Consider a projective measurement of a superposition of states:

$$\frac{1}{\sqrt{2}} |\Psi_1\rangle + \frac{1}{\sqrt{2}} |\Psi_2\rangle \xrightarrow[measurement]{projective} |\Psi_1\rangle, \qquad (|\Psi_1\rangle \neq |\Psi_2\rangle)$$

Assuming an unitary evolution of the "universe = quantum system + measure apparatus" the state evolves as:

$$\left(\frac{1}{\sqrt{2}} |\Psi_1\rangle + \frac{1}{\sqrt{2}} |\Psi_2\rangle\right)\otimes |\chi_0\rangle \xrightarrow[measurement]{unitary} |\Psi_1\rangle \otimes |\chi_1\rangle$$

But:

• Can we somehow prove that the initial state information is contained in the final state, or is it just an assumption we make, difficult to prove in practice?
• Would it be possible to construct a very simple measuring device where this would be easier to verify?