Imagine there is a class of system such that a measurement can be performed on an exemplar of this class, each measurement producing 1 bit of information. There are no limitation on how many times the experiment can invoke new instance of a class, and there are no limitations on how many times measurement can be performed. However there are only N types of measurements that can be done.
To clarify, here are some examples:
- The measured systems are fixed 4-bit strings, all equal to “0100”; N = 4 and corresponds to measurements of each one of these bits
- The measured system is 4 independent random bits. N = 4 and correspond to measurements of these random bits.
- Measured system is a pair of entangled photons. N = 4 (there are 2 ways of measuring polarization of the first photon, and 2 ways of measurement the polarization of the second photon).
It’s clear that system as a black box looks the same, and whatever its inner workings, it can be judged by results obtained from the string of 0s and 1s obtained by the measurements. For example, in 1'st case 3 of 4 types of measurements would produce 0s for every instance of the system. In 2nd case all measurements would produce independent random bits. In the 3rd case there would also be random bits, however their distribution will violate Bell's inequality.
My question is – what are the most general classes of systems that can be described by this definition and discriminated by a limited pre-set amount of kinds of measurements?