Quantum Decision Theory: Analysis and Optimization

  • Gianfranco CariolaroEmail author
Part of the Signals and Communication Technology book series (SCT)


Decision is the heart of any quantum digital communication system and is concerned with the formulation of how a measurement must be taken to argue on the transmitted message in the presence of uncertainties (due to the nature of any quantum measurement). The state of the quantum system is considered as assigned through a constellation of pure states or of density operators, whereas the measurement operators must be found with the goal of achieving the “best decision” (optimization), usually obtained by minimizing the error probability. The chapter, after a mathematical formulation of decision, moves on to optimization, where the guidelines are given by Holevo’s and Kennedy’s theorems. These theorems determine the conditions that must be fulfilled by an optimal system of measurement operators, but do not provide any clue on how to identify it. In any case, the problem of optimization is very difficult, and exact solutions (nonnumerical) are only known in few cases, mainly in the binary systems and in general when the state constellation has a symmetry. The specific symmetry that simplifies detection and optimization is called geometrically uniform symmetry (GUS).


  1. 1.
    C.W. Helstrom, J.W.S. Liu, J.P. Gordon, Quantum-mechanical communication theory. Proc. IEEE 58(10), 1578–1598 (1970)CrossRefMathSciNetGoogle Scholar
  2. 2.
    K. Kraus, States, Effect and Operations: Fundamental Notions of Quantum Theory, Lecture Notes in Physics, vol. 190 (Springer, New York, 1983)Google Scholar
  3. 3.
    Y.C. Eldar, A. Megretski, G.C. Verghese, Optimal detection of symmetric mixed quantum states. IEEE Trans. Inf. Theory 50(6), 1198–1207 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    A.S. Holevo, Statistical decision theory for quantum systems. J. Multivar. Anal. 3(4), 337–394 (1973)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    H.P. Yuen, R. Kennedy, M. Lax, Optimum testing of multiple hypotheses in quantum detection theory. IEEE Trans. Inf. Theory 21(2), 125–134 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    M. Grant, S. Boyd, CVX: Matlab software for disciplined convex programming, version 2.1. March 2014,
  7. 7.
    M. Grant, S. Boyd, Graph implementations for nonsmooth convex programs, in: Recent Advances in Learning and Control, ed. by V. Blondel, S. Boyd, H. Kimura, (eds). Lecture Notes in Control and Information Sciences. (Springer, 2008), pp. 95–110, boyd/graph_dcp.html
  8. 8.
    R.S. Kennedy, A near-optimum receiver for the binary coherent state quantum channel. Massachusetts Institute of Technology, Cambridge (MA), Technical Report, January 1973. MIT Research Laboratory of Electronics Quarterly Progress Report 108Google Scholar
  9. 9.
    V. Vilnrotter and C.W. Lau, Quantum detection theory for the free-space channel. NASA, Technical Report, August 2001. Interplanetary Network Progress (IPN) Progress Report 42–146Google Scholar
  10. 10.
    V. Vilnrotter, C.W. Lau, Quantum detection and channel capacity using state-space optimization. NASA, Technical Report, February 2002. Interplanetary Network Progress (IPN) Progress Report 42–148Google Scholar
  11. 11.
    V. Vilnrotter, C.W. Lau, Binary quantum receiver concept demonstration. NASA, Technical Report, Interplanetary Network Progress (IPN) Progress Report 42–165, May 2006Google Scholar
  12. 12.
    K. Kato, M. Osaki, M. Sasaki, O. Hirota, Quantum detection and mutual information for QAM and PSK signals. IEEE Trans. Commun. 47(2), 248–254 (1999)CrossRefGoogle Scholar
  13. 13.
    R.A. Horn, C.R. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1998)Google Scholar
  14. 14.
    Y.C. Eldar, G.D. Forney, On quantum detection and the square-root measurement. IEEE Trans. Inf. Theory 47(3), 858–872 (2001)Google Scholar
  15. 15.
    A. Assalini, G. Cariolaro, G. Pierobon, Efficient optimal minimum error discrimination of symmetric quantum states. Phys. Rev. A 81, 012315 (2010)CrossRefGoogle Scholar
  16. 16.
    G. Cariolaro, R. Corvaja, G. Pierobon, Compression of pure and mixed states in quantum detection. in Global Telecommunications Conference, vol. 2011 (GLOBECOM, IEEE, 2011), pp. 1–5Google Scholar
  17. 17.
    A.S. Holevo, V. Giovannetti, Quantum channels and their entropic characteristics. Rep. Prog. Phys. 75(4), 046001 (2012)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Department of Information EngineeringUniversity of PadovaPadovaItaly

Personalised recommendations