Current Science

Open Access Open Access  Restricted Access Subscription Access

Unsharp Measurements and Joint Measurability


Affiliations
1 Raman Research Institute, Bengaluru 560 080, India
2 Department of Physics, Bangalore University, Bengaluru 560 056, India and Inspire Institute Inc., Alexandria, Virginia 22303, United States
3 Inspire Institute Inc., Alexandria, Virginia, 22303, USA; Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113, and Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India
 

We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is necessary to beat the uncertainty bound and hence predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to validate a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix - which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables - could be realized if and only if compatible POVMs are employed.

Keywords

Incompatibility, Joint Measurability, Positive Operator Valued Measures, Unsharp Measurements.
User
Notifications
Font Size

  • Ludwig, G., Versuch einer axiomatischen grundlegung der quantenmechanik und allgemeinerer physikalischer theorien. Z. Phys., 1964, 181(3), 233–260.

  • Busch, P. and Lahti, P. J., On various joint measurements of position and momentum observables in quantum theory. Phys. Rev. D, 1984, 29(8), 1634–1646.

  • Busch, P., Unsharp reality and joint measurements for spin observables. Phys. Rev. D, 1986, 33(8), 2253–2261.

  • Busch, P., Grabowski, M. and Lahti, P. J., Operational Quantum Physics, Springer, 1995.

  • Andersson, E., Barnett, S. M. and Aspect, A., Joint measurements of spin, operational locality, and uncertainty. Phys. Rev. A, 2005, 72(4), 042104-1 to 8.

  • Son, W., Andersson, E., Barnett, S. M. and Kim, M. S., Joint measurements and Bell inequalities. Phys. Rev. A, 2005, 72(5), 052116-1 to 7.

  • Heinosaari, T., Reitzner, D. and Stano, P., Notes on joint measurability of quantum observables. Found. Phys., 2008, 38(12), 1133–1147.

  • Wolf, M. M., Perez-Garcia, D. and Fernandez, C., Measurements incompatible in quantum theory cannot be measured jointly in any other no-signaling theory. Phys. Rev. Lett., 2009, 103(23), 230402-1 to 4.

  • Yu, S., Liu, N., Li, L. and Oh, C. H., Joint measurement of two unsharp observables of a qubit. Phys. Rev. A, 2010, 81(6), 0621161 to 7.

  • Liang, Y. C., Spekkens, R. W. and Wiseman, H. M., Specker’s parable of the overprotective seer: a road to contextuality, nonlocality and complementarity. Phys. Rep., 2011, 506(1), 1–39.

  • Reeb, D., Reitzner, D. and Wolf, M. M., Coexistence does not imply joint measurability. J. Phys. A, 2013, 46, 462002-1 to 4.

  • Kunjwal, R., Heunen, C. and Fritz, T., Quantum realization of arbitrary joint measurability structures. Phys. Rev. A, 2014, 89(5), 052126-1 to 5.

  • Quintino, M. T., Vértesi, T. and Brunner, N., Joint measurability, Einstein–Podolsky–Rosen steering, and Bell nonlocality. Phys. Rev. Lett., 2014, 113(16), 160402-1 to 5.

  • Uola, U., Moroder, T. and Gühne, O., Joint measurability of generalized measurements implies classicality. Phys. Rev. Lett., 2014, 113(16), 160403-1 to 5.

  • Karthik, H. S., Prabhu Tej, J., Usha Devi, A. R. and Rajagopal, A. K., Joint measurability and temporal steering. J. Opt. Soc. Am. B, 2015, 32(4), A34–A39.

  • Pusey, M., Verifying the quantumness of a channel with an untrusted device. J. Opt. Soc. Am. B, 2015, 32(4), A56–A63.

  • Heinosaari, T., Kiukas, J., Reitzner, D. and Schultz, J., Incompatibility breaking quantum channels. arXiv:1504.05768 [quant.ph].

  • Berta, M., Christandl, M., Colbeck, R., Renes, J. M. and Renner, R., The uncertainty principle in the presence of quantum memory. Nature Phys., 2010, 6(9), 659–662.

  • Karthik, H. S., Usha Devi, A. R. and Rajagopal, A. K., Joint measurability, steering, and entropic uncertainty. Phys. Rev. A, 2015, 91(1), 012115-1 to 6.

  • Kochen, S. and Specker, E. P., The problem of hidden variables in quantum mechanics. J. Math. Mech., 1967, 17(1), 59–87.

  • Kunjwal, R. and Ghosh, S., Minimal state-dependent proof of measurement contextuality for a qubit. Phys. Rev. A, 2014, 89(4), 042118-1 to 9.

  • Conventionally, quantum violation of Kochen–Specker noncontextuality20 (which employs three pairwise compatible, but not tripplewise compatible observables) could be illustrated for Hilbert spaces dimension d ≥ 3, when only projective measurements were employed.

  • Schrödinger, E., Discussion of probability relations between separated systems. Proc. Cambridge Philos. Soc., 1935, 31(4), 555–563.

