Foundations of Physics

, Volume 47, Issue 8, pp 1009–1030 | Cite as

Negativity Bounds for Weyl–Heisenberg Quasiprobability Representations

  • John B. DeBrota
  • Christopher A. FuchsEmail author


The appearance of negative terms in quasiprobability representations of quantum theory is known to be inevitable, and, due to its equivalence with the onset of contextuality, of central interest in quantum computation and information. Until recently, however, nothing has been known about how much negativity is necessary in a quasiprobability representation. Zhu (Phys Rev Lett 117 (12):120404, 2016) proved that the upper and lower bounds with respect to one type of negativity measure are saturated by quasiprobability representations which are in one-to-one correspondence with the elusive symmetric informationally complete quantum measurements (SICs). We define a family of negativity measures which includes Zhu’s as a special case and consider another member of the family which we call “sum negativity.” We prove a sufficient condition for local maxima in sum negativity and find exact global maxima in dimensions 3 and 4. Notably, we find that Zhu’s result on the SICs does not generally extend to sum negativity, although the analogous result does hold in dimension 4. Finally, the Hoggar lines in dimension 8 make an appearance in a conjecture on sum negativity.


Quasiprobability representations Symmetric informationally complete measurements Frame decompositions Negativity measures 



The authors would like to thank Marcus Appleby, Blake Stacey, and Huangjun Zhu for helpful discussions and suggestions. This research was supported by the Foundational Questions Institute Fund on the Physics of the Observer (Grant FQXi-RFP-1612), a Donor Advised Fund at the Silicon Valley Community Foundation.


