136 research outputs found
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
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
On the phase diagram of the anisotropic XY chain in transverse magnetic field
We investigate an explicite formula for ground state energy of the
anisotropic XY chain in transverse magnetic field. In particular, we examine
the smoothness properties of the expression, given in terms of elliptic
integrals. We confirm known 2d-Ising type behaviour in the neighbourhood of
certain lines of phase diagram and give more detailed information there,
calculating a few next-to-leading exponents as well as corresponding
amplitudes. We also explicitly demonstrate that the ground-state energy is
infinitely differentiable on the boundary between ferromagnetic and oscillatory
phases.Comment: 15 pages, 8 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
Does the Use of Electronic Gradebooks Affect Schooling Outcomes? A Study in Polish Primary Schools
The purpose of this study was to determine whether, in the context of Polish primary schools, the use of electronic gradebook raises teaching effectiveness and has altogether any observable educational effects when compared to older, paper forms of record keeping. Our analyses use data from a survey conducted in 2010-2015 on a nationwide representative sample of 4,500 students from 171 primary schools. Because the electronic gradebook was not yet widely used in Poland at that time, data from this period can be used to analyze the problem stated in the title. The key tools used in our research were standardized school achievement tests and scales examining a student’s emotional integration with the school, quality of peer relations and academic self-concept. Measurements were taken after the completion of grades 3 and 6. The results of intelligence tests and indicators of the economic and social status of the student’s family were considered as control variables. Statistical analyses were performed using hierarchical linear regression models. No significant correlation was revealed between the form of the school gradebook and teaching effectiveness in reading, writing or math. Only one of the educational effects included in the study was significantly related to the form of the school gradebook: the use of an electronic form correlated negatively with the level of students’ academic self-concept. These results show that the consequences of introducing this type of e-record into schools are more complex than previously considered
Multipartite quantum correlations: symplectic and algebraic geometry approach
We review a geometric approach to classification and examination of quantum
correlations in composite systems. Since quantum information tasks are usually
achieved by manipulating spin and alike systems or, in general, systems with a
finite number of energy levels, classification problems are usually treated in
frames of linear algebra. We proposed to shift the attention to a geometric
description. Treating consistently quantum states as points of a projective
space rather than as vectors in a Hilbert space we were able to apply powerful
methods of differential, symplectic and algebraic geometry to attack the
problem of equivalence of states with respect to the strength of correlations,
or, in other words, to classify them from this point of view. Such
classifications are interpreted as identification of states with `the same
correlations properties' i.e. ones that can be used for the same information
purposes, or, from yet another point of view, states that can be mutually
transformed one to another by specific, experimentally accessible operations.
It is clear that the latter characterization answers the fundamental question
`what can be transformed into what \textit{via} available means?'. Exactly such
an interpretations, i.e, in terms of mutual transformability can be clearly
formulated in terms of actions of specific groups on the space of states and is
the starting point for the proposed methods.Comment: 29 pages, 9 figures, 2 tables, final form submitted to the journa
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
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
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