Communications in Mathematical Physics

, Volume 357, Issue 3, pp 1253–1304 | Cite as

Measurement Uncertainty Relations for Discrete Observables: Relative Entropy Formulation

  • Alberto Barchielli
  • Matteo Gregoratti
  • Alessandro Toigo


We introduce a new information-theoretic formulation of quantum measurement uncertainty relations, based on the notion of relative entropy between measurement probabilities. In the case of a finite-dimensional system and for any approximate joint measurement of two target discrete observables, we define the entropic divergence as the maximal total loss of information occurring in the approximation at hand. For fixed target observables, we study the joint measurements minimizing the entropic divergence, and we prove the general properties of its minimum value. Such a minimum is our uncertainty lower bound: the total information lost by replacing the target observables with their optimal approximations, evaluated at the worst possible state. The bound turns out to be also an entropic incompatibility degree, that is, a good information-theoretic measure of incompatibility: indeed, it vanishes if and only if the target observables are compatible, it is state-independent, and it enjoys all the invariance properties which are desirable for such a measure. In this context, we point out the difference between general approximate joint measurements and sequential approximate joint measurements; to do this, we introduce a separate index for the tradeoff between the error of the first measurement and the disturbance of the second one. By exploiting the symmetry properties of the target observables, exact values, lower bounds and optimal approximations are evaluated in two different concrete examples: (1) a couple of spin-1/2 components (not necessarily orthogonal); (2) two Fourier conjugate mutually unbiased bases in prime power dimension. Finally, the entropic incompatibility degree straightforwardly generalizes to the case of many observables, still maintaining all its relevant properties; we explicitly compute it for three orthogonal spin-1/2 components.


