Abstract

Sequential weak measurements of non-commuting observables are not only fundamentally interesting in terms of quantum measurement but also show potential in various applications. Previously reported methods, however, can only make limited sequential weak measurements experimentally. In this article, we propose the realization of sequential measurements of non-commuting Pauli observables and experimentally demonstrate for the first time the measurement of sequential weak values of three non-commuting Pauli observables using genuine single photons.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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  9. J. Dressel, “Weak values as interference phenomena,” Phys. Rev. A 91, 032116 (2015).
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  10. Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
    [Crossref]
  11. J. S. Lundeen and A. M. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of Hardy’s paradox,” Phys. Rev. Lett. 102, 020404 (2009).
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  12. K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, “Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair,” New. J. Phys. 11033011 (2009).
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  17. G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117, 120401 (2016).
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  22. N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
    [Crossref] [PubMed]
  23. G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photonic polarization,” Phys. Rev. Lett. 94, 220405 (2005).
    [Crossref]
  24. L. A. Rosema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
    [Crossref]
  25. A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking photons where they have been,” Phys. Rev. Lett. 111, 240402 (2013).
    [Crossref]
  26. M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A. 108, 1256–1261 (2011).
    [Crossref] [PubMed]
  27. Y. Kim, Y. S. Kim, S. Y. Lee, S. W. Han, S. Moon, Y. H. Kim, and Y. W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Comm. 9, 192 (2018).
    [Crossref]
  28. G. Vallone and D Dequal, “Strong measurements give a better direct measurement of the quantum wave function,” Phys. Rev. Lett. 116, 040502 (2016).
    [Crossref] [PubMed]
  29. J. Dressel, T. A. Brun, and A. N. Korotkov, “Implementing generalized measurements with superconducting qubits,” Phys. Rev. A 90, 032302 (2014).
    [Crossref]
  30. J. Dressel, J. R. G. Alonso, M. Waegell, and N. Y. Halpern, “Strengthening weak measurement of qubit out-of-order correlators,” Phys. Rev. A 98, 012132 (2018).
    [Crossref]
  31. A. Brodutch and E. Cohen, “A scheme for performing strong and weak sequential measurements of non-commuting observables,” Quantum Stud.: Math. Found. 4, 13–27 (2017).
    [Crossref]
  32. Y. Kedem and L. Vaidman, “Modular values and weak values of quantum observables,” Phys. Rev. Lett. 105, 230401 (2010).
    [Crossref]
  33. K. J. Resch and A. M. Steinberg, “Extracting joint weak values with local, single-particle measurements,” Phys. Rev. Lett. 92, 130402 (2004).
    [Crossref] [PubMed]
  34. J. S. Lundeen and K. J. Resch, “Practical measurement of joint weak values and their connection to the annihilation operator,” Phys. Lett. A 334, 337–344 (2005).
    [Crossref]
  35. H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).
    [Crossref]
  36. M. J. Hu, Z. Y. Zhou, X. M. Hu, C. F. Li, G. C. Guo, and Y. S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” Quant. Inf. 4, 63 (2018).
    [Crossref]
  37. A. J. Leggett and A. Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?” Phys. Rev. Lett. 54, 857–860 (1985).
    [Crossref] [PubMed]
  38. A. Avella, F. Piacentini, M. Borsarelli, M. Barbieri, M. Gramegna, R. Lussana, F. Villa, A. Tosi, I. P. Degiovanni, and M. Genovese, “Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality,” Phys. Rev. A 96, 052123 (2017).
    [Crossref]
  39. F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102(R) (2017).
    [Crossref]
  40. F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Entangled systems are unbounded sources of nonlocal correlations and of certified random numbers,” ArXiv:1802.07962 (2018).
  41. R. Jozsa, “Quantum effects in algorithms,” Chaos, Solitons Fractals,  10, 1657–1664 (1999).
  42. G. Mitchison and R. Jozsa, “Counterfactual computation,” Proc. R. Soc. Lond. A 457, 1175–1193 (2001).
    [Crossref]
  43. R. Ber, S. Marcovitch, O. Kenneth, and B. Reznik, “Process tomography for systems in a thermal state,” New. J. Phys. 15, 013050 (2013).
    [Crossref]
  44. S. Kochen and E. P. Specker, “The problem of hidden variables in quantum mechanics,” J. Math. Mach. 17, 59–87 (1967).
  45. C. Budroni, T. Moroder, M. Kleinmann, and O. Gühne, “Bounding temporal quantum correlations,” Phys. Rev. Lett. 111, 020403 (2013).
    [Crossref] [PubMed]
  46. L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violations of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
    [Crossref]
  47. F. Kaneda, S. Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
    [Crossref] [PubMed]
  48. J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
    [Crossref] [PubMed]
  49. D. Georgiev and E. Cohen, “Sequential weak values probe finite coarse-grained virtual Feynman histories,” Phys. Rev. A 97, 052102 (2018).
    [Crossref]

