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arXiv:quant-ph/0701029

Characterizing the complete hierarchy of correlations in an nnn-party system

Di Zhou, You, L.

Published Jan 7, 2007• 2 citations
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Abstract

A characterization of the complete correlation structure in an $n$-party system is proposed in terms of a series of $(k,n)$ threshold classical secret sharing protocols ($2\le k\le n$). The total correlation is shown to be the sum of independent correlations of 2-, 3-,$...$, $n$-parties. Our result unifies several earlier scattered works, and shines new light at the important topic of multi-party quantum entanglement. As an application, we explicitly construct the hierarchy of correlations in an $n$-qubit graph state.

DOI
10.48550/arxiv.quant-ph/0701029

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A characterization of the complete correlation structure in an nnn-party system is proposed in terms of a series of (k,n)(k,n)(k,n) threshold classical secret sharing protocols (2≤k≤n2\le k\le n2≤k≤n). The total correlation is shown to be the sum of independent correlations of 2-, 3-,........., nnn-parties. Our result unifies several earlier scattered works, and shines new light at the important topic of multi-party quantum entanglement. As an application, we explicitly construct the hierarchy of correlations in an nnn-qubit graph state.
A characterization of the complete correlation structure in an n-party system is proposed in terms of a series of (k, n) threshold classical secret sharing protocols (2 ≤ k ≤ n). The total correlation is shown to be the sum of independent correlations of 2-, 3-,…, n-parties. Our result unifies several earlier scattered works, and shines new light at the important topic of multi-party quantum entanglement. As an application, we explicitly construct the hierarchy of correlations in an n-qubit graph state.
Bipartite correlations in multiqubit systems cannot be shared freely. The presence of entanglement or classical correlation on certain pairs of qubits may imply correlations on other pairs. We present a method of characterization of bipartite correlations in multiqubit systems using a concept of entangled graphs which has been introduced in our earlier work @M. Plesch and V. Bužek, Phys. Rev. A 67, 012322 ~2003!#. In entangled graphs, each qubit is represented by a vertex while the entanglement and classical correlations are represented by two types of edges. We prove by construction that any entangled graph with classical correlations can be represented by a mixed state of N qubits. However, not all entangled graphs with classical correlations can be represented by a pure state.
Bipartite correlations in multiqubit systems cannot be shared freely. The presence of entanglement or classical correlation on certain pairs of qubits may imply correlations on other pairs. We present a method of characterization of bipartite correlations in multiqubit systems using a concept of entangled graphs which has been introduced in our earlier work [M. Plesch and V. Bu\ifmmode \check{z}\else \v{z}\fi{}ek, Phys. Rev. A 67, 012322 (2003)]. In entangled graphs, each qubit is represented by a vertex while the entanglement and classical correlations are represented by two types of edges. We prove by construction that any entangled graph with classical correlations can be represented by a mixed state of N qubits. However, not all entangled graphs with classical correlations can be represented by a pure state.
Strong polygamy of multi-party qqq-expected quantum correlations
We show that the polygamous nature of multi-party quantum correlations can be characterized in a {\em stronger} form based on Tsallis qqq-entropy and qqq-expectation value. By considering the amount of entanglement that can be distributed in multi-party systems, we establish a class of strong polygamy inequalities of multi-party entanglement in terms of Tsallis qqq-entropy and qqq-expectation for q≥1q \geq 1q≥1. Our new class of inequalities is in fact tighter than the usual polygamy inequalities of multi-party entanglement, and the tightness is explicitly illustrated by an example. Moreover, our new class of inequalities is concerned with the qqq-expected entanglement distributed between a single party and any possible subsets of the rest parties whereas the usual polygamy inequality only considers the entanglement between a single party and another. We further establish the equivalence between strong polygamy inequalities of quantum entanglement and quantum discord distributed in multi-party quantum systems.