Does Quantum Mechanics Suggest a Conservation of Possibilities?

[entropy1]

Suppose we have a quantum object in superposition to some measurement basis, given by: $\frac{\sqrt{2}}{\sqrt{3}}|a \rangle + \frac{1}{\sqrt{3}}|b \rangle$. (1)

Suppose the measurement is made, and the system evolves, according to MWI, into $\frac{\sqrt{2}}{\sqrt{3}}|a \rangle|W_a \rangle + \frac{1}{\sqrt{3}}|b \rangle|W_b \rangle$, where $|W_x \rangle$ represents the state of the measurement with result x, or the world where the measurement result has become x. (2)

So my observation is that before the measurement (1) there are two possibilities, namely $|a \rangle$ will be measured, or $|b \rangle$ will be measured, and after the measurement (2) a world where $|a \rangle$ is measured is possible and a world where $|b \rangle$ is measured is possible. In both cases both possibilities exist simultaneously. So to me this seems to be a sort of conservation of possibility, namely $|a \rangle$ or $|b \rangle$, but transformed by measurement from one manifestation to a different one.

Is this view legitimate?

 

[Demystifier]

Is this view legitimate?

No. Before the measurement it is wrong to think that there are precisely two possibilities. Your initial wave function can also be written as a single state $|\psi\rangle$, which can be thought as "one possibility". The conserved number of possibilities makes sense only if you say that one basis (in your case, the basis $|a\rangle, |b\rangle$) is a preferred basis.