Ehtibar N. Dzhafarov and Janne V. Kujala (2017), in J. A. de Barros, B. Coecke and E. Pothos (eds), Quantum Interaction (QI 2016), Lecture Notes in Computer Science 10106, 16–32 — ARXIV-1604.04799
Key takeaways
- What changes from CbD 1.0 (LIT-777). The constraint on the copies of one content across contexts was a maximal coupling of the whole connection. It is now a multimaximal coupling: one in which every subset of the connection is maximally coupled (Definition 1). The stated reason is heredity. In CbD 1.0 a noncontextual inconsistently connected system can have a contextual subsystem. The paper asserts this without giving an example.
- Binary connections have exactly one multimaximal coupling (Theorem 1, Corollary 1). Sort the copies by p_i = Pr[R_q^i = 1]. The coupling puts mass p₁ on all-1s, p_{l+1} − p_l on the string with l leading 2s, and 1 − p_k on all-2s. Theorem 2, proved in the companion paper (LIT-836), says it suffices that adjacent pairs in this order are maximally coupled. Theorem 5 says that for any variables, multimaximal is the same as pairwise maximal.
- Consequences (Section 4). A noncontextual binary system has only noncontextual subsystems (Theorem 3). Partial and complete (non)contextuality coincide, because the connection couplings are unique. Consistently connected systems get the traditional verdict. Any connection of two variables is judged exactly as in CbD 1.0, so cyclic systems are unchanged. Kochen–Specker systems with three or more variables per connection (Peres’s proofs) “will yield different results”.
- The measure carries over (Theorem 4). A quasi-coupling agreeing with the multimaximal couplings always exists, and its least total variation is the measure, as in LIT-777’s Section 6.
- Non-binary variables break it (Section 6). In Example 1, three 3-valued variables have no multimaximal coupling. In Example 2, a 6-valued connection has two. In Example 3, coarse-graining Example 2 into Example 1 turns a noncontextual system into a contextual one. The proposed remedy is to replace every variable by all its dichotomizations.
Standing in the record
Filed on 2026-10-09 from the manuscript bibliography of 2026-10-09 (work
what-survives-translation) at the owner’s request: a work the manuscript
considered and dropped from its final reference list. See the curation entry
of that day.
Read on 2026-10-09 (NOTE-643). It is the definition LIT-777’s footnotes 1, 7 and 10 point to, and the record’s current reference for the binary case. Its argument for multimaximality is completed by the companion paper LIT-836 (Theorem 2’s proof) and generalized by the canonical-systems paper LIT-790. THEORY-168 draws on its Examples 1–3.