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

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?

• I'd remember that non-product states can be correlated. Absence of entanglement doesn't mean absence of correlations.
– glS
Nov 12, 2021 at 13:58

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.