2018 (4)

Y. Kim, Y. S. Kim, S. Y. Lee, S. W. Han, S. Moon, Y. H. Kim, and Y. W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Comm. 9, 192 (2018).
[Crossref]

J. Dressel, J. R. G. Alonso, M. Waegell, and N. Y. Halpern, “Strengthening weak measurement of qubit out-of-order correlators,” Phys. Rev. A 98, 012132 (2018).
[Crossref]

M. J. Hu, Z. Y. Zhou, X. M. Hu, C. F. Li, G. C. Guo, and Y. S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” Quant. Inf. 4, 63 (2018).
[Crossref]

D. Georgiev and E. Cohen, “Sequential weak values probe finite coarse-grained virtual Feynman histories,” Phys. Rev. A 97, 052102 (2018).
[Crossref]

2017 (3)

A. Avella, F. Piacentini, M. Borsarelli, M. Barbieri, M. Gramegna, R. Lussana, F. Villa, A. Tosi, I. P. Degiovanni, and M. Genovese, “Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality,” Phys. Rev. A 96, 052123 (2017).
[Crossref]

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102(R) (2017).
[Crossref]

A. Brodutch and E. Cohen, “A scheme for performing strong and weak sequential measurements of non-commuting observables,” Quantum Stud.: Math. Found. 4, 13–27 (2017).
[Crossref]

2016 (3)

G. Vallone and D Dequal, “Strong measurements give a better direct measurement of the quantum wave function,” Phys. Rev. Lett. 116, 040502 (2016).
[Crossref] [PubMed]

F. Piacentini, A. Avella, M. P. Levi, M. Gramegna, G. Brida, I. P. Degiovanni, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, and M. Genovese, “Measuring incompatible observables by exploiting sequential weak values,” Phys. Rev. Lett. 117, 170402 (2016).
[Crossref] [PubMed]

G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117, 120401 (2016).
[Crossref] [PubMed]

2015 (3)

A. Brodutch, “Comment on ‘How the result of a single coin toss can turn out to be 100 heads’,” Phys. Rev. Lett. 114, 118901 (2015).
[Crossref]

C. Ferrie and J. Combes, “Ferrie and Combes reply:,” Phys. Rev. Lett. 114, 118902 (2015).
[Crossref]

J. Dressel, “Weak values as interference phenomena,” Phys. Rev. A 91, 032116 (2015).
[Crossref]

2014 (3)

C. Ferrie and J. Combes, “How the result of a single coin toss can turn out to be 100 heads,” Phys. Rev. Lett. 113, 120404 (2014).
[Crossref] [PubMed]

J. Dressel, T. A. Brun, and A. N. Korotkov, “Implementing generalized measurements with superconducting qubits,” Phys. Rev. A 90, 032302 (2014).
[Crossref]

F. Kaneda, S. Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
[Crossref] [PubMed]

2013 (4)

C. Budroni, T. Moroder, M. Kleinmann, and O. Gühne, “Bounding temporal quantum correlations,” Phys. Rev. Lett. 111, 020403 (2013).
[Crossref] [PubMed]

R. Ber, S. Marcovitch, O. Kenneth, and B. Reznik, “Process tomography for systems in a thermal state,” New. J. Phys. 15, 013050 (2013).
[Crossref]

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking photons where they have been,” Phys. Rev. Lett. 111, 240402 (2013).
[Crossref]

X. Y. Xu, Y. Kedem, K Sun, L. Vaidman, C. F. Li, and G. C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

2012 (4)

J. S. Lundeen and C. Bamber, Phys. Rev. Lett. “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[Crossref] [PubMed]

L. A. Rosema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).
[Crossref]

L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violations of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

2011 (2)

