Abstract

We show how one can be led from considerations of quantum steering to Bell’s theorem. We begin with Einstein’s demonstration that, assuming local realism, quantum states must be in a many-to-one (“incomplete”) relationship with the real physical states of the system. We then consider some simple constraints that local realism imposes on any such incomplete model of physical reality, and show they are not satisfiable. In particular, we present a very simple demonstration for the absence of a local hidden variable incomplete description of nature by steering to two ensembles, one of which contains a pair of nonorthogonal states. Historically this is not how Bell’s theorem arose—there are slight and subtle differences in the arguments—but it could have been.

© 2015 Optical Society of America

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References

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  1. A. Einstein, Letter to E. Schrödinger, Einstein Archives Call Number EA 22-47 (1935), quoted translations from [3], http://alberteinstein.info/vufind1/Record/EAR000024019.
  2. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
    [Crossref]
  3. D. Howard, “Einstein on locality and separability,” Stud. Hist. Phil. Sci. 16, 171–201 (1985).
    [Crossref]
  4. D. Howard, “Einstein, Schopenhauer and the historical background of the conception of space as a ground for the individuation of physical systems,” in The Cosmos of Science, J. Earman and J. Norton, eds. (University of Pittsburgh, 1999). The present paper is based heavily around the commentary in this reference and [3].
  5. G. Bacciagaluppi and E. Crull, The Einstein Paradox: the Debate on Nonlocality and Incompleteness in 1935 (to be published).
  6. To be more concrete, paraphrasing the translation of the letter in [5], Einstein says consider an entangled state expanded in two ways: ψAB=∑mncmnψm(x1)χn(x2)=∑mncmn̲ψm(x1)̲χn(x2),where ψ(x1), ψ(x1)̲, χ(x2) are “eigen-ψ” of commuting systems of observables α,α̲,β. If one makes an α (resp. α̲) measurement on A the state on B reduces to ψB=∑ncmnχn(x2) (resp. ψ̲B=∑ncmn̲χn(x2)) and all that is required for us to conclude that the “ψ-description” is not in one-to-one correspondence to the real state is that ψB, ψ̲B are “at all different from each other.”
  7. E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
    [Crossref]
  8. E. Schrödinger, “Probability relations between separated systems,” Proc. Cambridge Philos. Soc. 32, 446–452 (1936).
    [Crossref]
  9. The term “steering” was originally used by Schrödinger in his study [7,8] of the set ensembles of quantum states that a remote system could be collapsed to, given some (pure) initial entangled quantum state. In fact Schrödinger proved the theorem only for ensembles of (possibly nonorthogonal) states that are linearly independent, but this will suffice for the results we present. The question of whether the ensembles one steers to are consistent with a specific restriction on the form of the real states—that they actually are one-to-one with quantum states (a local hidden quantum state model)—has been recently formalized in [10] into a criterion for “EPR-steerability.” Here, however, we are looking at how Einstein or Schrödinger could have examined the question of whether the ensembles one steers to are consistent with any local “complete description of reality” whatsoever, which connects Schrödinger’s steering to Bell’s theorem.
  10. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen Paradox,” Phys. Rev. Lett. 98, 140402 (2007).
    [Crossref]
  11. S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction,” Phys. Rev. A 86, 012103 (2012).
    [Crossref]
  12. Schrödinger considered that certain quantum states—entangled states or macroscopic superpositions—were unattainable, which may have nullified Einstein’s specific argument for incompleteness. However, given Schrödinger’s concerns about both quantum jumps and the probabilistic nature of quantum predictions, it is probably safe to say that he would still have assumed the theory incomplete in some sense even if entangled states were somehow excised.
  13. S. Kochen and E. P. Specker, “The problem of hidden variables in quantum mechanics,” J. Math. Mech. 17, 59–87 (1967).
  14. N. Harrigan and T. Rudolph, “Ontological models and the interpretation of contextuality,” arXiv:0709.4266 (2007).
  15. R. W. Spekkens, “Contextuality for preparations, transformations, and unsharp measurements,” Phys. Rev. A 71, 052108 (2005).
    [Crossref]
  16. N. Gisin, “Bells inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).
  17. S. Popescu and D. Rohrlich, “Generic quantum nonlocality,” Phys. Lett. A 166, 293–297 (1992).
  18. D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
    [Crossref]
  19. L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett. 71, 1665–1668 (1993).
    [Crossref]
  20. N. D. Mermin, “Bringing home the atomic world: quantum mysteries for anybody,” Am. J. Phys. 49, 940–943 (1981).
    [Crossref]
  21. N. Harrigan and R. W. Spekkens, “Einstein, incompleteness, and the epistemic view of quantum states,” Found. Phys. 40, 125–157 (2010).
    [Crossref]
  22. R. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
    [Crossref]
  23. J. Barrett, Dept. of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD (personal communication, 2014).
  24. P. G. Lewis, D. Jennings, J. Barrett, and T. Rudolph, “Distinct quantum states can be compatible with a single state of reality,” Phys. Rev. Lett. 109, 150404 (2012).
    [Crossref]
  25. M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum state,” Nat. Phys. 8, 476–479 (2012).
    [Crossref]

