Parametrization of quantum states based on the quantum state discrimination problem

  • Seyed Arash Ghoreishi
  • Seyed Javad AkhtarshenasEmail author
  • Mohsen Sarbishaei


A discrimination problem consists of N linearly independent pure quantum states \(\Phi =\{|\phi _i\rangle \}\) and the corresponding occurrence probabilities \(\eta =\{\eta _i\}\). With any such problem, we associate, up to a permutation over the probabilities \(\{\eta _i\}\), a unique pair of density matrices \(\varvec{\rho _{_{T}}}\) and \(\varvec{\eta _{{p}}}\) defined on the N-dimensional Hilbert space \(\mathcal {H}_N\). The first one, \(\varvec{\rho _{_{T}}}\), provides a new parametrization of a generic full-rank density matrix in terms of the parameters of the discrimination problem, i.e., the mutual overlaps \(\gamma _{ij}=\langle \phi _i|\phi _j\rangle \) and the occurrence probabilities \(\{\eta _i\}\). The second one, on the other hand, is defined as a diagonal density matrix \(\varvec{\eta _p}\) with the diagonal entries given by the probabilities \(\{\eta _i\}\) with the ordering induced by the permutation p of the probabilities. \(\varvec{\rho _{_{T}}}\) and \(\varvec{\eta _{{p}}}\) capture information about the quantum and classical versions of the discrimination problem, respectively. In this sense, when the set \(\Phi \) can be discriminated unambiguously with probability one, i.e., when the states to be discriminated are mutually orthogonal and can be distinguished by a classical observer, then \(\varvec{\rho _{_{T}}} \rightarrow \varvec{\eta _{{p}}}\). Moreover, if the set lacks its independency and cannot be discriminated anymore, the distinguishability of the pair, measured by the fidelity \(F(\varvec{\rho _{_{T}}}, \varvec{\eta _{{p}}})\), becomes minimum. This enables one to associate with each discrimination problem a measure of discriminability defined by the fidelity \(F(\varvec{\rho _{_{T}}}, \varvec{\eta _{{p}}})\). This quantity, though distinct from the maximum probability of success, has the advantage of being easy to calculate, and in this respect, it can find useful applications in estimating the extent to which the set is discriminable.


Parametrization of quantum states Quantum discrimination Discriminability Fidelity 



This work was supported by Ferdowsi University of Mashhad under Grant No. 3/42098 (1395/08/08).


