# How to "separate" a separable density matrix?

If I have a bipartite system of two qubits $$A$$ and $$B$$, and the density matrix $$\rho$$ is separable, how do I decompose it into its separable parts?

That is, give $$\rho$$, expand it as follows:

$$\rho = \sum_{i=0}^Np_i \ \rho_{i}^A \otimes \rho_{i}^B$$

Where $$0 \le p_i \le 1$$ and $$\rho^{A,\ B}_i$$ are density matrices on the two subsystems $$A$$ and $$B$$ for some $$N$$.

Cross-posted on physics.SE

• it's nontrivial in general. Though it depends a lot on whether further assumptions are considered (eg whether you're asking for a minimal decomposition, a decomposition using only pure states, etc). Numerically there should be a way to obtain a decomposition, following the proof of the minimum number of terms sufficient to decompose any state via Caratheodory (though it will probably scale badly with the state dimension). Related posts are quantumcomputing.stackexchange.com/q/11525/55, quantumcomputing.stackexchange.com/q/5956/55
– glS
Nov 6, 2022 at 19:10
• – glS
Nov 6, 2022 at 19:11
• I've offered a solution for the question on physics.SE, which you may like to check out (I'm not sure if it's appropriate to Answer here with a link to that answer).
– pip
Feb 8 at 8:05