From: jbaez@zermelo.mit.edu (John C. Baez)
Subject: Re: The It from Bit: Quantum Logic and Information Theory
Organization: MIT Department of Mathematics, Cambridge, MA
Date: Tue, 21 Jul 92 16:09:37 GMT

In article <54526@mentor.cc.purdue.edu> hrubin@pop.stat.purdue.edu (Herman Rubin) writes:
>In article <1992Jul19.202309.29157@galois.mit.edu> jbaez@cayley.mit.edu (John C. Baez) writes:

>>There is an extensively investigated, beautiful, and (IMHO) perfectly
>>satisfactory theory of quantum logic which physicists and mathematicians
>>might find worth learning.  A good introduction is Jauch's "Foundations
>>of Quantum Theory" (if I have the title right).  

>It is not clear that the theory of quantum logic cited is useful for 
>anything, or adequate.  I believe that a better view of the uncertainty
>principle is that P&Q, for example, is utter nonsense; even making
>things probabilistic does not help.

Well, it's clear to me that it's useful and adequate, though I certainly
don't want to oversell it.  I'm not saying that Joe Blow quantum
mechanic needs to learn this approach to do calculations better.  It's
just a different viewpoint on the usual Hilbert space/C*-algebra
approach to quantum theory, one which emphasizes *propositions* (aka
projections) rather than states or observables.  It emphasizes how
quantum theory may be regarded as a noncommutative generalization of
measure theory (or probability theory).

>Also, there is no satisfactory truth value system which makes truth
>values merely probabilities, but rather elements of a probability
>space.  If A has probability .5, so does ~A.  But A&A has probability
>.5, while A&~A has probability 0.

There is certainly no reason to expect to calculate the probability of
P&Q holding knowing only the probabilities of P and Q, since as you
point out P and Q may not be independent.  However, this is just a
well-known fact of life and not any obstruction to defining a state on a
lattice of propositions as a map from the lattice to [0,1] satisfying
certain conditions (crucially, monotonicity).  

>There have been many attempts to describe quantum mechanics within 
>probability, but I know of no successful ones.  Quantum processes
>are far more complicated than stochastic processes.

Again, I am saying that quantum theory may be regarded as a
generalization of probability theory, not that it can be reduced to it.
Probability theory is just quantum theory in the special case where all
observables commute.  

I have a feeling we're talking at cross purposes here.  The following
diagram should either make what I'm saying clear OR make it clear that
I'm talking about some weird branches of math you are not familiar with.

Boolean lattices    are to   Orthomodular lattices
AS
Compact Hausdorff spaces     are to   C*-algebras
AS
Measure spaces               are to   von Neumann algebras
AS
probability measures         are to   states
AS
measurable subsets           are to   projections

The left column is a whole bunch of related ways of discussing classical
logic and probability theory.  The right column consists of the
quantum-mechanical analogs of the things in the left column.   To the
extent to which "boolean lattices" deserves to be called classical logic
(more precisely, the propositional calculus), "orthomodular lattices"
deserves to be called quantum logic.