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A. 108, 1256–1261 (2011).
[Crossref] [PubMed]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref] [PubMed]

2010 (1)

Y. Kedem and L. Vaidman, “Modular values and weak values of quantum observables,” Phys. Rev. Lett. 105, 230401 (2010).
[Crossref]

2009 (3)

J. S. Lundeen and A. M. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of Hardy’s paradox,” Phys. Rev. Lett. 102, 020404 (2009).
[Crossref]

K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, “Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair,” New. J. Phys. 11033011 (2009).
[Crossref]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

2008 (1)

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

2007 (1)

G. Mitchison, R. Jozsa, and S. Popescu, “Sequential weak measurement,” Phys. Rev. A 76, 062105 (2007).
[Crossref]

2005 (2)

J. S. Lundeen and K. J. Resch, “Practical measurement of joint weak values and their connection to the annihilation operator,” Phys. Lett. A 334, 337–344 (2005).
[Crossref]

G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photonic polarization,” Phys. Rev. Lett. 94, 220405 (2005).
[Crossref]

2004 (1)

K. J. Resch and A. M. Steinberg, “Extracting joint weak values with local, single-particle measurements,” Phys. Rev. Lett. 92, 130402 (2004).
[Crossref] [PubMed]

2002 (1)

Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
[Crossref]

2001 (1)

G. Mitchison and R. Jozsa, “Counterfactual computation,” Proc. R. Soc. Lond. A 457, 1175–1193 (2001).
[Crossref]

1999 (1)

R. Jozsa, “Quantum effects in algorithms,” Chaos, Solitons Fractals,  10, 1657–1664 (1999).

1991 (1)

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[Crossref] [PubMed]

1989 (3)

A. J. Leggett, “Comment on ‘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. 62, 2325 (1989).
[Crossref]

Asher Peres, “Quantum measurements with postselection,” Phys. Rev. Lett. 62, 2326 (1989).
[Crossref] [PubMed]

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

1988 (2)

Y. Aharonov, D. Z. Albert, and L. Vaidman, “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).
[Crossref] [PubMed]

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[Crossref] [PubMed]

1985 (1)

A. J. Leggett and A. Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?” Phys. Rev. Lett. 54, 857–860 (1985).
[Crossref] [PubMed]

1967 (1)

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

Acín, A.

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102(R) (2017).
[Crossref]

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Entangled systems are unbounded sources of nonlocal correlations and of certified random numbers,” ArXiv:1802.07962 (2018).

Aharonov, Y.

Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
[Crossref]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “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).
[Crossref] [PubMed]

Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley-VCH, 2005).
[Crossref]

Albert, D. Z.

Y. Aharonov, D. Z. Albert, and L. Vaidman, “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).
[Crossref] [PubMed]

Almeida, M. P.

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A. 108, 1256–1261 (2011).
[Crossref] [PubMed]

Alonso, J. R. G.

J. Dressel, J. R. G. Alonso, M. Waegell, and N. Y. Halpern, “Strengthening weak measurement of qubit out-of-order correlators,” Phys. Rev. A 98, 012132 (2018).
[Crossref]

Augusiak, R.

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102(R) (2017).
[Crossref]

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Entangled systems are unbounded sources of nonlocal correlations and of certified random numbers,” ArXiv:1802.07962 (2018).

Avella, A.

A. Avella, F. Piacentini, M. Borsarelli, M. Barbieri, M. Gramegna, R. Lussana, F. Villa, A. Tosi, I. P. Degiovanni, and M. Genovese, “Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality,” Phys. Rev. A 96, 052123 (2017).
[Crossref]

F. Piacentini, A. Avella, M. P. Levi, M. Gramegna, G. Brida, I. P. Degiovanni, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, and M. Genovese, “Measuring incompatible observables by exploiting sequential weak values,” Phys. Rev. Lett. 117, 170402 (2016).
[Crossref] [PubMed]

Baek, S. Y.

F. Kaneda, S. Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
[Crossref] [PubMed]

Bamber, C.

J. S. Lundeen and C. Bamber, Phys. Rev. Lett. “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[Crossref] [PubMed]

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref] [PubMed]

Banker, J.

G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117, 120401 (2016).
[Crossref] [PubMed]

Bar-Ad, S.

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking photons where they have been,” Phys. Rev. Lett. 111, 240402 (2013).
[Crossref]

Barbieri, M.