  1. 1.
    Fano, U.: Description of states in quantum mechanics by density matrix and operator techniques. Rev. Mod. Phys. 29, 74–93 (1957)ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Boya, L.J., Byrd, M., Mims, M., Sudarshan, E.C.G.: Density matrices and geometric phases for \(n\)-state systems. arXiv:quant-ph/9810084
  3. 3.
    Byrd, M.S., Slater, P.: Bures measures over the spaces of two-and three-dimensional density matrices. Phys. Lett. A 283, 152–156 (2001)ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    Tilma, T., Byrd, M., Sudarshan, E.C.G.: A parametrization of bipartite systems based on \(SU(4)\) Euler angles. J. Phys. A: Math. Gen. 35, 10445–10465 (2002)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Tilma, T., Sudarshan, E.C.G.: Generalized Euler angle parametrization for \(SU(N)\). J. Phys. A: Math. Gen. 35, 10467–10501 (2002)ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    Tilma, T., Sudarshan, E.C.G.: Generalized Euler angle parameterization for \(U(N)\) with applications to \(SU(N)\) coset volume measures. J. Geom. Phys. 52, 263–283 (2004)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Życzkowski, K., Słomczyński, W.: The Monge metric on the sphere and geometry of quantum states. J. Phys. A: Math. Gen. 34, 6689–6722 (2001)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Diţă, P.: Finite-level systems, Hermitian operators, isometries and a novel parametrization of Stiefel and Grassmann manifolds. J. Phys. A: Math. Gen. 38, 2657–2668 (2005)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Diţă, P.: Factorization of unitary matrices. J. Phys. A: Math. Gen. 36, 2781–2790 (2003)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Akhtarshenas, S.J.: Canonical coset parametrization and the Bures metric of the three-level quantum systems. J. Math. Phys. 48, 012102 (2007)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Akhtarshenas, S.J.: An explicit computation of the Bures metric over the space of \(N\)-dimensional density matrices. J. Phys. A: Math. Theor. 40, 11333–11341 (2007)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Ilin, N., Shpagina, E., Uskov, F., Lychkovskiy, O.: Squaring parametrization of constrained and unconstrained sets of quantum states. J. Phys. A: Math. Theor. 51, 085301 (2018)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Holevo, A.S.: Probabilistic and Statistical Aspects of Quantum Theory. North-Holland, New York (1982)zbMATHGoogle Scholar
  14. 14.
    Helstrom, C.W.: Quantum Detection and Estimation Theory. Academic Press, New York (1976)zbMATHGoogle Scholar
  15. 15.
    Gisin, N., Ribordy, G., Tittel, W., Zbinden, H.: Quantum cryptography. Rev. Mod. Phys. 74, 145–195 (2002)ADSCrossRefGoogle Scholar
  16. 16.
    Ollivier, H., Zurek, W.H.: Quantum discord: a measure of the quantumness of correlations. Phys. Rev. Lett. 88, 017901 (2001)ADSCrossRefGoogle Scholar
  17. 17.
    Henderson, L., Vedral, V.: Classical, quantum and total correlations. J. Phys. A: Math. Gen. 34, 6899–6905 (2001)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Wootters, W.K., Zurek, W.H.: A single quantum cannot be cloned. Nature 299, 802–803 (1982)ADSCrossRefGoogle Scholar
  19. 19.
    Buzek, V., Hillery, M.: Quantum copying: beyond the no-cloning theorem. Phys. Rev. A 54, 1844–1852 (1996)ADSMathSciNetCrossRefGoogle Scholar
  20. 20.
    Piani, M., Horodecki, P., Horodecki, R.: No-local-broadcasting theorem for multipartite quantum correlations. Phys. Rev. Lett. 100, 090502 (2008)ADSCrossRefGoogle Scholar
  21. 21.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2010)CrossRefGoogle Scholar
  22. 22.
    Barnett, S.M., Croke, S.: Quantum state discrimination. Adv. Opt. Photon. 1, 238–278 (2009)CrossRefGoogle Scholar
  23. 23.
    Barnett, S.M., Croke, S.: On the conditions for discrimination between quantum states with minimum error. J. Phys. A: Math. Theor. 42, 062001 (2009)ADSMathSciNetCrossRefGoogle Scholar
  24. 24.
    Ban, M., Kurokawa, K., Momose, R., Hirota, O.: Optimum measurements for discrimination among symmetric quantum states and parameter estimation. Int. J. Theor. Phys. 36, 1269–1288 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Andersson, E., Barnett, S.M., Gilson, C.R., Hunter, K.: Minimum-error discrimination between three mirror-symmetric states. Phys. Rev. A 65, 052308 (2002)ADSCrossRefGoogle Scholar
  26. 26.
    Chou, C.L.: Minimum-error discrimination among mirror-symmetric mixed quantum states. Phys. Rev. A 70, 062316 (2004)ADSCrossRefGoogle Scholar
  27. 27.
    Herzog, U., Bergou, J.A.: Minimum-error discrimination between subsets of linearly dependent quantum states. Phys. Rev. A 65, 050305 (2002)ADSCrossRefGoogle Scholar
  28. 28.
    Bae, J.: Structure of minimum-error quantum state discrimination. New J. Phys. 15, 073037 (2013)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Ivanovic, I.D.: How to differentiate between non-orthogonal states. Phys. Lett. A 123, 257–259 (1987)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Peres, A.: How to differentiate between non-orthogonal states. Phys. Lett. A 128, 19–19 (1988)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Dieks, D.: Overlap and distinguishability of quantum states. Phys. Lett. A 126, 303–306 (1988)ADSMathSciNetCrossRefGoogle Scholar
  32. 32.
    Chefles, A.: Unambiguous discrimination between linearly independent quantum states. Phys. Lett. A 239, 339–347 (1998)ADSMathSciNetCrossRefGoogle Scholar
  33. 33.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, New York (2013)zbMATHGoogle Scholar
  34. 34.
    Eldar, Y.C.: A semidefinite programming approach to optimal unambiguous discrimination of quantum states. IEEE Trans. Inf. Theory 49, 446–456 (2003)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Pang, S., Wu, S.: Optimum unambiguous discrimination of linearly independent pure states. Phys. Rev. A 80, 052320 (2009)ADSCrossRefGoogle Scholar
  36. 36.
    Bergou, J.A., Futschik, U., Feldman, E.: Optimal unambiguous discrimination of pure quantum states. Phys. Rev. Lett. 108, 250502 (2012)ADSCrossRefGoogle Scholar
  37. 37.
    Bandyopadhyay, S.: Unambiguous discrimination of linearly independent pure quantum states: optimal average probability of success. Phys. Rev. A 90, 030301(R) (2014)ADSCrossRefGoogle Scholar
  38. 38.
    Arfken, G.B., Weber, H.J., Harris, F.E.: Mathematical Methods for Physicists. Elsevier, Amsterdam (2013)zbMATHGoogle Scholar
  39. 39.
    Jaeger, G., Shimony, A.: Optimal distinction between two non-orthogonal quantum states. Phys. Lett. A 197, 83–87 (1995)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Chefles, A., Barnett, S.M.: Optimum unambiguous discrimination between linearly independent symmetric states. Phys. Lett. A 250, 223–229 (1998)ADSCrossRefGoogle Scholar
  41. 41.
    Eldar, Y.C., Stojnic, M., Hassibi, B.: Optimal quantum detectors for unambiguous detection of mixed states. Phys. Rev. A 69, 062318 (2004)ADSCrossRefGoogle Scholar
  42. 42.
    Jafarizadeh, M.A., Rezaei, M., Karimi, N., Amiri, A.R.: Optimal unambiguous discrimination of quantum states. Phys. Rev. A 77, 042314 (2008)ADSCrossRefGoogle Scholar
  43. 43.
    Markham, D., Miszczak, J.A., Puchala, Z., Życzkowski, K.: Quantum state discrimination: a geometric approach. Phys. Rev. A 77, 042111 (2008)ADSMathSciNetCrossRefGoogle Scholar
  44. 44.
    Bengtsson, I., Życzkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement. Cambridge University Press, Cambridge (2006)CrossRefGoogle Scholar
  45. 45.
    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)ADSMathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of PhysicsFerdowsi University of MashhadMashhadIran

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