2014 (1)

R. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
[Crossref]

2012 (3)

P. G. Lewis, D. Jennings, J. Barrett, and T. Rudolph, “Distinct quantum states can be compatible with a single state of reality,” Phys. Rev. Lett. 109, 150404 (2012).
[Crossref]

M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum state,” Nat. Phys. 8, 476–479 (2012).
[Crossref]

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction,” Phys. Rev. A 86, 012103 (2012).
[Crossref]

2010 (1)

N. Harrigan and R. W. Spekkens, “Einstein, incompleteness, and the epistemic view of quantum states,” Found. Phys. 40, 125–157 (2010).
[Crossref]

2007 (1)

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen Paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

2005 (1)

R. W. Spekkens, “Contextuality for preparations, transformations, and unsharp measurements,” Phys. Rev. A 71, 052108 (2005).
[Crossref]

1993 (1)

L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett. 71, 1665–1668 (1993).
[Crossref]

1992 (1)

S. Popescu and D. Rohrlich, “Generic quantum nonlocality,” Phys. Lett. A 166, 293–297 (1992).

1991 (1)

N. Gisin, “Bells inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).

1990 (1)

D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

1985 (1)

D. Howard, “Einstein on locality and separability,” Stud. Hist. Phil. Sci. 16, 171–201 (1985).
[Crossref]

1981 (1)

N. D. Mermin, “Bringing home the atomic world: quantum mysteries for anybody,” Am. J. Phys. 49, 940–943 (1981).
[Crossref]

1967 (1)

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

1936 (1)

E. Schrödinger, “Probability relations between separated systems,” Proc. Cambridge Philos. Soc. 32, 446–452 (1936).
[Crossref]

1935 (2)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[Crossref]

Bacciagaluppi, G.

G. Bacciagaluppi and E. Crull, The Einstein Paradox: the Debate on Nonlocality and Incompleteness in 1935 (to be published).

Barrett, J.

P. G. Lewis, D. Jennings, J. Barrett, and T. Rudolph, “Distinct quantum states can be compatible with a single state of reality,” Phys. Rev. Lett. 109, 150404 (2012).
[Crossref]

M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum state,” Nat. Phys. 8, 476–479 (2012).
[Crossref]

J. Barrett, Dept. of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD (personal communication, 2014).

Bartlett, S. D.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction,” Phys. Rev. A 86, 012103 (2012).
[Crossref]

Crull, E.

G. Bacciagaluppi and E. Crull, The Einstein Paradox: the Debate on Nonlocality and Incompleteness in 1935 (to be published).

Doherty, A. C.

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen Paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

Einstein, A.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

A. Einstein, Letter to E. Schrödinger, Einstein Archives Call Number EA 22-47 (1935), quoted translations from [3], http://alberteinstein.info/vufind1/Record/EAR000024019.

Gisin, N.

N. Gisin, “Bells inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).

Greenberger, D.

D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

Hardy, L.

