Optimal classical and quantum real and complex dimension witness

TL;DR

This paper develops a method to certify the dimension of classical and quantum systems, including real and complex spaces, using determinant-based tests to optimize detection of higher-dimensional spaces.

Contribution

It introduces a determinant-based dimension witness and identifies optimal preparations and measurements for certifying system dimensions, including real subspaces.

Findings

Determined minimal preparations and measurements for dimension certification.

Maximized detection probability for larger spaces with minimal extra contribution.

Discussed practical applications in quantum computer logical operation certification.

Abstract

We find the minimal number of independent preparations and measurements certifying the dimension of a classical or quantum system limited to $d$ states, optionally reduced to the real subspace. As a dimension certificate, we use the linear independence tested by a determinant. We find the sets of preparations and measurements that maximize the chance to detect larger space if the extra contribution is very small. We discuss the practical application of the test to certify the space logical operations on a quantum computer.