Abstract
In quantum information theory, symmetric informationally complete positive operator-valued measures (SIC-POVMs) are relevant to quantum state tomography [8], quantum cryptography [15], and foundational studies [16]. In general, it is hard to construct SIC-POVMs and only a few classes of them existed, as we know. Moreover, we do not know whether there exists an infinite class of them. Many researchers tried to construct approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs). In this paper, we propose two new constructions of ASIC-POVMs for prime power dimensions only by using multiplicative characters over finite fields.
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.References
Appleby, D.: SIC-POVMs and the extended Clifford group. J. Math. Phys. 46, 052107 (2005)
Appleby, D., Bengtsson, I., Brierley, S., Ericsson, A., Grassl, M., Larsson, J.: Systems of imprimitivity for the Clifford group. Quantum Inf. Comput. 14(3–4), 0339–0360 (2014)
Appleby, D., Bengtsson, I., Brierley, S., Grassl, M., Gross, D., Larsson, J.: The monomial representations of the Clifford group. Quantum Inf. Comput. 12(5–6), 0404–0431 (2012)
Appleby, D., Ericsson, Å., Fuchs, C.: Properties of QBist state spaces. Found. Phys. 41(3), 564–579 (2011)
Appleby, M., Chien, T., Flammia, S., Waldron, S.: Constructing exact symmetric informationally complete measurements from numerical solutions, arXiv:1703.05981 (2017)
Berndt, B., Evans, R., Williams, K.: Gauss and Jacobi Sums. Wiley, New York (1990)
Chen, B., Li, T., Fei, M.: General SIC measurement-based entanglement detection. Quantum Inf. Process. 14(6), 2281–2290 (2015)
Caves, C., Fuchs, C., Schack, R.: Unknown quantum states: the quantum de Finetti representation. J. Math. Phys. 43(9), 4537–4559 (2004)
Candes, E., Wakin, M.: An introduction to compressive sampling. IEEE Signal Process. 25(2), 21–30 (2008)
Conway, J., Harding, R., Sloane, N.: Packing lines, planes, etc.: packings in grassmannian spaces. Exp. Math. 5(2), 139–159 (1996)
Cao, X., Mi, J., Xu, S.: Two constructions of approximately symmetric information complete positive operator-valued measures. J. Math. Phys 58, 062201 (2017). https://doi.org/10.1063/1.4985153
Ding, C.: Complex codebooks from combinatorial designs. IEEE Trans. Inf. Theory 52(9), 4229–4235 (2006)
D’Ariano, G., Perinotti, P., Sacchi, M.: Informationally complete measurements and groups representation. J. Opt. B Quantum Semiclass. Opt. 6, S487–S491 (2004)
Fuchs, C., Hoang, M., Stacey, B.: The SIC question: history and state of play. Axioms 6(3), 21 (2017)
Fuchs, C., Sasaki, M.: Squeezing quantum information through a classical channel: measuring the quantumness of a set of quantum states. Quantum Inf. Comput. 3, 377–404 (2003)
Fuchs, C.: Quantum mechanics as quantum information (and only a little more), arXiv preprint arXiv:quant-ph/0205039 (2002)
Grassl, M.: On SIC-POVMs and MUBs in dimension 6. In: Proceedings ERATO Conference on Quantum Information Science EQIS (2004)
Grassl, M., Scott, A.: Fibonacci-Lucas SIC-POVMs, arXiv:1707.02944 (2017)
Hoggar, S.: 64 lines from a quaternionic polytope. Geom. Dedic. 69, 287–289 (1998)
Klappenecker, A., Rötteler, M., Spharlinski, I., Winterhof, A.: On approximately symmetric informationally complete positive operator-valued measures and related systems of quantum states. J. Math. Phys. A 46, 082104 (2005)
Katz, N.: An estimate for character sums. J. Am. Math. Soc. 2(2), 197–200 (1989)
Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)
Massey, J., Mittelholzer, T.: Welch’s Bound and Sequence Sets for Code-Division Multiple-Access Systems, Sequences II, pp. 63–78. Springer, New York (1999)
Oreshkov, O., Calsamiglia, J., Munoztapia, R., Bagan, E.: Optimal signal states for quantum detectors. New J. Phys. 13(7), 73032–73053 (2011)
Peres, A.: Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, Dordrecht (1993)
Renes, J.: Equiangular spherical codes in quantum cryptography. Physics 5(1), 81–92 (2004)
Renes, J., Blume, R., Scott, A., Caves, C.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45(6), 2171–2180 (2004)
Scott, A.: Tight informationally complete quantum measurements. J. Phys. A Math. Gen. 39(43), 13507–13530 (2006)
Scott, A.: SICs: Extending the list of solutions, arXiv:1703.03993 (2017)
Scott, A., Grassl, M.: Symmetric informationally complete positive-operator-valued measures: a new computer study. J. Math. Phys. 51, 1–16 (2010)
Sarwate, D.: Meeting the Welch Bound with Equality, Sequences and Their Applications, pp. 79–102. Springer, London (1999)
Wootters, W., Fields, B.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191(2), 363–381 (2005)
Wang, W., Zhang, A., Feng, K.: Constructions of approximately mutually unbiased bases and symmetric informationally complete positive operator-valued measures by Gauss and Jacobi sums (in Chinese). Sci. Sin. Math. 42, 971–984 (2012)
Xu, G., Xu, Z.: Compressed sensing matrices from fourier matrices. IEEE Trans. Inf. Theory 61(1), 469–478 (2015)
Zauner, G.: Quanten designs-grundzuge einer nichtkommutativen designtheorie (in German). Ph.D. thesis, Universitat Wien, (1999)
Acknowledgements
We are grateful to the two anonymous referees and the editor for useful comments and suggestions that improved the presentation and quality of this paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was supported by the National Natural Science Foundation of China (Grant Nos. 11371011, 11771007 and 61572027).
Rights and permissions
About this article
Cite this article
Luo, G., Cao, X. Two new constructions of approximately SIC-POVMs from multiplicative characters. Quantum Inf Process 16, 313 (2017). https://doi.org/10.1007/s11128-017-1767-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11128-017-1767-8