149 research outputs found
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
-qubit system accumulate in the distance 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 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 . The
framework involves a study of isomorphism classes of flat complex vector
bundles over the configuration space of 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
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 th tensor power of any irreducible
representation of 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
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
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
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