Abstract

We show that metrological resolution in the detection of small phase shifts provides a suitable generalization of the degrees of coherence and polarization. Resolution is estimated via Fisher information. Besides the standard two-beam Gaussian case, this approach provides also good results for multiple field components and nonGaussian statistics. This works equally well in quantum and classical optics.

© 2012 OSA

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  2. J. W. Goodman, Statistical Optics (John Wiley and Sons Inc., 1985).
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    [CrossRef]
  4. H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563–1571 (2002).
    [CrossRef]
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  6. J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields” Opt. Express 11, 1137–1143 (2003).
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  11. F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
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  13. A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063–1068 (2007).
    [CrossRef]
  14. A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
    [CrossRef] [PubMed]
  15. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [CrossRef]
  16. I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” arXiv:0807.2171v1 [physics.optics].
  17. P. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008).
    [CrossRef] [PubMed]
  18. A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
    [CrossRef]
  19. P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25, 2749–2757 (2008).
    [CrossRef]
  20. R. Martínez-Herrero and P.M. Mejías, “Maximizing Youngs fringe visibility under unitary transformations for mean-square coherent light,” Opt. Express 17, 603–610 (2009).
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  21. A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. A 27, 1764–1769 (2010).
    [CrossRef]
  22. A. Luis, “Coherence versus interferometric resolution,” Phys. Rev. A 81, 065802 (2010).
    [CrossRef]
  23. A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics, International Commission for Optics, vol. VI, A. T. Friberg and R. Dändliker, eds. (SPIE, 2009) pp. 171–188.
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    [CrossRef]
  26. A. Luis, “Degree of polarization of type-II unpolarized light,” Phys. Rev. A 75, 053806 (2007)
    [CrossRef]
  27. A. Luis, “Polarization distributions and degree of polarization for quantum Gaussian light fields,” Opt. Commun. 273, 173–181 (2007).
    [CrossRef]
  28. A. Luis, “Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference,” Phys. Rev. A 79, 053855 (2009).
    [CrossRef]
  29. A. Picozzi, “Entropy and degree of polarization for nonlinear optical waves,” Opt. Lett. 29, 1653–1655 (2004).
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  31. P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. A 23, 671–678 (2006).
    [CrossRef]
  32. A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
    [CrossRef]
  33. A. Luis, “Quantum-limited metrology with nonlinear detection schemes,” SPIE Reviews 1, 018006 (2010).
  34. H. Cramér, Mathematical Methods of Statistics (Asia Publishing House, 1962).
  35. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
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    [CrossRef]
  37. B. R. Frieden, Physics from Fisher information: A Unification, (Cambridge U. Press, 1999).
  38. A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995).
    [CrossRef]
  39. T. M. Cover and J. A. Thomas, Elements of Information Theory, (Wiley Interscience, 1991).
    [CrossRef]
  40. A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
    [CrossRef]
  41. F. Goudail, P. Réfrégier, and G. Delyon, “Bhattacharyya distance as a contrast parameter for statistical processing of noisy optical images,” J. Opt. Soc. Am. A 21, 1231–1240 (2004).
    [CrossRef]
  42. G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
    [CrossRef]
  43. P. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117–3119 (2005).
    [CrossRef] [PubMed]
  44. P. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366–1368 (2007).
    [CrossRef] [PubMed]
  45. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
    [CrossRef] [PubMed]
  46. H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
    [CrossRef]
  47. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [CrossRef]
  48. A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
    [CrossRef] [PubMed]
  49. Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
    [CrossRef]
  50. A. Luis and L. L. Sánchez-Soto, “Quantum phase difference, phase measurements and Stokes operators,” in Progress in Optics, vol. 41, E. Wolf, ed. (Elsevier, Amsterdam, 2000), pp. 421–482.
    [CrossRef]
  51. A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
    [CrossRef]
  52. A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
    [CrossRef]
  53. R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
    [CrossRef] [PubMed]
  54. A. Luis, “Quantum mechanics as a geometric phase: phase-space interferometers,” J. Phys. A 34, 7677–7684 (2001).
    [CrossRef]
  55. A. Luis, “Classical mechanics and the propagation of the discontinuities of the quantum wave function,” Phys. Rev. A 67, 024102 (2003).
    [CrossRef]
  56. B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
    [CrossRef] [PubMed]