  • Apart from Bell non-locality, yet another manifestation of nonlocality called ’steering’ has attracted attention recently. Though the striking feature that it is possible to remotely steer the state of a party (Alice) through local measurements on the state of another distant party (Bob) was noticed by Erwin Schrödinger23 as early as 1935. It was Reid (Demonstration of the Einstein–Podolsky– Rosen paradox. Phys. Rev. A, 1989, 40(2), 913–923.), who proposed the first experimentally testable critera of nonlocal steering to show that steering and Einstein–Podolsky–Rosen paradox are equivalent notions of non-locality. Further, Wiseman et al. (Steering, entanglement, nonlocality, and the Einstein–Podolsky–Rosen paradox. Phys. Rev. Lett., 2007, 98(14), 140402-1 to 4.) formally introduced the notion of local hidden state to investigate steering phenomena. They showed that steering constitutes a different class of non-locality that lies between entanglement and Bell nonlocality. For a comprehensive historical outline and for the modern formalism of quantum steering, see Cavalcanti, E. G., Jones, S. J., Wiseman, H. M. and Reid, M. D., Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox. Phys. Rev. A, 2009, 80(3), 032112-1 to 16.

  • Bell, J. S., On the Einstein–Podolski–Rosen paradox. Physics, 1964, 1(3), 195–200.

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

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

  • Wehner, S. and Winter, A., New J. Phys., 2010, 12(2), 025009-1 to 22.

  • Schneeloch, J., Broadbent, C. J., Walborn, S. P., Cavalcanti, E. G. and Howell, J. C., Einstein–Podolsky–Rosen steering inequalities from entropic uncertainty relations. Phys. Rev. A, 2013, 87(6), 062103-1 to 9.

  • The first entropic criteria of steering was formulated for position and momentum by Walborn, S. P., Salles, A., Gomes, R. M., Toscano, F. and Souto Ribeiro, P. H., Revealing hidden Einstein– Podolsky–Rosen nonlocality. Phys. Rev. Lett., 2011, 106(13), 130402-1 to 4. Entropic steering inequalities for discrete observables were developed more recently by Schneeloch et al.29.

  • Leggett, A. J. and Garg, A., Quantum mechanics versus macroscopic realism: is the flux there when nobody looks? Phys. Rev. Lett., 1985, 54(9), 857–860.

  • Fine, A., Hidden variables, joint probability, and the Bell inequalities. Phys. Rev. Lett., 1982, 48(5), 291–295; Joint distributions, quantum correlations, and commuting observables. J. Math. Phys., 1982, 23(7), 1306–1310.

  • Usha Devi, A. R., Karthik, H. S., Sudha and Rajgaopal, A. K., Macrorealism from entropic Leggett–Garg inequalities. Phys. Rev. A, 2013, 87(5), 052103-1 to 5.

  • Karthik, H. S., Katiyar, H., Shukla, A., Mahesh, T. S., Usha Devi, A. R. and Rajagopal, A. K., Inversion of moments to retrieve joint probabilities in quantum sequential measurements. Phys. Rev. A, 2013, 87(5), 052118-1 to 6.

  • Karthik, H. S., Usha Devi, A. R., Rajagopal, A. K., Sudha, Prabhu Tej, J. and Narayanan, A., Quantifying nonclassical correlations via moment matrix positivity. In Proceedings of the 13th Asian Quantum Information Science Conference (AQIS’13), IMSc, Chennai, 2013, pp. 38–39.

  • Shohat, J. A. and Tamarkin, J. D., The Problem of Moments, American Mathematical Society, 1943.

  • Akhiezer, N. I., The Classical Moment Problem, Hafner Publishing, Edinburgh, London, 1965.

  • Budroni, C., Moroder, T., Kleinmann, M. and Gühne, O., Bounding temporal quantum correlations. Phys. Rev. Lett., 2013, 111(2), 020403-1 to 5.


Abstract Views: 365

PDF Views: 131




  • Unsharp Measurements and Joint Measurability

Abstract Views: 365  |  PDF Views: 131

Authors

H. S. Karthik
Raman Research Institute, Bengaluru 560 080, India
A. R. Usha Devi
Department of Physics, Bangalore University, Bengaluru 560 056, India and Inspire Institute Inc., Alexandria, Virginia 22303, United States
A. K. Rajagopal
Inspire Institute Inc., Alexandria, Virginia, 22303, USA; Institute of Mathematical Sciences, C.I.T. Campus, Taramani, Chennai 600 113, and Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad 211 019, India

Abstract


We give an overview of joint unsharp measurements of non-commuting observables using positive operator valued measures (POVMs). We exemplify the role played by joint measurability of POVMs in entropic uncertainty relation for Alice's pair of non-commuting observables in the presence of Bob's entangled quantum memory. We show that Bob should record the outcomes of incompatible (non-jointly measurable) POVMs in his quantum memory so as to beat the entropic uncertainty bound. In other words, in addition to the presence of entangled Alice-Bob state, implementing incompatible POVMs at Bob's end is necessary to beat the uncertainty bound and hence predict the outcomes of non-commuting observables with improved precision. We also explore the implications of joint measurability to validate a moment matrix constructed from average pairwise correlations of three dichotomic non-commuting qubit observables. We prove that a classically acceptable moment matrix - which ascertains the existence of a legitimate joint probability distribution for the outcomes of all the three dichotomic observables - could be realized if and only if compatible POVMs are employed.

Keywords


Incompatibility, Joint Measurability, Positive Operator Valued Measures, Unsharp Measurements.

References





DOI: https://doi.org/10.18520/cs%2Fv109%2Fi11%2F2061-2068