# Approximation of quasi-states on manifolds

## TL;DR

This paper introduces a numerical algorithm to approximate median quasi-states on the 2-sphere, based on metric properties of quasi-states and probability measures, with implications for understanding quantum mechanics.

## Contribution

It presents the first algorithm for numerically computing median quasi-states on the 2-sphere, leveraging Wasserstein metrics for error estimation.

## Key findings

- Algorithm achieves arbitrary accuracy in computing median quasi-states.
- Error estimates depend on metric continuity properties.
- Non-approximation results are provided for symplectic quasi-states.

## Abstract

Quasi-states are certain not necessarily linear functionals on the space of continuous functions on a compact Hausdorff space. They were discovered as a part of an attempt to understand the axioms of quantum mechanics due to von Neumann. A very interesting and fundamental example is given by the so-called median quasi-state on the 2-sphere. In this paper we present an algorithm which numerically computes it to any specified accuracy. The error estimate of the algorithm crucially relies on metric continuity properties of a map, which constructs quasi-states from probability measures, with respect to appropriate Wasserstein metrics. We close with non-approximation results, particularly for symplectic quasi-states.