Stories in the two-state vector formalism

Abstract: The two-state vector formalism of quantum mechanics is a time-symmetrized approach to standard quantum theory. In our work, we aim to establish rigorous foundations for the future investigation within this formalism. We introduce the concept of a story - a compatible pair consisting of a two-state vector and an ideal measurement. Using this concept, we examine the structure of the space comprising all two-state vectors. We conclude that some pairs of two-state vectors or their statistical mixtures cannot be physically distinguished. In particular, we demonstrate that not every non-separable two-state vector is distinguishable from a statistical mixture of separable two-state vectors. This leads us to formulate the definition of a strictly non-separable two-state vector as a genuine manifestation of entanglement between the past and the future.