur die Differenz der gestorten und der ungestorten Resolvente, welche eine Wronski-Determinante beinhaltet. Dies fuhrt wiederum zu einer Darstellung der Storungsdeterminante und zu einem Analogon der Levinson-Formel. Uber diese drei Hauptteile hinaus beinhaltet diese Arbeit mehrere weitere Resultate. Mit der Methode der Summenformeln leiten wir fur den modifizierten Schrodingeroperator mit variablen Koeffizienten eine Lieb-Thirring-Ungleichung mit einer optimalen Konstante her. Desweiteren werden mit Hilfe von Variationsargumenten und der Symmetriezerlegung des entsprechenden Hilbertraumes Lieb-Thirring Ungleichungen fur den sternformigen Graphen bewiesen. In zahlreichen Fallen erhalten wir dabei die optimale Konstante.

We derive a number of upper and lower bounds for the first nontrivial eigenvalue of Laplacians on combinatorial and quantum graph in terms of the edge connectivity, i.e. the minimal number of edges which need to be removed to make the graph disconnected. On combinatorial graphs, one of the bounds corresponds to a well-known inequality of Fiedler, of which we give a new variational proof. On quantum graphs, the corresponding bound generalizes a recent result of Band and Lévy. All proofs are general enough to yield corresponding estimates for the p-Laplacian and allow us to identify the minimizers.Based on the Betti number of the graph, we also derive upper and lower bounds on all eigenvalues which are ‘asymptotically correct’, i.e. agree with the Weyl asymptotics for the eigenvalues of the quantum graph. In particular, the lower bounds improve the bounds of Friedlander on any given graph for all but finitely many eigenvalues, while the upper bounds improve recent results of Ariturk. Our estimates are also used to derive bounds on the eigenvalues of the normalized Laplacian matrix that improve known bounds of spectral graph theory.

The purpose of this article is to show that even the most elementary problems in asymptotic extremal graph theory can be highly non-trivial. We study linear inequalities between graph homomorphism densities. In the language of quantum graphs the validity of such an inequality is equivalent to the positivity of a corresponding quantum graph. Similar to the setting of polynomials, a quantum graph that can be represented as a sum of squares of labeled quantum graphs is necessarily positive. Lovasz asks whether the opposite is also true. We answer this question and also a related question of Razborov in the negative by introducing explicit valid inequalities that do not satisfy the required conditions. Our solution to these problems is based on a reduction from real multivariate polynomials and uses the fact that there are positive polynomials that cannot be expressed as sums of squares of polynomials. It is known that the problem of determining whether a multivariate polynomial is positive is decidable. Hence it is very natural to ask Is the problem of determining the validity of a linear inequality between homomorphism densities decidable? We give a negative answer to this question which shows that such inequalities are inherently difficult in their full generality. Furthermore we deduce from this fact that the analogue of Artin's solution to Hilbert's seventeenth problem does not hold in the setting of quantum graphs.

On square roots of the Haar state on compact quantum groups

We define a category containing the discrete quantum groups (and hence the discrete groups and the duals of compact groups) and the compact quantum groups (and hence the compact groups and the duals of discrete groups). The dual of an object can be defined within the same category and we have a biduality theorem. This theory extends the duality between compact quantum groups and discrete quantum groups (and hence the one between compact abelian groups and discrete abelian groups). The objects in our category are multiplier Hopf algebras, with invertible antipode, admitting invariant functionals (integrals), satisfying some extra condition (to take care of the non-abelianness of the underlying algebras). If we start with a multiplier Hopf ∗-algebra with positive invariant functionals, then also the dual is a multiplier Hopf ∗-algebra with positive invariant functionals. This makes it possible to formulate this duality also within the framework of C∗-algebras.

Constructing a useful tool for characterizing amino acid conformers by means of quantum chemical and graph theory indices

We present an approach to describe state-of-the-art photonic quantum experiments using graph theory. There, the quantum states are given by the coherent superpositions of perfect matchings. The crucial observation is that introducing complex weights in graphs naturally leads to quantum interference. This viewpoint immediately leads to many interesting results, some of which we present here. First, we identify an experimental unexplored multiphoton interference phenomenon. Second, we find that computing the results of such experiments is #P-hard, which means it is a classically intractable problem dealing with the computation of a matrix function Permanent and its generalization Hafnian. Third, we explain how a recent no-go result applies generally to linear optical quantum experiments, thus revealing important insights into quantum state generation with current photonic technology. Fourth, we show how to describe quantum protocols such as entanglement swapping in a graphical way. The uncovered bridge between quantum experiments and graph theory offers another perspective on a widely used technology and immediately raises many follow-up questions.

The purpose of this paper is to present a newly discovered link between three seemingly unrelated subjects—quantum graphs, the theory of random matrix ensembles and combinatorics. We discuss the nature of this connection, and demonstrate it in a special case pertaining to simple graphs, and to the random ensemble of ${2\times 2}$ unitary matrices. The corresponding combinatorial problem results in a few identities, which, to the best of our knowledge, were not proven previously.

We say that an element g of a group G is almost right Engel if there is a finite set R(g) such that for every x∈G, there is a positive integer n(x,g) such that […[[g,x],x],…,x]∈R(g) if x is repeated at least n(x,g) times. Thus, g is a right Engel element precisely when we can choose R(g)={1}. We prove that if all elements of a compact (Hausdorff) group G are almost right Engel, then G has a finite normal subgroup N such that G/N is locally nilpotent. If in addition there is a uniform bound ∣R(g)∣⩽m for the orders of the corresponding sets, then the subgroup N can be chosen of order bounded in terms of m. The proofs use the Wilson–Zelmanov theorem saying that profinite Engel groups are locally nilpotent and previous results of the authors about compact groups in which all elements are almost left Engel.

Constructive Ackermann's interpretation