2012

2010

A. Luis, “Quantum-limited metrology with nonlinear detection schemes,” SPIE Reviews 1, 018006 (2010).

A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
[CrossRef]

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[CrossRef]

A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. A 27, 1764–1769 (2010).
[CrossRef]

A. Luis, “Coherence versus interferometric resolution,” Phys. Rev. A 81, 065802 (2010).
[CrossRef]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef] [PubMed]

A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
[CrossRef] [PubMed]

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef] [PubMed]

2009

R. Martínez-Herrero and P.M. Mejías, “Maximizing Youngs fringe visibility under unitary transformations for mean-square coherent light,” Opt. Express 17, 603–610 (2009).
[CrossRef] [PubMed]

A. Luis, “Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference,” Phys. Rev. A 79, 053855 (2009).
[CrossRef]

2008

A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
[CrossRef]

P. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008).
[CrossRef] [PubMed]

A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
[CrossRef]

P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25, 2749–2757 (2008).
[CrossRef]

2007

2006

P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. A 23, 671–678 (2006).
[CrossRef]

2005

2004

2003

E. Wolf, “Unified theory of coherence and polarization of random electromagnetic beams,” Phys. Lett. A 312, 263–267 (2003).
[CrossRef]

J. Tervo, T. Setälä, and A. T. Friberg, “Degree of coherence for electromagnetic fields” Opt. Express 11, 1137–1143 (2003).
[CrossRef] [PubMed]

A. Luis, “Classical mechanics and the propagation of the discontinuities of the quantum wave function,” Phys. Rev. A 67, 024102 (2003).
[CrossRef]

2002

A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
[CrossRef]

H. M. Ozaktas, S. Yüksel, and M. A. Kutay, “Linear algebraic theory of partial coherence: discrete fields and measures of partial coherence,” J. Opt. Soc. Am. A 19, 1563–1571 (2002).
[CrossRef]

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

2001

A. Luis, “Quantum mechanics as a geometric phase: phase-space interferometers,” J. Phys. A 34, 7677–7684 (2001).
[CrossRef]

1997

Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
[CrossRef]

1995

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995).
[CrossRef]

1994

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[CrossRef] [PubMed]

1987

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

1981

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

1963

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef] [PubMed]

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef] [PubMed]

Björk, G.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[CrossRef]

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[CrossRef]

Borghi, R.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[CrossRef] [PubMed]

Braunstein, S. L.

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[CrossRef] [PubMed]

Caves, C. M.

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[CrossRef] [PubMed]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[CrossRef]

Cintra, R. J.

A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
[CrossRef]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory, (Wiley Interscience, 1991).
[CrossRef]

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Asia Publishing House, 1962).

Delyon, G.

Espinoza, P.

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[CrossRef]

Frery, A. C.

A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
[CrossRef]

Friberg, A. T.

Frieden, B. R.

B. R. Frieden, Physics from Fisher information: A Unification, (Cambridge U. Press, 1999).

Ghiu, I.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[CrossRef]

Gil, J. J.

J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
[CrossRef]

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” arXiv:0807.2171v1 [physics.optics].

Goodman, J. W.

J. W. Goodman, Statistical Optics (John Wiley and Sons Inc., 1985).

Gori, F.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[CrossRef] [PubMed]

Goudail, F.

Karczewski, B.

B. Karczewski, “Degree of coherence of the electromagnetic field,” Phys. Lett. 5, 191–192 (1963).
[CrossRef]

Klimov, A. B.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[CrossRef]

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[CrossRef]

Kutay, M. A.

Lee, H.-W.

H.-W. Lee, “Theory and application of the quantum phase-space distribution functions,” Phys. Rep. 259, 147–211 (1995).
[CrossRef]

Luis, A.