  1. 1.
    Zhu, H.: Quasiprobability representations of quantum mechanics with minimal negativity. Phys. Rev. Lett. 117 (12), 120404 (2016). arXiv:1604.06974 [quant-ph]
  2. 2.
    Weinberg, S.: The trouble with quantum mechanics. N. Y. Rev. Books 64(1). (2017)
  3. 3.
    Fuchs, C.A., Stacey, B.C.: QBist Quantum Mechanics: Quantum Theory as a Hero’s Handbook. Preprint (2016). arXiv:1612.07308 [quant-ph]
  4. 4.
    Fuchs, C.A.: QBism, the Perimeter of Quantum Bayesianism. Preprint (2010). arXiv:1003.5209 [quant-ph]
  5. 5.
    Fuchs, C.A., Schack, R.: Quantum-Bayesian Coherence. Rev. Mod. Phys. 85, 1693 (2013). arXiv:0906.2187 [quant-ph]
  6. 6.
    de Finetti, B.: Theory of Probability. Wiley, Chichester (1990)zbMATHGoogle Scholar
  7. 7.
    Lad, F.: Operational Subjective Statistical Methods: A Mathematical, Philosophical, and Historical Introduction. Wiley-Interscience, New York (1996)zbMATHGoogle Scholar
  8. 8.
    Bernardo, J.M., Smith, A.F.M.: Bayesian Theory. Wiley, Chichester (1994)CrossRefzbMATHGoogle Scholar
  9. 9.
    Caves, C.M., Fuchs, C.A., Schack, R.: Unknown quantum states: the quantum de Finetti representation. J. Math. Phys. 43, 4537 (2002)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Spekkens, R.W.: Negativity and contextuality are equivalent notions of nonclassicality. Phys. Rev. Lett. 101, 020401 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ferrie, C., Emerson, J.: Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations. J. Phys. A 41, 352001 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Appleby, D.M., Dang, H.B., Fuchs, C.A.: Symmetric Informationally-Complete Quantum States as Analogues to Orthonormal Bases and Minimum Uncertainty States. Entropy 16(3), 1484–92 (2014). arXiv:0707.2071 [quant-ph]ADSCrossRefGoogle Scholar
  13. 13.
    Appleby, D.M.: Symmetric informationally complete measurements of arbitrary rank. Opt. Spectrosc. 103, 416–428 (2007)ADSCrossRefGoogle Scholar
  14. 14.
    Zauner, G.: Quantendesigns. Grundzüge einer nichtkommutativen Designtheorie. PhD Thesis, University of Vienna, 1999. Published in English translation: Zauner, G.: Quantum designs: foundations of a noncommutative design theory. Int. J. Quantum Inf. 9, 445–508 (2011)
  15. 15.
    Caves, C.M.: Symmetric informationally complete POVMs. (2002)
  16. 16.
    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
  17. 17.
    Fuchs, C.A., Hoang, M.C., Stacey, B.C.: The SIC question: history and state of play. Preprint (2017). arXiv:1703.07901 [quant-ph]
  18. 18.
    Scott, A.J.: SICs: Extending the list of solutions. Preprint (2017). arXiv:1703.03993 [quant-ph]
  19. 19.
    Appleby, M., Chien, T.Y., Flammia, S., Waldron, S.: Constructing exact symmetric informationally complete measurements from numerical solutions. Preprint (2017). arXiv:1703.05981 [quant-ph]
  20. 20.
    Appleby, M., Fuchs, C.A., Stacey, B.C., Zhu, H.: Introducing the Qplex: a novel arena for quantum theory. arXiv:1612.03234 [quant-ph]
  21. 21.
    Khrennikov, A.: External observer reflections on QBism. Preprint (2016). arXiv:1512.07195 [quant-ph]
  22. 22.
    Khrennikov, A.: Towards better understanding QBism. Found. Sci. 18, 1–15 (2017)Google Scholar
  23. 23.
    Schleich, W.P.: Quantum Optics in Phase Space. Wiley-VCH, Berlin (2001)CrossRefzbMATHGoogle Scholar
  24. 24.
    Prugovecki, E.: Simultaneous measurement of several observables. Found. Phys. 3, 3–18 (1973)ADSCrossRefGoogle Scholar
  25. 25.
    Feynman, R.P.: Negative probability. In: Hiley, B.J., Peat, F.D. (eds.) Quantum Implications, Essays in Honour of David Bohm, pp. 235–246. Routledge and Kegan Paul, London (1987)Google Scholar
  26. 26.
    Khrennikov, A.: On the physical interpretation of negative probabilities in Prugovecki’s empirical theory of measurement. Can. J. Phys. 75, 291–298 (1997)ADSCrossRefGoogle Scholar
  27. 27.
    Khrennikov, A.: Interpretations of Probability, 2nd edn. De Gruyter, Berlin (2009)CrossRefzbMATHGoogle Scholar
  28. 28.
    Burgin, M.: Interpretations of negative probabilities. Preprint (2010). arXiv:1008.1287 []
  29. 29.
    Ferrie, C., Morris, R., Emerson, J.: Necessity of negativity in quantum theory. Phys. Rev. A 82, 044103 (2010)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Wootters, W.K.: A Wigner-function formulation of finite-state quantum mechanics. Ann. Phys. 176, 1 (1987)ADSMathSciNetCrossRefGoogle Scholar
  31. 31.
    Gibbons, K.S., Hoffman, M.J., Wootters, W.K.: Discrete phase space based on finite fields. Phys. Rev. A 70, 062101 (2004)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ferrie, C., Emerson, J.: Framed Hilbert space: hanging the quasi-probability pictures of quantum theory. N. J. Phys. 11, 063040 (2009)CrossRefGoogle Scholar
  33. 33.
    Ferrie, C.: Quasi-probability representations of quantum theory with applications to quantum information science. Rep. Prog. Phys. 74, 116001 (2011)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    Veitch, V., Ferrie, C., Gross, D., Emerson, J.: Negative quasi-probability as a resource for quantum computation. N. J. Phys. 14, 113011 (2012)CrossRefGoogle Scholar
  35. 35.
    Howard, M., Wallman, J., Veitch, V., Emerson, J.: Contextuality supplies the ‘magic’ for quantum computation. Nature 510, 351–355 (2014)ADSGoogle Scholar
  36. 36.
    Pashayan, H., Wallman, J.J., Bartlett, S.D.: Estimating outcome probabilities of quantum circuits using quasiprobabilities. Phys. Rev. Lett. 115, 070501 (2015)ADSCrossRefGoogle Scholar
  37. 37.
    Veitch, V., Mousavian, S.A.H., Gottesman, D., Emerson, J.: The resource theory of stabilizer quantum computation. N. J. Phys. 16, 013009 (2014)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Bhatia, R.: Matrix Analysis. Graduate Texts in Mathematics. Springer, New York (1997)Google Scholar
  39. 39.
    Bertsekas, D.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)zbMATHGoogle Scholar
  40. 40.
    Tabia, G.N.M., Appleby, D.M.: Exploring the geometry of qutrit state space using symmetric informationally complete probabilities. Phys. Rev. A 88, 012131 (2013)ADSCrossRefGoogle Scholar
  41. 41.
    Wootters, W.K., Fields, B.D.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–381 (1989)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Szymusiak, A.: Maximally informative ensembles for SIC-POVMs in dimension 3. J. Phys. A 47, 445301 (2014)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Stacey, B.C.: Geometric and information-theoretic properties of the Hoggar lines. Preprint (2016). arXiv:1609.03075 [quant-ph]
  44. 44.
    Appleby, D.M., Flammia, S., McConnell, G., Yard, J.: Generating ray class fields of real quadratic fields via complex equiangular lines. Preprint (2016). arXiv:1604.06098 [math.NT]
  45. 45.
    Appleby, M., Flammia, S., McConnell, G., Yard, J.: SICs and algebraic number theory. Preprint (2017). arXiv:1701.05200 [quant-ph]
  46. 46.
    Bengtsson, I.: The number behind the simplest SIC-POVM. Preprint (2016). arXiv:1611.09087 [quant-ph]
  47. 47.
    Ramirez, C., Sanchez, R., Kreinovich, V., Argaez, M.: \(\sqrt{x^2+c}\) is the most computationally efficient smooth approximation to \(|x|.\). J. Uncertain Syst. 8, 205–210 (2014)Google Scholar
  48. 48.
    Pike, R.: Optimization for Engineering Systems. Van Nostrand Reinhold, New York (1986)Google Scholar
  49. 49.
    Szymusiak, A., Słomczyński, W.: Informational power of the Hoggar symmetric informationally complete positive operator-valued measure. Phys. Rev. A 94, 012122 (2016)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.University of Massachusetts BostonBostonUSA

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