Measurements of Negative Joint Probabilities in Optical Quantum System

  • Masataka Iinuma
  • Yutaro Suzuki
  • Holger F. Hofmann
Part of the Mathematics for Industry book series (MFI, volume 14)


Quantum information science is based on the superposition principle and the non-local quantum correlation of quantum states. These non-classical properties are still mysterious and inadequately understood as physical phenomena. The biggest reason to un-resolving such problems is that we can directly not get all bare information of the quantum states since an action by any measurement absolutely changes the quantum states. Recently, we realized a variable strength measurement of photon polarization, which is capable of controlling the measurement strength from zero (no measurement) to fully projection (completely destructive measurement). This apparatus makes it possible to perform a sequential measurement of two non-commuting observables with an error and the back-action effects produced by the measurement, which never gives exact values simultaneously. We investigated the role of measurement uncertainties of the first variable strength measurement. The experimentally-obtained joint probabilities can be recognized as statistical mixture obtained by random polarization flips arising the measurement uncertainties from an intrinsic joint probability distribution. This natural assumption provided a removal of the back-action effect from the experimental probabilities and the obtained intrinsic probabilities resulted in negative. This analysis also shows how the negative probabilities are converted to observable positive statistics by variable combinations of resolution and back-action uncertainties.


Quantum state Negative joint probability Photon polarization Sequential measurement Variable strength measurement Non-commuting observable Measurement resolution Measurement back-action 



This work was supported by JSPS through KAKENHI grant numbers 24540428, 24540427 and 21540409.


  1. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum information, Cambridge University Press, Cambridge (2000)Google Scholar
  2. Hofmann, H.F.: Complex joint probabilities as expressions of reversible transformations in quantum mechanics. New J. Phys. 14, 043031 (2012). doi: 10.1088/1367-2630/14/4/043031 CrossRefGoogle Scholar
  3. Hofmann, H.F.: Derivation of quantum mechanics from a single fundamental modification of the relations between physical properties. Phys. Rev. A 89, 42115 (2014). doi: 10.1103/PhysRevA.89.042115 MATHCrossRefGoogle Scholar
  4. Iinuma, M., Suzuki, Y., Taguchi, G., Kadoya, Y., Hofmann, H.F.: Weak measurement of photon polarization by back-action-induced path interference. New J. Phys. 13, 33041 (2011). doi: 10.1088/1367-2630/13/3/033041 CrossRefGoogle Scholar
  5. Suzuki, Y., Iinuma, M., Hofmann, H.F.: Violation of Leggett-Garg inequalities in quantum measurements with variable resolution and back-action. New J. Phys. 14, 103022 (2012). doi: 10.1088/1367-2630/14/10/103022 CrossRefGoogle Scholar
  6. Kirkwood, J.G.: Quantum statistics of almost classical assemblies, Phys. Rev. 44, 31–37 (1933)Google Scholar
  7. Dirac, P.A.M.: On the analogy between classical and quantum mechanics. Rev. Mod. Phys. 17, 195–199 (1945)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Japan 2016

Authors and Affiliations

  • Masataka Iinuma
    • 1
  • Yutaro Suzuki
    • 1
  • Holger F. Hofmann
    • 1
    • 2
  1. 1.Graduate School of Advanced Sciences of MatterHiroshima UniversityHigashi-hiroshimaJapan
  2. 2.JST, CrestChiyoda-ku, TokyoJapan

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