Abstract

This paper constitutes an application of the polarization optics in the problem of quantum measurement. The non-Hermitian operators of the nonorthogonal multilayer optical polarizers represent observables in the sense of the generalized quantum theory of measurement. The intimate spectral structure of these polarizers can be disclosed in the frame of skew-angular vector bases and biorthonormal vector systems. We show that these polarizers correspond to skew projectors; their operators are “generated” by skew projectors in the sense of the spectral theorem of linear operators theory. Thus the common feature of all the polarizers (Hermitian and non-Hermitian) is that their “nuclei” are (orthogonal or skew) projectors—the generating projectors.

© 2014 Optical Society of America

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  1. P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).
  2. P. Busch, M. Grabowski, and P. J. Lahti, Operatorial Quantum Mechanics (Springer, 1995).
  3. E. B. Davies, Quantum Theory of Open Systems (Academic, 1976).
  4. W. M. de Muynck, Foundations of Quantum Mechanics, an Empiricist Approach (Kluwer Academic, 2002).
  5. J. Preskill, “Lectures notes for Physics 229: quantum information and computation,” 2001, http://www.theory.caltech.edu/people/preskill/ph229/#lecture .
  6. E. B. Davies and J. T. Lewis, “An operatorial approach to quantum probability,” Commun. Math. Phys. 17, 239–260 (1970).
    [CrossRef]
  7. P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, 1974).
  8. C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. I.
  9. A. Peres, “Quantum theory: concept and methods,” in Fundamental Theories of Physics, 57, (Kluwer Academic, 1993).
  10. J. W. Simmons and M. J. Guttmann, States, Waves and Photons—A Modern Introduction to Light (Addison-Wesley, 1970).
  11. J. J. Gil, “Polarimetric characterization of light and media,” Eur. Phys. J. Appl. Phys. 40, 1–47 (2007).
    [CrossRef]
  12. T. Tudor, “Dirac-algebraic approach to the theory of device operators in polarization optics,” J. Opt. Soc. Am. A 20, 728–732 (2003).
    [CrossRef]
  13. S.-Y. Lu and R. A. Chipman, “Homogeneous and inhomogeneous Jones matrices,” J. Opt. Soc. Am. A 11, 766–773 (1994).
    [CrossRef]
  14. T. Tudor, “Generalized observables in polarization optics,” J. Phys. A 36, 9577–9590 (2003).
    [CrossRef]
  15. S. Pancharatnam, “Generalized theory of interference and its applications. I—coherent pencils,” Proc. Indian Acad. Sci. 44A, 247–262 (1956).
  16. M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. Roy. Soc. London A 459, 1261–1292 (2003).
    [CrossRef]
  17. W. M. de Muynck, “An alternative to the Lüders generalization of the von Neumann projection, and its interpretation,” J. Phys. A 31, 431–444 (1998).
    [CrossRef]
  18. B. Higman, Applied Group-Theoretic and Matrix Methods (Clarendon, 1955).
  19. M. V. Berry and S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
    [CrossRef]
  20. O. V. Angelsky, S. G. Hanson, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology of the degree of coherence of optical waves,” Opt. Express 17, 15623–15634 (2009).
    [CrossRef]
  21. H. P. Yuen, “Generalized quantum measurements and approximate simultaneous measurement of noncommuting observables,” Phys. Lett. 91A, 101–104 (1982).
    [CrossRef]

2009 (1)

2007 (1)

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

2003 (3)

T. Tudor, “Generalized observables in polarization optics,” J. Phys. A 36, 9577–9590 (2003).
[CrossRef]

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. Roy. Soc. London A 459, 1261–1292 (2003).
[CrossRef]

T. Tudor, “Dirac-algebraic approach to the theory of device operators in polarization optics,” J. Opt. Soc. Am. A 20, 728–732 (2003).
[CrossRef]

1998 (1)

W. M. de Muynck, “An alternative to the Lüders generalization of the von Neumann projection, and its interpretation,” J. Phys. A 31, 431–444 (1998).
[CrossRef]

1996 (1)

M. V. Berry and S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
[CrossRef]

1994 (1)

1982 (1)

H. P. Yuen, “Generalized quantum measurements and approximate simultaneous measurement of noncommuting observables,” Phys. Lett. 91A, 101–104 (1982).
[CrossRef]

1970 (1)

E. B. Davies and J. T. Lewis, “An operatorial approach to quantum probability,” Commun. Math. Phys. 17, 239–260 (1970).
[CrossRef]

1956 (1)

S. Pancharatnam, “Generalized theory of interference and its applications. I—coherent pencils,” Proc. Indian Acad. Sci. 44A, 247–262 (1956).

