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Stories in the two-state vector formalism

Patryk Michalski, Andrzej Dragan

TL;DR

The paper develops a rigorous foundation for the time-symmetric two-state vector formalism by introducing the story, a compatible pair of a two-state vector $|\Psi\rrangle \in \mathcal{H} \otimes \mathcal{H}^*$ and a measurement, and then studies the structure of the twin space $\mathcal{H} \otimes \mathcal{H}^*$ in this context. It proves that every TSV forms a story with some measurement and analyzes the fixed-measurement case, showing the non-story set is a zero-measure linear subspace, which clarifies how boundary conditions can be systematically selected. The work investigates the distinguishability landscape: time-reversal symmetry can render certain TSVs indistinguishable from their reversed partners or from mixtures, while it also identifies strictly non-separable TSVs that are experimentally distinguishable from any mixture, thereby giving a genuine notion of past–future entanglement. These results set the stage for further development of detection methods (e.g., entanglement witnesses) and motivate connections to indefinite causal structures and relativistic generalizations, with potential implications for broader physics including tachyonic field theories and quantum gravity.

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.

Stories in the two-state vector formalism

TL;DR

The paper develops a rigorous foundation for the time-symmetric two-state vector formalism by introducing the story, a compatible pair of a two-state vector and a measurement, and then studies the structure of the twin space in this context. It proves that every TSV forms a story with some measurement and analyzes the fixed-measurement case, showing the non-story set is a zero-measure linear subspace, which clarifies how boundary conditions can be systematically selected. The work investigates the distinguishability landscape: time-reversal symmetry can render certain TSVs indistinguishable from their reversed partners or from mixtures, while it also identifies strictly non-separable TSVs that are experimentally distinguishable from any mixture, thereby giving a genuine notion of past–future entanglement. These results set the stage for further development of detection methods (e.g., entanglement witnesses) and motivate connections to indefinite causal structures and relativistic generalizations, with potential implications for broader physics including tachyonic field theories and quantum gravity.

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.
Paper Structure (7 sections, 4 theorems, 28 equations)

This paper contains 7 sections, 4 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Formal description of two-state vectors
  3. Story
  4. Structure of the twin space
  5. Distinguishability of two-state vectors
  6. Separable and non-separable two-state vectors
  7. Discussion and conclusions

Key Result

theorem 1

For every vector $|\Psi\rrangle \in \mathcal{H} \otimes \mathcal{H}^\star$, there exists a measurement $\{\hat{P}_i\}$ such that the pair $(|\Psi\rrangle,\{\hat{P}_i\})$ forms a story.

Theorems & Definitions (7)

  • definition 1
  • theorem 1
  • proposition 1
  • proposition 2
  • proposition 3
  • definition 2
  • definition 3