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  1. 1.
    Ozawa M.: Position measuring interactions and the Heisenberg uncertainty principle. Phys. Lett. A 299, 1–7 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ozawa M.: Physical content of Heisenberg’s uncertainty relation: limitation and reformulation. Phys. Lett. A 318, 21–29 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ozawa M.: Universally valid reformulation of the Heisenberg uncertainty principle on noise and disturbance in measurement. Phys. Rev. A 67, 042105 (2003)ADSCrossRefGoogle Scholar
  4. 4.
    Ozawa M.: Uncertainty relations for joint measurements of noncommuting observables. Phys. Lett. A 320, 367–374 (2004)ADSMathSciNetCrossRefGoogle Scholar
  5. 5.
    Ozawa M.: Heisenberg’s original derivation of the uncertainty principle and its universally valid reformulations. Curr. Sci. 109, 2006–2016 (2015)CrossRefGoogle Scholar
  6. 6.
    Werner R.F.: The uncertainty relation for joint measurement of position and momentum. Quantum Inf. Comput. 4, 546–562 (2004)ADSMathSciNetMATHGoogle Scholar
  7. 7.
    Busch P., Lahti P., Werner R.F.: Measurement uncertainty relations. J. Math. Phys. 55, 042111 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Busch P., Lahti P., Werner R.F.: Quantum root-mean-square error and measurement uncertainty relations. Rev. Mod. Phys. 86, 1261–1281 (2014)ADSCrossRefGoogle Scholar
  9. 9.
    Busch P., Lahti P., Werner R.F.: Heisenberg uncertainty for qubit measurements. Phys. Rev. A 89, 012129 (2014)ADSCrossRefGoogle Scholar
  10. 10.
    Werner R.F.: Uncertainty relations for general phase spaces. Front. Phys. 11, 110305 (2016)CrossRefGoogle Scholar
  11. 11.
    Busch P., Heinonen T., Lahti P.: Heisenberg’s uncertainty principle. Phys. Rep. 452, 155–176 (2007)ADSCrossRefGoogle Scholar
  12. 12.
    Dammeier L., Schwonnek R., Werner R.F.: Uncertainty relations for angular momentum. New J. Phys. 17, 093046 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Abbott A.A., Alzieu P.-L., Hall M.J.W., Branciard C.: Tight state-independent uncertainty relations for qubits. Mathematics 4, 8 (2016)CrossRefMATHGoogle Scholar
  14. 14.
    Heisenberg W.: Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik. Zeitschr. Phys. 43, 172–198 (1927)ADSCrossRefMATHGoogle Scholar
  15. 15.
    Robertson H.: The uncertainty principle. Phys. Rev. 34, 163–164 (1929)ADSCrossRefGoogle Scholar
  16. 16.
    Kraus K.: Complementary observables and uncertainty relations. Phys. Rev. D 35, 3070–3075 (1987)ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    Maassen H., Uffink J.B.M.: Generalized entropic uncertainty relations. Phys. Rev. Lett. 60, 1103–1106 (1988)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Krishna M., Parthasarathy K.R.: An entropic uncertainty principle for quantum measurements. Sankhya Indian J. Stat. 64, 842–851 (2002)MathSciNetMATHGoogle Scholar
  19. 19.
    Wehner S., Winter A.: Entropic uncertainty relations—a survey. New J. Phys. 12, 025009 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kaniewski J., Tomamichel M., Wehner S.: Entropic uncertainty from effective anticommutators. Phys. Rev. A 90, 012332 (2014)ADSCrossRefGoogle Scholar
  21. 21.
    Abdelkhalek K., Schwonnek R., Maassen H., Furrer F., Duhme J., Raynal P., Englert B-G., Werner R.F.: Optimality of entropic uncertainty relations. Int. J. Quantum Inf. 13, 1550045 (2015)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Coles P.J., Berta M., Tomamichel M., Whener S.: Entropic uncertainty relations and their applications. Rev. Mod. Phys. 89, 015002 (2017)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Holevo A.S.: Statistical Structure of Quantum Theory, Lecture Notes in Physics. Springer, Berlin (2001)CrossRefGoogle Scholar
  24. 24.
    Busch P., Grabowski M., Lahti P.: Operational Quantum Physics. Springer, Berlin (1997)MATHGoogle Scholar
  25. 25.
    Busch P., Lahti P., Pellonpää J.-P., Ylinen K.: Quantum Measurement. Springer, Berlin (2016)CrossRefMATHGoogle Scholar
  26. 26.
    Busch P., Heinosaari T.: Approximate joint measurements of qubit observables. Quantum Inf. Comp. 8, 797–818 (2008)MathSciNetMATHGoogle Scholar
  27. 27.
    Heinosaari T., Wolf M.M.: Nondisturbing quantum measurements. J. Math. Phys. 51, 092201 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Heinosaari T., Miyadera T.: Universality of sequential quantum measurements. Phys. Rev. 91, 022110 (2015)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    Appleby D.M.: Error principle. Int. J. Theoret. Phys. 37, 2557–2572 (1998)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Appleby D.M.: Quantum Errors and Disturbances: Response to Busch, Lahti and Werner, Entropy 18, 174 (2016)CrossRefGoogle Scholar
  31. 31.
    Buscemi F., Hall M.J.W., Ozawa M., Wilde M.M.: Noise and disturbance in quantum measurements: an information-theoretic approach. Phys. Rev. Lett. 112, 050401 (2014)ADSCrossRefGoogle Scholar
  32. 32.
    Abbot A.A., Branciard C.: Noise and disturbance of Qubit measurements: An information-theoretic characterisation. Phys. Rev. A 94, 062110 (2016)ADSCrossRefGoogle Scholar
  33. 33.
    Coles P.J., Furrer F.: State-dependent approach to entropic measurement–disturbance relations. Phys. Lett. A 379, 105–112 (2015)ADSCrossRefMATHGoogle Scholar
