# How do you mix two pure states to obtain a mixed state?

If we have the following two states $$$$|\psi\rangle_1 = \frac{1}{\sqrt{2}}|0\rangle_A|0\rangle_B + \frac{1}{\sqrt{2}} |1\rangle_A |1\rangle_B$$$$ $$$$|\psi\rangle_2 = \frac{1}{\sqrt{2}}|0\rangle_A|0\rangle_B - \frac{1}{\sqrt{2}} |1\rangle_A |1\rangle_B$$$$ How do you mix them with the same proportion to create a mixed state? What would be the resulting density operator?

You can prepare the mixed state as follows. Flip a perfect coin. If it comes up heads, prepare $$|\psi\rangle_1$$, otherwise prepare $$|\psi\rangle_2$$. Finally, forget the result of flipping the coin.
$$\rho = \frac{1}{2} |\psi\rangle_1\langle\psi|_1 + \frac{1}{2} |\psi\rangle_2\langle\psi|_2.$$
Either it will be in $$|\psi\rangle_1$$ or $$|\psi\rangle_2$$ with 50/50 probability, which (after collapse) will not have any "special quantum weirdness" which would be observed via interference and can only be described with mixed states (classical probability).