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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?

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