- Question 12 A Give A Mathematical Criterion That Can Be Used To Decide Whether A Given Two Photon Polarization State I 1 (145.02 KiB) Viewed 179 times
Question 12 (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: 1 |V) = (|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 s0|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 |Vo, 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 ) and [0) 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 [4) 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, = 45°, 0₂ = 22.5° and 0₂2-22.5°. Using your result, explain how the results of the famous experiment carried out by Aspect and colleagues conflict with local hidden variable theories. [3] [4] [8] [2]