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answerhappygod
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Need steps
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(a) Give a mathematical criterion that can be used to decide whether a given two-photon polarization state is entangled or not. Illustrate your answer by explaining whether the following states are entangled: (4) = (VV) + |VH)) |$) = (VV) + |HH)) √2 where the positional convention is used and V) and H) represent vertical and horizontal polarization states relative to the z-axis. (b) The kets |Vo) = cos 0V) + sin 0|H) and Ho) = sin 0|V) + cos |H) represent states of vertical and horizontal polarization relative to an axis in the direction defined by the polar angle and the azimuthal angle = 0. Suppose that observer A measures the polarization of photon 1 relative to the axis with 0 = 0₁, while observer B measures the polarization of photon 2 relative to the axis with 0 0₂. Express the kets Vo₁ Vo₂) and Ho, Ho₂) in terms of |VV), [VH), HV) and HH). (c) Use the results from part (b) to calculate the probability, Pvv(01,02), that both observers A and B obtain vertical polarization for the states V) and ) defined in part (a). What feature of your answers further illustrates the distinction between entangled and non-entangled states in these two cases? (d) The correlation function, C(0₁-02) associated with the ket ) is: C(0₁-0₂) = cos 2(0₁-0₂). The Clauser, Horne, Shimony and Holt (CHSH) quantity is defined as: Σ = C(0₁-0₂) + C(0₁-0₂) + C(0₁-0₂) - C(0₁ - 0₂). Evaluate Σ for the situations where the angles are 0₁ = 0, 0₁ = 45°, 0₂ = 22.5° and 0₂ = -22.5°. Using your result, explain how the results of the famous experiment carried out by Aspect and colleagues conflict with local hidden variable theories.
(a) Give a mathematical criterion that can be used to decide whether a given two-photon polarization state is entangled or not. Illustrate your answer by explaining whether the following states are entangled: (4) = (VV) + |VH)) |$) = (VV) + |HH)) √2 where the positional convention is used and V) and H) represent vertical and horizontal polarization states relative to the z-axis. (b) The kets |Vo) = cos 0V) + sin 0|H) and Ho) = sin 0|V) + cos |H) represent states of vertical and horizontal polarization relative to an axis in the direction defined by the polar angle and the azimuthal angle = 0. Suppose that observer A measures the polarization of photon 1 relative to the axis with 0 = 0₁, while observer B measures the polarization of photon 2 relative to the axis with 0 0₂. Express the kets Vo₁ Vo₂) and Ho, Ho₂) in terms of |VV), [VH), HV) and HH). (c) Use the results from part (b) to calculate the probability, Pvv(01,02), that both observers A and B obtain vertical polarization for the states V) and ) defined in part (a). What feature of your answers further illustrates the distinction between entangled and non-entangled states in these two cases? (d) The correlation function, C(0₁-02) associated with the ket ) is: C(0₁-0₂) = cos 2(0₁-0₂). The Clauser, Horne, Shimony and Holt (CHSH) quantity is defined as: Σ = C(0₁-0₂) + C(0₁-0₂) + C(0₁-0₂) - C(0₁ - 0₂). Evaluate Σ for the situations where the angles are 0₁ = 0, 0₁ = 45°, 0₂ = 22.5° and 0₂ = -22.5°. Using your result, explain how the results of the famous experiment carried out by Aspect and colleagues conflict with local hidden variable theories.