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Mutually Unbiased Property of Maximally Entangled Bases and Product Bases in \(\mathbb {C}^{d}\otimes \mathbb {C}^{d}\)

  • Ling-Shan Xu
  • Gui-Jun Zhang
  • Yi-Yang Song
  • Yuan-Hong Tao
Article

Abstract

We investigate mutually unbiased property between maximally entangled bases and product bases in bipartite systems \(\mathbb {C}^{d} \otimes \mathbb {C}^{d}\). We first visualize the description of \(p_{1}^{a_{1}}-1\)-member mutually unbiased maximally entangled bases(MUMEBs) in \(\mathbb {C}^{d} \otimes \mathbb {C}^{d}\ \), while \(d=p_{1}^{a_{1}}p_{2}^{a_{2}}...p_{s}^{a_{s}}\), \(3\leq p_{1}^{a_{1}}\leq p_{2}^{a_{2}}\leq ...\leq p_{s}^{a_{s}}\), \(p_{1}^{a_{1}},...,p_{s}^{a_{s}}\) are distinct primes, which was proposed by Liu et al. (Quantum Inf. Process. 16(6), 159, 2017). We then establish two more mutually unbiased product bases which are also mutually unbiased to the above \(p_{1}^{a_{1}}-1\) MUMEBs, thus we present \(p_{1}^{a_{1}}+ 1\) mutually unbiased bases(MUBs) in \(\mathbb {C}^{d} \otimes \mathbb {C}^{d}\ \). We also show the concrete construction of those MUBs in bipartite systems \(\mathbb {C}^{3} \otimes \mathbb {C}^{3}\ \), \(\mathbb {C}^{4} \otimes \mathbb {C}^{4}\ \), \(\mathbb {C}^{5} \otimes \mathbb {C}^{5}\ \) and \(\mathbb {C}^{12} \otimes \mathbb {C}^{12}\).

Keywords

Mutually unbiased bases Maximally entangled state Product state Product basis 

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Ling-Shan Xu
    • 1
  • Gui-Jun Zhang
    • 1
  • Yi-Yang Song
    • 1
  • Yuan-Hong Tao
    • 1
  1. 1.Department of Mathematics, College of SciencesYanbian UniversityYanjiChina

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