A. Avella, F. Piacentini, M. Borsarelli, M. Barbieri, M. Gramegna, R. Lussana, F. Villa, A. Tosi, I. P. Degiovanni, and M. Genovese, “Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality,” Phys. Rev. A 96, 052123 (2017).
[Crossref]

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A. 108, 1256–1261 (2011).
[Crossref] [PubMed]

Ber, R.

R. Ber, S. Marcovitch, O. Kenneth, and B. Reznik, “Process tomography for systems in a thermal state,” New. J. Phys. 15, 013050 (2013).
[Crossref]

Bhandari, R.

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[Crossref] [PubMed]

Borsarelli, M.

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[Crossref]

K. J. Resch and A. M. Steinberg, “Extracting joint weak values with local, single-particle measurements,” Phys. Rev. Lett. 92, 130402 (2004).
[Crossref] [PubMed]

Reznik, B.

R. Ber, S. Marcovitch, O. Kenneth, and B. Reznik, “Process tomography for systems in a thermal state,” New. J. Phys. 15, 013050 (2013).
[Crossref]

Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
[Crossref]

Ritchie, N. W. M.

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[Crossref] [PubMed]

Rohrlich, D.

Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley-VCH, 2005).
[Crossref]

Rosema, L. A.

L. A. Rosema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

Rozema, L. A.

L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violations of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

Samuel, J.

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[Crossref] [PubMed]

Shikano, Y.

H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).
[Crossref]

Soudagar, Y.

L. A. Rosema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violations of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

Specker, E. P.

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

Starling, D. J.

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

Steinberg, A. M.

L. A. Rosema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violations of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

J. S. Lundeen and A. M. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of Hardy’s paradox,” Phys. Rev. Lett. 102, 020404 (2009).
[Crossref]

K. J. Resch and A. M. Steinberg, “Extracting joint weak values with local, single-particle measurements,” Phys. Rev. Lett. 92, 130402 (2004).
[Crossref] [PubMed]

Stevenson, P. M.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

Stewart, C.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref] [PubMed]

Story, J. G.

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[Crossref] [PubMed]

Sudarshan, E. C. G.

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

Sun, K

X. Y. Xu, Y. Kedem, K Sun, L. Vaidman, C. F. Li, and G. C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

Sutherland, B.

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref] [PubMed]

Thekkadath, G. S.

G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117, 120401 (2016).
[Crossref] [PubMed]

Tollaksen, J.

Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
[Crossref]

Tosi, A.

A. Avella, F. Piacentini, M. Borsarelli, M. Barbieri, M. Gramegna, R. Lussana, F. Villa, A. Tosi, I. P. Degiovanni, and M. Genovese, “Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality,” Phys. Rev. A 96, 052123 (2017).
[Crossref]

F. Piacentini, A. Avella, M. P. Levi, M. Gramegna, G. Brida, I. P. Degiovanni, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, and M. Genovese, “Measuring incompatible observables by exploiting sequential weak values,” Phys. Rev. Lett. 117, 170402 (2016).
[Crossref] [PubMed]

Vaidman, L.

X. Y. Xu, Y. Kedem, K Sun, L. Vaidman, C. F. Li, and G. C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking photons where they have been,” Phys. Rev. Lett. 111, 240402 (2013).
[Crossref]

Y. Kedem and L. Vaidman, “Modular values and weak values of quantum observables,” Phys. Rev. Lett. 105, 230401 (2010).
[Crossref]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “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).
[Crossref] [PubMed]

Vallone, G.

G. Vallone and D Dequal, “Strong measurements give a better direct measurement of the quantum wave function,” Phys. Rev. Lett. 116, 040502 (2016).
[Crossref] [PubMed]

Villa, F.

A. Avella, F. Piacentini, M. Borsarelli, M. Barbieri, M. Gramegna, R. Lussana, F. Villa, A. Tosi, I. P. Degiovanni, and M. Genovese, “Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality,” Phys. Rev. A 96, 052123 (2017).
[Crossref]

F. Piacentini, A. Avella, M. P. Levi, M. Gramegna, G. Brida, I. P. Degiovanni, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, and M. Genovese, “Measuring incompatible observables by exploiting sequential weak values,” Phys. Rev. Lett. 117, 170402 (2016).
[Crossref] [PubMed]

Waegell, M.

