Dominika Burešová presents Quantum Logics with Symmetric Difference: Algebraic and causal-principle perspectives

The seminar presented by Dominika Burešová and Pavel Pták combines two
parts.


Firstly, let us present a note that adds to the investigation of Abbott
algebras
in relation to quantum logics. We introduce a variety of modular Abbott
algebras
and show that they
are isomorphic to the variety of modular quantum logics. We extend this
isomorphism for
the varieties endowed with a symmetric difference.

In the second part, let us present a note about Reichenbach’s common cause
completeness in symmetric-difference-closed quantum logics. One of the lines of
recent quantum logic investigations concern the quantum logics that are endowed
with an operation of symmet-
ric difference (with a XOR operation). It is natural to ask whether these
logics might be relevant to the study of Reichenbach’s common cause
principle (RCCP). The principle states that any two positively correl-
ated events allow for a common cause. In the mathematical formulation,
one asks if there are symmetric-difference-closed (non-Boolean) quantum
logics that enable a positive answer as regards the RCCP. In this note
we show that there are such logics.

References:
Pták, P., Burešová, D.: Modular Abbott Algebras. Axioms 14 (2025), 10. ISSN
2075-1680.
Burešová, D.: Reichenbach’s Common Cause Completeness in
Symmetric-Difference-Closed Quantum Logics. Mathematica Slovaca (to appear).