Asymptotically Disjoint Quantum States

Abstract

A clarification of the heuristic concept of decoherence requires a consistent description of the classical behavior of some quantum systems. We adopt algebraic quantum mechanics since it includes not only classical physics, but also permits a judicious concept of a classical mixture and explains the possibility of the emergence of a classical behavior of quantum systems. A nonpure quantum state can be interpreted as a classical mixture if and only if its components are disjoint. Here, two pure quantum states are called disjoint if there exists an element of the center of the algebra of observables such that its expectation values with respect to these states are different. An appropriate automorphic dynamics can transform a factor state into a classical mixture of asymptotically disjoint final states. Such asymptotically disjoint quantum states lead to regular decision problems while exactly disjoint states evoke singular problems which engineers reject as improperly posed.

Compare for example BOHR (1949), p.209.

For example, Bohr argued that the description of a meauring instrument cannot be included in the realm of quantum mechanics. Most clearly Bohr stated his view in a letter of October 26, 1935 to Schrödinger: “Das Argument its ja dabei vor allem, dass die Messinstrumente, wenn sie als solche dienen sollen, nicht in den eigentlichen Anwendungsbereich der Quantenmechanik einbezogen werden konnen.” Quoted on p.510 in Kalckar (1996).