Strategies for optimal single-shot discrimination of quantum measurements

In this work we study the problem of single-shot discrimination of von Neumann measurements, which we associate with measure-and-prepare channels. There are two possible approaches to this problem. The first one is simple and does not utilize entanglement. We focus only on the discrimination of classical probability distributions, which are outputs of the channels. We find necessary and sufficient criterion for perfect discrimination in this case. A more advanced approach requires the usage of entanglement. We quantify the distance between two measurements in terms of the diamond norm (called sometimes the completely bounded trace norm). We provide an exact expression for the optimal probability of correct distinction and relate it to the discrimination of unitary channels. We also state a necessary and sufficient condition for perfect discrimination and a semidefinite program which checks this condition. Our main result, however, is a cone program which calculates the distance between the measurements and hence provides an upper bound on the probability of their correct distinction. As a by-product, the program finds a strategy (input state) which achieves this bound. Finally, we provide a full description for the cases of Fourier matrices and mirror isometries.Comment: 13 pages, 4 figure

Experimental and Simulation Study of the Superstructure and Its Components

The issues discussed in this chapter are of interest of both the manufacturers and the experts responsible for condition of the track superstructure. In general, stress in steel elements may affect the energy state, phase changes, and corrosion. It may reduce fatigue strength and cause damage and cracks of the rails. It is one of the causes of accelerated development of standard railhead defects. Proper selection of, e.g., bending process parameters provides uniform distribution and acceptable level of residual stresses in the bent components. Residual stresses that develop during manufacturing process in the railway turnout steel components can change their strength properties. The first part of this chapter presents ultrasonic measurement method and computer simulation that allowed to develop a method to diagnose state and distribution of residual stresses in steel components of the railway turnout (wing rails and switch blades) in the production process. The second part of this chapter includes experimental and simulation studies of superstructure in operational conditions. A track substructure with a crashed stone composite is a solution of reinforced standard track substructure. The results are used to draw conclusions concerning further development and possible modifications of a proposed solution. A significant number of simulation calculations also allow to determine the duration of guaranteed functionality of a reinforced track substructure

Vertices cannot be hidden from quantum spatial search for almost all random graphs

In this paper we show that all nodes can be found optimally for almost all random Erd\H{o}s-R\'enyi ${\mathcal G}(n,p)$ graphs using continuous-time quantum spatial search procedure. This works for both adjacency and Laplacian matrices, though under different conditions. The first one requires $p=\omega(\log^8(n)/n)$, while the seconds requires $p\geq(1+\varepsilon)\log (n)/n$, where $\varepsilon>0$. The proof was made by analyzing the convergence of eigenvectors corresponding to outlying eigenvalues in the $\|\cdot\|_\infty$ norm. At the same time for $p<(1-\varepsilon)\log(n)/n$, the property does not hold for any matrix, due to the connectivity issues. Hence, our derivation concerning Laplacian matrix is tight.Comment: 18 pages, 3 figur

Piecing together the structural organisation of lipid exchange at membrane contact sites.

Membrane contact sites (MCSs) are areas of close proximity between organelles, implicated in transport of small molecules and in organelle biogenesis. Lipid transfer proteins at MCSs facilitate the distribution of lipid species between organelle membranes. Such exchange processes rely on the apposition of two different membranes delimiting distinct compartments and a cytosolic intermembrane space. Maintaining organelle identity while transferring molecules therefore implies control over MCS architecture both on the ultrastructural and molecular levels. Factors including intermembrane distance, density of resident proteins, and contact surface area fine-tune MCS function. Furthermore, the structural arrangement of lipid transfer proteins and associated proteins underpins the molecular mechanisms of lipid fluxes at MCSs. Thus, the architecture of MCSs emerges as an essential aspect of their function

The e-property of asymptotically stable Markov semigroups

The relations between asymptotic stability and the e-property of Markov semigroups acting on measures defined on general (Polish) metric spaces are studied. While usually much attention is paid to asymptotic stability (and the e-property has been for years verified only to establish it), it should be noted that the e-property itself is also important as it, e.g., ensures that numerical errors in simulations are negligible. Here, it is shown that any asymptotically stable Markov-Feller semigroup with an invariant measure such that the interior of its support is non-empty satisfies the eventual e-property. Moreover, we prove that any Markov-Feller semigroup, which is strongly stochastically continuous, and which possesses the eventual e-property, also has the e-property. We also present an example highlighting that strong stochastic continuity cannot be replaced by its weak counterpart, unless a state space of a process corresponding to a Markov semigroup is a compact metric space.Comment: 19 page

The e-property of asymptotically stable Markov-Feller operators

In this work, we prove that any asymptotically stable Markov-Feller operator possesses the e-property everywhere outside at most a meagre set. We also provide an example showing that this result is tight. Moreover, an equivalent criterion for the e-property is proposed.Comment: 17 page

Molecular visualization of cellular complexity.

Structural biology has paved the way for a ground-up description of biological systems, contributing atomic structures of proteins amenable to crystallography, uncovering high-resolution maps of ‘difficult’ proteins with the cryo-electron microscopy revolution, and filling knowledge gaps regarding dynamic and disordered proteins using nuclear magnetic resonance. From the very beginning, the cellular context of a protein of interest was considered; John Kendrew chose sperm whale myoglobin for crystallization because of myoglobin’s importance and abundance within the dark red tissues of diving animals and thereby solved the first three-dimensional protein structure1. Together, cell and structural biology work synergistically towards a common goal: to build a mechanistic description of biological systems