A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
[CrossRef] [PubMed]

A. Luis, “Quantum-limited metrology with nonlinear detection schemes,” SPIE Reviews 1, 018006 (2010).

A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. A 27, 1764–1769 (2010).
[CrossRef]

A. Luis, “Coherence versus interferometric resolution,” Phys. Rev. A 81, 065802 (2010).
[CrossRef]

A. Luis, “Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference,” Phys. Rev. A 79, 053855 (2009).
[CrossRef]

A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
[CrossRef]

A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
[CrossRef]

P. Réfrégier and A. Luis, “Irreversible effects of random unitary transformations on coherence properties of partially polarized electromagnetic fields,” J. Opt. Soc. Am. A 25, 2749–2757 (2008).
[CrossRef]

A. Luis, “Degree of coherence for vectorial electromagnetic fields as the distance between correlation matrices,” J. Opt. Soc. Am. A 24, 1063–1068 (2007).
[CrossRef]

A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
[CrossRef] [PubMed]

A. Luis, “Degree of polarization of type-II unpolarized light,” Phys. Rev. A 75, 053806 (2007)
[CrossRef]

A. Luis, “Polarization distributions and degree of polarization for quantum Gaussian light fields,” Opt. Commun. 273, 173–181 (2007).
[CrossRef]

A. Luis, “Classical mechanics and the propagation of the discontinuities of the quantum wave function,” Phys. Rev. A 67, 024102 (2003).
[CrossRef]

A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
[CrossRef]

A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
[CrossRef]

A. Luis, “Quantum mechanics as a geometric phase: phase-space interferometers,” J. Phys. A 34, 7677–7684 (2001).
[CrossRef]

A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995).
[CrossRef]

A. Luis, “An overview of coherence and polarization properties for multicomponent electromagnetic waves,” in Advances in Information Optics and Photonics, International Commission for Optics, vol. VI, A. T. Friberg and R. Dändliker, eds. (SPIE, 2009) pp. 171–188.

A. Luis and L. L. Sánchez-Soto, “Quantum phase difference, phase measurements and Stokes operators,” in Progress in Optics, vol. 41, E. Wolf, ed. (Elsevier, Amsterdam, 2000), pp. 421–482.
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Marian, P.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[CrossRef]

Marian, T. A.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[CrossRef]

Martínez-Herrero, R.

Mejías, P.M.

Mukunda, N.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef] [PubMed]

R. Simon, E. C. G. Sudarshan, and N. Mukunda, “Gaussian-Wigner distributions in quantum mechanics and optics,” Phys. Rev. A 36, 3868–3880 (1987).
[CrossRef] [PubMed]

Nascimento, A. D. C.

A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
[CrossRef]

Ou, Z. Y.

Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
[CrossRef]

Ozaktas, H. M.

Picozzi, A.

Réfrégier, P.

Rivas, A.

A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
[CrossRef] [PubMed]

A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
[CrossRef]

Roueff, A.

San José, I.

I. San José and J. J. Gil, “Invariant indices of polarimetric purity. Generalized indices of purity for nxn covariance matrices,” arXiv:0807.2171v1 [physics.optics].

Sánchez-Soto, L. L.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
[CrossRef]

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[CrossRef]

A. Luis and L. L. Sánchez-Soto, “A quantum description of the beam splitter,” Quantum Semiclass. Opt. 7, 153–160 (1995).
[CrossRef]

A. Luis and L. L. Sánchez-Soto, “Quantum phase difference, phase measurements and Stokes operators,” in Progress in Optics, vol. 41, E. Wolf, ed. (Elsevier, Amsterdam, 2000), pp. 421–482.
[CrossRef]

Santarsiero, M.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
[CrossRef] [PubMed]

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
[CrossRef] [PubMed]

Sehat, A.

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
[CrossRef]

Setälä, T.

Silberberg, Y.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef] [PubMed]

Simon, B. N.

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Simon, S.

B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
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Soderholm, J.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
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Söderholm, J.

