# How many tensor product terms are necessary to express a separable state? [duplicate]

Wikipedia (https://en.wikipedia.org/wiki/Separable_state) defines a separable state, as a state $$\rho$$ which can be written as:

$$\rho = \sum_{k=1}^l p_k \rho_1^k \otimes \rho_2^k$$

where $$\sum_{k=1}^l p_k = 1$$ and $$\rho_1^k, \rho_2^k$$ are all mixed states of the respective subsystems ("Alice's" and "Bob's" systems). Later it is stated that it can be assumed without loss of generality that $$\rho_1^k, \rho_2^k$$ are all rank-1 projections, i.e. pure states of the respective subsystems.

My question is as follows: in the general case, if I demand that all $$\rho_1^k, \rho_2^k$$ are pure, how many terms are required? Simply put, what is the minimal necessary $$l$$? Even for the simplest case (2 qubits) I can't find a naive bound.

Also, given a separable state $$\psi$$, is there a way to find the "minimal" decomposition? And how may one prove that some given $$\psi$$ cannot be written with less than so-and-so terms?

• Thanks, somehow missed that. – smitke6 May 14 '19 at 8:44