Abstract

The advent of quantum computers threatens the security of conventional encryption schemes (e.g., those based upon the excessive amount of computational time that might be required to guess your password). Quantum encryption is intended to restore the security by basing it instead upon the impossibility of the simultaneous measurement of two noncommuting operators. I derive a measurement associated with the angular momentum lowering operator, which describes a simultaneous (yet realizable) measurement of two noncommuting spin-vector components. Correlations between two such detectors are also discussed and compared with the conventional Stern–Gerlach results.

© 2006 Optical Society of America

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References

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  1. J. J. Sakurai, Modern Quantum Mechanics (Benjamin, 1985).
  2. R. J. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
    [CrossRef]
  3. H. Yuen and J. Shapiro, "Optical communication with two-photon coherent states-Part III: Quantum measurements re photoemissive detectors," IEEE Trans. Inf. Theory IT-26, 78-92 (1980).
  4. Y. Ozhigov, and M. V. Lomonosov eds., First International Symposium on Quantum Informatics, Proc. SPIE 5128 (2003).
  5. C. Williams, ed., First NASA International Conference on Quantum Computing and Quantum Communications (Springer, 1998).
  6. J. S. Bell, "On the Einstein Podolsky Rosen theory," Physics 1, 195-200 (1964).
  7. C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).
  8. P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173-179 (1986).
    [CrossRef]
  9. M. A. Naimark, "Spectral functions of a symmetric operator," Izu. Akad. Nauk SSSR Ser. Mat. 4, 277-318 (1940), in Russian.
  10. S. R. Shepard, "A quantum theory of angle," Ann. N.Y. Acad. Sci. 755, 812-821 (1995).
    [CrossRef]
  11. P. Busch, "Unsharp reality and joint measurements for spin observables," Phys. Rev. D 33, 2253-2261 (1986).
    [CrossRef]

1995

S. R. Shepard, "A quantum theory of angle," Ann. N.Y. Acad. Sci. 755, 812-821 (1995).
[CrossRef]

1986

P. Busch, "Unsharp reality and joint measurements for spin observables," Phys. Rev. D 33, 2253-2261 (1986).
[CrossRef]

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173-179 (1986).
[CrossRef]

1964

J. S. Bell, "On the Einstein Podolsky Rosen theory," Physics 1, 195-200 (1964).

1963

R. J. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

1940

M. A. Naimark, "Spectral functions of a symmetric operator," Izu. Akad. Nauk SSSR Ser. Mat. 4, 277-318 (1940), in Russian.

Aspect, A.

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173-179 (1986).
[CrossRef]

Bell, J. S.

J. S. Bell, "On the Einstein Podolsky Rosen theory," Physics 1, 195-200 (1964).

Busch, P.

P. Busch, "Unsharp reality and joint measurements for spin observables," Phys. Rev. D 33, 2253-2261 (1986).
[CrossRef]

Glauber, R. J.

R. J. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Grangier, P.

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173-179 (1986).
[CrossRef]

Helstrom, C. W.

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

Lomonosov, M. V.

Y. Ozhigov, and M. V. Lomonosov eds., First International Symposium on Quantum Informatics, Proc. SPIE 5128 (2003).

Naimark, M. A.

M. A. Naimark, "Spectral functions of a symmetric operator," Izu. Akad. Nauk SSSR Ser. Mat. 4, 277-318 (1940), in Russian.

Ozhigov, Y.

Y. Ozhigov, and M. V. Lomonosov eds., First International Symposium on Quantum Informatics, Proc. SPIE 5128 (2003).

Roger, G.

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173-179 (1986).
[CrossRef]

Sakurai, J. J.

J. J. Sakurai, Modern Quantum Mechanics (Benjamin, 1985).

Shapiro, J.

H. Yuen and J. Shapiro, "Optical communication with two-photon coherent states-Part III: Quantum measurements re photoemissive detectors," IEEE Trans. Inf. Theory IT-26, 78-92 (1980).

Shepard, S. R.

S. R. Shepard, "A quantum theory of angle," Ann. N.Y. Acad. Sci. 755, 812-821 (1995).
[CrossRef]

Williams, C.