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
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A. D. C. Nascimento, R. J. Cintra, and A. C. Frery, “Hypothesis testing in speckled data with stochastic distances,” IEEE Trans. Geos. Remot. Sens. 48, 373–385 (2010).
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J. Opt. Soc. A

A. Luis, “Coherence and visibility for vectorial light,” J. Opt. Soc. A 27, 1764–1769 (2010).
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P. Réfrégier and F. Goudail, “Kullback relative entropy and characterization of partially polarized optical waves,” J. Opt. Soc. A 23, 671–678 (2006).
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J. Phys. A

A. Luis, “Quantum mechanics as a geometric phase: phase-space interferometers,” J. Phys. A 34, 7677–7684 (2001).
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J. Phys. A: Math. Gen.

A. Luis, “Visibility for anharmonic fringes,” J. Phys. A: Math. Gen. 35, 8805–8815 (2002).
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Opt. Commun.

G. Björk, J. Soderholm, L. L. Sánchez-Soto, A. B. Klimov, I. Ghiu, P. Marian, and T. A. Marian, “Quantum degrees of polarization,” Opt. Commun. 283, 4440–4447 (2010).
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A. Luis, “Polarization distributions and degree of polarization for quantum Gaussian light fields,” Opt. Commun. 273, 173–181 (2007).
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Opt. Express

Opt. Lett.

F. Gori, M. Santarsiero, and R. Borghi, “Maximizing Young’s fringe visibility through reversible optical transformations,” Opt. Lett. 32, 588–590 (2007).
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R. Martínez-Herrero and P.M. Mejías, “Maximum visibility under unitary transformations in two-pinhole interference for electromagnetic fields,” Opt. Lett. 32, 1471–1473 (2007).
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A. Luis, “Maximum visibility in interferometers illuminated by vectorial waves,” Opt. Lett. 32, 2191–2193 (2007).
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P. Réfrégier, “Mean-square coherent light,” Opt. Lett. 33, 1551–1553 (2008).
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P. Réfrégier, “Mutual information-based degrees of coherence of partially polarized light with Gaussian fluctuations,” Opt. Lett. 30, 3117–3119 (2005).
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P. Réfrégier and A. Roueff, “Visibility interference fringes optimization on a single beam in the case of partially polarized and partially coherent light,” Opt. Lett. 32, 1366–1368 (2007).
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Phys. Rev. A

A. Luis, “Classical mechanics and the propagation of the discontinuities of the quantum wave function,” Phys. Rev. A 67, 024102 (2003).
[CrossRef]

A. Sehat, J. Söderholm, G. Björk, P. Espinoza, A. B. Klimov, and L. L. Sánchez-Soto, “Quantum polarization properties of two-mode energy eigenstates,” Phys. Rev. A 71, 033818 (2005).
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A. Luis, “Quantum-classical correspondence for visibility, coherence, and relative phase for multidimensional systems,” Phys. Rev. A 78, 025802 (2008).
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A. Rivas and A. Luis, “Characterization of quantum angular-momentum fluctuations via principal components,” Phys. Rev. A 77, 022105 (2008).
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A. Luis, “Degree of polarization in quantum optics,” Phys. Rev. A 66, 013806 (2002).
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A. Luis, “Degree of polarization of type-II unpolarized light,” Phys. Rev. A 75, 053806 (2007)
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A. Luis, “Ensemble approach to coherence between two scalar harmonic light vibrations and the phase difference,” Phys. Rev. A 79, 053855 (2009).
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A. Luis, “Coherence versus interferometric resolution,” Phys. Rev. A 81, 065802 (2010).
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B. N. Simon, S. Simon, F. Gori, M. Santarsiero, R. Borghi, N. Mukunda, and R. Simon, “Nonquantum entanglement resolves a basic issue in polarization optics,” Phys. Rev. Lett. 104, 023901 (2010).
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I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
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[CrossRef]

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Equations (60)

