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Locally maximally mixed states

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Abstract

Preparing the locally maximally mixed (LMM) states is a physically operational work. We investigate the set \(\mathcal{P}_d\) containing two-qudit LMM states. We show that the point with a canonical decomposition (CD) has either the unique or infinitely many CDs. Next we show that the point in \(\mathcal{P}_2\) has infinitely many CDs. Further we construct the necessary and sufficient condition by which the non-extreme point of rank two has the unique CD. We also show that the maximally correlated state of rank d is not an extreme point of \(\mathcal{P}_d\). As an application, we show that if the range of rank-three \(\rho \in \mathcal{P}_3\) is spanned by product vectors, then \(\rho \) is not an extreme point of \(\mathcal{P}_3\). Moreover, \(\rho \) is realizable by unitary channels as a method of constructing a family of two-qutrit LMM states. We also prove that Conjecture 1 in [C. King et al., J. Phys. A: Math. Theor 40, 7939 (2007)] holds for \(\rho \).

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Acknowledgements

This work was supported by the NNSF of China (Grant No. 11871089), and the Fundamental Research Funds for the Central Universities (Grant No. ZG216S2005).

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  1. School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Lin Chen & Mengyao Hu

  2. International Research Institute for Multidisciplinary Science, Beihang University, Beijing, 100191, China

    Lin Chen

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  1. Lin Chen
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Correspondence to Mengyao Hu.

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Chen, L., Hu, M. Locally maximally mixed states. Quantum Inf Process 19, 305 (2020). https://doi.org/10.1007/s11128-020-02804-4

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