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Supposing I want to make a quantum joke, like writing this on a coffee machine:

$$| \text{Status}\rangle = \frac{1}{\sqrt{2}}\ \big( | \text{Working}\rangle \color{red}{\pm} | \text{Down}\rangle \big) \, ,$$

should I choose the $\color{red}{+}$ or the $\color{red}{-}$ sign, or is it the same? Why?

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    $\begingroup$ The sign indicates singlet/triplet states. $\endgroup$ – Kyle Kanos Nov 29 '15 at 15:43
  • $\begingroup$ Well, if all you're going to do is make a direct measurement of whether you're working or down, the sign doesn't matter for the outcome. $\endgroup$ – march Nov 29 '15 at 15:46
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    $\begingroup$ States don't have to be normalised. So you don't need the root two. But it adds a nice touch, a note of misplaced precision.... By the same token, you could multiply |working> by the imaginary unit $i$ to suggest that the status is a complicated affair, and the idea that it would be working is purely imaginary.... $\endgroup$ – joseph f. johnson Nov 29 '15 at 19:12
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    $\begingroup$ As someone who is not familiar with quantum physics, this joke gives me a headache, along with the answers. Could someone give me a <500 chars explanation of this joke? $\endgroup$ – Nzall Nov 29 '15 at 22:16
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    $\begingroup$ @NateKerkhofs Understanding quantum physics doesn't make it funny either. $\endgroup$ – Norbert Schuch Nov 29 '15 at 23:32
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The signs are important for fixing an out of order machine. Define the states $|\pm\rangle$ as:

$$|\pm\rangle = \frac{1}{\sqrt{2}}\left[\left |\text{Working}\right\rangle\pm \left |\text{Down}\right\rangle\right]$$

And we define the observable $O$ as:

$$O = |+\rangle\langle + |\ - \ |-\rangle\, \langle -|$$

Suppose then that coffee machine is out of order. To fix it, you measure $O$ and then you measure if it is working, if not you repeat the procedure of measuring $O$ and then checking if it is working. At each step you have 50% probability that it will be found to be working.

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    $\begingroup$ quantum machines seem to be really easy to fix via brute force! $\endgroup$ – diffeomorphism Nov 29 '15 at 18:14
  • $\begingroup$ Awesome answer! Ahaha nice! $\endgroup$ – Les Adieux Nov 29 '15 at 19:00
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    $\begingroup$ How does the protocol for fixing the machine depend on whether the machine starts in state $|+\rangle$ or $|-\rangle$? $\endgroup$ – Vectornaut Nov 30 '15 at 4:38
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Perhaps you don't want a quantum superposition, but just a statistical mixture:

$$\rho = \begin{pmatrix}1/2 & 0 \\ 0 & 1/2\end{pmatrix}$$

Although I'm not 100% sure that this will describe your situation any better...

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If you want to declare indeterminacy and a probability of being either working or down you should use the vector notation:

       (working )

Status> =

       ( down)

Status being the column vector analogous to the column state vector of the wavefunction in a matrix representation

The user would be the operator :)

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