Sinkhorn–Knopp theorem for PPT states

Daniel Cariello

In this work, we adapt Sinkhorn–Knopp theorem for rectangular positive maps. We extend their concepts of support and total support to these maps. We show that a positive map T:Mk→Mm is equivalent to a doubly stochastic map if and only if T:Mk→Mm is equivalent to a positive map with total support. Moreover, if k and m are coprime then support is sufficient for the equivalence with a doubly stochastic map. This result provides a necessary and sufficient condition for the filter normal form, which is commonly used in Quantum Information Theory. Let A=∑i=1nAi⊗Bi∈Mk⊗Mm be a state and GA:Mk→Mm be the positive map GA(X)=∑i=1nBitr(AiX). We show that A can be put in the filter normal form if and only if GA:Mk→Mm is equivalent to a positive map with total support. We prove that any state A∈Mk⊗Mm≃Mkm such that dim⁡(ker⁡(A))<k−1, if k=m, and dim⁡(ker⁡(A))<min{k,m}, if k≠m, can be put in the filter normal form.

Takashi Matsuoka, Joint work with W. A. Majewski and M. Ohya

John Farfán Rodríguez

Finalmente agradeço à Universidade Federal de Uberlândia e à Coordenação de aperfeiçoamento de pessoal de nível superior-CAPES pelo apoio financeiro no decorrer de meu curso de mestrado.vii MORA, R. J. A. Aplicações do Teorema de Sinkhorn-Knopp ao Emaranhamento Quântico.2022.81 p. Dissertação de Mestrado, Universidade Federal de Uberlândia, Uberlândia-MG. ResumoO problema da separabilidade dos estados quânticos pede um critério determinístico para distinguir os estados quânticos emaranhados daqueles que não estão emaranhados (separáveis).Esse problema foi resolvido somente para estados 2 M k ⌦ M m com km  6. Para k, m arbitrários sabe-se que esse problema é NP-difícil.Neste trabalho apresentamos a solução do problema para estados em M 2 ⌦ M 2 como aplicação do Teorema de Sinkhorn-Knopp para mapas positivos.Além disso, mostramos que podemos reduzir o problema da separabilidade em M 2 ⌦ M n para os estados que podem ser postos na forma normal de filtro.

XIAO-HONG WANG, Shao-Ming Fei, Ke Wu

Ajit Iqbal Singh

Power symmetric matrices defned and studied by R. Sinkhorn (1981) and their generalization by R.B. Bapat, S.K. Jain and K. Manjunatha Prasad (1999) have been utilized to give positive block matrices with trace one possessing positive partial transpose, the so-called PPT states. Another method to construct such PPT states is given, it uses the form of a matrix unitarily equivalent to its transpose obtained by S.R. Garcia and J.E. Tener (2012). Evolvement or suppression of separability or entanglement of various levels for a quantum dynamical semigroup of completely positive maps has been studied using Choi-Jamiolkowsky matrix of such maps and the famous Horodecki's criteria (1996). A Trichotomy Theorem has been proved, and examples have been given that depend mainly on generalized Choi maps and clearly distinguish the levels of entanglement breaking.

Daniel Cariello