arXiv:1403.6590

On some entropy inequalities

Lin Zhang

Published Mar 26, 2014• 3 citations

Abstract

In this short report, we give some new entropy inequalities based on Rényi relative entropy and the observation made by Berta {\em et al} [arXiv:1403.6102]. These inequalities obtained extends some well-known entropy inequalities. We also obtain a condition under which a tripartite operator becomes a Markov state.

DOI: 10.48550/arxiv.1403.6590

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arXiv:1403.6590

On some entropy inequalities

Lin Zhang

Published Mar 26, 2014• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.1403.6590

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β

, R{é}nyi entropy.

Characterizing the relations among the three bipartite reduced density operators$ \rho _{AB}

,

\rho _{AC}

and

\rho _{BC}

of a tripartite mixed state

\rho _{ABC}

has been an open problem in quantum information. One of such relations has been reduced by [Cadney et al, LAA. 452, 153, 2014] to a conjectured inequality in terms of matrix rank, namely

r(\rho _{AB}) \cdot r( \rho _{AC})\ge r( \rho _{BC})

for any

\rho _{ABC}$. It is denoted as open problem 41 in the website “Open quantum problems-IQOQI Vienna”. We prove the inequality, and thus establish a complete picture of the four-party linear inequalities in terms of the 0-entropy. Our proof is based on the construction of a novel canonical form of bipartite matrices under local equivalence. We extend our result to inequalities in multipartite systems, as well as the condition when the inequality is saturated.