C. Williams, ed., First NASA International Conference on Quantum Computing and Quantum Communications (Springer, 1998).

Yuen, H.

H. Yuen and J. Shapiro, "Optical communication with two-photon coherent states-Part III: Quantum measurements re photoemissive detectors," IEEE Trans. Inf. Theory IT-26, 78-92 (1980).

Ann. N.Y. Acad. Sci.

S. R. Shepard, "A quantum theory of angle," Ann. N.Y. Acad. Sci. 755, 812-821 (1995).
[CrossRef]

Europhys. Lett.

P. Grangier, G. Roger, and A. Aspect, "Experimental evidence for photon anticorrelation effect on a beam splitter: a new light on single-photon interferences," Europhys. Lett. 1, 173-179 (1986).
[CrossRef]

Izu. Akad. Nauk SSSR

M. A. Naimark, "Spectral functions of a symmetric operator," Izu. Akad. Nauk SSSR Ser. Mat. 4, 277-318 (1940), in Russian.

Phys. Rev.

R. J. Glauber, "The quantum theory of optical coherence," Phys. Rev. 130, 2529-2539 (1963).
[CrossRef]

Phys. Rev. D

P. Busch, "Unsharp reality and joint measurements for spin observables," Phys. Rev. D 33, 2253-2261 (1986).
[CrossRef]

Physics

J. S. Bell, "On the Einstein Podolsky Rosen theory," Physics 1, 195-200 (1964).

Other

C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, 1976).

H. Yuen and J. Shapiro, "Optical communication with two-photon coherent states-Part III: Quantum measurements re photoemissive detectors," IEEE Trans. Inf. Theory IT-26, 78-92 (1980).

Y. Ozhigov, and M. V. Lomonosov eds., First International Symposium on Quantum Informatics, Proc. SPIE 5128 (2003).

C. Williams, ed., First NASA International Conference on Quantum Computing and Quantum Communications (Springer, 1998).

J. J. Sakurai, Modern Quantum Mechanics (Benjamin, 1985).

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Equations (21)

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m ̃ x m ̃ x m ̃ x P m ̃ x m ̃ y m ̃ y m ̃ y P m ̃ y
m ̃ x = J ^ x / 2 m ̃ y = J ^ y / 2 ,
ψ ( ϕ ) m ψ m e i m ϕ ,
ψ ( ϕ x , ϕ y ) m x m y ψ ( e ) e i m x ϕ x e i m y ϕ y ,
P ( ϕ ˜ x ) π π | ψ ( ϕ x , ϕ y ) | 2 d ϕ ˜ y
P ( ϕ ˜ y ) π π | ψ ( ϕ x , ϕ y ) | 2 d ϕ ˜ x ,
ϕ ˜ x π π ϕ ˜ x P ( ϕ ˜ x ) d ϕ ˜ x ϕ ˜ y π π ϕ ˜ y P ( ϕ ˜ y ) d ϕ ˜ y
ϕ ˜ x = ϕ x / 2 ϕ ˜ y = ϕ y / 2 ,
P ( m x     a , m y     b ) m y     a m x     b P ( e a , e b ) = 1 / 4 ,
P ( e ϕ     a , e b ) = 1 / 16 [ 1 2 ( m ζ     a m x     b + m η     a m y     b ) cos ( ϕ ) 2 ( m ζ     a m y     b m η     a m x     b ) sin ( ϕ ) ] ,
P ( η + , y + ) = 1 4 [ 1 1 2 cos ( ϕ ) ]
P ( η + , x + ) = 1 4 [ 1 1 2 sin ( ϕ ) ] ,
P ( η + , y + ) P ( η + , x + ) + P ( x + , y + )
J ^ | + = | , J ^ | = 0.
E ^ | E = e | E = e ( | m + | e ) ,
ψ + ( e ) = ( e / ) ψ - ( e ) ,
e = ζ ( m x i m y ) ,
e | e e | = ζ 1 I ^ ,
e | e = 1 ( e ) ,
| e = 1 / 2 ( e | + + | )     e = 2 ( m x i m y )
P ( e ) = | e | ψ | 2 2 ( | ψ on    H s ) .

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