Generating quantum-measurement probabilities from an optimality principle

Abstract

An alternative formulation to the (generalized) Born rule is presented. It involves estimating an unknown model from a finite set of measurement operators on the state. An optimality principle is given that relates to achieving bounded solutions by regularizing the unknown parameters in the model. The objective function maximizes a lower bound on the quadratic Renyi classical entropy. The unknowns of the model in the primal are interpreted as transition witnesses. An interpretation of the Born rule in terms of fidelity is given with respect to transition witnesses for the pure state and the case of positive operator-valued measures (POVMs). The models for generating quantum-measurement probabilities apply to orthogonal projective measurements and POVM measurements, and to isolated and open systems with Kraus maps. A straightforward and constructive method is proposed for deriving the probability rule, which is based on Lagrange duality. An analogy is made with a kernel-based method for probability mass function estimation, for which similarities and differences are discussed. These combined insights from quantum mechanics, statistical modeling, and machine learning provide an alternative way of generating quantum-measurement probabilities.