Restrictions on the existence of weak values in quantum mechanics: weak quantum evolution concept

Abstract: It is shown, that the Aharonov-Albert-Vaidman concept of weak values appears to be a consequence of a more general quantum phenomenon of weak quantum evolution. Here the concept of weak quantum evolution is introduced and discussed for the first time. In particular, it is shown on the level of quantum evolution that there exist restrictions on the applicability of weak quantum evolution- and, hence, weak values approach. These restrictions connect the size of given quantum ensemble with the parameters of pre- and post-selected quantum states. It is shown, that the latter requirement can be fulfilled for the model system, where the concept of weak values was initially introduced by Aharonov,Vaidman and Albert. Moreover, a deep connection between weak quantum evolution and conventional probability of quantum transition between two non-orthogonal quantum states is established for the first time. It is found that weak quantum evolution of quantum system between its two non-orthogonal quantum states is inherently present in the measurement-determined definition of quantum transition probability between these two quantum states.