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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?

Qmechanic
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QGR
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1 Answers1

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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 $Ψ^*Ψ$.

anna v
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    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 $\mathcal{O}$. – fqq Jun 19 '20 at 10:14
  • 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. – fqq Jun 19 '20 at 10:18
  • I mean it as the meaning of the first paragraph here https://chem.libretexts.org/Bookshelves/Physical_and_Theoretical_Chemistry_Textbook_Maps/Book%3A_Time_Dependent_Quantum_Mechanics_and_Spectroscopy_(Tokmakoff)/05%3A_The_Density_Matrix/5.01%3A_Introduction_to_the_Density_Matrix – anna v Jun 19 '20 at 10:32
  • also see relation 9.18 here https://homepage.univie.ac.at/reinhold.bertlmann/pdfs/T2_Skript_Ch_9corr.pdf . the result is always real because the wavefunctions are projected on each other – anna v Jun 19 '20 at 10:38
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    You still cannot describe mixed states with a wavefunction, I don't see what 9.18 or reality have to do with my comments. – fqq Jun 19 '20 at 15:09
  • @fqq , that is why the density matrix formalism was invented, when one gets to mixed states its really a statistical study towards classical dimensions, but the fact is that one reduces complex numbers to real observable values .It is the wavefunctions that the are fundamental mathematical frame. – anna v Jun 19 '20 at 15:16
  • That's one way to see it at the introductory level, see the various great answers to the duplicate. – fqq Jun 19 '20 at 16:57