Current Science

Open Access Open Access  Restricted Access Subscription Access

Uncertainty Trade-Off and Disturbance Trade-Off for Quantum Measurements


Affiliations
1 Centre for Policy Studies, Mylapore, Chennai 600 004, India
2 Department of Physics, Indian Institute of Technology Madras, Chennai 600 036, India
 

An important non-classical feature of quantum measurements is the celebrated uncertainty trade-off, namely that the uncertainties in the outcomes of measurements performed on distinct yet identically prepared ensembles of systems cannot all be made arbitrarily small. Recently, we have shown that quantum measurements also exhibit another non-classical feature of disturbance trade-off namely, that the disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. In this article, we review the known results on uncertainty trade-off and disturbance trade-off for projective and non-projective measurements.

Keywords

Disturbance, Entropy, Projective Measurement, Uncertainty.
User
Notifications
Font Size

  • Heisenberg, W., Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik. Z. Phys., 1927, 43(3-4), 172–198.

  • Robertson, H. P., The uncertainty principle. Phys. Rev., 1929, 34(1), 163–164.

  • Schrödinger, E., Zum Heisenbergschen unschärferprinzip. Sitzung. Preuss. Akad. Wiss., Physik.-Math. Klasse, 1930, 14, 296–303.

  • Wehner, S. and Winter, A., Entropic uncertainty relations – a survey. New J. Phys., 2010, 12(2), 025009-1 to 025009-22.

  • Mandayam, P. and Srinivas, M. D., Disturbance trade-off principle for quantum measurements. Phys. Rev. A, 2014, 90(6), 062128-1 to 062128-8.

  • Nielsen, M. A. and Chuang, I. L., Quantum Computation and Quantum Information, Cambridge University Press, 2000.

  • Mandayam, P. and Srinivas, M. D., Measures of disturbance and incompatibility for quantum measurements. Phys. Rev. A, 2014, 89(6), 062112-1 to 062112-9.

  • Tsallis, C., Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys., 1988, 52(1–2), 479–487.

  • Deutsch, D., Uncertainty in quantum measurements. Phys. Rev. Lett., 1983, 50(9), 631–633.

  • Białynicki-Birula, I. and Mycielski, J., Uncertainty relations for information entropy in wave mechanics. Comm. Math. Phys., 1975, 44(2), 129–132.

  • Kraus, K., Complementary observables and uncertainty relations. Phys. Rev. D, 1987, 35(10), 3070–3075.

  • Maassen, H. and Uffink, J. B. M., Generalized entropic uncertainty relations. Phys. Rev. Lett., 1988, 60(12), 1103–1106.

  • Ivanovic, I. D., An inequality for the sum of entropies of unbiased quantum measurements. J. Phys. A, 1992, 25(7), L363–L364.

  • Sanchez, J., Entropic uncertainty and certainty relations for complementary observables. Phys. Lett. A, 1993, 173(3), 233–239.

  • Sanchez-Ruiz, J., Optimal entropic uncertainty relation in two-dimensional Hilbert space. Phys. Lett. A, 1998, 244(4), 189– 195.

  • Białynicki-Birula, I., Formulation of the uncertainty relations in terms of the Rényi entropies. Phys. Rev. A, 2006, 74(5), 052101-1 to 052101-6.

  • Wehner, S. and Winter, A., Higher entropic uncertainty relations for anti-commuting observables. J. Math. Phys., 2008, 49(6), 062105-1 to 062105-11.

  • Rajagopal, A. K., The Sobolev inequality and the Tsallis entropic uncertainty relation. Phys. Lett. A, 1995, 205(1), 32–36.

  • Majerník, V. and Majerníková, E., Rep. Math. Phys., 2001, 47(3), 381–392.

  • Rastegin, A. E., Notes on entropic uncertainty relations beyond the scope of Riesz’s theorem. Int. J. Theor. Phys., 2012, 51(4), 1300–1315.

  • Rastegin, A. E., Uncertainty and certainty relations for complementary qubit observables in terms of Tsallis’ entropies. Quant. Inf. Process, 2013, 12(9), 2947–2963.

  • Krishna, M. and Parthasarathy, K. R., An entropic uncertainty principle for quantum measurements. Sankhyā: Indian J. Stat., 2002, 64(3), 842–851.

  • Srinivas, M. D., Optimal entropic uncertainty relation for successive measurements in quantum information theory. Pramana, 2003, 60(6), 1137–1152.

  • Ghirardi, G., Marinatto, L. and Romano, R., An optimal entropic uncertainty relation in a two-dimensional Hilbert space. Phys. Lett. A, 2003, 317(1–2), 32–36.

  • Heinosaari, T. and Ziman, M., The Mathematical Language of Quantum Theory: From Uncertainty to Entanglement, Cambridge University Press, Cambridge, 2012.

  • Srinivas, M. D., Collapse postulate for observables with continuous spectra. Comm. Math. Phys., 1980, 71(2), 131–158.

  • Srinivas, M. D., Measurements and Quantum Probabilities, Universities Press, 2001.

  • Ozawa, M., Quantum measuring processes of continuous observables. J. Math. Phys., 1984, 25(1), 79–87.

  • Umegaki, H., Conditional expectation in an operator algebra. Tohoku Math. J., 1954, 6, 177; Nakamura, M. and Turumuru, T., Tohoku Math. J., 1954, 6, 182; Tomiyama, J., Proc. Jpn. Acad., 1957, 33, 608.

  • Arveson, W. B., Am. J. Math., 1967, 89, 572; Stormer, E., In Foundations of Quantum Mechanics and Ordered Linear Spaces (eds Hartkamper, A. and Neumann, H.), Springer, Berlin, 1976.

  • Davies, E. B., Quantum Theory of Open Systems, Academic Press, 1976.

  • Busch, P. and Singh, J., Lüders theorem for unsharp quantum measurements. Phys. Lett. A, 1998, 249(1), 10–12.

  • Arias, A., Gheondea, A. and Gudder, S., Fixed points of quantum operations. J. Math. Phys., 2002, 43, 5872–5881.

  • Wu, S., Yu, S. and Mølmer, K., Entropic uncertainty relation for mutually unbiased bases. Phys. Rev. A, 2009, 79(2), 022104-1 to 022104-5.

  • Bosyk, G. M., Portesi, M. and Plastino, A., Collision entropy and optimal uncertainty. Phys. Rev. A, 2012, 85(1), 012108-1 to 012108-6.


Abstract Views: 431

PDF Views: 136




  • Uncertainty Trade-Off and Disturbance Trade-Off for Quantum Measurements

Abstract Views: 431  |  PDF Views: 136

Authors

M. D. Srinivas
Centre for Policy Studies, Mylapore, Chennai 600 004, India
Prabha Mandayam
Department of Physics, Indian Institute of Technology Madras, Chennai 600 036, India

Abstract


An important non-classical feature of quantum measurements is the celebrated uncertainty trade-off, namely that the uncertainties in the outcomes of measurements performed on distinct yet identically prepared ensembles of systems cannot all be made arbitrarily small. Recently, we have shown that quantum measurements also exhibit another non-classical feature of disturbance trade-off namely, that the disturbances associated with measurements performed on distinct yet identically prepared ensembles of systems in a pure state cannot all be made arbitrarily small. In this article, we review the known results on uncertainty trade-off and disturbance trade-off for projective and non-projective measurements.

Keywords


Disturbance, Entropy, Projective Measurement, Uncertainty.

References





DOI: https://doi.org/10.18520/cs%2Fv109%2Fi11%2F2044-2051