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Maximizing Complementary Quantities by Projective Measurements

  • General and Applied Physics
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Abstract

In this work, we study the so-called quantitative complementarity quantities. We focus in the following physical situation: two qubits (q A and q B ) are initially in a maximally entangled state. One of them (q B ) interacts with a N-qubit system (R). After the interaction, projective measurements are performed on each of the qubits of R, in a basis that is chosen after independent optimization procedures: maximization of the visibility, the concurrence, and the predictability. For a specific maximization procedure, we study in detail how each of the complementary quantities behave, conditioned on the intensity of the coupling between q B and the N qubits. We show that, if the coupling is sufficiently “strong,” independent of the maximization procedure, the concurrence tends to decay quickly. Interestingly enough, the behavior of the concurrence in this model is similar to the entanglement dynamics of a two qubit system subjected to a thermal reservoir, despite that we consider finite N. However, the visibility shows a different behavior: its maximization is more efficient for stronger coupling constants. Moreover, we investigate how the distinguishability, or the information stored in different parts of the system, is distributed for different couplings.

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References

  1. N. Bohr, The quantum postulate and the recent development of atomic theory. Nature. 121, 580 (1928)

    Article  ADS  MATH  Google Scholar 

  2. W.K. Wootters, W.H. Zurek, Complementarity in the double-slit experiment: Quantum nonseparability and a quantitative statement of Bohr’s principle. Phys. Rev. D. 19, 473 (1979)

    Article  ADS  Google Scholar 

  3. J. Summhammer, H. Rauch, D. Tuppinger, Stochastic and deterministic absorption in neutron-interference experiments. Phys. Rev. A. 36, 4447 (1987)

    Article  ADS  Google Scholar 

  4. D.M. Greenberger, A. Yasin, Simultaneous wave and particle knowledge in a neutron interferometer. Phys. Lett. A. 128, 391 (1988)

    Article  ADS  Google Scholar 

  5. L. Mandel, Coherence and indistinguishability. Opt. Lett. 16, 1882 (1991)

    Article  ADS  Google Scholar 

  6. G. Jaeger, M.A. Horne, A. Shimony, Complementarity of one-particle and two-particle interference. Phys. Rev. A. 48, 1023 (1993)

    Article  ADS  Google Scholar 

  7. G. Jaeger, A. Shimony, L. Vaidman, Two interferometric complementarities. Phys. Rev. A. 51, 54 (1995)

    Article  ADS  Google Scholar 

  8. B.-G. Englert, Fringe visibility and which-way information: an inequality. Phys. Rev. Lett. 77, 2154 (1996)

    Article  ADS  Google Scholar 

  9. B.-G. Englert, J.A. Bergou, Quantitative quantum erasure. Opt. Commun. 179, 337 (2000)

    Article  ADS  Google Scholar 

  10. M.O. Scully, H. Walther, Quantum optical test of observation and complementarity in quantum mechanics. Phys. Rev. A. 39, 5229 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  11. M.O. Scully, B.-G. Englert, H. Walther, Quantum optical tests of complementarity. Nature. 351, 111 (1991)

    Article  ADS  Google Scholar 

  12. L. Mandel, Indistinguishability in one-photon and two-photon interference. Found. Phys. 25, 211 (1995)

    Article  ADS  Google Scholar 

  13. T.E. Tessier, Complementarity Relations for Multi-Qubit Systems. Found. Phys. Lett. 18, 107 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. M. Jakob, J.A. Bergou, Quantitative complementarity relations in bipartite systems: Entanglement as a physical reality. Opt. Commun. 283, 827 (2010)

    Article  ADS  Google Scholar 

  15. F.M. Miatto, K. Piché, T. Brougham, R.W. Boyd, Nonlinear susceptibility of composite optical materials in the Maxwell Garnett model. Phys. Rev. A. 92, 062331 (2015)

    Article  ADS  Google Scholar 

  16. E. Bagan, J.A. Bergou, S.S. Cottrell, M. Hillery, Relations between coherence and path information. Phys. Rev. Lett. 116, 160406 (2016)

    Article  ADS  Google Scholar 

  17. P.J. Coles, Entropic framework for wave-particle duality in multipath interferometers. Phys. Rev. A. 93, 062111 (2016)

    Article  ADS  Google Scholar 

  18. ed. by D. Greenberger, W. L. Reiter, A. Zeilinger. Epistemological and experimental perspectives on quantum physics (Springer, Netherlands, 1999)

  19. M.O. Scully, K. Drühl, Quantum eraser: A proposed photon correlation experiment concerning observation and “delayed choice” in quantum mechanics. Phys. Rev. A. 25, 2208 (1982)

    Article  ADS  Google Scholar 

  20. P. Storey, S. Tan, M. Collett, D. Walls, Path detection and the uncertainty principle. Nature. 367, 626 (1994)

    Article  ADS  Google Scholar 

  21. H. Wiseman, F. Harrison, Uncertainty over complementarity? (Note: the question mark is included in the title). Nature. 377, 584 (1995)

    Article  ADS  Google Scholar 

  22. R. Mir, J.S. Lundeen, M.W. Mitchell, A.M. Steinberg, J.L. Garretson, H.M. Wiseman, A double-slit ‘which-way’ experiment on the complementarity-uncertainty debate. New J. Phys. 9, 287 (2007)

    Article  ADS  Google Scholar 

  23. A. Luis, L. L. Sánchez-Soto, Complementarity enforced by random classical phase kicks. Phys. Rev. Lett. 81, 4031 (1998)

