Wigner distributions and the joint measurement of incompatible observables

A theory of joint nonideal measurement of incompatible observables is used in order to assess the relative merits of quantum tomography and certain measurements of generalized observables, with respect to completeness of the obtained information. A method is studied for calculating a Wigner distribution from the joint probability distribution obtained in a joint measurement. 1 Quantum tomography The Wigner distribution W (q, p) has the well-known Fourier representation W (q, p) = 1 2π2 ∫ ∞ −∞ dξ1 ∫ ∞ −∞ dξ2W̃ (ξ)e i √ 2(pξ1−qξ2), W̃ (ξ) = Trρ̂e †−ξ∗â), ξ = ξ1 + iξ2 (carets denote operators). Putting ξ = iηe/ √ 2, it was observed by Vogel and Risken [1] that W̃ (ξ) satisfies W̃ (iηe/ √ 2) = Trρ̂e, which is the characteristic function of the rotated quadrature phase operator Q̂(θ) = 1 √ 2 (â†eiθ + âe−iθ), measured in homodyne optical detection. Since the characteristic function is the Fourier transform of the probability distribution, it was found [1] that the Wigner distribution can be given as the integral W (q, p) = 1 4π2 ∫ ∞ −∞ dx ∫ 2π