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What are the bearers of hidden states? On an important ambiguity in the formulation of Bell's theorem

Joanna Luc

Abstract

One of the conclusions that Bell drew from his famous inequality was that any hidden variable theory that satisfies Local Causality is incompatible with the predictions of Quantum Mechanics for Bell's Experiment. However, Local Causality does not appear in the derivation of Bell's inequality. Instead, two other assumptions are used, namely Factorizability and Settings Independence. Therefore, in order to establish the mentioned Bell's conclusion, we need to relate these two assumptions to Local Causality. The prospects for doing so turn out to depend on the assumed location of the hidden states that appear in Bell's inequality. In this paper, I consider the following two views on such states: (1) that they are states of the two-particle system at the moment of preparation, and (2) that they are states of thick slices of the past light cones of measurements. I argue that straightforward attempts to establish Bell's conclusion fail in both approaches. Then, I consider three refined attempts, which I also criticise, and I propose a new way of establishing Bell's conclusion that combines intuitions underlying several previous approaches.

What are the bearers of hidden states? On an important ambiguity in the formulation of Bell's theorem

Abstract

One of the conclusions that Bell drew from his famous inequality was that any hidden variable theory that satisfies Local Causality is incompatible with the predictions of Quantum Mechanics for Bell's Experiment. However, Local Causality does not appear in the derivation of Bell's inequality. Instead, two other assumptions are used, namely Factorizability and Settings Independence. Therefore, in order to establish the mentioned Bell's conclusion, we need to relate these two assumptions to Local Causality. The prospects for doing so turn out to depend on the assumed location of the hidden states that appear in Bell's inequality. In this paper, I consider the following two views on such states: (1) that they are states of the two-particle system at the moment of preparation, and (2) that they are states of thick slices of the past light cones of measurements. I argue that straightforward attempts to establish Bell's conclusion fail in both approaches. Then, I consider three refined attempts, which I also criticise, and I propose a new way of establishing Bell's conclusion that combines intuitions underlying several previous approaches.

Paper Structure

This paper contains 24 sections, 6 theorems, 41 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Some basic concepts and notation
  3. Two examples of HVTs: True Spin Theory and Bohmian mechanics
  4. A typology of HVTs
  5. Locally determinate and locally indeterminate HVTs
  6. Deterministic, indeterministic and locally deterministic HVTs
  7. Assumptions of Bell's theorem
  8. Hidden state as a state of the two-particle system at the moment of preparation
  9. Hidden state as a state of thick slices of the past light cones of measurements
  10. Late Bell's concept of Local Causality
  11. Locally deterministic theories vs. Local Causality$_{C(\Sigma, \Sigma)}$ and Settings Independence$_{C(\Sigma, \Sigma)}$
  12. Further discussion of the violation of Settings Independence$_{C(\Sigma, \Sigma)}$ in locally deterministic theories
  13. Attempts at improvement
  14. Local causality is Factorizability$_{t_P, 1, 2}$
  15. The idea of an incomplete but sufficient specification of hidden states
  16. ...and 9 more sections

Key Result

Theorem 1

Any locally determinate theory that is locally deterministic is also deterministic.

Figures (5)

  • Figure 1: My notation for regions (top) and states (bottom); $C_{\Sigma_{R, t}, \Sigma_{R,t'}}$ covers only the shaded region.
  • Figure 2: Bell's (Bell1990) depiction of regions appearing in his formulation of Local Causality (left) and the derivation of the Bell-CHSH inequality (right).
  • Figure 3: Relations between the regions mentioned in the formulation of Local Causality$_{C(\Sigma, \Sigma)}$. In the proof of Factorizability$_{C(\Sigma, \Sigma)}$, we substitute for $R$ either region of the choice of settings (i.e., $R_a$/$R_b$) or the region of measurement together with the region of the choice of settings (i.e., $R_A \cup R_a$/$R_B \cup R_b$). For details see Appendix \ref{['app:Fact-from-LC']}.
  • Figure 4: Comparison of two different choices of $\lambda$ in Bell's theorem, $\lambda_{t_P, 1, 2}$ vs. $\lambda_{C(\Sigma, \Sigma)}$.
  • Figure 5: Regions used in the formulation of Local Causality$_\text{H-S}$ and the derivation of Bell's theorem by hofer-szabo-SU. Regions $R$ and $R'$ are within the purple lines.

Theorems & Definitions (15)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 2
  • Theorem 3
  • Theorem 1
  • proof
  • ...and 5 more