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Strong polygamy of multi-party q-expected quantum correlations

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Abstract

We show that the polygamous nature of multi-party quantum correlations can be characterized in a stronger form based on Tsallis q-entropy and q-expectation value; we establish a class of strong polygamy inequalities of multi-party entanglement in terms of Tsallis q-entropy and q-expectation value for \(q \ge 1\). Our new class of inequalities is 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 entanglement distributed between a single party and any possible subsets of the rest parties, whereas the usual polygamy inequality only considers the entanglement between one party and another. We further establish the equivalence between strong polygamy of quantum entanglement and quantum discord distributed in multi-party quantum systems.

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Notes

  1. Equivalently, for a quantum state \(\rho \) with a spectral decomposition \(\rho =\sum _i \lambda _i \left| e_i\right\rangle \left\langle e_i\right| \).

  2. Without loss of generality, we may assume the (\(n+1\))-party state \(\rho _{AB_1\ldots B_n}\) is an (\(n+1\))-qudit state by taking d as the largest dimension of the subsystems.

  3. That is, \(\{{{\mathbb {X}}}| \varnothing \ne {{\mathbb {X}}}\subset {{\mathbb {B}}}\}=\{{{\mathbb {X}}}^c| \varnothing \ne {{\mathbb {X}}}\subset {{\mathbb {B}}}\}\).

  4. For \(q>0\) and any three-party pure state \(\left| \psi \right\rangle _{ABC}\) with two-party reduced density matrices \(\rho _{AB}=\text {Tr}_C\left| \psi \right\rangle _{ABC}\left\langle \psi \right| \) and \(\rho _{AC}=\text {Tr}_B\left| \psi \right\rangle _{ABC}\left\langle \psi \right| \),

    $$\begin{aligned} S_q\left( \rho _{A|B}\right) +S_q\left( \rho _{A|C}\right) =0. \end{aligned}$$

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Acknowledgements

This work was supported by Basic Science Research Program (NRF-2020R1F1A1A010501270) and Quantum Computing Technology Development Program (NRF-2020M3E4A1080088) through the National Research Foundation of Korea (NRF) grant funded by the Korea government (Ministry of Science and ICT).

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  1. Department of Applied Mathematics and Institute of Natural Sciences, Kyung Hee University, Yongin-si, Gyeonggi-do, 446-701, Korea

    Jeong San Kim

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Kim, J.S. Strong polygamy of multi-party q-expected quantum correlations. Quantum Inf Process 20, 34 (2021). https://doi.org/10.1007/s11128-020-02974-1

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