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V ( E ) = 2 | E 1 E 2 * | | E 1 | 2 + | E 2 | 2 | μ ( E ) | = | E 1 E 2 * | ( | E 1 | 2 | E 2 | 2 ) 1 / 2 ,
𝒫 = | I M I m | I M + I m = [ 1 4 det Γ ( tr Γ ) 2 ] 1 / 2 , Γ ( E ) = ( | E 1 | 2 E 1 E 2 * E 1 * E 2 | E 2 | 2 ) ,
V 2 + R 2 = 𝒫 2 , R ( E ) = | | E 1 | 2 | E 2 | 2 | E 1 | 2 + | E 2 | 2 | .
| μ ( U E ) | 𝒫 .
P ( ϕ ˜ | θ ) Π j = 1 N P ( θ j | ϕ ˜ ) Φ ( ϕ ˜ ) ,
( Δ ϕ ) 2 1 N F , F = d θ 1 P ( θ | ϕ ) [ d P ( θ | ϕ ) d ϕ ] 2 .
H = d θ [ P ( θ | ϕ + δ ϕ ) P ( θ | ϕ ) ] 2 ( δ ϕ ) 2 4 F ,
K = d θ P ( θ | ϕ + δ ϕ ) ln P ( θ | ϕ + δ ϕ ) P ( θ | ϕ ) ( δ ϕ ) 2 2 F ,
C s = ln d θ P s ( θ | ϕ + δ ϕ ) P 1 s ( θ | ϕ ) s ( 1 s ) 2 ( δ ϕ ) 2 F ,
χ 2 = F F + 4 tr ( G 2 ) .
P ( θ | ϕ ) = d 2 k E W ( U ϕ E ) δ [ θ θ ( E ) ] .
P ( θ | ϕ ) = d ϑ 𝒲 ( θ , ϑ ) ,
F F C = d 2 k E 1 W ( U ϕ E ) [ d W ( U ϕ E ) d ϕ ] 2 .
χ 2 = F C F C + 4 tr ( G 2 ) ,
W ( E ) = det M π k exp ( E M E ) ,
F C = 2 [ tr ( Γ G Γ 1 G ) tr ( G 2 ) ] .
F C = 4 i , j = 1 k | G i , j | 2 𝒫 i , j 2 1 𝒫 i , j 2 4 tr ( G 2 ) 𝒫 max 2 1 𝒫 max 2 ,
𝒫 i , j = | I i I j | I i + I j 𝒫 max = I M I m I m + I m ,
χ 𝒫 max
Γ = ( | E 1 | 2 E 1 E 2 * E 1 * E 2 | E 2 | 2 ) , G ( 1 0 0 1 )
F C = 4 tr ( G 2 ) | μ | 2 1 | μ | 2 , χ = | μ | ,
G ( I 0 0 I ) ,
Γ = ( Γ 1 ϒ ϒ Γ 2 ) = Γ 0 1 / 2 U ( I Ω Ω I ) U Γ 0 1 / 2 ,
Γ 0 = ( Γ 1 0 0 Γ 2 ) , Γ j = ( | E j , x | 2 E j , x E j , y * E j , x * E j , y | E j , y | 2 )
U = ( U 1 0 0 U 2 ) , Ω = ( μ S 0 0 μ I ) ,
F C = 2 tr ( G 2 ) ( 1 1 μ S 2 + 1 1 μ I 2 2 ) , χ 2 = μ S 2 + μ I 2 2 μ S 2 μ I 2 2 μ S 2 μ I 2 .
F F Q = 2 i , j ( p i p j ) 2 p i + p j | ψ i | G ^ | ψ j | 2 ,
ρ = i p i | ψ i ψ i | .
F Q = 4 ( Δ G ^ ) 2 .
χ 2 = F Q F Q + 4 tr ( G 2 ) .
ρ i = 1 1 + n ¯ i n = 0 ( n ¯ i 1 + n ¯ i ) n | n i i n | ,
F Q = 4 tr ( G 2 ) 𝒫 2 1 𝒫 2 + 2 / n ¯ , χ = 𝒫 ( n ¯ n ¯ + 2 ) 1 / 2 ,
𝒫 = | n ¯ 1 n ¯ 2 | n ¯ 1 + n ¯ 2 , n ¯ = n ¯ 1 + n ¯ 2 .
