This question is based on a scenario that is partly hypothetical and partly based on the experimental features of molecule-based quantum devices, which often present a quantum evolution and have some potential to be scalable, but are generally extremely challenging to characterize in detail (a relevant but not unique example is a series of works related to this electrical control of nuclear spin qubits in single molecules).
The scenario: Let us say we have a variety of black boxes, each of which is are able to process information. We don't control the quantum evolution of the boxes; in the language of the quantum circuit model, we do not control the sequence of quantum gates. We know each black box is hardwired to a different algorithm, or, more realistically, to a different time-dependent Hamiltonian, including some incoherent evolution. We don't know the details of each black box. In particular, we don't know whether their quantum dynamics are coherent enough to produce a useful implementation of a quantum algorithm (let us herein call this "quantumness"; the lower bound for this would be "it's distinguishable from a classical map"). To work with our black boxes towards this goal, we only know how to feed them classical inputs and obtain classical outputs. Let us here distinguish between two sub-scenarios:
- We cannot perform entanglement ourselves: we employ product states as inputs, and single qubit measurements on the outputs. However, we can choose the basis of our input preparation and of our measurements (at minimum, between two orthogonal bases).
- As above, but we cannot choose the bases and have to worked on some fixed, "natural" base.
The goal: to check, for a given black box, the quantumness of its dynamics. At least, for 2 or 3 qubits, as a proof-of-concept, and ideally also for larger input sizes.
The question: in this scenario, is there a series of correlation tests, in the style of Bell's inequalities, which can achieve this goal?