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In my course, we had an example: give the density matrix of a system arising from a proces that generates a [0> with probability 1/2 and a [1> with prob. 1/2. The answer is $1/2[0\rangle\langle0] +1/2[1\rangle\langle1]$.

But it seems to me that $\sqrt{1/2}[0\rangle + \sqrt{1/2}[1\rangle$ does the same, which has a different density function, $1/2[0\rangle\langle0]+1/2[1\rangle\langle0]+1/2[0\rangle\langle1]+1/2[1\rangle\langle1]$. What is my mistake?

Qmechanic
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ericj
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    Does this answer your question? How is a quantum superposition different from a mixed state? – Tobias Fünke Feb 13 '22 at 11:02
  • No, probably I don't see it. Those answers are too general and difficult. Let's stick to this example. – ericj Feb 13 '22 at 12:11
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    But what do you mean with "this example" - you're asking about the difference between the density matrix and a wave function. This is exactly what the linked post deals with, i.e. it explains the difference between the superposition and density matrices. Actually, the answers, as far as I see, also deal with two state systems, which corresponds to your example. In general, for mixed states, there is no single wave function yielding this density matrix. – Tobias Fünke Feb 13 '22 at 13:04
  • Ask yourself what happens if you measure either of the two states in the {|+>,|->} basis, and see which of the two matches what you expect from that process. – Norbert Schuch Feb 13 '22 at 13:43
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    The question is a duplicate and in fact deals with the same example as the duplicate. Any answers to your question would do as well for the linked duplicate. If you want a hint, try a superposition with complex coefficients. – ZeroTheHero Feb 13 '22 at 14:06

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