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

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## Abstract

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*(*d*, *d*) 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.

### Keywords

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

### Acknowledgements

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).

### References

- 1.Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, F.: A new proof for the existence of mutually unbiased bases. Algorithmica
**34**, 512–528 (2002)MathSciNetCrossRefMATHGoogle Scholar - 2.Brierley, S.: Mutually unbiased bases in low dimensions. Ph.D. thesis. University of York Department of Mathematics (2009)Google Scholar
- 3.Carlet, C.: One-weight \(\mathbb{Z}_4\)-linear codes. In: Buchmann, J., Høholdt, T., Stichtenoth, H., Tapia-Recillas, H. (eds.) Coding Theory, Cryprography and Related Areas, pp. 57–72. Springer, New York (2000)CrossRefGoogle Scholar
- 4.Durt, T., Englert, B.-G., Bengtsson, I., Zyczkowski, K.: On mutually unbiased bases. Int. J. Quantum Inf.
**8**, 535–640 (2010)CrossRefMATHGoogle Scholar - 5.Haagerup, U.: Orthogonal maximal abelian \(*\)-subalgebras of the \(n\times n\) matrices and cyclic \(n\)-roots. Operator Algebras and Quantum Fild Theory (Rome), 1996, pp. 296–322. International press, Cambridge (1996)Google Scholar
- 6.Klappenecker, A., RÖtteler, M.: Constructions of mutually unbiased bases. arXiv:quant-ph/0309120v1
- 7.Liu, J.Y., Yang, M.H., Feng, K.Q.: Mutually unbiased maximally entangled bases in \(\mathbb{C}^d\times \mathbb{C}^{d}\). arXiv:1609.02674v1 (2016)
- 8.Lidl, R., Niederreiter, H.: Introduction to Finite Fields and their Applications, 2nd edn. Cambridge University Press, Cambridge (1994)CrossRefMATHGoogle Scholar
- 9.Tadej, W., Zyczkowski, K.: A concise guide to complex Hadamard matrices. Open Syst. Inf. Dyn.
**13**, 133–177 (2006)MathSciNetCrossRefMATHGoogle Scholar - 10.Tao, Y.H., Nan, H., Zhang, J., Fei, S.M.: Mutually unbiased maximally entangled bases in \({\mathbb{C}}^d\times {\mathbb{C}}^{kd}\). Quantum Inf. Process.
**14**, 2291–2300 (2015)ADSCrossRefMATHGoogle Scholar - 11.Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys.
**191**, 363–381 (1989)ADSMathSciNetCrossRefGoogle Scholar - 12.Wan, Z.X.: Lectures Notes on Finite Fields and Galois Rings. Word Scientific, Singapore (2003)CrossRefMATHGoogle Scholar
- 13.Zhang, J., Tao, Y.H., Nan, H., Fei, S.M.: Construction of mutually unbiased bases in \(\mathbb{C}^d\times \mathbb{C}^{2^ld^{\prime }}\). Quantum Inf. Process.
**14**, 2625–2644 (2015)Google Scholar