> However, there is a rather simple proof that shows that the maximum amount of amplitude which you can transfer from one component of a quantum state to another using a control qubit is fundamentally limited by the Unitarian nature of quantum operations.

The "weird trick" is that they do non unitary things via partial measurement to beat that bound.

There might well be hidden costs here. Or judging from what others have said, it might just be a more straightforward mistake.

> The "weird trick" is that they do non unitary things via partial measurement to beat that bound.

Heh? Unitarity is basically the defining characteristic of quantum mechanics. You can't just casually violate that.

I'm not an expert in this subject area, but I know that I've sometimes seen the phenomena that are sometimes called "wavefunction collapse" described as non-unitary (and for good reason). Even if you buy into an interpretation of QM that doesn't include collapse as a separate process (most non-Copenhagen interpretations, in other words), measurements of the quantum state will still have to somehow look non-unitary to any given observer. (Something something projection operators, in the formulation I learned in grad school.)

There are definitely some interesting tricks you can do to exploit these behaviors. (I'm thinking for example of the "quantum Zeno effect", in which one can "find an answer without ever asking the question".[0]) But I don't know remotely enough about quantum computing to know what's actually possible in this context.

[0] A fun writeup on this idea is here (with an example involving an attempt not to wake any adorable sleeping puppies): http://www.preposterousuniverse.com/blog/2006/02/27/quantum-...

That is not correct. Quantum mechanics has both a unitary and a non-unitary evolution. The non-unitary evolution is quantum measurement/collapse:

http://www.wikiwand.com/en/Wave_function_collapse

How to reconcile them in a satisfactory theory is an open issue, but non-unitary measurements play a crucial role in many quantum protocols, including, for example, quantum teleportation.

Basically if we would look at the wave function of the whole universe it would always behave in a Unitarian way. Non-unitarian behavior (such as qubit readouts) are the result of coupling our -initially isolated- quantum system with an external system that has many degrees of freedom and causes decoherence in the individual components of the original system's wave function. A measurement operation in that sense is an entanglement of the original system with an external quantum system, followed by decoherence of the wavefunction due to the coupling of the external system to another one with a large number of degrees of freedom (which destroys the interference in the components of the wave function and thus turns a quantum state into a "classical" state). The role of decoherence and entanglemenr was not well understood for a long time and led to many of the seeming paradoxes of quantum mechanics, but today the theory of "open quantum systems" explains that behavior quite well and one can even simulate such systems using e.g. a so-called "Master equation" approach. What's important to remember here is that the whole system (so initial quantum system + environment) is always behaving in a Unitarian way, but this must not be true when looking at individual parts of the wave function in isolation.

> Basically if we would look at the wave function of the whole universe it would always behave in a Unitarian way.

There is absolutely no consensus on this. This is merely saying that you believe in Everett style interpretations, which have so far failed to explain the appearance of non-linear processes in quantum mechanics satisfactorily.

Decoherence does _not_ solve the quantum measurement problem. You can not derive the Born rule.

Basically what decoherence does for you is translate a quantum amplitude on the space of operators into a classical distribution on the spectrum of your coupling. It does not tell you why you are allowed to interpret this distribution as a probability distribution. It does not tell you why you should be able to say that the state with higher amplitude is more likely to occur. In decoherence all states occur. So it is even unclear a priori what should be meant by the probability of a state occurring.

This is not my private opinion, this is the opinion of people like Zeh, who was instrumental in developing our current understanding of decoherence:

"[Decoherence] would explain why we never observe an apparatus pointing, say, to two different results, i.e. decoherence would provide a solution to the measurement problem of quantum mechanics. As pointed out by many authors, however (e.g. Adler 2003; Zeh 1995, pp. 14–15), this claim is not tenable."

http://plato.stanford.edu/entries/qm-decoherence/#SolMeaPro

Edit: Note that this is not in defense of the correctness of the above paper.

> which have so far failed to explain the appearance of non-linear processes in quantum mechanics satisfactorily.

There's a famous theorem that any non-linearity in quantum mechanics (however small) would permit FTL information transfer. Could you link me one paper (either from a reputable journal or a highly cited article on the arXiv) that demonstrates non-linearity other than the exception of instantaneous collapse via measurement?

I wasn't talking about non-linearity other than the one you mention. That should be abundantly obvious from what I posted.

You can't have quantum computation without measurement, and thus without non-linearity. Quantum computation is not simply "doing the computation in each possible world", it also is some trick to extract (collapse) the information into one particular world, and I don't think we currently understand fully what that means, but it's certainly non-linear. After all the information is often in the amplitude.

Conversely there are measurement based models for quantum computation that rely purely on the non-linear process.

As far as I know it, in any recent QM lecture or textbooks a measurement operation is described as an entanglement of the original system with an external quantum system with very large number of degrees of freedom. In older texts you would just find Born's rule.

^Yep, basically what I was going to say, but I forgot to reply more promptly. Thanks.

So, I don't have the theory background a PhD candidate might, so forgive my ignorance, but why would a theory be expected to build on the already proven partial results? I was under the impression their use was as a quality heuristic, not something a proof of a lower bound of the general case could make much use of.

Generally because that’s how mathematics tends to work - people publish a partial result & then they or others build on it to improve that result in various ways[1]. The authors are also usually embedded in a wider community of mathematicians who act as a pre-publication bullshit filter by asking pointed questions at seminars & that kind of thing.

It’s quite rare for a genuinely novel solution to a major problem in mathematics to spring up out of nowhere in the literature.

[1] Think of, eg, the Twin Prime conjecture (that there are infinitely many pairs of primes only 2 integers apart). For decades there was no progress on this problem, before Yitang Zhang published a paper last year proving that there were infinitely many primes less than a finite bound apart (70 million in his case). Then a horde of matheaticians attacked the problem using his & other new approaches to progressively reduce that large bound down to the point where it’s now been proven that there are infinitely many primes < 246 apart (according to the polymath group - I’m not sure whether this is a published result yet). The twin prime conjecture is in sight!

Yea, but to my understanding the partial work done thus far does not apply to an understanding of the general case solution. Let's say there is a linear solution for some form--how does that help establish a lower bound on the general form?

Recognition of past work I can understand. But it seems like occam would argue with me against the use of some of the partial results without understanding why they would be useful.