During my regular* sit down with Nature magazine today I stumbled upon an intriguing quote with regards to attempts to reformulate quantum mechanics as a theory of observation under uncertainty:

“*A lot of features we think of as uniquely quantum are generic to many probabilistic theories*.”

Apparently various quantum phenomena, such as entanglement, can be shown to arise ‘naturally’ as the consequence of applying a few simple axioms formulated to describe the preparation and observation of a physical system. These so-called General Probablistic Theories (GPTs)—e.g. Hardy—are certainly at lot more physics-y than stats-y (for one thing, they allow only linear transformations on finite-dimensional vector spaces and their topological dual) but they do seem sufficiently powerful to recover a lot of observable quantum effects. I’ll need to read more of the GPT literature but at face value it seems like there could be some low hanging fruit for a more mathematical statistics based reformulation of these quantum theoretical reformulations!

* I’m a vegetarian!

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When I learned quantum in college, it was done out of the blue, and, so, expectation operators were just given as fact. It was only much later, after I learned linear algebra, that I realized these could be interpreted as inner products and, later still, that many measurements in, say, signal processing could be seen the same way. This gave me incredible freedom because long gone was the need to be a slave to orthogonal basis sets when doing measurements.

So I now think that it is much more useful for undergraduate physicists and electrical engineers to learn linear algebra and perhaps even a bit of linear operators and Hilbert space before being introduced to quantum and theory of signal processing (respectively).

It feeds back, too. Why not complex manifolds or support? That’s more obvious in EE with spectra, but there may be reasons knowing that are good. I suspect there are more of these kinds of things which do not make it into basic courses. For example negative probabilities … which I just learned about this year:

http://johncarlosbaez.wordpress.com/2013/07/19/negative-probabilities/

http://drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm

So, I’m not surprised.

(*) Claire and I are vegetarians, too. We each are for different reasons. We have three cats, who, of course, are not. (Or should I say, the three cats have us?)