J. Dressel, J. R. G. Alonso, M. Waegell, and N. Y. Halpern, “Strengthening weak measurement of qubit out-of-order correlators,” Phys. Rev. A 98, 012132 (2018).
[Crossref]

White, A. G.

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A. 108, 1256–1261 (2011).
[Crossref] [PubMed]

G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photonic polarization,” Phys. Rev. Lett. 94, 220405 (2005).
[Crossref]

Wiseman, H. M.

G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photonic polarization,” Phys. Rev. Lett. 94, 220405 (2005).
[Crossref]

Wittek, P.

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102(R) (2017).
[Crossref]

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Entangled systems are unbounded sources of nonlocal correlations and of certified random numbers,” ArXiv:1802.07962 (2018).

Xu, X. Y.

X. Y. Xu, Y. Kedem, K Sun, L. Vaidman, C. F. Li, and G. C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

Yamamoto, T.

K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, “Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair,” New. J. Phys. 11033011 (2009).
[Crossref]

Yokota, K.

K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, “Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair,” New. J. Phys. 11033011 (2009).
[Crossref]

Zappa, F.

F. Piacentini, A. Avella, M. P. Levi, M. Gramegna, G. Brida, I. P. Degiovanni, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, and M. Genovese, “Measuring incompatible observables by exploiting sequential weak values,” Phys. Rev. Lett. 117, 170402 (2016).
[Crossref] [PubMed]

Zhang, Y. S.

M. J. Hu, Z. Y. Zhou, X. M. Hu, C. F. Li, G. C. Guo, and Y. S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” Quant. Inf. 4, 63 (2018).
[Crossref]

Zhou, Z. Y.

M. J. Hu, Z. Y. Zhou, X. M. Hu, C. F. Li, G. C. Guo, and Y. S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” Quant. Inf. 4, 63 (2018).
[Crossref]

Chaos, Solitons Fractals (1)

R. Jozsa, “Quantum effects in algorithms,” Chaos, Solitons Fractals,  10, 1657–1664 (1999).

J. Math. Mach. (1)

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

Nat. Comm. (1)

Y. Kim, Y. S. Kim, S. Y. Lee, S. W. Han, S. Moon, Y. H. Kim, and Y. W. Cho, “Direct quantum process tomography via measuring sequential weak values of incompatible observables,” Nat. Comm. 9, 192 (2018).
[Crossref]

Nature (1)

J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, “Direct measurement of the quantum wavefunction,” Nature 474, 188–191 (2011).
[Crossref] [PubMed]

New. J. Phys. (2)

K. Yokota, T. Yamamoto, M. Koashi, and N. Imoto, “Direct observation of Hardy’s paradox by joint weak measurement with an entangled photon pair,” New. J. Phys. 11033011 (2009).
[Crossref]

R. Ber, S. Marcovitch, O. Kenneth, and B. Reznik, “Process tomography for systems in a thermal state,” New. J. Phys. 15, 013050 (2013).
[Crossref]

Phys. Lett. A (2)

Y. Aharonov, A. Botero, S. Popescu, B. Reznik, and J. Tollaksen, “Revisiting Hardy’s paradox: counterfactual statements, real measurements, entanglement and weak values,” Phys. Lett. A 301, 130–138 (2002).
[Crossref]

J. S. Lundeen and K. J. Resch, “Practical measurement of joint weak values and their connection to the annihilation operator,” Phys. Lett. A 334, 337–344 (2005).
[Crossref]

Phys. Rev. A (8)

H. Kobayashi, G. Puentes, and Y. Shikano, “Extracting joint weak values from two-dimensional spatial displacements,” Phys. Rev. A 86, 053805 (2012).
[Crossref]

J. Dressel, T. A. Brun, and A. N. Korotkov, “Implementing generalized measurements with superconducting qubits,” Phys. Rev. A 90, 032302 (2014).
[Crossref]

J. Dressel, J. R. G. Alonso, M. Waegell, and N. Y. Halpern, “Strengthening weak measurement of qubit out-of-order correlators,” Phys. Rev. A 98, 012132 (2018).
[Crossref]

G. Mitchison, R. Jozsa, and S. Popescu, “Sequential weak measurement,” Phys. Rev. A 76, 062105 (2007).
[Crossref]

J. Dressel, “Weak values as interference phenomena,” Phys. Rev. A 91, 032116 (2015).
[Crossref]

D. Georgiev and E. Cohen, “Sequential weak values probe finite coarse-grained virtual Feynman histories,” Phys. Rev. A 97, 052102 (2018).
[Crossref]