Angelsky, O. V.

Berry, M. V.

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. Roy. Soc. London A 459, 1261–1292 (2003).
[CrossRef]

M. V. Berry and S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
[CrossRef]

Busch, P.

P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

P. Busch, M. Grabowski, and P. J. Lahti, Operatorial Quantum Mechanics (Springer, 1995).

Chipman, R. A.

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. I.

Davies, E. B.

E. B. Davies and J. T. Lewis, “An operatorial approach to quantum probability,” Commun. Math. Phys. 17, 239–260 (1970).
[CrossRef]

E. B. Davies, Quantum Theory of Open Systems (Academic, 1976).

de Muynck, W. M.

W. M. de Muynck, “An alternative to the Lüders generalization of the von Neumann projection, and its interpretation,” J. Phys. A 31, 431–444 (1998).
[CrossRef]

W. M. de Muynck, Foundations of Quantum Mechanics, an Empiricist Approach (Kluwer Academic, 2002).

Dennis, M. R.

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. Roy. Soc. London A 459, 1261–1292 (2003).
[CrossRef]

Dirac, P. A. M.

P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, 1974).

Diu, B.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. I.

Gil, J. J.

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

Gorodyns’ka, N. V.

Gorsky, M. P.

Grabowski, M.

P. Busch, M. Grabowski, and P. J. Lahti, Operatorial Quantum Mechanics (Springer, 1995).

Guttmann, M. J.

J. W. Simmons and M. J. Guttmann, States, Waves and Photons—A Modern Introduction to Light (Addison-Wesley, 1970).

Hanson, S. G.

Higman, B.

B. Higman, Applied Group-Theoretic and Matrix Methods (Clarendon, 1955).

Klein, S.

M. V. Berry and S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
[CrossRef]

Lahti, P. J.

P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

P. Busch, M. Grabowski, and P. J. Lahti, Operatorial Quantum Mechanics (Springer, 1995).

Laloë, F.

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. I.

Lewis, J. T.

E. B. Davies and J. T. Lewis, “An operatorial approach to quantum probability,” Commun. Math. Phys. 17, 239–260 (1970).
[CrossRef]

Lu, S.-Y.

Mittelstaedt, P.

P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference and its applications. I—coherent pencils,” Proc. Indian Acad. Sci. 44A, 247–262 (1956).

Peres, A.

A. Peres, “Quantum theory: concept and methods,” in Fundamental Theories of Physics, 57, (Kluwer Academic, 1993).

Simmons, J. W.

J. W. Simmons and M. J. Guttmann, States, Waves and Photons—A Modern Introduction to Light (Addison-Wesley, 1970).

Tudor, T.

Yuen, H. P.

H. P. Yuen, “Generalized quantum measurements and approximate simultaneous measurement of noncommuting observables,” Phys. Lett. 91A, 101–104 (1982).
[CrossRef]

Zenkova, C. Yu.

Commun. Math. Phys. (1)

E. B. Davies and J. T. Lewis, “An operatorial approach to quantum probability,” Commun. Math. Phys. 17, 239–260 (1970).
[CrossRef]

Eur. Phys. J. Appl. Phys. (1)

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

J. Mod. Opt. (1)

M. V. Berry and S. Klein, “Geometric phases from stacks of crystal plates,” J. Mod. Opt. 43, 165–180 (1996).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Phys. A (2)

W. M. de Muynck, “An alternative to the Lüders generalization of the von Neumann projection, and its interpretation,” J. Phys. A 31, 431–444 (1998).
[CrossRef]

T. Tudor, “Generalized observables in polarization optics,” J. Phys. A 36, 9577–9590 (2003).
[CrossRef]

Opt. Express (1)

Phys. Lett. (1)

H. P. Yuen, “Generalized quantum measurements and approximate simultaneous measurement of noncommuting observables,” Phys. Lett. 91A, 101–104 (1982).
[CrossRef]

Proc. Indian Acad. Sci. (1)

S. Pancharatnam, “Generalized theory of interference and its applications. I—coherent pencils,” Proc. Indian Acad. Sci. 44A, 247–262 (1956).

Proc. Roy. Soc. London A (1)

M. V. Berry and M. R. Dennis, “The optical singularities of birefringent dichroic chiral crystals,” Proc. Roy. Soc. London A 459, 1261–1292 (2003).
[CrossRef]

Other (10)

B. Higman, Applied Group-Theoretic and Matrix Methods (Clarendon, 1955).

P. A. M. Dirac, The Principles of Quantum Mechanics, 3rd ed. (Clarendon, 1974).

C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics (Wiley, 1977), Vol. I.

