Optimal Multi-port-based Teleportation Schemes

In this paper, we introduce optimal versions of a multi-port based teleportation scheme allowing to send a large amount of quantum information. We fully characterise probabilistic and deterministic case by presenting expressions for the average probability of success and entanglement fidelity. In the probabilistic case, the final expression depends only on global parameters describing the problem, such as the number of ports N, the number of teleported systems k, and local dimension d. It allows us to show square improvement in the number of ports with respect to the non-optimal case. We also show that the number of teleported systems can grow when the number N of ports increases as o(N) still giving high efficiency. In the deterministic case, we connect entanglement fidelity with the maximal eigenvalue of a generalised teleportation matrix. In both cases the optimal set of measurements and the optimal state shared between sender and receiver is presented. All the results are obtained by formulating and solving primal and dual SDP problems, which due to existing symmetries can be solved analytically. We use extensively tools from representation theory and formulate new results that could be of the separate interest for the potential readers.

Correlation inequalities and monotonicity properties of the Ruelle operator

In this paper, we provide sufficient conditions for the validity of the FKG Inequality, on Thermodynamic Formalism setting, for a class of eigenmeasures of the dual of the Ruelle operator. We use this correlation inequality to study the maximal eigenvalue problem for the Ruelle operator associated to low regular potentials. As an application, we obtain explicit upper bounds for the main eigenvalue (consequently, for the pressure) of the Ruelle operator associated to Ising models with a power law decay interaction energy.

The Aα-Spectral Radii of Graphs with Given Connectivity

The A α -matrix is A α ( G ) = α D ( G ) + ( 1 − α ) A ( G ) with α ∈ [ 0 , 1 ] , given by Nikiforov in 2017, where A ( G ) is adjacent matrix, and D ( G ) is its diagonal matrix of the degrees of a graph G. The maximal eigenvalue of A α ( G ) is said to be the A α -spectral radius of G. In this work, we determine the graphs with largest A α ( G ) -spectral radius with fixed vertex or edge connectivity. In addition, related extremal graphs are characterized and equations satisfying A α ( G ) -spectral radius are proposed.

Two Proofs and One Algorithm Related to the Analytic Hierarchy Process

36 years ago, Thomas Saaty introduced a new mathematical methodology, called Analytic Hierarchy Process (AHP), regarding the decision-making processes. The methodology was widely applied by Saaty and by other authors in the different human activity areas, like planning, business, education, healthcare, etc. but, in general, in the area of management. In this paper, we provide two new proofs for well-known statement that the maximal eigenvalue λmax is equal to n for the eigenvector problem Aw=λw, where A is, so-called, the consistent matrix of pairwise comparisons of type n×n (n≥ 2) with the solution vector w that represents the probability components of disjoint events. Moreover, we suggest an algorithm for the determination of the eigenvalue problem solution Aw=nw as well as the corresponding flowchart. The algorithm for arbitrary consistent matrix A can be simply programmed and used.

An Efficient Algorithm for Finding the Maximal Eigenvalue of Zero Symmetric Nonnegative Matrices

In this paper, we propose an improved power algorithm for finding maximal eigenvalues. Without any partition, we can get the maximal eigenvalue and show that the modified power algorithm is convergent for zero symmetric reducible nonnegative matrices. Numerical results are reported to demonstrate the effectiveness of the modified power algorithm. Finally, a modified algorithm is proposed to test the positive definiteness (positive semidefiniteness) of Z-matrices.

On the maximal numerical range of some matrices

The maximal numerical range $W_0(A)$ of a matrix $A$ is the (regular) numerical range $W(B)$ of its compression $B$ onto the eigenspace $\mathcal L$ of $A^*A$ corresponding to its maximal eigenvalue. So, always $W_0(A)\subseteq W(A)$. Conditions under which $W_0(A)$ has a non-empty intersection with the boundary of $W(A)$ are established, in particular, when $W_0(A)=W(A)$. The set $W_0(A)$ is also described explicitly for matrices unitarily similar to direct sums of $2$-by-$2$ blocks, and some insight into the behavior of $W_0(A)$ is provided when $\mathcal L$ has codimension one.

The best constants in the Wirtinger inequality

This study aimed to determine the best constants in the Wirtinger inequality [Formula: see text] where [Formula: see text] is defined on [Formula: see text] with [Formula: see text]. First, we referred the computation of [Formula: see text] to the maximal eigenvalue of an integral type operator [Formula: see text]. Second, we proved that the computation of the eigenvalues of [Formula: see text] is equivalent to the solution of a Strum–Liouville problem with some boundary conditions and hence we referred the computation of [Formula: see text] to finding the minimal zero of a function with one variable. Third, by comparing with a result of Lifshits, Papageorgiou, Woźniakowski, we obtained that the strong asymptotic order of [Formula: see text]: [Formula: see text].

A criterion for compactness in Lp(ℝ) of the resolvent of the maximal general form Sturm–Liouville operator

We consider the equationwhere andWe assume that this equation is correctly solvable in Lp(ℝ). Under these assumptions, we study the problem of compactness of the resolvent of the maximal continuously invertible Sturm–Liouville operator . HereIn the case p = 2, for the compact operator , we obtain two-sided sharp-by-order estimates of the maximal eigenvalue.