As is known, when discussing Bell inequalities, it is very important to identify all (explicit and implicit) assumptions that are important for deriving these inequalities. Our goal is to draw attention to one assumption, which plays a key role in the analysis of these inequalities, but at the same time are erroneous from our point of view. The fact is that at the present time it is customary to believe that modern quantum mechanics is a complete, internally consistent theory. In particular, it is considered as an indisputable truth its provision according to which all vector states of particles, including superpositions of single-particle states localized in the disjoint (macroscopically distinguishable) spatial regions, as well as vector states of the EPR-pair (the pair of electrons separated by a space-like interval) are always pure quantum states. Practically speaking, this is precisely why the question of the physical interpretation of quantum theory has been considered for a long time, since the formulation of such a problem is reasonable only in relation to a physical theory, the mathematical apparatus of which is completed and internally consistent.

Our idea is that modern quantum theory does not satisfy this requirement and the formulation of the problem of its interpretation is premature. Namely, there is reason to believe that the question of the legality of the unlimited use of the superposition principle in all single-particle quantum processes, as well as in the theory of EPR pairs, is not properly well-regulated in modern quantum theory. In other words, we suggest to reconsider the question of the “purity” of vector states in these cases. Hence, it is necessary to reaxamine the question of the collapse of the wave function (vector state) in these cases.

Using the example of a complete scattering of a particle on a one-dimensional symmetric potential barrier [1] (arXiv: 1805.03952v5), we showed that the quantum dynamics of a particle in this process is governed by the asymptotic superselection rule, which limits the action of the superposition principle in this one-particle process . According to this rule, the wave function describing the (vector) state of a scattering particle — the scattering state — is a mixed vector state. But, unlike ‘ordinary’ mixed states described by the density operator (whose square is not equal to the operator itself), observables cannot be introduced for this vector state (that is, for the whole quantum ensemble of scattering particles). This can be done only for sub-ensembles of transmitted and reflected particles, the individual dynamics of which at all stages of scattering are reconstructed in [1] from the in- and out-asymptotes of the scattering state (their dynamics can be experimentally investigated only indirectly). The theory [1] of mixed vector states arising in a one-dimensional completed scattering process contains some features of well-known approaches to fundamental problems of quantum mechanics, but does not coincide with any of them.
Examples:

(1) The well-known asymptotic superselection rules destinated to convert pure quantum states into mixed ones, were obtained for open quantum systems. Whereas, the asymptotic rule of superselection [1] was obtained for a closed quantum system.
(2) In the theory of the collapse of the wave function, non-unitary theories play a key role. In [1], the dynamics of the subprocesses of transmission and reflection are non-unitary too (the dynamics of the process itself are unitary, but the superselection rule forbids introducing observables for it).
(3) In the Girardi theory, nonlinear effects play a key role in the transformation of pure quantum states into mixed ones. In [1], although the dynamics of the process itself are linear, the dynamics of each of its two subprocesses are nonlinear. The point is that the incoming and outgoing waves, for each subprocess’ (stationary) scattering state, are “stitched” together in the center of the spatial region (where a symmetric barrier is specified) with making use of nonlinear conditions of continuity — they assume the continuity of the wave function itself and the probability flow dencity (but do not imply the continuity of the derivative of the wave function at this point).

We note that in the case of an “uncompleted scattering process” of a particle on a one-dimensional potential barrier, the asymptotic superselection rule does not appear.

Bottom line: the proposed approach takes all the “good” of the known approaches to the fundamental problems of quantum mechanics, but respects the very quantum mechanics with its basic concept of ‘closed systems’ and linear unitary dynamics; it shows that the notion of mixed vector states is that tool which could reconcile the linear unitary quantum theory of closed systems with classical physics.