A. Avella, F. Piacentini, M. Borsarelli, M. Barbieri, M. Gramegna, R. Lussana, F. Villa, A. Tosi, I. P. Degiovanni, and M. Genovese, “Anomalous weak values and the violation of a multiple-measurement Leggett-Garg inequality,” Phys. Rev. A 96, 052123 (2017).
[Crossref]

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Unbounded randomness certification using sequences of measurements,” Phys. Rev. A 95, 020102(R) (2017).
[Crossref]

Phys. Rev. D (1)

I. M. Duck, P. M. Stevenson, and E. C. G. Sudarshan, “The sense in which a ‘weak measurement’ of a spin-1/2 particle’s spin component yields a value 100,” Phys. Rev. D 40, 2112–2117 (1989).
[Crossref]

Phys. Rev. Lett. (24)

C. Ferrie and J. Combes, “How the result of a single coin toss can turn out to be 100 heads,” Phys. Rev. Lett. 113, 120404 (2014).
[Crossref] [PubMed]

A. Brodutch, “Comment on ‘How the result of a single coin toss can turn out to be 100 heads’,” Phys. Rev. Lett. 114, 118901 (2015).
[Crossref]

C. Ferrie and J. Combes, “Ferrie and Combes reply:,” Phys. Rev. Lett. 114, 118902 (2015).
[Crossref]

Y. Aharonov, D. Z. Albert, and L. Vaidman, “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).
[Crossref] [PubMed]

A. J. Leggett, “Comment on ‘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. 62, 2325 (1989).
[Crossref]

Asher Peres, “Quantum measurements with postselection,” Phys. Rev. Lett. 62, 2326 (1989).
[Crossref] [PubMed]

J. S. Lundeen and A. M. Steinberg, “Experimental joint weak measurement on a photon pair as a probe of Hardy’s paradox,” Phys. Rev. Lett. 102, 020404 (2009).
[Crossref]

J. S. Lundeen and C. Bamber, Phys. Rev. Lett. “Procedure for direct measurement of general quantum states using weak measurement,” Phys. Rev. Lett. 108, 070402 (2012).
[Crossref] [PubMed]

G. S. Thekkadath, L. Giner, Y. Chalich, M. J. Horton, J. Banker, and J. S. Lundeen, “Direct measurement of the density matrix of a quantum system,” Phys. Rev. Lett. 117, 120401 (2016).
[Crossref] [PubMed]

G. Vallone and D Dequal, “Strong measurements give a better direct measurement of the quantum wave function,” Phys. Rev. Lett. 116, 040502 (2016).
[Crossref] [PubMed]

Y. Kedem and L. Vaidman, “Modular values and weak values of quantum observables,” Phys. Rev. Lett. 105, 230401 (2010).
[Crossref]

K. J. Resch and A. M. Steinberg, “Extracting joint weak values with local, single-particle measurements,” Phys. Rev. Lett. 92, 130402 (2004).
[Crossref] [PubMed]

P. B. Dixon, D. J. Starling, A. N. Jordan, and J. C. Howell, “Ultrasensitive beam deflection measurement via interferometric weak value amplification,” Phys. Rev. Lett. 102, 173601 (2009).
[Crossref] [PubMed]

X. Y. Xu, Y. Kedem, K Sun, L. Vaidman, C. F. Li, and G. C. Guo, “Phase estimation with weak measurement using a white light source,” Phys. Rev. Lett. 111, 033604 (2013).
[Crossref] [PubMed]

F. Piacentini, A. Avella, M. P. Levi, M. Gramegna, G. Brida, I. P. Degiovanni, E. Cohen, R. Lussana, F. Villa, A. Tosi, F. Zappa, and M. Genovese, “Measuring incompatible observables by exploiting sequential weak values,” Phys. Rev. Lett. 117, 170402 (2016).
[Crossref] [PubMed]

N. W. M. Ritchie, J. G. Story, and R. G. Hulet, “Realization of a measurement of a ‘weak value’,” Phys. Rev. Lett. 66, 1107–1110 (1991).
[Crossref] [PubMed]

