# Dimensionality Distinguishers

## TL;DR

This paper generalizes the CHSH game to include all Boolean functions and measurement bases, creating dimension distinguishers that can differentiate between quantum state dimensions, notably between 2 and 3.

## Contribution

It introduces a comprehensive framework for dimension distinguishers based on generalized CHSH game variants, expanding the tools for quantum dimension verification.

## Key findings

- Constructed equivalence classes for success probabilities
- Developed dimension distinguishers for quantum states
- Demonstrated distinguishing between dimensions 2 and 3

## Abstract

The celebrated Clauser, Horne, Shimony and Holt (CHSH) game model helps to perform the security analysis of many two-player quantum protocols. This game specifies two Boolean functions whose outputs have to be computed to determine success or failure. It also specifies the measurement bases used by each player. In this paper, we generalize the CHSH game by considering all possible non-constant Boolean functions and all possible measurement basis (up to certain precision). Based on the success probability computation, we construct several equivalence classes and show how they can be used to generate three classes of dimension distinguishers. In particular, we demonstrate how to distinguish between dimensions 2 and 3 for a special form of maximally entangled state.