| ψ = 1 2 ( | n 1 | 0 2 + | 0 1 | n 2 ) .
P ( θ ) = | θ | ψ | 2 , | θ = 1 2 π m = 0 N e im θ | n m 1 | m 2 ,
P ( θ | ϕ ) = 1 π cos 2 [ n ( θ ϕ ) / 2 ] ,
P j = k W j , k ,
F = j P j 2 P j = j P j ( ln P j ) = j , k W j , k ( ln P j ) , F C = j , k W j , k 2 W j , k = j , k W j , k ( ln W j , k ) ,
F C F = j , k W j , k ( ln W j , k P j ) = j P j k Λ j , k ( ln Λ j , k ) ,
Λ j , k = W j , k P j , k Λ j , k = 1 .
F C F = j P j k Λ j , k 2 Λ j , k 0 .
H P = 2 ( 1 j P j P ˜ j ) , H W = 2 ( 1 j , k W j , k W ˜ j , k ) .
P j P ˜ j = k , W j , k W ˜ j , = 1 2 k , ( W j , k W ˜ j , + W j , W ˜ j , k ) ,
( k W j , k W ˜ j , k ) 2 = k , W j , k W ˜ j , W j , W ˜ j , k ,
W ( U ϕ E ) = det M π k exp ( E U ϕ M U ϕ E ) ,
W ( U ϕ E ) d ϕ | ϕ = 0 = E M E W ( E ) ,
M = d ( U ϕ M U ϕ ) d ϕ | ϕ = 0 = i [ G , M ] .
F C = d 2 k E ( E M E ) 2 W ( E ) = i , j , , m = 1 k M i , j M , m d 2 k E E i * E j E * E m W ( E ) .
E j E m E i * E * = E j E i * E m E * + E m E i * E j E * ,
F C = [ tr ( M M 1 ) ] 2 + tr [ ( M M 1 ) 2 ] = tr [ ( M M 1 ) 2 ] ,
F C = 2 [ tr ( M 1 G M G ) tr ( G 2 ) ] .
F C = i , j ( I i I j ) 2 I i I j | G i , j | 2 ,
F C = 4 i , j = 1 k | G i , j | 2 𝒫 i , j 2 1 𝒫 i , j 2 4 tr ( G 2 ) 𝒫 max 2 1 𝒫 max 2 ,
𝒫 i , j = | I i I j | I i + I j 𝒫 max = I M I m I M + I m ,
| ψ i | G ^ | ψ j | 2 = | n 1 | n 2 | [ exp ( i φ ) a 1 a 2 + exp ( i φ ) a 1 a 2 ] | n 1 | n 2 | 2 .
n 1 | n 2 | [ exp ( i φ ) a 1 a 2 + exp ( i φ ) a 1 a 2 ] | n 1 | n 2 = exp ( i φ ) ( n 1 + 1 ) n 2 δ n 1 = n 1 + 1 δ n 2 = n 2 1 + exp ( i φ ) n 1 ( n 2 + 1 ) δ n 1 = n 1 1 δ n 2 = n 2 + 1 .
F Q = 2 n 1 , n 2 = 0 ( p n 1 + 1 p n 2 1 p n 1 p n 2 ) 2 p n 1 + 1 p n 2 1 + p n 1 p n 2 ( n 1 + 1 ) n 2 + exchange 1 2 ,
p n i = 1 1 + n ¯ i ( n ¯ i 1 + n ¯ i ) n i .
F Q = 4 ( n ¯ 1 n ¯ 2 ) 2 2 n ¯ 1 n ¯ 2 + n ¯ 1 + n ¯ 2 .
F Q = 4 𝒫 2 n ¯ 2 n ¯ 2 ( 1 𝒫 2 ) / 2 + n ¯ = 4 tr ( G 2 ) 𝒫 2 1 𝒫 2 + 2 / n ¯ ,

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