This means that choosing ∆θ ξ according to Eq. 61 as the bounds of the uniform probability distribution of θ ξ will satisfy the expectation value relations in all previous equations. Since Eq. 61 is a transcendental equation, no general closed form exists for the solutions to ∆θ ξ . To find ∆θ ξ , a numerical interpolation function of the first positive solution of sin(x) = ax can be pre-computed for the values 0 ≤ a ≤ 1 and then used during the simulation. Up to this point, the only unspecified variables are the individual phases θ 0,ξ and θ 1,ξ . To determine these variables, Eq. 38a and Eq. 38b can be used. Using the conditions outlined in Eq. 39 and Eq. 58, the left hand side of Eq. 38a can be rewritten as