Quantum Information Processing

, Volume 15, Issue 8, pp 3421–3441 | Cite as

Influence of environmental noise on the weak value amplification

  • Xuannmin Zhu
  • Yu-Xiang Zhang


Quantum systems are always disturbed by environmental noise. We have investigated the influence of the environmental noise on the amplification in weak measurements. Three typical quantum noise processes are discussed in this article. The maximum expectation values of the observables of the measuring device decrease sharply with the strength of the depolarizing and phase damping channels, while the amplification effect of weak measurement is immune to the amplitude damping noise. To obtain significantly amplified signals, we must ensure that the preselection quantum systems are kept away from the depolarizing and phase damping processes.


Weak measurement Weak value Quantum noise Amplification 



This work was financially supported by the National Natural Science Foundation of China (Grants Nos. 11305118, and 11475084), and the Fundamental Research Funds for the Central Universities.


  1. 1.
    Aharonov, Y., Albert, D.Z., Vaidman, L.: How the result of a measurement of a component of the spin of a spin 1/2 particle can turn out to be 100. Phys. Rev. Lett. 60, 1351–1354 (1988)ADSCrossRefGoogle Scholar
  2. 2.
    Ritchie, N.W.M., Story, J.G., Hulet, R.G.: Realization of a measurement of a “weak value”. Phys. Rev. Lett. 66, 1107–1110 (1991)ADSCrossRefGoogle Scholar
  3. 3.
    Pryde, G.J., O’Brien, J.L., White, A.G., Ralph, T.C., Wiseman, H.M.: Measurement of quantum weak values of photon polarization. Phys. Rev. Lett. 94, 220405 (2005)ADSCrossRefGoogle Scholar
  4. 4.
    Hosten, O., Kwiat, P.: Observation of the spin Hall effect of light via weak measurements. Science 319, 787–790 (2008)ADSCrossRefGoogle Scholar
  5. 5.
    Dixon, P.B., Starling, D.J., Jordan, A.N., Howell, J.C.: Ultrasensitive beam deflection measurement via interferometric weak value amplification. Phys. Rev. Lett. 102, 173601 (2009)ADSCrossRefGoogle Scholar
  6. 6.
    Brunner, N., Simon, C.: Measuring small longitudinal phase shifts: weak measurements or standard interferometry? Phys. Rev. Lett. 105, 010405 (2010)ADSCrossRefGoogle Scholar
  7. 7.
    Zilberberg, O., Romito, A., Gefen, Y.: Charge sensing amplification via weak values measurement. Phys. Rev. Lett. 106, 080405 (2011)ADSCrossRefGoogle Scholar
  8. 8.
    Feizpour, A., Xing, X., Steinberg, A.M.: Amplifying single-photon nonlinearity using weak measurements. Phys. Rev. Lett. 107, 133603 (2011)ADSCrossRefGoogle Scholar
  9. 9.
    Strübi, G., Bruder, C.: Measuring ultrasmall time delays of light by joint weak measurements. Phys. Rev. Lett. 110, 083605 (2013)ADSCrossRefGoogle Scholar
  10. 10.
    Dressel, J., Malik, M., Miatto, F.M., Jordan, A.N., Boyd, R.W.: Colloquium: understanding quantum weak values: basics and applications. Rev. Mod. Phys. 86, 307–316 (2014)ADSCrossRefGoogle Scholar
  11. 11.
    Chen, S., Zhou, X., Mi, C., Luo, H., Wen, S.: Modified weak measurements for the detection of the photonic spin Hall effect. Phys. Rev. A 91, 105 (2015)Google Scholar
  12. 12.
    Nishizawa, A.: Weak-value amplification beyond the standard quantum limit in position measurements. Phys. Rev. A 92, 032123 (2015)ADSCrossRefGoogle Scholar
  13. 13.
    Pang, S., Brun, T.A.: Improving the precision of weak measurements by postselection measurement. Phys. Rev. Lett. 115, 120401 (2015)ADSCrossRefGoogle Scholar
  14. 14.
    Lundeen, J.S., Sutherland, B., Patel, A., Stewart, C., Bamber, C.: Direct measurement of the quantum wavefunction. Nature 474, 188–191 (2011)CrossRefGoogle Scholar
  15. 15.
    Shpitalnik, V., Gefen, Y., Romito, A.: Tomography of many-body weak values: Mach–Zehnder interferometry. Phys. Rev. Lett. 101, 226802 (2008)ADSCrossRefGoogle Scholar
  16. 16.
    Hofmann, H.F.: Complete characterization of post-selected quantum statistics using weak measurement tomography. Phys. Rev. A 81, 012103 (2010)ADSCrossRefGoogle Scholar
  17. 17.
    Wu, S.: State tomography via weak measurements. Sci. Rep. 3, 1193 (2013)ADSGoogle Scholar
  18. 18.
    Salvail, J.Z., Agnew, M., Johnson, A.S., Bolduc, E., Leach, J., Boyd, R.W.: Full characterization of polarization states of light via direct measurement. Nat. Photon. 7, 316–321 (2013)ADSCrossRefGoogle Scholar
  19. 19.
    Maccone, L., Rusconi, C.C.: State estimation: a comparison between direct state measurement and tomography. Phys. Rev. A 89, 022122 (2014)ADSCrossRefGoogle Scholar
  20. 20.
