Comment by obastani | Hacker News Reader

Comment by obastani | Hacker News Reader

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This seems to be a key reference (article titled "Complementarity and uncertainty"):

http://www.nature.com/nature/journal/v375/n6530/pdf/375367b0...

In particular, they have a description of why they find the traditional connection between uncertainty and wave-particle duality to be unsatisfactory. Unfortunately the technical details are beyond me, but hopefully someone else can sort them out?

Edit: Okay, some more digging suggests the following story: (i) We have wave-particle duality => Heisenberg uncertainty, but not the other way around (ii) With entropic uncertainty [1] (stronger than Heisenberg uncertainty), the authors show that we can obtain wave-particle duality

Again, can somebody more knowledgeable can clarify? :)

[1] http://en.wikipedia.org/wiki/Entropic_uncertainty

@obastani : very good comment.

Aside: If |Psi> is a state with wavefunction <x|Psi>, then is there a linear operator A such that <x|A|Psi> = log(<x|Psi>)?

Note: Logarithm of a complex function.

Well, the logarithm is all but linear. If there was such an A you'd have

   <x|A(|Psi> + |Chi>) = log(<x|(|Psi> + |Chi>))
                       = log(<x|Psi> + <x|Chi>)
                       = log(<x|Psi>) log(<x|Chi>)
                      /= <x|A|Psi> + <x|A|Chi>

I don't think that

  log(a + b) = log(a) log(b)

is a rule I'm familiar with!

You're right, I messed that one up. But as

  log(a + b) /= log(a) + log(b)

in general, the result is still correct.

Yes, agree that log() wouldn't result from a linear operator.

The idea behind my question: does the Shannon entropic integral correspond to the L2 length of some projected state? Leading to a prescription to prepare a state with minimum uncertainty product of canonically conjugate physical quantities.

Afaik, in classical statistical physics the log() shows up when the N! in a binomial probability distribution limits to large N via the Stirling approximation. It would be interesting to find a different route for log() to enter the picture from a quantum standpoint.

All admittedly vague and hand waving speculation.

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