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It has been argued the exponential size of the wavefunction can be interpreted as many parallel worlds, and this explains how quantum computers can factor large integers and compute discrete logarithms (although on closer examination, Shor's algorithm doesn't work at all by trying out a superposition of all candidate solutions. It works by using number theory to find a different function with periodicity, and period finding.).

However, unless we can examine specific individual branches of our choice, why should we argue for the ontological reality of the branches? Measuring which branch we are in "collapses", i.e. picks out a branch sampled at random, and not a branch of our choice.

To be more specific, let's say we wish to invert a one-way function f with no periodicity properties, i.e. solve $f(x)=y$ for a specific y. We can easily prepare the state $C\sum_x |x\rangle\otimes |f(x)\rangle$ where C is an overall normalization factor. However, unless we can also postselect for the second register being y, how can we interpret this as many parallel worlds?

Doesn't this inability argue for a more "collectivist" interpretation where the individual branches don't have "individual" existence, but only the collective relative phases of all the branches taken together have any real existence?

To take a less quantum computing example, consider Schroedinger's cat. Suppose we have N cats in N boxes, and we perform independent cat experiments on each of them. Let's suppose I pass you a prespecified N bit string in advance, and I ask you to "subjectively" take me to the branch where the life/death status of the cat in the ith box matches the value of the ith bit. That, you can't do unless you perform this experiment over and over again for an order of $2^N$ times.

PS Actually, there might be a way using quantum suicide. Unless the life/death status of each cat matches that of the string, kill me. However, this rests on the dubious assumption that I will still find myself alive after this experiment, which rests upon the dubious assumption of the continuity of consciousness over time, and that it can't ever end subjectively.

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MWI doesn't argue for parallelism at all. MWI is a probability distribution with a preferred basis problem. If after sampling a quantum system, you find yourself in a given branch, you can't access the other branches any more. You can't access two or more branches at the same time. It's or, not and.

It might appear as if interference between two or more branches can occur, and so, more than one branch can exist at any given time. On closer examination however, this requires taking incompatible bases at different times. Take the Mach-Zehnder interferometer as an example. It appears as if there is a split into two worlds with the photon taking a different path in each world, and then, both worlds interfere so that the photon only comes out one way, and never the other. It certainly looks as if two branches interact, which means both of them have to exist simultaneously. However, all this tells us is that the preferred basis is that given by the direction the photon takes after leaving the interferometer. The basis given by the path it took inside is the wrong basis. In the "correct" preferred basis, you can't access both branches simultaneously. This also suggests for coherent systems, you can't simultaneously take preferred bases at different times if the system is evolving. It's only possible to compare bases at different times if they are compatible. This is what consistent histories is all about.

If two branches decouple, their weights as given by their Born amplitudes can't change in time. It's not possible for one weight to keep increasing in absolute value. This is what you would expect from probabilities, but not from physical branches. This property is unitarity. In a hypothetical world with nonhermitian Hamiltonians, it's possible to have time evolutions like $$c_1 |\psi_1\rangle + c_2 |\psi_2\rangle \to c_1 |\psi_1\rangle + c_2 e^{\lambda t}|\psi_2\rangle$$ where $\lambda$ is real and positive. Such a hypothetical world can also be given an MWI. Imagine applying this to querying a database, with $\psi_2$ corresponding to the matching entry. Then, all that's necessary is to wait for some time of order $\ln N/\lambda $ before the branch for the right entry dominates all the other entries. We don't live in such a world, and this weakens the case for the physical nature of the individual branches.

If only one branch can be accessed, there is no need to assume all possible branches exist, unless you wish to subscribe to a possible worlds semantics of all possible worlds existing out there, in addition to the "actual" world we find ourselves in.

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It only argues against the "ontological" aspects of the other branches. It is pointless to argue ontology, because there is no logical positivist meaning you can attach to it. But you have given a logical positivist meaning to it, by saying you would consider the many-worlds real when you can post-select to one of them by an efficient procedure.

The lack of efficient procedure means that your definition makes the worlds unreal. So what. Other people might put the cutoff for real at a different point, like being able to prepare the cat-states in an exponentially growing amount of time, and then they become more real. The collection of predictions is the invariant thing, so that the philosophical words you attach to the stuff is not so important.

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You're right. The issue is Grover's algorithm. Suppose you have a quantum database with N entries, and a selection criteria. You know only exactly one of the entries satisfy your chosen criteria, but you have absolutely no idea which entry it is. If quantum mechanics works by an underlying ontology of highly massive parallelism, then why do you still need to search $\mathcal{O}(\sqrt N)$ times to find the right entry? Why not far fewer times?

The answer is interesting: some wavefunction describing a superposition of all possible entries is created. This might appear to be parallelism in action at first sight. After a number of database calls, the wavefunction can be described as a superposition of a part which is absolutely ignorant of the right entry, and a term which knows what the right entry is. As we're initially ignorant of the right entry, initially, the coefficient for the right entry can only be around $1/\sqrt N$ at most at first. So, each call to the database, we can only increase the coefficient of the part which knows by around $1/\sqrt N$. If each call, the knowing term experiences constructive interference, we can push up its coefficient to order unity after $\mathcal{O}(\sqrt N)$ calls, but not any faster. The point is, if the part which knows only has a tiny coefficient, any measurement of the wavefunction at that stage will only be able to access that part with a probability of at most the absolute square of that coefficient. There is no random access to specific MWI branches of our choice.

Even more interestingly, the database has to be a quantum database, and the database queries have to be reversible so that no trace of which entry was called remains within the database or its environment. Otherwise, if traces remain, we have decoherence, and instead of constructive interference of the part which knows each query, we get a decohered sum instead and now, $\mathcal{O}(N)$ queries are needed. If Grover's works by massive parallelism, how come it breaks down if traces of which entry was called remains anywhere? No, it works by constructive and destructive interference, which is an entirely different thing from massive parallelism.

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It is quite evident why no traces should remain in database: if the external medium knows which entry you queried, the parallelism disappears. It would be surprising if it was different. –  Anixx Sep 14 '12 at 22:29
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