A rather obvious question perhaps but if I have a Hilbert space of dimension $d$ and $d$ is prime, I cannot possibly write my state as $$\rho = \sum_i p_i\rho_a\otimes\rho_b$$ simply because the dimensions don't add up correctly, yes?
In general, is the number of factors of $d$ useful if one wants to check if a state is entangled or not?
Answer
If the dimension is prime, there is no point in talking of two subsystems (which would be entangled), and thus no point in talking about entanglement (which is a property between two subsystems). Since, if subsystem 1 has dimension $d_1$ and subsystem 2 has dimension $d_2$, the total Hilbert space dimension would be $d_1d_2$, which is not prime.
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