AIP conference proceedings

Maximum rank of reduced density matrix of m x n-dimensional bipartite states

Bayu Dwi Hatmoko, Agus Purwanto

Published Jan 1, 2019• 0 citations

Abstract

The entanglement is the main requirement of a quantum channel to carry out the quantum task successfully. Quantum information task is transmitting the quantum information from the sender to the recipient. Many methods are used to characterize entanglement, one of which is by using the rank of the reduced density matrix. In this paper, we have shown that for a m x n bipartite state and m < n that the maximum rank of the reduced density matrix is always equal to m. In other words, the maximum rank of the reduced density matrix always satisfied the smallest dimensional order of the bipartite state.

DOI: 10.1063/1.5141635

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AIP conference proceedings

Maximum rank of reduced density matrix of m x n-dimensional bipartite states

Bayu Dwi Hatmoko, Agus Purwanto

Published Jan 1, 2019• 0 citations

Publisher

Abstract

DOI: 10.1063/1.5141635

Related papers

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Mary Beth Ruskai, Elisabeth M. Werner

We show that a bipartite state on a tensor product of two matrix algebras is almost surely entangled if its rank is not greater than that of one of its reduced density matrices.

arXiv:0812.0405Journal of Physics A Mathematical and TheoreticalFeb 5, 200914 citations

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Yize Sun, Lin Chen

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\mathop{\rm rank}(\sum_i R_i \otimes S_i)\leq K \mathop{\rm rank}(\sum_i R_i^T \otimes S_i)

holds for any bipartite matrix

M=\sum_i R_i \otimes S_i

and Schmidt rank

K

. We prove that the conjecture holds for

M

with

K=3

and some special

M

with arbitrary

K

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.