Quantum Computing since Democritus by Aaronson Scott

Author:Aaronson, Scott [Aaronson, Scott]
Language: eng
Format: mobi, epub, pdf
Publisher: Cambridge University Press
Published: 2013-02-27T16:00:00+00:00

Exercise (harder): Prove that it's possible to choose the “canonical” maximal flows in such a way that making a small change to U or |ψ produces only a small change in the matrix (pij) of transition probabilities.

The Schrödinger theory

So that was one cute example of a hidden-variable theory. I now want to show you an example that I think is even cuter. When I started thinking about hidden-variable theories, this was actually the first idea I came up with. Later I found out that Schrödinger had the same idea in a nearly forgotten 1931 paper.10

Specifically, Schrödinger's idea was to define transition probabilities in quantum mechanics by solving a system of coupled nonlinear equations. The trouble is that Schrödinger couldn't prove that his system had a solution (let alone a unique one); that had to wait for the work of Masao Nagasawa11 in the 1980s. Luckily for me, I only cared about finite-dimensional quantum systems, where everything was much simpler, and where I could give a reasonably elementary proof that the equation system was solvable.

So what's the idea? Well, recall that, given a unitary matrix U, we want to “convert” it somehow into a stochastic matrix S that maps the initial distribution to the final one. This is basically equivalent to asking for a matrix P of transition probabilities: that is, a nonnegative matrix whose ith column sums to |αi|2 and whose jth row sums to |βj|2. (This is just the requirement that the marginal probabilities should be the usual quantum-mechanical ones.)

Since we want to end up with a nonnegative matrix, a reasonable first step would be to replace every entry of U by its absolute value:

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