Cost of postselection in decision theory

TL;DR

This paper analyzes the impact of explicitly assigning costs to postselection in decision theory, deriving optimal measurement strategies that interpolate between unambiguous discrimination and minimum error, with implications for quantum measurement design.

Contribution

It introduces a decision-theoretic framework incorporating explicit costs for postselection, bridging existing quantum measurement approaches.

Findings

Derives optimal decision rules based on postselection costs

Connects postselection cost to measurement strategies in quantum systems

Provides a unified view of unambiguous and minimum error discrimination

Abstract

Postselection is the process of discarding outcomes from statistical trials that are not the event one desires. Postselection can be useful in many applications where the cost of getting the wrong event is implicitly high. However, unless this cost is specified exactly, one might conclude that discarding all data is optimal. Here we analyze the optimal decision rules and quantum measurements in a decision theoretic setting where a prespecified cost is assigned to discarding data. Our scheme interpolates between unambiguous state discrimination (when the cost of postselection is zero) and a minimum error measurement (when the cost of postselection is maximal). We also relate our formulation to previous approaches which focus on minimizing the probability of indecision.