Quantum logic based on closed Hilbert space subspaces

[nomadreid]

TL;DR Summary

Given a complex Hilbert space, the set of closed subspaces under inclusion form an orthomodular lattice. But how to use this lattice to generate a lattice of truth values for propositions representing events is unclear to me.

One proposal that I have read (but cannot re-find the source, sorry) was to identify a truth value for a proposition (event) with the collection of closed subspaces in which the event had a probability of 1. But as I understand it, a Hilbert space is a framework which, unless trivial, keeps open the possibility of many values.

As well, if each such set of subspaces consisted of a single subspace, then I could imagine how to order (partial order) the collection of sets of subspaces, i.e., by the same order used to order the subspaces. But if there is more than one subspace in a set?

There are indeed sources on the internet which discuss this quantum logic, but the discussions of the assignments of truth values concentrate on the connectives (meet and join, orthocomplementation, <) , but if they discuss the basics of the lattice, it escapes me. I would be grateful for some clarification on the basics here.

 

[meopemuk]

Quantum logic has two major ingredients: propositions and states. As you said correctly, propositions can be identified with subspaces in a Hilbert space. By Gleason theorem, (pure) states can be identified with vectors of unit length in the same space. For each state vector x and each proposition A, we can find "the probability that A is true in the state x" by calculating the projection of x on A and taking the square of its length.
In contrast to classical logic, where the truth value of each proposition can be either 1 (true) or 0 (false), in quantum logic the truth value is replaced by probability -- a number in the interval [0,1].
Eugene.

 

[nomadreid]

Thank you, Eugene (aka meopemuk).

Allow me to try to formulate this in a couple of ways to expose my misunderstanding of the relationships involved.

The state is associated with the probability distribution of the entire state of that point in spacetime, which represents a collection of subspaces, each of which is a projection of the state. With respect to the full state, the event will have a probability 0<p<1 , but with respect to each subspace, the probability is one or zero. Otherwise put, one looks at the probability (overall) that the event will have a probability of one (when restricted).

Or, to put it in terms of models: each subspace is a world, and the full state is a Kripke-type or possible worlds-type frame; the truth values are associated with the worlds in which is the proposition is true. Thus, in each world the proposition is either satisfied or not satisfied, no in-between; however, in the entire frame, some measure would quantify the relative frequency of the set of worlds (subspaces) in which the proposition is satisfied (the event occurs) to the set of all possible events (in the full Hilbert space) -- that is, the position of this proposition in the lattice -- and this measure would be the probability, i.e., the truth value.

From the same galaxy in a time zone far far away...