The connection between quantum mechanics and logic

Scott at Shtetl-Optimized is forced to write a post (which, I think is a good thing; otherwise, how else are we to get such nice posts?):

… I see colleagues who I respect and admire enormously—in this case, several who have done pioneering experiments that tested quantum mechanics in whole new regimes—making statements that can be so easily misinterpreted by a public and a science press hungry to misinterpret, that I find my fingers rushing to type even as my brain struggles in vain to stop them.

Now, to the more interesting parts of the post — about quantum logic:

So is there a connection between quantum mechanics and logic?  There is—and it was worked out by Birkhoff and von Neumann in 1936.  Recall that Paterek et al. identify propositions with projective measurements, and axioms with states.  But in logic, an axiom is just any proposition you assume; otherwise it has the same form as any other proposition.  So it seems to me that we ought to identify both propositions and axioms with projective measurements.  States that are eigenstates of all the axioms would then correspond to models of those axioms.  (Interestingly, the notion of a model never appears in the Paterek et al. paper.)  Also, logical inferences would derive some propositions from other propositions, like so: “any state that is an eigenstate of both X and Y is also an eigenstate of Z.”  As it turns out, this is precisely the approach Birkhoff and von Neumann took.  The field they started is called “quantum logic.”

A nice post!


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