G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, “Measurement of quantum weak values of photonic polarization,” Phys. Rev. Lett. 94, 220405 (2005).
[Crossref]

L. A. Rosema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violation of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

A. Danan, D. Farfurnik, S. Bar-Ad, and L. Vaidman, “Asking photons where they have been,” Phys. Rev. Lett. 111, 240402 (2013).
[Crossref]

C. Budroni, T. Moroder, M. Kleinmann, and O. Gühne, “Bounding temporal quantum correlations,” Phys. Rev. Lett. 111, 020403 (2013).
[Crossref] [PubMed]

L. A. Rozema, A. Darabi, D. H. Mahler, A. Hayat, Y. Soudagar, and A. M. Steinberg, “Violations of Heisenberg’s measurement-disturbance relationship by weak measurements,” Phys. Rev. Lett. 109, 100404 (2012).
[Crossref]

F. Kaneda, S. Y. Baek, M. Ozawa, and K. Edamatsu, “Experimental test of error-disturbance uncertainty relations by weak measurement,” Phys. Rev. Lett. 112, 020402 (2014).
[Crossref] [PubMed]

J. Samuel and R. Bhandari, “General setting for Berry’s phase,” Phys. Rev. Lett. 60, 2339–2342 (1988).
[Crossref] [PubMed]

A. J. Leggett and A. Garg, “Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks?” Phys. Rev. Lett. 54, 857–860 (1985).
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Proc. Natl. Acad. Sci. U.S.A. (1)

M. E. Goggin, M. P. Almeida, M. Barbieri, B. P. Lanyon, J. L. O’Brien, A. G. White, and G. J. Pryde, “Violation of the Leggett-Garg inequality with weak measurements of photons,” Proc. Natl. Acad. Sci. U.S.A. 108, 1256–1261 (2011).
[Crossref] [PubMed]

Proc. R. Soc. Lond. A (1)

G. Mitchison and R. Jozsa, “Counterfactual computation,” Proc. R. Soc. Lond. A 457, 1175–1193 (2001).
[Crossref]

Quant. Inf. (1)

M. J. Hu, Z. Y. Zhou, X. M. Hu, C. F. Li, G. C. Guo, and Y. S. Zhang, “Observation of non-locality sharing among three observers with one entangled pair via optimal weak measurement,” Quant. Inf. 4, 63 (2018).
[Crossref]

Quantum Stud.: Math. Found. (1)

A. Brodutch and E. Cohen, “A scheme for performing strong and weak sequential measurements of non-commuting observables,” Quantum Stud.: Math. Found. 4, 13–27 (2017).
[Crossref]

Science (1)

O. Hosten and P. Kwiat, “Observation of the spin Hall effect of light via weak measurements,” Science 319, 787–790 (2008).
[Crossref] [PubMed]

Other (3)

Y. Aharonov and D. Rohrlich, Quantum Paradoxes: Quantum Theory for the Perplexed (Wiley-VCH, 2005).
[Crossref]

W. Heisenberg, The Physical Principles of the Quantum Theory, (Dover Publications, 2003).

F. J. Curchod, M. Johansson, R. Augusiak, M. J. Hoban, P. Wittek, and A. Acín, “Entangled systems are unbounded sources of nonlocal correlations and of certified random numbers,” ArXiv:1802.07962 (2018).

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Figures (2)

Fig. 1
Fig. 1 Experimental setup for realizing sequential weak measurements of three non-commuting polarization observables of photons. The single photons are produced by generating a pair of photons via spontaneous parametric down conversion (SPDC) with idler photons used as triggers. The signal photons, after initial state preparation, are sent into weak measurement modules a, b, and c sequentially. After sequential weak measurements, post-selection and coincidence counting are performed. Modules a, b and c respectively realize weak measurements of polarization observables σ̂y, σ̂z and σ̂φ. Q1 is rotated at 0, Q3 is rotated at π/2, H1, H3 and H10 are rotated at π/8, H4, H8, H9, H11, H15, H16, H18, H22 and H23 are rotated at π/4, H17 and H24 are rotated at φ/2 (in our experiment, we take φ = π/3). H16, H12 and H20 are rotated at γ/2, while H15, H13 and H19 are rotated at −γ/2 (in our experiment, we measure SWV with coupling parameter γ = 25° and γ = 30°, respectively.) Q2, H7, Q4, H14, Q5, and H21 combining PBSs after them in the corresponding modules are used to make projective measurements on the corresponding pointers, and these wave plates are rotated at some angles according to which observables of these pointers would be measured.
Fig. 2
Fig. 2 (a) and (b) show the results of our experiment to measure SWV with coupling parameter γ = 25° and γ = 30°, respectively. The θ corresponds to the parameter of the post-selected state |ψf〉 = cosθ|H〉 + sinθ|V〉. Red and blue parts respectively represent the real and imaginary parts of SWVs that we measured. The boxes represent experimental data and dashed lines represent theoretical predictions rather than fitted curves. Error bars are evaluated on the basis of Poissonian counting statistical.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

