# Random positive operator valued measures

## TL;DR

This paper explores the properties of random positive operator valued measures (POVMs), analyzing their eigenvalue distributions, system limits, and compatibility criteria using free probability theory and quantum information concepts.

## Contribution

It introduces new notions of random POVMs, establishes their equivalences, and applies free probability to analyze their spectral and informational properties.

## Key findings

- Eigenvalue distributions follow specific limiting laws
- Large system limits for quantum informational quantities are derived
- Compatibility criteria are compared for generic POVMs

## Abstract

We introduce several notions of random positive operator valued measures (POVMs), and we prove that some of them are equivalent. We then study statistical properties of the effect operators for the canonical examples, obtaining limiting eigenvalue distributions with the help of free probability theory. Similarly, we obtain the large system limit for several quantities of interest in quantum information theory, such as the sharpness, the noise content, and the probability range. Finally, we study different compatibility criteria, and we compare them for generic POVMs.