scholarly journals Dynamical Semigroups in the Birkhoff Polytope of Order 3 as a Tool for Analysis of Quantum Channels

2020 ◽  
Vol 27 (01) ◽  
pp. 2050001
Author(s):  
Mateusz Snamina ◽  
Emil J. Zak
Keyword(s):  
Time Evolution ◽  
Positive Definite ◽  
Conserved Quantity ◽  
Quantum Channels ◽  
Markov Semigroups ◽  
Multiplicative Structure ◽  
Positive Definite Matrices ◽  
Birkhoff Polytope

In the present paper we show a link between bistochastic quantum channels and classical maps. The primary goal of this work is to analyse the multiplicative structure of the Birkhoff polytope of order 3 (the simplest nontrivial case). A suitable complex parametrization of the Birkhoff polytope is proposed, which reveals several its symmetries and characteristics, in particular: (i) the structure of Markov semigroups inside the Birkhoff polytope, (ii) the relation between the set of Markov time evolutions, the set of positive definite matrices and the set of divisible matrices. A condition for Markov time evolution of semigroups in the set of symmetric bistochastic matrices is derived, which leads to an universal conserved quantity for all Markov evolutions. Finally, the complex parametrization is extended to the Birkhoff polytope of order 4.

scholarly journals Multi-variable weighted geometric means of positive definite matrices

2011 ◽  
Vol 435 (2) ◽  
pp. 307-322 ◽  
Author(s):  
Hosoo Lee ◽  
Yongdo Lim ◽  
Takeaki Yamazaki
Keyword(s):  
Positive Definite ◽  
Positive Definite Matrices ◽  
Geometric Means

Identifying graph clusters using variational inference and links to covariance parametrization

2009 ◽  
Vol 367 (1906) ◽  
pp. 4407-4426
Author(s):  
David Barber
Keyword(s):  
Mean Field ◽  
Matrix Decomposition ◽  
Difficult Problem ◽  
Positive Definite ◽  
Variational Inference ◽  
Field Theories ◽  
Variational Approximation ◽  
Positive Definite Matrices ◽  
The Matrix ◽  
Non Gaussian

Finding clusters of well-connected nodes in a graph is a problem common to many domains, including social networks, the Internet and bioinformatics. From a computational viewpoint, finding these clusters or graph communities is a difficult problem. We use a clique matrix decomposition based on a statistical description that encourages clusters to be well connected and few in number. The formal intractability of inferring the clusters is addressed using a variational approximation inspired by mean-field theories in statistical mechanics. Clique matrices also play a natural role in parametrizing positive definite matrices under zero constraints on elements of the matrix. We show that clique matrices can parametrize all positive definite matrices restricted according to a decomposable graph and form a structured factor analysis approximation in the non-decomposable case. Extensions to conjugate Bayesian covariance priors and more general non-Gaussian independence models are briefly discussed.

scholarly journals On a geometric property of positive definite matrices cone

2009 ◽  
Vol 3 (2) ◽  
pp. 64-76 ◽  
Author(s):  
Masatoshi Ito ◽  
Yuki Seo ◽  
Takeaki Yamazaki ◽  
Masahiro Yanagida
Keyword(s):  
Geometric Property ◽  
Positive Definite ◽  
Positive Definite Matrices
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