Mutually unbiased unextendible maximally entangled bases in some systems of higher dimension
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Mutually unbiased unextendible maximally entangled bases in some systems of higher dimension Zong-Xing Xiong1
· Zhu-Jun Zheng1 · Shao-Ming Fei2,3
Received: 20 May 2020 / Accepted: 29 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We study the construction of mutually unbiased bases such that all the bases are unextendible maximally entangled ones. By using some results from the theory of finite fields, we construct mutually unbiased unextendible maximally entangled bases in some bipartite systems of higher dimension: C4 ⊗ C5 , C6 ⊗ C7 , C10 ⊗ C11 and C12 ⊗C13 , which extend the known result of C2 ⊗C3 . We also generalize these results to more bipartie systems of specific dimension. Keywords Mutually unbiased bases · Unextendible maximally entangled bases · Finite fields · Additive character
1 Introduction Mutually unbiased bases (MUBs) became an essential feature of quantum mechanics since the work of Schwinger [1] who revealed that no information can be retrieved when a quantum state prepared in a basis state is measured with respect to the basis mutually unbiased with the prepared one. It is well known to us nowadays that mutually unbiased bases play an important roles in many quantum information processing such as quantum state tomography [2–4], cryptographic protocols [5,6], and the mean kings problem [7]. In [8], the authors showed that a complete set of mutually unbiased bases of a bipartite system contains a fixed amount of entanglement, independent of the choice of the set. This shed lights on the connection between mutually unbiased bases and quantum entanglement, another essential feature of quantum mechanics who plays an
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Zhu-Jun Zheng [email protected]
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Department of Mathematics, South China University of Technology, Guangzhou 510640, China
2
School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
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Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany 0123456789().: V,-vol
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important role in quantum information, such as teleportation, quantum error correction, and quantum secret sharing [9]. Maximally entangled states (MES) plays a very important roles in quantum information processing tasks [10]. The notion of unextendible maximally entangled bases (UMEBs) was first introduced in [11], which represents a set of less than d 2 orthonormal maximally entangled states in Cd ⊗ Cd such that no additional maximally entangled vectors are orthogonal to them. The UMEB in arbitrary bipartite system Cd ⊗ Cd has been investigated in [12,13]. In [12], the authors constructed two mutually unbiased UMEBs in C2 ⊗C3 . Later, a generic way of constructing a pair of UMEBs in C2 ⊗C3 which are mutually unbiased was further presented in [14]. Since then, there have been efforts constructing maximally entangled bases or unextendible maximally entangled bases which are mutually unbiased [15–20]. In this paper, inspired by the approach of constructing mutually unbiased unext
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