L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett. 71, 1665–1668 (1993).
[Crossref]

Harrigan, N.

N. Harrigan and R. W. Spekkens, “Einstein, incompleteness, and the epistemic view of quantum states,” Found. Phys. 40, 125–157 (2010).
[Crossref]

N. Harrigan and T. Rudolph, “Ontological models and the interpretation of contextuality,” arXiv:0709.4266 (2007).

Horne, M.

D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

Howard, D.

D. Howard, “Einstein on locality and separability,” Stud. Hist. Phil. Sci. 16, 171–201 (1985).
[Crossref]

D. Howard, “Einstein, Schopenhauer and the historical background of the conception of space as a ground for the individuation of physical systems,” in The Cosmos of Science, J. Earman and J. Norton, eds. (University of Pittsburgh, 1999). The present paper is based heavily around the commentary in this reference and [3].

Jennings, D.

P. G. Lewis, D. Jennings, J. Barrett, and T. Rudolph, “Distinct quantum states can be compatible with a single state of reality,” Phys. Rev. Lett. 109, 150404 (2012).
[Crossref]

Jones, S. J.

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen Paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

Kochen, S.

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

Lewis, P. G.

P. G. Lewis, D. Jennings, J. Barrett, and T. Rudolph, “Distinct quantum states can be compatible with a single state of reality,” Phys. Rev. Lett. 109, 150404 (2012).
[Crossref]

Mermin, N. D.

N. D. Mermin, “Bringing home the atomic world: quantum mysteries for anybody,” Am. J. Phys. 49, 940–943 (1981).
[Crossref]

Podolsky, B.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Popescu, S.

S. Popescu and D. Rohrlich, “Generic quantum nonlocality,” Phys. Lett. A 166, 293–297 (1992).

Pusey, M. F.

M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum state,” Nat. Phys. 8, 476–479 (2012).
[Crossref]

Rohrlich, D.

S. Popescu and D. Rohrlich, “Generic quantum nonlocality,” Phys. Lett. A 166, 293–297 (1992).

Rosen, N.

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Rudolph, T.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction,” Phys. Rev. A 86, 012103 (2012).
[Crossref]

P. G. Lewis, D. Jennings, J. Barrett, and T. Rudolph, “Distinct quantum states can be compatible with a single state of reality,” Phys. Rev. Lett. 109, 150404 (2012).
[Crossref]

M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum state,” Nat. Phys. 8, 476–479 (2012).
[Crossref]

N. Harrigan and T. Rudolph, “Ontological models and the interpretation of contextuality,” arXiv:0709.4266 (2007).

Schrödinger, E.

E. Schrödinger, “Probability relations between separated systems,” Proc. Cambridge Philos. Soc. 32, 446–452 (1936).
[Crossref]

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[Crossref]

A. Einstein, Letter to E. Schrödinger, Einstein Archives Call Number EA 22-47 (1935), quoted translations from [3], http://alberteinstein.info/vufind1/Record/EAR000024019.

Shimony, A.

D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

Specker, E. P.

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

Spekkens, R. W.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction,” Phys. Rev. A 86, 012103 (2012).
[Crossref]

N. Harrigan and R. W. Spekkens, “Einstein, incompleteness, and the epistemic view of quantum states,” Found. Phys. 40, 125–157 (2010).
[Crossref]

R. W. Spekkens, “Contextuality for preparations, transformations, and unsharp measurements,” Phys. Rev. A 71, 052108 (2005).
[Crossref]

Werner, R.

R. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
[Crossref]

Wiseman, H. M.

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen Paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

Zeilinger, A.

