Construction of mutually unbiased maximally entangled bases through permutations of Hadamard matrices

  • Dengming xu


We construct mutually unbiased maximally entangled bases (MUMEBs) in bipartite system \({\mathbb {C}}^d\otimes {\mathbb {C}}^d (d\ge 3)\) with d a power of a prime number. Precisely, by means of permutation matrices and Hadamard matrices, we construct \(2(d-1)\) MUMEBs in \({\mathbb {C}}^d\otimes {\mathbb {C}}^d\). It follows that \(M(d,d)\ge 2(d-1)\), which is twice the number given in Liu et al. (2016), where M(dd) denotes the maximal size of all sets of MUMEBs in \({\mathbb {C}}^d\otimes {\mathbb {C}}^d\). In addition, let q be another power of a prime number, we construct MUMEBs in \({\mathbb {C}}^d\otimes {\mathbb {C}}^{qd}\) from those in \({\mathbb {C}}^d\otimes {\mathbb {C}}^d\) by the use of the tensor product of unitary matrices.


Mutually unbiased bases Mutually unbiased maximally entangled states Pauli operators Permutation matrices Hardamard matrices Galois rings 



The author would like to express his special gratitude to You Gao for his guidance and to Keqing Feng for introducing this topic to him. The author thanks Jie Lin and Yong Jin for useful discussions and also thanks the anonymous referees for helpful comments. The work is supported by NSFC (No. 11501564).


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© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Sino-European Institute of Aviation EngineeringCivil Aviation University of ChinaTianjinPeople’s Republic of China

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