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arXiv:1602.07514 · International Journal of Theoretical Physics

A Quantum Computational Semantics for Epistemic Logical Operators. Part I: Epistemic Structures

Enrico G. Beltrametti, María Luisa Dalla Chiara, Roberto Giuntini, Roberto Leporini, Giuseppe Sergioli

Published Jun 1, 2013• 9 citations
arXivHTMLPDF
DOI
10.1007/s10773-013-1642-z

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In this paper we analyze classical and quantum correlations using the tools of epistemic logic. Our main contribution consists of two new logical systems. The first one is called General Epistemic Logic (GEL), it extends traditional epistemic logic with operators that allow us to reason about the information carried by a complex system composed of several parts. The second system is called the Logic of Correlated Knowledge (LCK), which extends GEL with sentences that describe the observational capabilities of an agent. On the semantic side we introduce correlation models, as a generalization of the “interpreted systems” semantics. We use this setting to investigate several types of informational correlations (e.g. distributed information, quantum correlated information) that complex systems can exhibit. We also provide an informational-logical characterization of the notion of “quantum entanglement”.
Quantum computation theory has inspired new forms of quantum logic, called quantum computational logics, where formulas are supposed to denote pieces of quantum information, while logical connectives are interpreted as special examples of quantum logical gates. The most natural semantics for these logics is a form of holistic semantics, where meanings behave in a contextual way. In this framework, the concept of quantum probability can assume different forms. We distinguish an absolute concept of probability, based on the idea of quantum truth, from a relative concept of probability (a form of transition-probability, connected with the notion of fidelity between quantum states). Quantum information has brought about some intriguing epistemic situations. A typical example is represented by teleportation-experiments. In some previous works we have studied a quantum version of the epistemic operations “to know”, “to believe”, “to understand”. In this article, we investigate another epistemic operation (which is informally used in a number of interesting quantum situations): the operation “being probabilistically informed”.