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Does many-worlds interpretation of QM make the process of measurement (wavefunction collapse) be an unitary transform? If so, how does it do this?

If we have an "object" qubit in state

$$\left|q_1\right>$$

and a "meter" qubit in state

$$\left|q_2\right>$$

then what is the measurement matrix here?

Since measurement should erase an old state of "meter" qubit, then how can it be unitary?

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  • $\begingroup$ Comments are not for extended discussion; this conversation has been moved to chat. $\endgroup$ – David Z Jan 5 '15 at 21:51
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Measurement is not the same as wavefunction collapse. A measurement is an interaction that produces information about a system that can be copied to multiple other systems. Such an interaction does not copy all of the information in the state of a system, only the information instantiated in some set of orthogonal projectors, see

http://arxiv.org/abs/1212.3245.

This does not contradict the no-cloning theorem, which prohibits copying the whole content of the information in the state of a system, but does not prohibit copying the specific information cited above.

You ask what the measurement operator for one qubit being measured onto another. If they are being measured in the computation basis $|0\rangle,|1\rangle$ then a perfect measurement is represented by controlled not $U = |00\rangle\langle 00|+|01\rangle\langle 01|+|10\rangle\langle 11|+|11\rangle\langle 10|$, which is unitary.

If the measured qubit is sharp in the computation basis and the meter qubit is in $|0\rangle$ then the final state of the measurement qubit is the same as that of the measured qubit.

If the state of the measured qubit is not sharp in the computation basis and the meter qubit is in $|0\rangle$, then we get $$ (\alpha|0\rangle+\beta|1\rangle)|0\rangle \to\alpha|00\rangle+\beta|11\rangle. $$

The way to interpret this is that there are two versions of each qubit, one in state $|0\rangle$, the other in state $|1\rangle$. As the meter qubit interacts with other systems they too will differentiate into two versions, one for each possible outcome. These different versions of each system don't interact with each other, so you can't see them. Since the resulting versions each act approximately like the universe as described by classical physics they are called universes. This all takes place unitarily. See

http://arxiv.org/abs/quant-ph/0104033.

The old state is not erased, rather it was initially prepared in such a way that the measurement gives you the information you want.

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    $\begingroup$ No. Saying that there is information about one system that can be copied to others does not imply that all the information about that system is copied and in fact it is not all copied. The information that is copied is the information instantiated in a particular set of orthogonal projectors. $\endgroup$ – alanf Jan 6 '15 at 9:39
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    $\begingroup$ I deleted my post about many words because I am sick with this. We can know only what happens in our world. If someone believes that the unitarity can be rescued by inventing worlds where we cannot get in, I wish him to enjoy that. The collapse cannot be rescued by many worlds, but it's of no use to hold this, because who believes in it would invent more and more ad-hoc arguments, like kids that want smth. that the parents disagree. $\endgroup$ – Sofia Feb 22 '15 at 16:04
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    $\begingroup$ @Hipnosifl I don't know who is obsessed by the MI. I deleted my answer because I am sick with this. The collapse cannot be solved by many worlds, because it is not known what causes it. It is not sure that the decoherence at the contact with the macroscopic apparatus, produces the collapse - see entanglements where we have more than 1 apparatus, and we can't even say at which time are the worlds generated. But it won't help, because who wants those worlds at any price would argue endlessly, and invent new and new ad-hoc arguments. So, pleasant journey in other worlds. $\endgroup$ – Sofia Feb 22 '15 at 16:20
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    $\begingroup$ @alanf did you see the skeletons of those many worlds? An ad-hoc assumptions, is an assumption supported by no evidence, and only made for explaining your theory. A plausible assumption is one that also explains other experiments. $\endgroup$ – Sofia Feb 23 '15 at 11:37
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    $\begingroup$ @sofia: The existence of the multiverse is entailed by the only known explanation for single particle interference, entanglement and so on, just as the existence of dinosaurs is entailed by the only known explanation of their skeletons. The MWI does not introduce ad hoc qualifications of quantum mechanics. Also support does not make sense. Any given evidence is compatible with an infinite set of theories, so it can't support any particular theory in that set. But it can refute a theory, see "Logic of Scientific Discovery" and "Realism and the Aim of Science" by Karl Popper. $\endgroup$ – alanf Feb 23 '15 at 12:07

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