How do I convert this separable state into a product state?

Question

I know two particles in a Bell state cannot be written as a product state as they are entangled. But what if I had a classically correlated state$$\rho = \frac{1}{2}(|11\rangle\langle 11| + |00\rangle\langle00|)$$How do I write this as a product state?

Answer 1

After a discussion in comments, it turns out this cannot be written as a product state.

The reduced density matrix of each subsystem is $\frac{1}{2}(|0\rangle\langle 0| + |1\rangle\langle1|)$ and so $$\rho_1\otimes\rho_2 = \frac{1}{4}(|00\rangle\langle00|+|11\rangle\langle11| + |01\rangle\langle01| + |10\rangle\langle10|)\neq\rho$$

Answer 2

"Product state" here refers to a pure state. The given state, however, is mixed (as can be seen immediately from its eigenvalues). Thus, it cannot be written as a single pure state, be it a product state or not.

Answer 3

As you write yourself, the state is classically correlated. Thus, it cannot be of product form: A product does not have any kind of correlations, whether quantum or classical.