  34. 34.
    Barchielli A., Gregoratti M., Toigo A.: Measurement uncertainty relations for position and momentum: Relative entropy formulation. Entropy 19, 301 (2017)ADSMathSciNetCrossRefGoogle Scholar
  35. 35.
    Burnham K.P., Anderson D.R.: Model Selection and Multi-Model Inference. 2nd edn. Springer, New York (2002)MATHGoogle Scholar
  36. 36.
    Cover T.M., Thomas J.A.: Elements of Information Theory. 2nd edn. Wiley, Hoboken (2006)MATHGoogle Scholar
  37. 37.
    Ohya M., Petz D.: Quantum Entropy and Its Use. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  38. 38.
    Barchielli A., Lupieri G.: Instruments and channels in quantum information theory. Opt. Spectrosc. 99, 425–432 (2005)ADSCrossRefMATHGoogle Scholar
  39. 39.
    Barchielli A., Lupieri G.: Quantum measurements and entropic bounds on information transmission. Quantum Inf. Comput. 6, 16–45 (2006)MathSciNetMATHGoogle Scholar
  40. 40.
    Barchielli A., Lupieri G.: Instruments and mutual entropies in quantum information. Banach Center Publ. 73, 65–80 (2006)MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Maccone L.: Entropic information-disturbance tradeoff. Europhys. Lett. 77, 40002 (2007)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    Davies E.B.: Quantum Theory of Open Systems. Academic, London (1976)MATHGoogle Scholar
  43. 43.
    Holevo A.S.: Quantum Systems, Channels, Information. de Gruiter, Berlin (2012)CrossRefMATHGoogle Scholar
  44. 44.
    Heinosaari T., Ziman M.: The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement. Cambridge University Press, Cambridge (2012)MATHGoogle Scholar
  45. 45.
    Heinosaari T., Miyadera T., Ziman M.: An invitation to quantum incompatibility. J. Phys. A Math. Theor. 49, 123001 (2016)ADSMathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Topsøe F.: Basic concepts, identities and inequalities—the toolkit of information theory. Entropy 3, 162–190 (2001)ADSMathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Pedersen G.K.: Analysis Now. Springer, New York (1989)CrossRefMATHGoogle Scholar
  48. 48.
    Busch P., Heinosaari T., Schultz J., Stevens N.: Comparing the degrees of incompatibility inherent in probabilistic physical theories. Europhys. Lett. 103, 10002 (2013)ADSCrossRefGoogle Scholar
  49. 49.
    Heinosaari T., Schultz J., Toigo A., Ziman M.: Maximally incompatible quantum observables. Phys. Lett. A 378, 1695–1699 (2014)MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Keyl M., Werner R.F.: Optimal cloning of pure states, testing single clones. J. Math. Phys. 40, 3283–3299 (1999)ADSMathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Werner R.F.: Optimal cloning of pure states. Phys. Rev. A 58, 1827–1832 (1998)ADSCrossRefGoogle Scholar
  52. 52.
    Lahti P.: Coexistence and joint measurability in quantum mechanics. Int. J. Theor. Phys. 42, 893–906 (2003)MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Wootters W.K., Fields D.B.: Optimal state-determination by mutually unbiased measurements. Ann. Phys. 191, 363–381 (1989)ADSMathSciNetCrossRefGoogle Scholar
  54. 54.
    Durt T., Englert B.-G., Bengtsson I., Zyczkowsky K.: On mutually unbiased bases. Int. J. Quantum Inf. 8, 535–640 (2010)CrossRefMATHGoogle Scholar
  55. 55.
    Bandyopadhyay S., Boykin P.O., Roychowdhury V., Vatan F.: A new proof for the existence of mutually unbiased bases. Algorithmica 34, 512–528 (2002)MathSciNetCrossRefMATHGoogle Scholar
  56. 56.
    Appleby, D.M.: Properties of the extended Clifford group with applications to SIC-POVMs and MUBs. arXiv:0909.5233
  57. 57.
    Carmeli C., Schultz J., Toigo A.: Covariant mutually unbiased bases. Rev. Math. Phys. 28, 1650009 (2016)MathSciNetCrossRefMATHGoogle Scholar
  58. 58.
    Lang, S.: Algebra, 3rd edition, Graduate Texts in Mathematics, 211 Springer, New York (2002)Google Scholar
  59. 59.
    Carmeli C., Heinosaari T., Toigo A.: Informationally complete joint measurements on finite quantum systems. Phys. Rev. A 85, 012109 (2012)ADSCrossRefGoogle Scholar
  60. 60.
    Heinosaari T., Jivulescu M.A., Reitzner D., Ziman M.: Approximating incompatible von Neumann measurements simultaneously. Phys. Rev. A 82, 032328 (2010)ADSCrossRefGoogle Scholar
  61. 61.
    Berta M., Christandl M., Colbeck R., Renes J.M., Renner R.: The uncertainty principle in the presence of quantum memory. Nat. Phys. 6, 659 (2010)CrossRefGoogle Scholar
  62. 62.
    Frank R. L., Lieb E.H.: Extended quantum conditional entropy and quantum uncertainty inequalities. Commun. Math. Phys. 323, 487–495 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  63. 63.
    Weyl H.: Symmetry. Princeton University Press, Princeton (1952)CrossRefMATHGoogle Scholar
  64. 64.
    Carmeli C., Heinosaari T., Toigo A.: Sequential measurements of conjugate observables. J. Phys. A Math. Theor. 44, 285304 (2011)MathSciNetCrossRefMATHGoogle Scholar
  65. 65.
    Carmeli C., Heinosaari T., Schultz J., Toigo A.: Tasks and premises in quantum state determination. J. Phys. A Math. Theor. 47, 075302 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanItaly
  2. 2.Istituto Nazionale di Alta Matematica (INDAM-GNAMPA)RomeItaly
  3. 3.Istituto Nazionale di Fisica Nucleare (INFN), Sezione di MilanoMilanItaly

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