    Zou, P., Zhang, Z.-M., Song, W.: Direct measurement of general quantum states using strong measurement. Phys. Rev. A 91, 052109 (2015)ADSCrossRefGoogle Scholar
  21. 21.
    Hardy, L.: Quantum mechanics, local realistic theories, and Lorentz-invariant realistic theories. Phys. Rev. Lett. 68, 2981–2984 (1992)ADSMathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Aharonov, Y., Botero, A., Popescu, S., Reznik, B., Tollaksen, J.: Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values. Phys. Lett. A 301, 130–138 (2002)ADSMathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Lundeen, J.S., Steinberg, A.M.: Experimental joint weak measurement on a photon pair as a probe of Hardy’s Paradox. Phys. Rev. Lett. 102, 020404 (2009)ADSCrossRefGoogle Scholar
  24. 24.
    Yokota, K., Yamamoto, T., Koashi, M., Imotoar, N.: Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair. New J. Phys. 11, 033011 (2009)ADSCrossRefGoogle Scholar
  25. 25.
    Kocsis, S., Braverman, B., Ravets, S., Stevens, M.J., Mirin, R.P., Shalm, L.K., Steinberg, M.: Observing the average trajectories of single photons in a two-slit interferometer. Science 332, 1170–1173 (2011)ADSCrossRefGoogle Scholar
  26. 26.
    Cho, A.: Furtive approach rolls back the limits of quantum uncertainty. Science 333, 690–693 (2011)ADSMathSciNetCrossRefGoogle Scholar
  27. 27.
    Howland, G.A., Schneeloch, J., Lum, D.J., Howell, J.C.: Simultaneous measurement of complementary observables with compressive sensing. Phys. Rev. Lett. 112, 253602 (2014)ADSCrossRefGoogle Scholar
  28. 28.
    Ferrie, C., Combes, J.: How the result of a single coin toss can turn out to be 100 heads. Phys. Rev. Lett. 113, 120404 (2014)ADSCrossRefGoogle Scholar
  29. 29.
    Pati, A.K., Singh, U., Sinha, U.: Measuring non-Hermitian operators via weak values. Phys. Rev. A 92, 052120 (2015)ADSCrossRefGoogle Scholar
  30. 30.
    Jozsa, R.: Complex weak values in quantum measurement. Phys. Rev. A 76, 044103 (2007)ADSCrossRefGoogle Scholar
  31. 31.
    Taylor, J.R.: An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. University Science Books, Sausalito (1999)Google Scholar
  32. 32.
    Ferrie, C., Combes, J.: Weak value amplification is suboptimal for estimation and detection. Phys. Rev. Lett. 112, 040406 (2014)ADSCrossRefGoogle Scholar
  33. 33.
    Knee, G.C., Gauger, E.M.: When amplification with weak values fails to suppress technical noise. Phys. Rev. X 4, 011032 (2014)Google Scholar
  34. 34.
    Kedem, Y.: Using technical noise to increase the signal-to-noise ratio of measurements via imaginary weak values. Phys. Rev. A 85, 060102(R) (2012)ADSCrossRefGoogle Scholar
  35. 35.
    Pang, S., Dressel, J., Brun, T.A.: Entanglement-assisted weak value amplification. Phys. Rev. Lett. 113, 030401 (2014)ADSCrossRefGoogle Scholar
  36. 36.
    Jordan, A.N., Martínez-Rincón, J., Howell, J.C.: Technical advantages for weak-value amplification: when less is more. Phys. Rev. X 4, 011031 (2014)Google Scholar
  37. 37.
    Zhu, X., Zhang, Y., Pang, S., Qiao, C., Liu, Q., Wu, S.: Quantum measurements with preselection and postselection. Phys. Rev. A 84, 052111 (2011)ADSCrossRefGoogle Scholar
  38. 38.
    Wu, S., Li, Y.: Weak measurements beyond the Aharonov–Albert–Vaidman formalism. Phys. Rev. A 83, 052106 (2011)ADSCrossRefGoogle Scholar
  39. 39.
    Kofman, A.G., Ashhab, S., Nori, F.: Nonperturbative theory of weak pre- and post-selected measurements. Phys. Rep. 520, 43–133 (2012)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    Susa, Y., Shikano, Y., Hosoya, A.: Optimal probe wave function of weak-value amplification. Phys. Rev. A 85, 052110 (2012)ADSCrossRefGoogle Scholar
  41. 41.
    Pang, S., Brun, T.A., Wu, S., Chen, Z.-B.: Amplification limit of weak measurements: a variational approach. Phys. Rev. A 90, 012108 (2014)ADSCrossRefGoogle Scholar
  42. 42.
    Shikano, Y., Hosoya, A.: Weak values with decoherence. J. Phys. A Math. Theor. 43, 025304 (2010)ADSMathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Nielsen, M.A., Chuang, I.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)MATHGoogle Scholar
  44. 44.
    Chen, S., Zhou, X., Mi, C., Luo, H., Wen, S.: Modified weak measurements for the detection of the photonic spin Hall effect. Phys. Rev. A 91, 062105 (2015)ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.School of Physics and Optoelectronic EngineeringXidian UniversityXi’anChina
  2. 2.Hefei National Laboratory for Physical Sciences at MicroscaleUniversity of Science and Technology of ChinaHefeiChina

Personalised recommendations