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Suppose a blind man builds a machine that paints three apples with three colors, either red, blue or green. Once the machine has done this, are the three apples in the following superposition:

\begin{align} &|R\rangle_1\otimes |G\rangle_2\otimes |B\rangle_3 + |R\rangle_1\otimes |B\rangle_2\otimes |G\rangle_3\\ +&|G\rangle_1\otimes |R\rangle_2\otimes |B\rangle_3+|G\rangle_1\otimes |B\rangle_2\otimes |R\rangle_3\\ +&|B\rangle_1\otimes |R\rangle_2\otimes |G\rangle_3+|B\rangle_1\otimes |G\rangle_2\otimes |R\rangle_3 \end{align}

or is the wavefunction just one of $|R\rangle_1\otimes |G\rangle_2\otimes |B\rangle_3, |R\rangle_1\otimes |B\rangle_2\otimes |G\rangle_3, etc...$?

It feels like because the man is blind, the apples should be in a superposition as any measurement (say by asking someone else to check the colour of the apple) will lead to the wavefunction collapsing to one of the states in the superposition. However, on the other hand it feels like because the machine definitely assigns a colour to each apple, the wave-function should not be in a superposition. So which is it?

glS
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Kouta Dagnino
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    it is not the same thing to create a pure state like R> G> B> + B> R> G> and a mixed state RGB><RGB + BRG><BRG. in the first case it is a superposition and in the second case a statistical mixture. it depends on the way used to create the state (no question about blindness) – Naima Jun 17 '21 at 09:31

2 Answers2

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Based on your wording, the created state is a mixture of the component ones, not a superposition.

If you engineer a state that is either $|\psi\rangle$ or $|\phi\rangle$ but you are not sure which one it is, then the overall state oughts to be described as a mixture of the form $p|\psi\rangle\!\langle\psi|+(1-p)|\phi\rangle\!\langle\phi|$, with weights $p,1-p$ depending on your prior (that is, on the probabilities with which you estimate each component state to have been created).

A superposition $\alpha|\psi\rangle+\beta|\phi\rangle$ is a completely different beast. See e.g. Is "quantum superposition" just a fancy way of saying that a system is in one state or another with some probabilities?, What's an atomic superstate/superposition, and how is it possible?, Are superposition quantum systems really in two pure states at once?, How is quantum superposition different from mixed state?, and links therein.

glS
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Talking of superpositions only makes sense for objects which you can measure in more than one basis - say, the color basis and a basis of "red plus green" vs. "red minus green". This is not possible for apples.

So no, this has nothing to do with superposition. (More generally, incomplete knowledge has nothing to do with superposition.)

Norbert Schuch
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