D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

Am. J. Phys. (2)

D. Greenberger, M. Horne, A. Shimony, and A. Zeilinger, “Bell’s theorem without inequalities,” Am. J. Phys. 58, 1131–1143 (1990).
[Crossref]

N. D. Mermin, “Bringing home the atomic world: quantum mysteries for anybody,” Am. J. Phys. 49, 940–943 (1981).
[Crossref]

Found. Phys. (1)

N. Harrigan and R. W. Spekkens, “Einstein, incompleteness, and the epistemic view of quantum states,” Found. Phys. 40, 125–157 (2010).
[Crossref]

J. Math. Mech. (1)

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

J. Phys. A (1)

R. Werner, “Steering, or maybe why Einstein did not go all the way to Bell’s argument,” J. Phys. A 47, 424008 (2014).
[Crossref]

Nat. Phys. (1)

M. F. Pusey, J. Barrett, and T. Rudolph, “On the reality of the quantum state,” Nat. Phys. 8, 476–479 (2012).
[Crossref]

Phys. Lett. A (2)

N. Gisin, “Bells inequality holds for all non-product states,” Phys. Lett. A 154, 201–202 (1991).

S. Popescu and D. Rohrlich, “Generic quantum nonlocality,” Phys. Lett. A 166, 293–297 (1992).

Phys. Rev. (1)

A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47, 777–780 (1935).
[Crossref]

Phys. Rev. A (2)

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reconstruction of Gaussian quantum mechanics from Liouville mechanics with an epistemic restriction,” Phys. Rev. A 86, 012103 (2012).
[Crossref]

R. W. Spekkens, “Contextuality for preparations, transformations, and unsharp measurements,” Phys. Rev. A 71, 052108 (2005).
[Crossref]

Phys. Rev. Lett. (3)

P. G. Lewis, D. Jennings, J. Barrett, and T. Rudolph, “Distinct quantum states can be compatible with a single state of reality,” Phys. Rev. Lett. 109, 150404 (2012).
[Crossref]

L. Hardy, “Nonlocality for two particles without inequalities for almost all entangled states,” Phys. Rev. Lett. 71, 1665–1668 (1993).
[Crossref]

H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen Paradox,” Phys. Rev. Lett. 98, 140402 (2007).
[Crossref]

Proc. Cambridge Philos. Soc. (2)

E. Schrödinger, “Discussion of probability relations between separated systems,” Proc. Cambridge Philos. Soc. 31, 555–563 (1935).
[Crossref]

E. Schrödinger, “Probability relations between separated systems,” Proc. Cambridge Philos. Soc. 32, 446–452 (1936).
[Crossref]

Stud. Hist. Phil. Sci. (1)

D. Howard, “Einstein on locality and separability,” Stud. Hist. Phil. Sci. 16, 171–201 (1985).
[Crossref]

Other (8)

D. Howard, “Einstein, Schopenhauer and the historical background of the conception of space as a ground for the individuation of physical systems,” in The Cosmos of Science, J. Earman and J. Norton, eds. (University of Pittsburgh, 1999). The present paper is based heavily around the commentary in this reference and [3].

G. Bacciagaluppi and E. Crull, The Einstein Paradox: the Debate on Nonlocality and Incompleteness in 1935 (to be published).

To be more concrete, paraphrasing the translation of the letter in [5], Einstein says consider an entangled state expanded in two ways: ψAB=∑mncmnψm(x1)χn(x2)=∑mncmn̲ψm(x1)̲χn(x2),where ψ(x1), ψ(x1)̲, χ(x2) are “eigen-ψ” of commuting systems of observables α,α̲,β. If one makes an α (resp. α̲) measurement on A the state on B reduces to ψB=∑ncmnχn(x2) (resp. ψ̲B=∑ncmn̲χn(x2)) and all that is required for us to conclude that the “ψ-description” is not in one-to-one correspondence to the real state is that ψB, ψ̲B are “at all different from each other.”

The term “steering” was originally used by Schrödinger in his study [7,8] of the set ensembles of quantum states that a remote system could be collapsed to, given some (pure) initial entangled quantum state. In fact Schrödinger proved the theorem only for ensembles of (possibly nonorthogonal) states that are linearly independent, but this will suffice for the results we present. The question of whether the ensembles one steers to are consistent with a specific restriction on the form of the real states—that they actually are one-to-one with quantum states (a local hidden quantum state model)—has been recently formalized in [10] into a criterion for “EPR-steerability.” Here, however, we are looking at how Einstein or Schrödinger could have examined the question of whether the ensembles one steers to are consistent with any local “complete description of reality” whatsoever, which connects Schrödinger’s steering to Bell’s theorem.

