arXiv:2105.07679 · IEEE Transactions on Information Theory

Proof of a Conjectured 0-Rényi Entropy Inequality With Applications to Multipartite Entanglement

Zhiwei Song, Lin Chen, Yize Sun, Mengyao Hu

Published Dec 9, 2022• 7 citations

Abstract

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.

DOI: 10.1109/tit.2022.3227914

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arXiv:2105.07679 · IEEE Transactions on Information Theory

Proof of a Conjectured 0-Rényi Entropy Inequality With Applications to Multipartite Entanglement

Zhiwei Song, Lin Chen, Yize Sun, Mengyao Hu

Published Dec 9, 2022• 7 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.1109/tit.2022.3227914

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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}

\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

M

-partite quantum states with the dimension ${N}^{\ensuremath{\bigotimes}M}$, and (iii) compatibility conditions for the two-party reduced density matrixes of $(2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2)$-dimensional tripartite mixed quantum state.

ρ_{AA^*}^{(2)}

,which controls the Rényi-2 reflected entropy. We show that

ρ_{AA^*}^{(2)}

in a chaotic eigenstate is well-described by this ensemble average which, unlike an ordinary reduced density matrix, takes three qualitatively distinct forms across subsystem-size regimes, which may be viewed as a tripartite generalization of subsystem ETH. We benchmark our analytical predictions against exact diagonalization of the mixed-field Ising model.

q

concurrence, which manifests the relationship among all the ``one-to-group'' marginal entanglements in any multipartite qudit system. This implies lower and upper bounds for the marginal entanglement of any three-qudit system. We further extend our study to general entanglement distribution inequalities for high-dimensional entanglement in terms of the unified-

(r,s)

entropy entanglement, including Tsallis entropy, R\'enyi entropy, and von Neumann entropy entanglement as special cases. These results provide insights into characterizing bipartite high-dimensional entanglement in quantum information processing.