# Unentangled Measurements and Frame Functions

## TL;DR

This paper extends Gleason's theorem to unentangled frame functions in multi-partite quantum systems, revealing their structure across various dimensions and uncovering a combinatorial pattern with potential foundational significance.

## Contribution

It classifies unentangled frame functions for multi-qubit and mixed-dimension systems, including infinite-dimensional spaces, generalizing previous results and revealing new structural insights.

## Key findings

- Classified unentangled frame functions for multi-qubit systems.
- Extended the classification to systems with varying and infinite dimensions.
- Discovered a combinatorial structure suggesting fundamental interpretations.

## Abstract

Gleason's theorem asserts the equivalence of von Neumann's density operator formalism of quantum mechanics and frame functions, which are functions on the pure states that sum to 1 on any orthonormal basis of Hilbert space of dimension at least 3. The unentangled frame functions are initially only defined on unentangled (that is, product) states in a multi-partite system. The third author's Unentangled Gleason's Theorem shows that unentangled frame functions determine unique density operators if and only if each subsystem is at least 3-dimensional. In this paper, we determine the structure of unentangled frame functions in general. We first classify them for multi-qubit systems, and then extend the results to factors of varying dimensions including countably infinite dimensions (separable Hilbert spaces). A remarkable combinatorial structure emerges, suggesting possible fundamental interpretations.