The convex set containing two-qutrit maximally entangled states

  • Lin ChenEmail author


We construct the convex set \(\mathcal{M}\) of two-qutrit states, and the subset \(\mathcal{P}\subset \mathcal{M}\). We characterize the extremal points of rank one and rank two of \(\mathcal{M}\). We further show that the extremal points of \(\mathcal{P}\) have rank not equaling to two, and at most six. We apply our results to a long-standing conjecture on the locally distinguishable subspace under local operations and classical communications. We prove that rank-one or rank-two matrices in \(\mathcal{M}\) hold for the conjecture. We further simplify the conjecture, by showing that the conjecture holds if and only if it holds for states in \(\mathcal{P}\).


Two-qutrit system Maximally entangled state Convex set Extremal point 



We thank the anonymous referee for improving an earlier version of this paper. This work was supported by the NNSF of China (Grant No. 11871089), Beijing Natural Science Foundation (4173076), and the Fundamental Research Funds for the Central Universities (Grant Nos. KG12040501, ZG216S1810 and ZG226S18C1).


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Authors and Affiliations

  1. 1.School of Mathematics and Systems ScienceBeihang UniversityBeijingChina
  2. 2.International Research Institute for Multidisciplinary ScienceBeihang UniversityBeijingChina

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