University of Colorado, USA, mcj@colorado.edu
This work extends my previous results [1] characterizing contextuality in terms of probabilistic causal models, paralleling [2] but applicable to cases of inconsistent connectedness (i.e., violation of no-disturbance). There, I proved the equivalence of three definitions for contextuality of any measurement system M: (1) there exists no causal model for M that simultaneously minimizes all direct influences of context on measurement outcomes, (2) any causal model for M contains hidden influences (influences that go in opposite directions for different latent states, or equivalently signaling that carries no information), and (3) contextuality as defined in the Contextuality-by-Default (CbD) theory [3]. These previous results were limited to a class of causal models having a particular canonical structure. Here I generalize the results to arbitrary causal models.
[1] Jones, M. (2019). Relating causal and probabilistic approaches to contextuality. Philosophical Transactions of the Royal Society A, 377, 20190133. (Presented at QCQMB19)
[2] Cavalcanti, E. G. (2018). Classical causal models for Bell and Kochen-Specker inequality violations require fine-tuning. Physical Review X, 8, 021018.
[3] Dzhafarov, E.N., & Kujala, J.V. (2017). Contextuality-by-Default 2.0: Systems with binary random variables. In J.A. de Barros, B. Coecke, & E. Pothos (Eds.), Lecture Notes in Computer Science 10106, 16-32.