Where Was I When They Were Passing Out the Wit?

Scott Aaronson has a new comment policy:

If you reject an overwhelming consensus on some issue in the hard sciences — whether it’s evolution, or general relativity, or climate change, or anything else — this blog is an excellent place to share your concerns with the world. Indeed, you’re even welcome to derail discussion of completely unrelated topics by posting lengthy rants against the academic orthodoxy — the longer and angrier the better! However, if you wish to do this, I respectfully ask that you obey the following procedure:

1. Publish a paper in a peer-reviewed journal setting out the reasons for your radical departure from accepted science.
2. Reference the paper in your rant.

If you attempt to skip to the “rant” part without going through this procedure, your comments will be deleted without warning. Repeat offenders will be permanently banned from the blog. Life is short. I make no apologies.

It looks like Dave Bacon can now talk about time travel, but my own conspiracy theories will have to wait. But soon, I promise, the real meaning behind supersymmetric quantum mechanics will be made clear. They laughed at me when I suggested that the BPS interpretation of shape invariance may have a non-topological origin. The fools — I’ll show them all!

Incidentally, I highly recommend Aaronson’s “Quantum Computing Since Democritus” lecture notes, particularly Lecture 11: decoherence and hidden variables and Lecture 10.5: Penrose. From the latter:

By exploiting Penrose’s insights, I was able to create a completely unbreakable CAPTCHA. How does it work? It simply asks whether you believe the Gödel sentence G(F) for some reasonable formal system F! Assuming you answer yes, it then (and this is a minor security hole I should really patch sometime) asks whether you’re a human or a machine. If you say you’re a human, you pass. If, on the other hand, you say you’re a machine, the program informs you that, while your answer happened to be correct in this instance, you clearly couldn’t have arrived at it via a knowably sound procedure, since you don’t possess the requisite microtubules. Therefore your request for an email account must unfortunately be denied.

And from the former:

When I was talking before about the fragility of quantum states — how they’re so easy to destroy, so hard to put back together — you might have been struck by a parallel with the Second Law of Thermodynamics. Obviously that’s just a coincidence, right? Duhhh, no. The way people think about it today, decoherence is just one more manifestation of the Second Law.

Let’s see how this works. Given a probability distribution D=(p1,…,pN), recall that the entropy of D is

[tex]H(D) = -\sum_{i} p_i \log p_i . [/tex]

Then given a quantum mixed state ρ, the von Neumann entropy of ρ is defined to be the minimum, over all unitary transformations U, of the entropy of the probability distribution that results from measuring UρU-1 in the standard basis. To illustrate, every pure state has an entropy of 0, whereas the one-qubit maximally mixed state has an entropy of 1.

Now, if we assume that the universe is always in a pure state, then the “entropy of the universe” starts out 0, and remains 0 for all time! On the other hand, the entropy of the universe isn’t really what we care about — we care about the entropy of this or that region. And we saw before that, as previously-separate physical systems interact with each other, they tend to evolve from pure states into mixed states — and therefore their entropy goes up. In the decoherence perspective, this is simply the Second Law at work.

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