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

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