Singular value decomposition and matrix reorderings in quantum information theory

We review Schmidt and Kraus decompositions in the form of singular value decomposition using operations of reshaping, vectorization and reshuffling. We use the introduced notation to analyse the correspondence between quantum states and operations with the help of Jamiolkowski isomorphism. The presented matrix reorderings allow us to obtain simple formulae for the composition of quantum channels and partial operations used in quantum information theory. To provide examples of the discussed operations we utilize a package for the Mathematica computing system implementing basic functions used in the calculations related to quantum information theory.Comment: 11 pages, no figures, see http://zksi.iitis.pl/wiki/projects:mathematica-qi for related softwar

Flat Connections and Quantum Groups

We review the Kohno-Drinfeld theorem as well as a conjectural analogue relating quantum Weyl groups to the monodromy of a flat connection D on the Cartan subalgebra of a complex, semi-simple Lie algebra g with poles on the root hyperplanes and values in any g-module V. We sketch our proof of this conjecture when g=sl(n) and when g is arbitrary and V is a vector, spin or adjoint representation. We also establish a precise link between the connection D and Cherednik's generalisation of the KZ connection to finite reflection groups.Comment: 20 pages. To appear in the Proceedings of the 2000 Twente Conference on Lie Groups, in a special issue of Acta Applicandae Mathematica

Irreducible Decomposition of Products of 10D Chiral Sigma Matrices

We review the enveloping algebra of the 10 dimensional chiral sigma matrices. To facilitate the computation of the product of several chiral sigma matrices we have developed a symbolic program. Using this program one can reduce the multiplication of the sigma matrices down to linear combinations of irreducilbe elements. We are able to quickly derive several identities that are not restricted to traces. A copy of the program written in the Mathematica language is provided for the community.Comment: 28 pages, Mathematica Program sigmavector10D.nb is included. Submitted ot CP

Why the naïve Derivation Recipe model cannot explain how mathematicians’ proofs secure mathematical knowledge

This is a pre-copyedited, author-produced PDF of an article accepted for publication in Philosophia Mathematica following peer review. Under embargo. Embargo end date: 7 July 2018 The version of record [Lavor, B., 'Why the Naive Derivation Recipe Model Cannot Explain How Mathematician's Proofs Secure Mathematical Knowledge', Philosophia Mathematica (2016) 24(3): 401-404, is available online at: https://doi.org/10.1093/philmat/nkw012. © The Author [2016]. Published by Oxford University Press. All rights reserved.The view that a mathematical proof is a sketch of or recipe for a formal derivation requires the proof to function as an argument that there is a suitable derivation. This is a mathematical conclusion, and to avoid a regress we require some other account of how the proof can establish it.Peer reviewedFinal Accepted Versio

Effective pseudopotential for energy density functionals with higher order derivatives

We derive a zero-range pseudopotential that includes all possible terms up to sixth order in derivatives. Within the Hartree-Fock approximation, it gives the average energy that corresponds to a quasi-local nuclear Energy Density Functional (EDF) built of derivatives of the one-body density matrix up to sixth order. The direct reference of the EDF to the pseudopotential acts as a constraint that divides the number of independent coupling constants of the EDF by two. This allows, e.g., for expressing the isovector part of the functional in terms of the isoscalar part, or vice versa. We also derive the analogous set of constraints for the coupling constants of the EDF that is restricted by spherical, space-inversion, and time-reversal symmetries.Comment: 18 LaTeX pages, 2 EPS Figures, 27 Tables, and 18 files of the supplemental material (LaTeX, Mathematica, and Fortran), introduction rewritten, table XXVII and figure 2 corrected, in press in Physical Review