Quantum Information Processing

, 17:232 | Cite as

A minimal set of measurements for qudit-state tomography based on unambiguous discrimination

  • Donghoon Ha
  • Younghun Kwon


Quantum-state tomography (QST) is a fundamental task for reconstructing unknown quantum state from statistics of measurements. We propose a qudit-state tomography based on unambiguous discrimination (UD) of d linearly independent pure states. We then prove that our proposal for QST provides a minimal set of measurements. Our proposal can be used in any finite dimension, and our strategy can be realized by a projective measurement on a system combined with a d-dimensional auxiliary system. In addition, we present another method to improve previously known UD QST of pure quantum state.


Quantum-state tomography Unambiguous discrimination Measurement 



This work is supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF2015R1D1A1A01060795).


  1. 1.
    Stokes, G.C.: On the composition and resolution of streams of polarized light from different sources. Trans. Camb. Philos. Soc. 9, 399 (1852)ADSGoogle Scholar
  2. 2.
    Peres, A.: Quantum Theory: Concepts and Methods. Kluwer, Amsterdam (1995)zbMATHGoogle Scholar
  3. 3.
    James, D.F.V., Kwiat, P.G., Munro, W.J., White, A.G.: Measurement of qubits. Phys. Rev. A 64, 052312 (2001)ADSCrossRefGoogle Scholar
  4. 4.
    Thew, R.T., Nemoto, K., White, A.G., Munro, W.J.: Qudit quantum-state tomography. Phys. Rev. A 66, 012303 (2002)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Leonhardt, U.: Measuring the Quantum State of Light. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
  6. 6.
    Paris, M., Řeháček, J.: Quantum State Estimation. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  7. 7.
    Ivanović, I.D.: Geometrical description of quantal state determination. J. Phys. A Math. Gen. 14, 3241 (1981)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363 (1989)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Bandyopadhyay, S., Boykin, P.O., Roychowdhury, V., Vatan, F.: A new proof for the existence of mutually unbiased bases. Algorithmica 34, 512 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Prugovečki, E.: Information-theoretical aspects of quantum measurement. Int. J. Theor. Phys. 16, 321 (1977)CrossRefzbMATHGoogle Scholar
  11. 11.
    Caves, C.M., Fuchs, C.A., Schack, R.: Unknown quantum states: the quantum de Finetti representation. J. Math. Phys. 43, 4537 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lemmens, P.W.H., Seidel, J.J.: Equiangular lines. J. Alegebra 24, 494 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Scott, A.J., Grassl, M.: Symmetric informationally complete positive-operator-valued measures: a new computer study. J. Math. Phys. 51, 042203 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Gour, G., Kalev, A.: Construction of all general symmetric informationally complete measurements. J. Phys. A Math. Theor. 47, 335302 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kalev, A., Gour, G.: Mutually unbiased measurements in finite dimensions. New J. Phys. 16, 053038 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    Salazar, R., Delgado, A.: Quantum tomography via unambiguous state discrimination. Phys. Rev. A 86, 012118 (2012)ADSCrossRefGoogle Scholar
  18. 18.
    Ivanovic, I.D.: How to differentiate between non-orthogonal states. Phys. Lett. A 123, 257 (1987)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Dieks, D.: Overlap and distinguishability of quantum states. Phys. Lett. A 126, 303 (1988)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Peres, A.: How to differentiate between non-orthogonal states. Phys. Lett. A 128, 19 (1988)ADSMathSciNetCrossRefGoogle Scholar
  21. 21.
    Jaeger, G., Shimony, A.: Optimal distinction between two non-orthogonal quantum states. Phys. Lett. A 197, 83 (1995)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Chefles, A.: Unambiguous discrimination between linearly independent quantum states. Phys. Lett. A 239, 339 (1998)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Rudolph, T., Spekkens, R.W., Turner, P.S.: Unambiguous discrimination of mixed states. Phys. Rev. A 68, 010301(R) (2003)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Herzog, U., Bergou, J.A.: Optimum unambiguous discrimination of two mixed quantum states. Phys. Rev. A 71, 050301(R) (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Pang, S., Wu, S.: Optimum unambiguous discrimination of linearly independent pure states. Phys. Rev. A 80, 052320 (2009)ADSCrossRefGoogle Scholar
  26. 26.
    Kleinmann, M., Kampermann, H., Bruß, D.: Structural approach to unambiguous discrimination of two mixed quantum states. J. Math. Phys. 51, 032201 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Sugimoto, H., Hashimoto, T., Horibe, M., Hayashi, A.: Complete solution for unambiguous discrimination of three pure states with real inner products. Phys. Rev. A 82, 032338 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Bergou, J.A., Futschik, U., Feldman, E.: Optimal Unambiguous Discrimination of Pure Quantum States. Phys. Rev. Lett. 108, 250502 (2012)ADSCrossRefGoogle Scholar
  29. 29.
    Ha, D., Kwon, Y.: Analysis of optimal unambiguous discrimination of three pure quantum states. Phys. Rev. A 91, 062312 (2015)ADSCrossRefGoogle Scholar
  30. 30.
    Adams, W.W., Goldstein, L.J.: Introduction to Number Theory. Prentice-Hall, Englewood Cliffs (1976)zbMATHGoogle Scholar
  31. 31.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, New York (2010)CrossRefzbMATHGoogle Scholar
  32. 32.
    Baumgratz, T., Cramer, M., Plenio, M.B.: Quantifying coherence. Phys. Rev. Lett. 113, 140401 (2014)ADSCrossRefGoogle Scholar
  33. 33.
    Finkelstein, J.: Pure-state informationally complete and “really” complete measurements. Phys. Rev. A 70, 052107 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Applied PhysicsHanyang UniversityAnsanRepublic of Korea

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