System 1: This has a Hilbert space of dimension $N$.
System 2: This has a Hilbert space of dimension $N'$, with the condition that $N' \ll N$. We want to simulate system 1 using the system 2, and so we use a "nearly orthogonal" basis of $N$ vectors to "span" the Hilbert space, with the vectors satisfying the following relations:
$$ \langle V_i | V_i \rangle = 1, \quad |\langle V_i | V_j \rangle| \leq \epsilon \, \forall \, i \neq j.$$
Note that $\epsilon$ is a small number and can be adjusted so as to create the basis of $N$ vectors in the System 2's space.
Now we know that if we probe these two systems using $N'$ number of operators then one can easily distinguish between these two systems, e.g. using $N'$ number of operations. However say if $N = 10^{23}$ and $N' = 10^5$, it isn't feasible to use $10^5$- point correlation functions. So we look for simpler observables to distinguish between the two systems. Which observable(s) / series of operations can demonstrate whether System 2 has a smaller Hilbert space?
As an example, let's say the system we want to understand is a gas at a temperature $T$ and is our System 1. What exactly goes wrong if we try to simulate it using a smaller Hilbert space, i.e. System 2? Do laws of thermodynamics hold?