Projection postulate and superposition principle in non-lattice quantum logics

Sylvia Pulmannová

S. Pulmannov�

Geoffrey Paul Hellman

This paper explores the status of the von Neumann-Lüders state transition rule (the “projection postulate”) within “real-logic” quantum logic. The entire discussion proceeds from a reading of the Lüders rule according to which, although idealized in applying only to “minimally disturbing” measurements, it nevertheless makes empirical claims and is not a purely mathematical theorem. An argument (due to Friedman and Putnam) is examined to the effect that QL has an explanatory advantage over Copenhagen and other interpretations which relativize truth-value assignments to experimental arrangements. Two versions of QL, the lattice-theoretic (LT) and partial-Boolean-algebra (PBA), are considered. It turns out that the projection postulate is intimately connected with the choice of conditional connective for QL. The effect of the projection postulate is obtained with the Sasaki conditional. However, this choice is found to require extra assumptions, on both the LT and PBA versions, which are either just as ad hoc as the projection postulate itself or indefensible from within the real-logic QL framework.

Viktoras Rimsha, Laimontas Rimsha

We develop a statistical interpretation of quantum interference in which probabilities, rather than wave amplitudes as primitive objects, form the fundamental descriptive level of quantum phenomena. Interference effects are treated as structural relations between alternative probabilistic contributions, governed by a principle of global consistency (priority of the whole over parts). Within this framework, quantum probabilities are non-Kolmogorovian but remain objectively defined. We show that this interpretation naturally leads to the lattice-theoretic structure of quantum propositions introduced by Birkhoff and von Neumann. Quantum logic thus emerges not as an abstract postulate, but as a structural consequence of quantum probability itself. Finally, we illustrate how this viewpoint provides a clear conceptual reading of quantum algorithms. As an appendix, the Grover search algorithm is analyzed as an example where interference-based probability redistribution can be understood without invoking metaphysical superposition, but purely through statistical structure.

Stanley Gudder

A superposition principle is considered both in classical mechanics and in the quantum logic approach to quantum mechanics. It is shown, roughly speaking, that in classical mechanics the only type of superposition of states is a mixture and that no pure state is a nontrivial superposition of other pure states. In quantum mechanics it is shown that, if a superposition principle holds, then the quantum logic is a complete atomic lattice.

PEKKA J. LAHTI