Quantum approach to epistemic semantics

Abstract

Quantum information has suggested new forms of quantum logic, called quantum computational logics, where meanings of sentences are represented by pieces of quantum information (generally, density operators of some Hilbert spaces), which can be stored and transmitted by means of quantum particles. This approach can be applied to a semantic characterization of epistemic logical operations, which may occur in sentences like “At time $\mathfrak t^{\prime }$ Bob knows that at time $\mathfrak t$ Alice knows that the spin-value is up”. Each epistemic agent (say, Alice, Bob,...) has a characteristic truth perspective, corresponding to a particular orthonormal basis of the Hilbert space $\mathbb {C}^2$. From a physical point of view, a truth perspective can be associated with an apparatus that allows one to measure a given observable. An important feature that characterizes the knowledge of any agent is the amount of information that is accessible to him/her (technically, a special set of density operators, which also represents the internal memory of the agent in question). One can prove that interesting epistemic operations are special examples of quantum channels, which generally are not unitary. The act of knowing may involve some intrinsic irreversibility due to possible measurement procedures or to a loss of information about the environment.We also illustrate some relativistic-like effects that arise in the behavior of epistemic agents.