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Some interpretations, like the many-worlds interpretation, treat the wavefunction (modulo an overall phase factor) as objective and fundamental.

But consider the following example for a qubit: a classical probability distribution over wavefunctions with a 1/2 probability of $|0\rangle$ and a 1/2 probability for $|1\rangle$. Then, consider another classical probability distribution with a 1/2 probability for $\frac{1}{\sqrt 2}\left(|0\rangle+|1\rangle\right)$ and a 1/2 probability for $\frac{1}{\sqrt 2}\left(|0\rangle-|1\rangle\right)$.

Both examples are described by the same density matrix $\left(\begin{array}{cc} \frac{1}{2} &0\\0&\frac{1}{2}\end{array}\right)$ and can't be distinguished empirically by any experiment. If wavefunctions are objective and fundamental, why can't we distinguish between both examples?

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If by "objective" you mean "real", a wave function, $Ψ$ is only mathematically fundamental because it is a postulate of quantum mechanics, a function of complex numbers, it cannot be measured independently.

Only $Ψ^*Ψ$ is a measurable prediction as the probability distribution. This allows different formats for $Ψ$, that can give the same real valued $Ψ^*Ψ$ .

The density matrix is another way of organizing the wavefunctions, each $ρ_{ij}$ is a part of the total $Ψ^*Ψ$.

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    $\begingroup$ The last sentence seems not true, a (non-pure-state) density matrix has more content than a single wavefunction. In the example in the question, there is no state vector in the 2d Hilbert space that reproduces the expectation values $tr(\rho \mathcal{O})=tr\mathcal{O}/2$ for all observables $\mathcal{O}$. $\endgroup$
    – fqq
    Commented Jun 19, 2020 at 10:14
  • $\begingroup$ If you are arguing that any density matrix emerges as the reduced density matrix from some "wavefunction of the universe" pure state $|\Psi><\Psi|$, that could be interesting but deserves a more detailed explanation. $\endgroup$
    – fqq
    Commented Jun 19, 2020 at 10:18
  • $\begingroup$ I mean it as the meaning of the first paragraph here chem.libretexts.org/Bookshelves/… $\endgroup$
    – anna v
    Commented Jun 19, 2020 at 10:32
  • $\begingroup$ also see relation 9.18 here homepage.univie.ac.at/reinhold.bertlmann/pdfs/… . the result is always real because the wavefunctions are projected on each other $\endgroup$
    – anna v
    Commented Jun 19, 2020 at 10:38
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    $\begingroup$ You still cannot describe mixed states with a wavefunction, I don't see what 9.18 or reality have to do with my comments. $\endgroup$
    – fqq
    Commented Jun 19, 2020 at 15:09

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