Quantum Logic

Quantum logic studies the structure of propositions associated with quantum measurements. In its traditional form, sharp yes–no questions correspond to projections, or equivalently closed Hilbert-space subspaces. Their lattice is not generally Boolean. This mathematical difference does not mean that ordinary reasoning becomes inconsistent.

Propositions as subspaces

A projection $P$ defines a yes–no test with probability $p(P)=\operatorname{Tr}(\rho P)$. For a pure state in its range the answer is certain; for a state in the orthogonal complement the answer is certainly no.

For subspaces, conjunction is represented by intersection, disjunction by closed linear span, and negation by orthogonal complement. These operations describe sharp-event structure. They do not mean that incompatible tests can be carried out simultaneously without disturbance.

The detailed operations are developed in Projection Lattices. Birkhoff and von Neumann introduced the program of studying this propositional structure.

A two-dimensional failure of distributivity

Let $P$ be the line spanned by $\lvert0\rangle$, $Q$ the line spanned by $\lvert+\rangle$, and $R$ the line spanned by $\lvert-\rangle$. Because $Q$ and $R$ span the whole space,

$$P\wedge(Q\vee R)=P.$$

But distinct one-dimensional lines have zero intersection, so

$$(P\wedge Q)\vee(P\wedge R)=0.$$

This is a failure of the distributive law. It arises because the quantum join is linear span, not set union. In a state belonging to $P$, the proposition $Q\vee R=I$ is certain although neither $Q$ nor $R$ is individually certain.

The example does not itself prove the Kochen–Specker theorem. Nondistributivity already appears for a qubit, while the usual sharp-projector Kochen–Specker obstruction requires Hilbert-space dimension at least three.

Classical contexts inside quantum theory

A commuting family of projections generates a Boolean event algebra. Within one such context, intersections and unions behave like ordinary events and the Born probabilities form an ordinary probability distribution.

The foundational difficulty concerns how different contexts fit together. One cannot replace their full structure by a single context-independent Boolean sample space while retaining all the quantum constraints. Contextuality explains the relevant no-go statements and their assumptions.

Sharp-projection logic is also not the whole theory of measurement. Generalized measurements use effects that need not be projections and have a partial addition law rather than an arbitrary lattice join. See Effect Algebras.

Exercises

Why is the sentence “a quantum proposition is both true and false” not a consequence of nondistributivity?

Solution

Orthocomplementation still gives $P\wedge P^\perp=0$ and $P\vee P^\perp=I$. What fails is a distributive relation among several propositions from incompatible contexts. It does not identify a proposition with its negation or license contradictions in the surrounding mathematical argument.

References

• Birkhoff, Garrett, and John von Neumann, “The logic of quantum mechanics,” Annals of Mathematics 37, 823–843 (1936). Original paper.
• Kochen, Simon, and Ernst P. Specker, “The problem of hidden variables in quantum mechanics,” Journal of Mathematics and Mechanics 17, 59–87 (1967). Paper.