A. Peres, “Quantum theory: concept and methods,” in Fundamental Theories of Physics, 57, (Kluwer Academic, 1993).

J. W. Simmons and M. J. Guttmann, States, Waves and Photons—A Modern Introduction to Light (Addison-Wesley, 1970).

P. Busch, P. J. Lahti, and P. Mittelstaedt, The Quantum Theory of Measurement (Springer, 1996).

P. Busch, M. Grabowski, and P. J. Lahti, Operatorial Quantum Mechanics (Springer, 1995).

E. B. Davies, Quantum Theory of Open Systems (Academic, 1976).

W. M. de Muynck, Foundations of Quantum Mechanics, an Empiricist Approach (Kluwer Academic, 2002).

J. Preskill, “Lectures notes for Physics 229: quantum information and computation,” 2001, http://www.theory.caltech.edu/people/preskill/ph229/#lecture .

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Figures (2)

Fig. 1.
Fig. 1.

Biorthonormal eigensystem corresponding to the operator P.

Fig. 2.
Fig. 2.

Skew projector T1 gives the projection of the state vector |S on |E1=|Pθ along |E2=|Py.

Equations (42)

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

i|SiSi|=I,
D=IDI=i,j=12|SiSi|D|SjSj|=i,j=12Si|D|Sj|SiSj|.
D|Si=λi|Si,
Si|Sj=δij,
D=i=12λi|SiSi|,
D=λM|EMEM|+λm|EmEm|,
P|Px=|PxPx|,
P|Pθ=|PθPθ|,
P|R=|RR|,
R|Pθ(δ)=eiδ/2|PθPθ|+eiδ/2|Pθ+90°Pθ+90°|.
Px|Pθ=Pθ|Px=cosθ,
Py|Pθ=Pθ|Py=sinθ,
L|P45°=P45°|L*=12(1+i)=12eiπ/4,
R|P45°=P45°|R*=12(1+i)=12eiπ/4,
P|Pθ=cos2θ|PxPx|+sinθcosθ[|PxPy|+|PyPx|]+sin2θ|PyPy|,
R|Pθ(δ)=[cos2θeiδ/2+sin2θeiδ/2]|PxPx|+2isinθcosθsinδ2[|PxPy|+|PyPx|]+[sin2θeiδ/2+cos2θeiδ/2]|PyPy|=(cosδ2+icos2θsinδ2)|PxPx|+isin2θsinδ2[|PxPy|+|PyPx|]+(cosδ2icos2θsinδ2)|PyPy|.
Fj|Ei=δij.
|EiFi|
D=λ1|E1F1|+λ2|E2F2|.
P=P|PθP|Px=|PθPθ|PxPx|=cosθ|PθPx|.
|E1=|Pθ,withλ1=cos2θ,
|E2=|Py,withλ2=0.
F1|E2=F1|Py=0F1|E1=F1|Pθ=1|F1=1cosθ|Px.
|F2=1cosθ|Pθ+90°.
Ti=|EiFi|,
T1=|E1F1|=1cosθ|PθPx|,
T2=|E2F2|=1cosθ|PyPθ+90°|.
T1T2=0,
T1+T2=1cosθ{|PθPx|+|PyPθ+90°|}=1cosθ{[cosθ|Px+sinθ|Py]Px|+|Py[sinθPx|+cosθPy|]}=|PxPx|+|PyPy|=I,
T1|S=1cosθ|PθPx|S=Px|Scosθ|Pθ.
T2|S=1cosθ|PyPθ+90°|S=Pθ+90°|Scosθ|Py.
P=λ1T1,
C=R|Px(π/2)P|P45°.
C=[eiπ4|PxPx|+eiπ4|PyPy|]12[1+|PxPy|+|PyPx|]=12[eiπ4|PxPx|+eiπ4|PyPy|+eiπ4|PxPy|+eiπ4|PyPx|].
C=12eiπ4|Px[Px|+Py|]+12eiπ4|Py[Px|+Py|]=12eiπ4[|Px+eiπ2|Py][Px|+Py|]=eiπ412[|Pxi|Py]12[Px|+Py|]=eiπ4|LP45°|.
|E1=|Lwithλ1=eiπ4P45°|L=12,
|E2=|P45°withλ2=0.
F1|E2=F1|P45°=0F1|E1=F1|L=1|F1=2eiπ/4|P45°,
F2|E1=F2|L=0F2|E2=F2|P45°=1|F2=2eiπ/4|R.
T1=|E1F1|=2eiπ4|LP45°|
T2=|E2F2|=2eiπ4|P45°R|.
C=λ1T1,

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