| ϕ ˜ p = ψ f | e i γ A ^ σ ^ y | ψ i | 0 p ,
| φ ˜ p ψ f | ( 1 i γ A ^ σ ^ y ) | ψ i | 0 p ψ f | ψ i e i γ A ^ w σ ^ y | 0 p ,
σ ^ + p = 2 γ Re A ^ w ,
σ ^ R p = 2 γ Im A ^ w ,
| Ψ s p 1 p N = e i γ N A ^ N σ ^ y e i γ 1 A ^ 1 σ ^ y | ψ i | 0 p 1 | 0 p N .
| Φ p 1 p N = ψ f | Ψ sp 1 p N .
σ ^ + σ ^ + σ ^ + p 1 p 2 p 3 = 2 γ 3 Re [ A ^ 1 A ^ 2 A ^ 3 w + A ^ 1 A ^ 2 w A ^ 3 w + A ^ 1 A ^ 3 w A ^ 2 w + A ^ 2 A ^ 3 w A ^ 1 w ] .
σ ^ R σ ^ R σ ^ R p 1 p 2 p 3 = 2 γ 3 Im [ A ^ 1 A ^ 2 A ^ 3 w + A ^ 1 A ^ 2 w A ^ 3 w + A ^ 1 A ^ 3 w A ^ 2 w + A ^ 2 A ^ 3 w A ^ 1 w ] .
σ ^ + p = sin ( 2 γ ) Re ( σ ^ A w ) cos 2 ( γ ) + sin 2 ( γ ) | σ ^ A w | 2 ,
σ ^ R p = sin ( 2 γ ) Im ( σ ^ A w ) cos 2 ( γ ) + sin 2 ( γ ) | σ ^ A w | 2 ,
σ ^ + σ ^ + p 1 p 2 = 1 2 Re ( A ^ 1 A ^ 2 w + A ^ 2 w A ^ 1 w * ) sin 2 2 γ ) cos 4 γ + ( | A ^ 1 w | 2 + | A ^ 2 w | 2 ) cos 2 γ sin 2 γ + | A ^ 1 A ^ 2 w | 2 sin 4 γ
σ ^ R σ ^ + p 1 p 2 = 1 2 Im ( A ^ 1 A ^ 2 w A ^ 1 w A ^ 2 w * ) sin 2 2 γ ) cos 4 γ + ( | A ^ 1 w | 2 + | A ^ 2 w | 2 ) cos 2 γ sin 2 γ + | A ^ 1 A ^ 2 w | 2 sin 4 γ .
σ ^ + σ ^ + σ ^ + p 1 p 2 p 3 = 2 Re ( A ^ 1 A ^ 2 A ^ 3 w + A ^ 3 w * A ^ 1 A ^ 2 w + A ^ 2 w * A ^ 1 A ^ 3 w + A ^ 1 w * A ^ 2 A ^ 3 w ) tan 3 γ K
σ ^ R σ ^ R σ ^ R p 1 p 2 p 3 = 2 Im ( A ^ 1 A ^ 2 A ^ 3 w + A ^ 3 w * A ^ 1 A ^ 2 w + A ^ 2 w * A ^ 1 A ^ 3 w + A ^ 1 w * A ^ 2 A ^ 3 w ) tan 3 γ K ,
K = 1 + ( | A ^ 1 w | 2 + | A ^ 2 w | 2 + | A ^ 3 w | 2 ) tan 2 γ + ( | A ^ 1 A ^ 2 w | 2 + | A ^ 1 A ^ 3 w | 2 + | A ^ 2 A ^ 3 w | 2 ) tan 4 γ + | A ^ 1 A ^ 2 A ^ 3 w | 2 tan 6 γ .

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