The Evolution of our Probability Image for the Spin Orientation of a Spin — 1/2 — Ensemble as Measurements are Made on Several Members of the Ensemble — Connections with Information Theory and Bayesian Statistics

Abstract

An analysis is made of the information acquired when measurements are made, of the component of spin along an arbitrarily chosen axis, (Θ, Φ), of each of several or many spin 1/2 systems, each prepared in the same way. Since the probability p↑ of the observed value of the spin being “up” is a unique function of the relative orientation of the axis of measurement, (Θ, Φ), and the quantization axis, (θ, φ), of the state into which the system was prepared, this probability function, p↑(θ, φ; Θ, Φ), may be interpreted as a likelihood function for the orientation, (θ, φ), of the state into which the system is prepared, conditioned by its having been measured with “up” spin along the axis with orientation (Θ, Φ). If one assumes that the preparation procedure prepares each system into the same unknown pure state, and further assumes that, a priori, each equal infinitisimal solid angle is equally likely to contain the orientation of this pure state, the likelihood function for the orientation, via Bayes’ theorem, becomes the (unnormalized) probability density function for the orientation of the pure state. Thus, from an ensemble of spin 1/2 systems, each prepared in the pure state with quantization axis (θ, φ), the probability for realizing n “up” values and m = N—n “down” values, in some specified order, in a sequence of N measurement events for the spin component along the (Θ, Φ) axis, is given by Π(θ, φ; Θ, Φ) = p ↑ n(1-p ↑)m = p ↑ n(1-p ↑]) N-n , (as in the Bernoulli coin-tossing problem). By the above arguments Π(θ, φ; Θ, Φ) dΩ, becomes, for fixed (Θ, Φ,n,N), the (unnormalized) probability for the system being prepared in the spin state whose direction is aligned to within dΩ of (θ, φ), conditioned by the measurement events which determined Π. If we take the density operator corresponding to this distribution function to be ρ = [∫dΩ,ρ0(θ, φ)Π(θ, φ; Θ, Φ)]/[∫dΩΠ(θ, φ; Θ, Φ)], where ρ0(θ, φ) is the density operator corresponding to the pure state aligned along (θ, φ), one finds that the probability of measuring “up” spin along (Θ, Φ), conditioned by a measurement of n “up” values in N trials is given by (n+1)/(N+2), which is Laplace’s rule of succession. Also presented is a geometrical parametrization of the density operator for a spin 1/2 system. The density operator for the pure state with spin orientation (θ, φ) is represented as the corresponding point on the unit sphere. Each point in the interior of the sphere represents a non-idempotent density operator, diagonal in the coordinate system in which the radial line to the representative point lies on the +2 axis, and such that the radius is equal to (σz) = λ↑ — λ↓. A measurement of n “up” values in TV trials, with a prior probability distribution that is uniform over the volume of this sphere, leads to (n+2)/(N+4) for the probability of measuring “up” spin; a more “cautious” rule of succession than that of Laplace.