Homology groups for particles on one-connected graphs

We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our approach is based on some fundamental combinatorial properties of the configuration spaces, Mayer-Vietoris sequences for different parts of configuration spaces and some limited use of discrete Morse theory. As one of the results, we derive a closed-form formulae for ranks of the homology groups for indistinguishable particles on tree graphs. We also give a detailed discussion of the second homology group of the configuration space of both distinguishable and indistinguishable particles. Our motivation is the search for new kinds of quantum statistics.Comment: 26 pages, 16 figure

Asymptotic properties of entanglement polytopes for large number of qubits

Entanglement polytopes have been recently proposed as the way of witnessing the SLOCC multipartite entanglement classes using single particle information. We present first asymptotic results concerning feasibility of this approach for large number of qubits. In particular we show that entanglement polytopes of $L$-qubit system accumulate in the distance $\frac{1}{2\sqrt{L}}$ from the point corresponding to the maximally mixed reduced one-qubit density matrices. This implies existence of a possibly large region where many entanglement polytopes overlap, i.e where the witnessing power of entanglement polytopes is weak. Moreover, the witnessing power cannot be strengthened by any entanglement distillation protocol as for large $L$ the required purity is above current capability.Comment: 5 pages, 4 figure

Critical points of the linear entropy for pure L-qubit states

We present a substancially improved version of the method proposed in Sawicki et al (2012, 2014) for finding critical points of the linear entropy for L-qubit system. The new approach is based on the corespondance between momentum maps for abelian and non-abelian groups, as described in Kirwan (1984). The proposed method can be implemented numerically much easier than the previous one.Comment: 16 pages, 3 figure

Non-abelian Quantum Statistics on Graphs

We show that non-abelian quantum statistics can be studied using certain topological invariants which are the homology groups of configuration spaces. In particular, we formulate a general framework for describing quantum statistics of particles constrained to move in a topological space $X$. The framework involves a study of isomorphism classes of flat complex vector bundles over the configuration space of $X$ which can be achieved by determining its homology groups. We apply this methodology for configuration spaces of graphs. As a conclusion, we provide families of graphs which are good candidates for studying simple effective models of anyon dynamics as well as models of non-abelian anyons on networks that are used in quantum computing. These conclusions are based on our solution of the so-called universal presentation problem for homology groups of graph configuration spaces for certain families of graphs.Comment: 50 pages, v3: updated to reflect the published version. Commun. Math. Phys. (2019

Układ 10-20 lokalizacji elektrod EEG czyli gdzie tkwi pewien błąd koncepcyjny

Opisano problemy związane z definiowaniem położenia elektrod do badań neurofizjologicznych w standardowym układzie 10-20. Przedstawiono możliwości skonstruowanego przez siebie solwera softw are’owego. Zaprezentowano propozycję zmodyfikowania zasad układu 10-20 - polegającą na zastosowaniu tych samych odległości kątowych (ty: 22,5°) na wszystkich osiach głowy XYZ.Zadanie pt. „Digitalizacja i udostępnienie w Cyfrowym Repozytorium Uniwersytetu Łódzkiego kolekcji czasopism naukowych wydawanych przez Uniwersytet Łódzki” nr 885/P-DUN/2014 dofinansowane zostało ze środków MNiSW w ramach działalności upowszechniającej naukę

Designing locally maximally entangled quantum states with arbitrary local symmetries

One of the key ingredients of many LOCC protocols in quantum information is a multiparticle (locally) maximally entangled quantum state, aka a critical state, that possesses local symmetries. We show how to design critical states with arbitrarily large local unitary symmetry. We explain that such states can be realised in a quantum system of distinguishable traps with bosons or fermions occupying a finite number of modes. Then, local symmetries of the designed quantum state are equal to the unitary group of local mode operations acting diagonally on all traps. Therefore, such a group of symmetries is naturally protected against errors that occur in a physical realisation of mode operators. We also link our results with the existence of so-called strictly semistable states with particular asymptotic diagonal symmetries. Our main technical result states that the $N$th tensor power of any irreducible representation of $\mathrm{SU}(N)$ contains a copy of the trivial representation. This is established via a direct combinatorial analysis of Littlewood-Richardson rules utilising certain combinatorial objects which we call telescopes.Comment: 49 pages, 18 figure

How many invariant polynomials are needed to decide local unitary equivalence of qubit states?

Given L-qubit states with the fixed spectra of reduced one-qubit density matrices, we find a formula for the minimal number of invariant polynomials needed for solving local unitary (LU) equivalence problem, that is, problem of deciding if two states can be connected by local unitary operations. Interestingly, this number is not the same for every collection of the spectra. Some spectra require less polynomials to solve LU equivalence problem than others. The result is obtained using geometric methods, i.e. by calculating the dimensions of reduced spaces, stemming from the symplectic reduction procedure.Comment: 22 page

Pilgrimages of the Polish Gentry to Holy Places in the 17th and the 18th Centuries

Pielgrzymki szlachty polskiej do miejsc świętych w XVII i XVIII wieku(streszczenie) W wiekach XVII i XVIII szlachta polska i litewska chętnie wyruszała w podróże. W przypadku podróży zagranicznych o charakterze religijnym największym zainteresowaniem cieszyły się wyprawy do Ziemi Świętej, grobu św. Jakuba Starszego w Santiago de Compostela i do Rzymu. W ich trakcie odwiedzano sanktuaria przechowujące relikwie świętych oraz sławne z cudów za sprawą Matki Boskiej lub świętych na terenie dzisiejszych Czech, Austrii, Bawarii czy północnych Włoch. Popularnymi celami pielgrzymek były też ośrodki pątnicze na terenie Rzeczypospolitej, których liczba wzrosła w XVIII wieku do około 150. Znaczenie miały zwłaszcza sanktuaria maryjne, w szczególności te posiadające ukoronowane w latach 1717-1792 wizerunki Najświętszej Marii Panny.Częstym powodem podjęcia pielgrzymki była przewlekła choroby. Powszechnie wierzono w lecznicze oddziaływanie miejsc świętych. Zwłaszcza, gdy za przyczynę choroby uznawano czary lub opętanie przez diabła. Jednym z takich miejsc był klasztor w Łagiewnikach (Sanktuarium Bożego Miłosierdzia w Krakowie-Łagiewnikach), którego kroniki pełne są świadectw uwolnienia z opętania.</p