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20 January 2021
 
  » arxiv » quant-ph/0111076

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The Sasaki Hook is not a [Static] Implicative Connective but Induces a Backward [in Time] Dynamic One that Assigns Causes
Bob Coecke ; Sonja Smets ;
Date 13 Nov 2001
Subject Quantum Physics; Mathematical Physics; Logic | quant-ph math-ph math.LO math.MP
AbstractIn this paper we argue that the Sasaki adjunction, which formally encodes the logicality that different authors tried to attach to the Sasaki hook as a `quantum implicative connective’, has a fundamental dynamic nature and encodes the so-called `causal duality’ (Coecke, Moore and Stubbe 2001; quant-ph/0009100) for the particular case of a quantum measurement with a projector as corresponding self-adjoint operator. In particular: The action of the Sasaki hook $(astackrel{S}{ o}-)$ for fixed antecedent $a$ assigns to some property ``the weakest cause before the measurement of actuality of that property after the measurement’’, i.e. ${(astackrel{S}{ o}b)}$ is the weakest property that guarantees actuality of $b$ after performing the measurement represented by the projector that has the `subspace $a$’ as eigenstates for eigenvalue 1, say, the measurement that `tests’ $a$ . From this we conclude that the logicality attributable to quantum systems contains a fundamentally dynamic ingredient: Causal duality actually provides a new dynamic interpretation of orthomodularity. We also reconsider the status of the Sasaki hook within `dynamic (operational) quantum logic’ (DOQL). We can derive two labeled dynamic hooks (forwardly and backwardly) that encode how quantum measurements act on properties. In an even more radical perspective one could say that the transition from either classical or constructive/intuitionistic logic to quantum logic entails besides the introduction of an additional unary connective `operational resolution’ (Coecke 2001a; math.LO/0011208) the shift from a binary connective implication to a ternary connective where two of the arguments refer to qualities of the system and the third, the new one, to an obtained outcome (in a measurement).
Source arXiv, quant-ph/0111076
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