Certification of qubits in the prepare-and-measure scenario with large input alphabet and connections with the Grothendieck constant

TL;DR

This paper investigates the quantumness of qubits in a prepare-and-measure setup with many preparations and measurements, linking the problem to the Grothendieck constant and deriving bounds on detection efficiency.

Contribution

It introduces new constants related to the Grothendieck constant for large input alphabets and connects them to noise resistance and detection efficiency in qubit certification.

Findings

Bound the white noise resistance of prepared qubits.

Determine the minimum detection efficiency for large-scale setups.

Establish connections between Grothendieck constants and quantum certification.

Abstract

We address the problem of testing the quantumness of two-dimensional systems in the prepare-and-measure (PM) scenario, using a large number of preparations and a large number of measurement settings, with binary outcome measurements. In this scenario, we introduce constants, which we relate to the Grothendieck constant of order 3. We associate them with the white noise resistance of the prepared qubits and to the critical detection efficiency of the measurements performed. Large-scale numerical tools are used to bound the constants. This allows us to obtain new bounds on the minimum detection efficiency that a setup with 70 preparations and 70 measurement settings can tolerate.