Research PaperNo Access

Two new constructions of approximately symmetric informationally complete positive operator-valued measures

Gang Wang

School of Science, Tianjin University of Technology, Tianjin 300384, P. R. China

E-mail Address: gwang06080923@mail.nankai.edu.cn

Corresponding author.

Search for more papers by this author

Min-Yao Niu

College of Science, Civil Aviation University of China, Tianjin 300300, P. R. China

Search for more papers by this author

, and

Fang-Wei Fu

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, P. R. China

Search for more papers by this author

https://doi.org/10.1142/S0219749919500217Cited by:1 (Source: Crossref)

Abstract

A symmetric informationally complete positive operator-valued measure (SIC-POVM) is a POVM in $ℂ^{K}$ consisting of $K^{2}$ positive operators of rank one such that all of whose Hermite inner products are equal. SIC-POVMs are important in quantum information theory, which have many applications in quantum state tomography, quantum cryptography and basic research in quantum mechanics. However, it is very difficult to construct SIC-POVMs. Therefore, many scholars have focused on approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs) for which the Hermite inner products are close to equal. In this paper, two new constructions of ASIC-POVMs are provided by using character sums and some special functions over finite fields.

Keywords:

References

1. A. Klappenecker, M. Rötteler, I. E. Spharlinski and A. Winterhof, J. Math. Phys. A 46(8) (2005) 082104. Web of Science, Google Scholar
2. A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic Publishers, Dordrecht, 1993). Google Scholar
3. C. M. Caves, C. A. Fuchs and R. Schack, J. Math. Phys. 43(9) (2004) 4537–4559. Web of Science, Google Scholar
4. G. M. D. Ariano, P. Perinotti and M. F. Sacchi, J. Opt. B: Quantum Semiclassical Opt. 6 (2004) S487–S491. Google Scholar
5. G. Zauner, Ph.D. thesis, Universitat Wien, 1999. Google Scholar
6. D. Appleby, D. Ericsson and C. Fuchs, Found. Phys. 41(3) (2011) 564–579. Web of Science, Google Scholar
7. C. A. Fuchs, Quantum Inf. Comput. 4(6) (2004) 467–478. Web of Science, Google Scholar
8. M. Grassl, in Proc. ERATO Conf. Quantum Information Science (EQIS 2004), Tokyo (Erato Conference on Quantum Information Science, 2004) (2004), pp. 60–61. Google Scholar
9. D. M. Appleby, J. Math. Phys. 46(5) (2005) 052107. Web of Science, Google Scholar
10. A. J. Scott and M. Grassl, J. Math. Phys. 51(4) (2010) 042203. Web of Science, Google Scholar
11. D. M. Appleby, I. Bengtsson, S. Brierley, M. Grassl, D. Gross and J. Larsson, Quantum Inf. Comput. 12(5) (2012) 404–431. Web of Science, Google Scholar
12. W. Wang, A. Zhang and K. Feng, Sci. Sin. Math. 42(10) (2012) 971–984. Google Scholar
13. X. Cao, J. Mi and S. Xu, J. Math. Phys. 58(6) (2017) 062201. Web of Science, Google Scholar
14. G. Luo and X. Cao, Quantum Inf. Process 16(313) (2017). Google Scholar
15. R. Lidl and H. Niederreiter, Finite Fields: Encyclopedia of Mathematics, Vol. 20 (Cambridge University Press, Cambridge, 1983). Google Scholar
16. K. Nyberg, Advances in Cryptology- EUROCRYPT93, Lecture Notes in Computer Science, Vol. 765, eds. T. Helleseth (Springer, Berlin, 1994), pp. 55–64. Google Scholar
17. J. Hall, Ph.D. thesis, RMIT University, Melbourne, Victoria, Australia, 2011. Google Scholar