    Article  ADS  Google Scholar 

  24. P. Busch, C. Shilladay, Complementarity and uncertainty in Mach-Zehnder interferometry and beyond. Phys. Rep. 435, 1 (2006)

    Article  ADS  Google Scholar 

  25. R. Rossi, J.P. Souza, L.A.M. de Souza, M.C. Nemes, Multipartite quantum eraser in cavity QED. Phys. Rev. A. 88, 062102 (2013)

  26. S.P. Walborn, M.O. Terra Cunha, S. Pádua, C.H. Monken, Double-slit quantum eraser. Phys. Rev. A. 65, 033818 (2002)

    Article  ADS  Google Scholar 

  27. G. Teklemariam, E. Fortunato, M. Pravia, T. Havel, D. Cory, NMR Analog of the quantum disentanglement eraser. Phys. Rev. Lett. 86, 5845 (2001)

    Article  ADS  Google Scholar 

  28. G. Teklemariam, E.M. Fortunato, M.A. Pravia, Y. Sharf, T.F. Havel, D.G. Cory, A. Bhattaharyya, J. Hou, Quantum erasers and probing classifications of entanglement via nuclear magnetic resonance. Phys. Rev. A. 66, 012309 (2002)

    Article  ADS  Google Scholar 

  29. Y.-H. Kim, R. Yu, S.P. Kulik, Y. Shih, M.O. Scully, Delayed “Choice” quantum eraser. Phys. Rev. Lett. 84, 1 (2000)

    Article  ADS  Google Scholar 

  30. A. Salles, F. de Melo, M.P. Almeida, M. Hor-Meyll, S.P. Walborn, P.H. Souto Ribeiro, L. Davidovich, Experimental investigation of the dynamics of entanglement: Sudden death, complementarity, and continuous monitoring of the environment. Phys. Rev. A. 78, 022322 (2008)

  31. A. Heuer, G. Pieplow, R. Menzel. arXiv:1501.00817 (2015)

  32. H.J. Carmichael, Vol. 1. Statistical methods in quantum optics (Springer, Berlin, 1999)

  33. H.P. Breuer, Physical review A—atomic, molecular, and optical Physics. arXiv:0611208 [quant-ph]

  34. K. Jacobs, Topics in quantum measurement and quantum noise. arXiv:9810015 [quant-ph] (1998)

  35. S. Haroche, J.-M. Raimond. Exploring the quantum (Oxford University Press, Oxford, 2006)

    Book  MATH  Google Scholar 

  36. G.H. Aguilar, A. Valdés-hernández, L. Davidovich, S.P. Walborn, P.H. Souto Ribeiro, Experimental entanglement redistribution under decoherence channels. Phys. Rev. Lett. 113, 240501 (2014)

    Article  ADS  Google Scholar 

  37. S. Gleyzes, S. Kuhr, C. Guerlin, J. Bernu, S. Deléglise, U. Busk Hoff, M. Brune, J.-M. Raimond, S. Haroche, Quantum jumps of light recording the birth and death of a photon in a cavity. Nature. 446, 297 (2007)

    Article  ADS  Google Scholar 

  38. S. Kuhr, S. Gleyzes, C. Guerlin, J. Bernu, U.B. Hoff, S. Deleéglise, S. Osnaghi, M. Brune, J.-M. Raimond, S. Haroche, E. Jacques, P. Bosland, B. Visentin, Ultrahigh finesse Fabry-Pérot superconducting resonator. Appl. Phys. Lett. 90, 164101 (2007)

    Article  ADS  Google Scholar 

  39. M. Brune, F. Schmidt-Kaler, A. Maali, J. Dreyer, E. Hagley, J.M. Raimond, S. Haroche, Quantum rabi oscillation: a direct test of field quantization in a cavity. Phys. Rev. Lett. 76, 1800 (1996)

    Article  ADS  MATH  Google Scholar 

  40. J. Kilian, in Founding cryptography on oblivious transfer. Proc. 20th ACM STOC, (1988), pp. 20–31

  41. N.H.Y. Ng, S.K. Joshi, C. Chen Ming, C. Kurtsiefer, S. Wehner, Experimental implementation of bit commitment in the noisy-storage model. Nat. Commun. 3, 1326 (2012)

    Article  ADS  Google Scholar 

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Acknowledgments

The authors would like to thank the support from the Brazilian agencies CNPq and CAPES (CAPES (6842/2014-03); CNPq (470131/2013-6)). L.A.M.S. also thanks the University of Nottingham for hospitality and support during part of this work preparation. The authors acknowledge useful discussions with P. Saldanha, M. F. Santos, and G. Murta.

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Authors and Affiliations

  1. Universidade Federal de Viçosa - Campus Florestal, LMG818 Km6, Minas Gerais, Florestal, 35690-000, Brazil

    Leonardo A. M. Souza & Romeu Rossi Jr.

  2. Departamento de Física, Universidade Federal de Minas Gerais, Caixa Postal 702, Belo Horizonte, 30161-970, Brazil

    Nadja K. Bernardes

Authors
  1. Leonardo A. M. Souza
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  2. Nadja K. Bernardes
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  3. Romeu Rossi Jr.
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Correspondence to Leonardo A. M. Souza.

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M. Souza, L.A., Bernardes, N.K. & Rossi, R. Maximizing Complementary Quantities by Projective Measurements. Braz J Phys 47, 157–166 (2017). https://doi.org/10.1007/s13538-016-0481-9

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