Stories in the two-state vector formalism

TL;DR

This paper develops a rigorous foundation for the two-state vector formalism in quantum mechanics, introducing the concept of a story and analyzing the distinguishability of two-state vectors, highlighting entanglement between past and future.

Contribution

It introduces the concept of a story, examines the structure of two-state vectors, and defines strictly non-separable vectors as genuine entanglement manifestations.

Findings

Some pairs of two-state vectors are indistinguishable from mixtures.

Examples of distinguishable and indistinguishable two-state vectors are provided.

Defines strictly non-separable vectors as entanglement between past and future.

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 analyze the problem of distinguishability and confirm that some pairs of two-state vectors or their statistical mixtures cannot be physically distinguished. In particular, we discuss an example of a two-state vector that is indistinguishable from a statistical mixture of separable two-state vectors and provide an example of a two-state vector that can be distinguished from every such mixture. 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.