arXiv:1407.2639 · Journal of Physics A Mathematical and Theoretical

Critical points of the linear entropy for pureL-qubit states

Tomasz Maciążek, Adam Sawicki

Published Jan 5, 2015• 10 citations

Abstract

We present a substantially improved version of the method proposed by Sawicki et al (2012 Phys. Rev. A 86 040304(R) , 2014 Rev. Math. Phys. 26 1450004 ) for finding critical points of the linear entropy for an L -qubit system. The new approach is based on the correspondence between the momentum maps for abelian and non-abelian groups, as described in Kirwan (1984 Cohomology of Quotients in Symplectic and Algebraic Geometry ( Mathematical Notes vol 31) (Princeton: Princeton Univ. Press)). The proposed method can be implemented numerically and is much easier than the previous method.

DOI: 10.1088/1751-8113/48/4/045305

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Previous article Next article Infinitely Divisible Measurements in Quantum Probability TheoryA. S. KholevoA. S. Kholevohttps://doi.org/10.1137/1131063PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. Sakai, $C\sp*$-algebras and $W\sp*$-algebras, Springer-Verlag, New York, 1971xii+253 56:1082 0219.46042 CrossrefGoogle Scholar[2] E. B. Davies, Quantum theory of open systems, Academic Press [Harcourt Brace Jovanovich Publishers], London, 1976x+171 58:8853 0388.46044 Google Scholar[3] V. V. Sazonov and , V. N. Tutubalin, Probability distributions on topological groups, Theory Probab. Appl., 11 (1966), 1–45 10.1137/1111001 0171.38701 LinkGoogle Scholar[4] A. Barchielli, , L. Lanz and , G. M. Prosperi, Statistics of continuous trajectories in quantum mechanics: operation-valued stochastic processes, Found. Phys., 13 (1983), 779–812 87b:81006 CrossrefGoogle Scholar[5] A. Barchielli and , G. Lupieri, Dilations of operation valued stochastic processesQuantum probability and applications, II (Heidelberg, 1984), Lecture Notes in Math., Vol. 1136, Springer, Berlin, 1985, 57–66 87g:46101 0572.60046 CrossrefGoogle Scholar[6] E. Schmidt, On multiplicative Lebesgue integration and families of evolution operators, Math. Scand., 29 (1971), 113–133 50:4885 0232.47024 Google Scholar[7] G. Lindblad, On the generators of quantum dynamical semigroups, Comm. Math. Phys., 48 (1976), 119–130 54:1990 0343.47031 CrossrefGoogle Scholar[8] D. E. Evans and , J. T. Lewis, Dilations of irreversible evolutions in algebraic quantum theory, Comm. Dublin Inst. Adv. Studies Ser. A, (1977), v+104 58:8915 0365.46059 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Matrix product states and projected entangled pair states: Concepts, symmetries, theorems17 December 2021 | Reviews of Modern Physics, Vol. 93, No. 4 Cross Ref Generalized ansatz for continuous matrix product states7 May 2020 | Physical Review A, Vol. 101, No. 5 Cross Ref Continuum limits of matrix product states19 November 2018 | Physical Review B, Vol. 98, No. 17 Cross Ref Measurement Uncertainty Relations for Position and Momentum: Relative Entropy Formulation24 June 2017 | Entropy, Vol. 19, No. 7 Cross Ref Lie-semigroup structures for reachability and control of open quantum systems: kossakowski-lindblad generators form lie wedge to markovian channelsReports on Mathematical Physics, Vol. 64, No. 1-2 Cross Ref Lévy Processes and Continuous Quantum Measurements Cross Ref A composite Fourier‐wavelet transform and square‐integrable group representationsJournal of Mathematical Physics, Vol. 35, No. 8 Cross Ref Convolution Semigroups in Noncommutative Probability TheoryL. V. Denisov17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 3AbstractPDF (636 KB)Infinitely Divisible Markov Mappings in Quantum Probability TheoryL. V. Denisov17 July 2006 | Theory of Probability & Its Applications, Vol. 33, No. 2AbstractPDF (521 KB) Volume 31, Issue 3| 1987Theory of Probability & Its Applications History Submitted:29 January 1986Published online:28 July 2006 InformationCopyright © 1987 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1131063Article page range:pp. 493-497ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

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Measurement and Filtering of Quantum Diffusion II

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It is known that relative entropy of entanglement for an entangled state $\ensuremath{\rho}$ is defined via its closest separable (or positive partial transpose) state $\ensuremath{\sigma}$. Recently, it has been shown how to find $\ensuremath{\rho}$ provided that $\ensuremath{\sigma}$ is given in a two-qubit system. In this article we study the reverse process, that is, how to find $\ensuremath{\sigma}$ provided that $\ensuremath{\rho}$ is given. It is shown that if $\ensuremath{\rho}$ is of a Bell-diagonal, generalized Vedral-Plenio, or generalized Horodecki state, one can find $\ensuremath{\sigma}$ from a geometrical point of view. This is possible due to the following two facts: (i) the Bloch vectors of $\ensuremath{\rho}$ and $\ensuremath{\sigma}$ are identical to each other; (ii) the correlation vector of $\ensuremath{\sigma}$ can be computed from a crossing point between a minimal geometrical object, in which all separable states reside in the presence of Bloch vectors, and a straight line, which connects the point corresponding to the correlation vector of $\ensuremath{\rho}$ and the nearest vertex of the maximal tetrahedron, where all two-qubit states reside. It is shown, however, that these properties are not maintained for the arbitrary two-qubit states.

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