Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Noncommuting common causes revisited

Gábor Hofer-Szabó, Szilárd Szalay

Abstract

In this paper, we revisit the concept of noncommuting common causes; refute two objections raised against them, the triviality objection and the lack of causal explanatory force; and explore how their existence modifies the EPR argument. More specifically, we show that 1) product states screening off all quantum correlations do not compromise noncommuting common causal explanations; 2) noncommuting common causes can satisfy the law of total probability; 3) perfect correlations can have indeterministic noncommuting common causes; and, as a combination of the above claims, 4) perfect correlations can have noncommuting common causes which are both nontrivial and satisfy the law of total probability.

Noncommuting common causes revisited

Abstract

In this paper, we revisit the concept of noncommuting common causes; refute two objections raised against them, the triviality objection and the lack of causal explanatory force; and explore how their existence modifies the EPR argument. More specifically, we show that 1) product states screening off all quantum correlations do not compromise noncommuting common causal explanations; 2) noncommuting common causes can satisfy the law of total probability; 3) perfect correlations can have indeterministic noncommuting common causes; and, as a combination of the above claims, 4) perfect correlations can have noncommuting common causes which are both nontrivial and satisfy the law of total probability.

Paper Structure

This paper contains 21 sections, 17 theorems, 98 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Noncommutative probability theory
  3. General theory
  4. Discrete finite systems
  5. Two-qubit systems
  6. Trivial and nontrivial common cause systems
  7. Triviality of common cause systems
  8. Examples
  9. Noncommuting common causes and the law of total probability
  10. Law of total probability
  11. Examples
  12. Indeterministic noncommuting common causes of perfect correlations
  13. Maximal and perfect correlations
  14. Examples
  15. Limitations of atomic common cause systems for perfect correlations
  16. ...and 6 more sections

Key Result

Lemma 1

For the weakly commuting partition $\{ C_k \}_k$, for the two projections $C:=\sum_{k:\phi(C_k)\neq0} C_k$ and $D:=\sum_{k:\phi(C_k)=0} C_k$, we have $\phi(DX)=\phi(XD)=0$ for $X\in\{ AB,AB^\perp,A^\perp B,A^\perp B^\perp \}$.

Figures (1)

  • Figure 1: Plots of the functions given implicitly in \ref{['eq:theta']}. (Blue: $\xi(\theta)$, green $a(\theta)=\cos(\xi(\theta)/2)$, red $b(\theta)=\sin(\xi(\theta)/2)$.)

Theorems & Definitions (32)

  • Lemma 1
  • proof
  • Proposition 2
  • proof
  • Corollary 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Corollary 6
  • ...and 22 more