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

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

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

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

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

The Mossbauer spectroscopy and analytical investigations of the polycrystaline compounds with general formula ZnxSnyCr2Se4

We present combined X-ray powder diffraction and Mossbauer 119Sn studies of polycrystalline compounds with a general formula ZnxSnyCrzSe4 (where x + y + z ¼ 3). The obtained single-phase compounds crystallize in the spinel cubic structure | Fd3m. Tin ions are found to occupy both tetrahedral and octahedral sublattices. On the contrary to the strong tetrahedral site preference energy of Sn, the presented data strongly suggest that the increase in lattice parameters with Sn doping is caused by Sn ions that incorporated into octahedral positions. A quadrupole and isomer shifts of 119Sn in (SnSe4)6¡ and (SnSe6)4¡ are also reported

Review of Vibroacoustic Analysis Methods of Electric Vehicles Motors

The dynamic development of electromobility has resulted in new directions of research, one of which is the analysis of the noise of traction motors. The designs of the motors used in electric vehicles are relatively new and often modified. In addition, strong competition also forces an increase in the power generated per unit mass of the motor, often at the expense of weakening the mechanical structure. This may result in an increase in the noise level generated by the electric drive, so this issue should be analyzed at the motor design stage. Different construction and operating conditions in relation to industrial or railway traction motors make it necessary to constantly develop methods for the noise analysis of the motors for electric vehicles. The aim of this article is to review the methods used so far in an analysis of the noise generated by the motors for electric vehicles. Three main methods are used by the authors of this paper: the analytical method, the hybrid method using two-dimensional models, and the hybrid method using three-dimensional models. In addition to the review of these methods, the paper also focuses on a synthetic summary of the most important factors determining the level and nature of the noise generated by electric vehicle motors