A. Einstein, Letter to E. Schrödinger, Einstein Archives Call Number EA 22-47 (1935), quoted translations from [3], http://alberteinstein.info/vufind1/Record/EAR000024019.

Schrödinger considered that certain quantum states—entangled states or macroscopic superpositions—were unattainable, which may have nullified Einstein’s specific argument for incompleteness. However, given Schrödinger’s concerns about both quantum jumps and the probabilistic nature of quantum predictions, it is probably safe to say that he would still have assumed the theory incomplete in some sense even if entangled states were somehow excised.

N. Harrigan and T. Rudolph, “Ontological models and the interpretation of contextuality,” arXiv:0709.4266 (2007).

J. Barrett, Dept. of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford OX1 3QD (personal communication, 2014).

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Figures (1)

Fig. 1.
Fig. 1. (a) Two ensembles of ρ . (b) The space of real states, with disjoint regions of support labeled 1–6 such that S x = S 1 S 2 S 3 , S X = S 4 S 5 S 6 , S a = S 1 S 2 S 4 S 5 , and S b = S 2 S 3 S 5 S 6 .

Equations (27)

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

ρ B = i p i | ϕ i ϕ i | ,
ρ B = 1 2 | x x | + 1 2 | X X | = 1 2 | y y | + 1 2 | Y Y | .
ν ( λ ) = 1 2 x ( λ ) + 1 2 X ( λ ) = 1 2 y ( λ ) + 1 2 Y ( λ ) ,
S ν = S x S X = S y S Y .
S x d λ y ( λ ) = | x | y | 2 α ,
μ j S j d λ μ ( λ ) , j = 1 , , 4 .
x 1 = y 1 = X 4 = Y 4 = α , x 2 = y 3 = X 3 = Y 2 = 1 α ,
ν j = 1 2 x j + 1 2 X j = 1 2 y j + 1 2 Y j , j = 1 , , 4 .
X 4 + X 5 + X 6 = 1 .
X 4 + X 5 = | a | X | 2 = sin 2 θ 2 , X 5 + X 6 = | b | X | 2 = sin 2 θ 2 ,
0 X 6 ( a ) S 6 d λ ξ a ( λ ) X ( λ ) S 6 d λ X ( λ ) = X 6 ,
X 4 + X 5 + X 6 ( a ) = sin 2 θ 2 X 4 ( b ) + X 5 + X 6 = sin 2 θ 2 .
X 6 = X 6 ( a ) + cos 2 θ 2 ,
X 4 = X 4 ( b ) + cos 2 θ 2 ,
X 4 + X 6 2 cos 2 θ 2 > 1 , for θ ( 0 , π / 2 ) ,
| z | x | 2 = | z | y | 2 = | Z | X | 2 = | Z | Y | 2 = 1 + α 2 β ,
z 1 + z 2 = z 1 + z 3 = Z 3 + Z 4 = Z 2 + Z 4 = β ,
z 3 + z 4 = z 2 + z 4 = Z 1 + Z 2 = Z 1 + Z 3 = 1 β .
ν j = 1 2 z j + 1 2 Z j j = 1 , , 4 .
z 1 + z 2 = Z 2 + Z 4 = β ,
z 2 + z 4 = Z 1 + Z 2 = 1 β ,
z 1 + Z 1 = α .
Z 1 = α z 1 = α ( β z 2 ) = α β + ( 1 β z 4 ) = 1 2 β + α z 4 ,
α | x | y | 2 , β | x | z | 2 , γ | y | z | 2 ,
ν ( λ ) = p x ( λ ) + ( 1 p ) X ( λ ) = 1 2 a ( λ ) + 1 2 b ( λ )
p x ( λ ) + ( 1 p ) X ( λ ) 1 2 a ( λ ) + 1 2 b ( λ ) ,
ψ A B = m n c m n ψ m ( x 1 ) χ n ( x 2 ) = m n c m n ̲ ψ m ( x 1 ) ̲ χ n ( x 2 ) ,

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