3.9 Quantum logic

Quantum logic pioneered by Birkhoff and von Neumann [20] (see also, Mackey [92], Kalmbach [78], Cohen [45], Pulmannová [112], Svozil [129]) deals with propositions expressing properties of quantum systems. To every physical property we can associate in a natural way the proposition
“the physical system has property ",
which means
“if the observable is measured, then the property is observed".
Such a proposition is always true or false.

Technically, quantum logic identifies logical entities with Hilbert space entities. In particular, elementary propositions are associated with closed linear subspaces of a Hilbert space through the origin (zero vector); the implication relation is associated with the set-theoretical subset relation , and the logical or , and , and not operations are associated with the set-theoretic intersection, the linear span of subspaces and the orthogonal subspace , respectively. The negation of is denoted by . The logical statement which is always true is identified with the entire Hilbert space , and its complement with the zero-dimensional subspace (zero vector).
neumann
     
                                                                         Figure 9: John von Neumann

Two propositions and are orthogonal if . Two propositions are co-measurable (compatible, commuting) if there exist three mutually orthogonal propositions such that

Intuitively, propositions and consist of an “identical part" and two orthogonal parts . Clearly, orthogonality implies co-measurability, since if and are orthogonal, we may take to be , respectively.

A simple example is the propositional structure encountered in the quantum mechanics of spin state measurements of a spin one-half particle. Assume that the associated Hilbert space is two-dimensional and real-valued, and consider measurements of the spin-component along one particular direction, say the -axis. There are two possible spin components of the particle, , which can be codified as and . The corresponding elementary propositions are:
= “the particle is in state " = one-dimensional subspace spanned by the vector , and
= “the particle is in state " = one-dimensional subspace spanned by the vector .
The tautology 1 is the proposition “the particle is in state or in state ", which is the whole space, and the absurdity 0 is the proposition “the particle is neither in state nor in state ", which is the zero-dimensional subspace of which is . The propositional structure obtained in this case is the classical Boolean algebra with two elements; its Hasse diagram appears in Figure 10 . Recall that in a Hasse diagram propositions are represented by dots, implication is represented “vertically", that is is drawn higher than if , and the propositions and are connected by a line.

So far we have discussed only about co-measurable observables. The corresponding propositions form a Boolean algebra with elements, so the so-called blocks in case of a Hilbert space of dimension . However, non-co-measurable observables should be treated as well! Of course, we may take the view that non-co-measurable observables make no physical sense, so we should forget about them (at least, with respect to current day knowledge; maybe, a “more complete" theory could make sense of them!). This legitimate, but somewhat “minority" position goes beyond our aim, so we will concentrate on the mainstream approach which considers that non-co-measurable observables make physical sense at least as theoretical constructions. Note that this attitude is common also in mathematics, where, for example, makes no “direct" sense, but proves to be extremely useful.

A simple formalism to deal with non-co-measurable observables is via the so-called pasting construction. Consider a collection of blocks and note that some of them may have a common non-trivial observable. A “logical" structure can be extracted as follows:


                                         Figure 10: Hasse diagram of the spin one-half state co-measurable propositions.
identify all tautologies in all blocks,
identify all absurdities in all blocks,
identify identical elements in different blocks,
keep intact the logical structure of all blocks.
As a simple example let us paste together observables of the spin one-half systems, see Figure 10. Then we have two propositional systems, the first corresponding to the outcomes of a measurement of the spin states along the –axis

and another one, corresponding to the outcomes of a measurement of the spin states along a different spatial direction, say (mod ), an identical propositional system,

So, we identify tautologies and absurdities and keep all other propositions intact. The result is the M0 propositional structure presented in Figure 11. It is easy to see that M0 is not any longer a Boolean algebra, since distributivity is not satisfied, as the following example shows:

Algebraically, M0 is an orthocomplemented lattice, that is, any two elements have a least upper bound and a greatest lower bound,

and for all , the modular law is satisfied:




   

                                                                      Figure 11: _Hasse diagram of M0. _

This can be obtained with a Stern-Gerlach type of experiment.