Geometrical picture of photocounting measurements

TL;DR

This paper introduces a geometric framework for representing quantum measurements via POVMs, enabling better interpretation of measurement outcomes and improved state estimation techniques, especially for array detectors and unbalanced homodyne detection.

Contribution

It establishes a geometric interpretation of POVMs and Born's rule, providing a new method for expectation value estimation and quantum-state reconstruction.

Findings

Geometric interpretation of POVMs and Born's rule.

Direct estimation of expectation values with array detectors.

Application to quantum-state reconstruction with unbalanced homodyne detection.

Abstract

We revisit the representation of generalized quantum observables by establishing a geometric picture in terms of their positive operator-valued measures (POVMs). This leads to a clear geometric interpretation of Born's rule by introducing the concept of contravariant operator-valued measures. Our approach is applied to the theory of array detectors, which is a challenging task as the finite dimensionality of the POVM substantially restricts the available information about quantum states. Our geometric technique allows for a direct estimation of expectation values of different observables, which are typically not accessible with such detection schemes. In addition, we also demonstrate the applicability of our method to quantum-state reconstruction with unbalanced homodyne detection.