Can LOCC operations take product states to non-product states?

Question

Given a product state $\rho^{(1)} \otimes \rho^{(2)}$, can this state become non-product state under LOCC? Can LOCC create correlations between two systems?

Answer 1

Sure, the classical communication allows you to generate the necessary correlations between two sites. For instance, suppose that Alice flips a coin and communicates the outcome to Bob. When the outcome is heads, they use their local operations to create the state $|0\rangle \langle 0| \otimes |0 \rangle \langle 0 |$. When the outcome is tails they use their local operations to create $|1\rangle \langle 1 | \otimes |1 \rangle \langle 1|$. Overall the state they generate (assuming the coin is fair) is $$ \frac12 |0\rangle \langle 0| \otimes |0 \rangle \langle 0 |+ \frac12 |1\rangle \langle 1| \otimes |1 \rangle \langle 1 | $$ which is not a product state. You can make this construction more general and create any separable state in this manner.