A quantum computational semantics for epistemic logical operators. Part I: epistemic structures

Some critical open problems of epistemic logics can be investigated in the framework of a quantum computational approach. The basic idea is to interpret sentences like “Alice knows that Bob does not understand that π is irrational” as pieces of quantum information (generally represented by density operators of convenient Hilbert spaces). Logical epistemic operators (to understand, to know. . .) are dealt with as (generally irreversible) quantum operations, which are, in a sense, similar to measurement-procedures. This approach permits us to model some characteristic epistemic processes, that concern both human and artificial intelligence. For instance, the operation of “memorizing and retrieving information” can be formally represented, in this framework, by using a quantum teleportation phenomenon

Epistemic quantum computational structures in a Hilbert-space environment

Some critical open problems of epistemic logics can be investigated in the framework of a quantum computational approach. The basic idea is to interpret sentences like “Alice knows that Bob does not understand that π is irrational” as pieces of quantum information (generally represented by density operators of convenient Hilbert spaces). Logical epistemic operators (to understand, to know ...) are dealt with as (generally irreversible) quantum operations, which are, in a sense, similar to measurement-procedures. This approach permits us to model some characteristic epistemic processes, that concern both human and artificial intelligence. For instance, the operation of “memorizing and retrieving information” can be formally represented, in this framework, by using a quantum teleportation phenomenon

Extended Representations of Observables and States for a Noncontextual Reinterpretation of QM

A crucial and problematical feature of quantum mechanics (QM) is nonobjectivity of properties. The ESR model restores objectivity reinterpreting quantum probabilities as conditional on detection and embodying the mathematical formalism of QM into a broader noncontextual (hence local) framework. We propose here an improved presentation of the ESR model containing a more complete mathematical representation of the basic entities of the model. We also extend the model to mixtures showing that the mathematical representations of proper mixtures does not coincide with the mathematical representation of mixtures provided by QM, while the representation of improper mixtures does. This feature of the ESR model entails that some interpretative problems raising in QM when dealing with mixtures are avoided. From an empirical point of view the predictions of the ESR model depend on some parameters which may be such that they are very close to the predictions of QM in most cases. But the nonstandard representation of proper mixtures allows us to propose the scheme of an experiment that could check whether the predictions of QM or the predictions of the ESR model are correct.Comment: 17 pages, standard latex. Extensively revised versio

Some Properties of Transforms in Culture Theory

It is shown that, in certain circumstances, systems of cultural rules may be represented by doubly stochastic matrices denoted called possibility transforms, and by certain real valued possibility densities with inner product. Using such objects we may characterize a certain problem of ethnographic and ethological description as a problem of prediction, in which observations are predicted by properties of fixed points of transforms of pure systems, or by properties of convex combinations of such pure systems. That is, ethnographic description is an application of the Birkhoff theorem regarding doubly stochastic matrices on a space whose vertices are permutations.Comment: Read at International Quantum Structures Association meetings, 200

Physical propositions and quantum languages

The word \textit{proposition} is used in physics with different meanings, which must be distinguished to avoid interpretational problems. We construct two languages $\mathcal{L}^{\ast}(x)$ and $\mathcal{L}(x)$ with classical set-theoretical semantics which allow us to illustrate those meanings and to show that the non-Boolean lattice of propositions of quantum logic (QL) can be obtained by selecting a subset of \textit{p-testable} propositions within the Boolean lattice of all propositions associated with sentences of $\mathcal{L}(x)$. Yet, the aforesaid semantics is incompatible with the standard interpretation of quantum mechanics (QM) because of known no-go theorems. But if one accepts our criticism of these theorems and the ensuing SR (semantic realism) interpretation of QM, the incompatibility disappears, and the classical and quantum notions of truth can coexist, since they refer to different metalinguistic concepts (\textit{truth} and \textit{verifiability according to QM}, respectively). Moreover one can construct a quantum language $\mathcal{L}_{TQ}(x)$ whose Lindenbaum-Tarski algebra is isomorphic to QL, the sentences of which state (testable) properties of individual samples of physical systems, while standard QL does not bear this interpretation.Comment: 15 pages, no figure, standard Late

On the lattice structure of probability spaces in quantum mechanics

Let C be the set of all possible quantum states. We study the convex subsets of C with attention focused on the lattice theoretical structure of these convex subsets and, as a result, find a framework capable of unifying several aspects of quantum mechanics, including entanglement and Jaynes' Max-Ent principle. We also encounter links with entanglement witnesses, which leads to a new separability criteria expressed in lattice language. We also provide an extension of a separability criteria based on convex polytopes to the infinite dimensional case and show that it reveals interesting facets concerning the geometrical structure of the convex subsets. It is seen that the above mentioned framework is also capable of generalization to any statistical theory via the so-called convex operational models' approach. In particular, we show how to extend the geometrical structure underlying entanglement to any statistical model, an extension which may be useful for studying correlations in different generalizations of quantum mechanics.Comment: arXiv admin note: substantial text overlap with arXiv:1008.416

Relational Quantum Mechanics and Probability

We present a derivation of the third postulate of Relational Quantum Mechanics (RQM) from the properties of conditional probabilities.The first two RQM postulates are based on the information that can be extracted from interaction of different systems, and the third postulate defines the properties of the probability function. Here we demonstrate that from a rigorous definition of the conditional probability for the possible outcomes of different measurements, the third postulate is unnecessary and the Born's rule naturally emerges from the first two postulates by applying the Gleason's theorem. We demonstrate in addition that the probability function is uniquely defined for classical and quantum phenomena. The presence or not of interference terms is demonstrated to be related to the precise formulation of the conditional probability where distributive property on its arguments cannot be taken for granted. In the particular case of Young's slits experiment, the two possible argument formulations correspond to the possibility or not to determine the particle passage through a particular path.Comment: Foundations of Physics, Springer Verlag, 201

`What is a Thing?': Topos Theory in the Foundations of Physics

The goal of this paper is to summarise the first steps in developing a fundamentally new way of constructing theories of physics. The motivation comes from a desire to address certain deep issues that arise when contemplating quantum theories of space and time. In doing so we provide a new answer to Heidegger's timeless question ``What is a thing?''. Our basic contention is that constructing a theory of physics is equivalent to finding a representation in a topos of a certain formal language that is attached to the system. Classical physics uses the topos of sets. Other theories involve a different topos. For the types of theory discussed in this paper, a key goal is to represent any physical quantity $A$ with an arrow \breve{A}_\phi:\Si_\phi\map\R_\phi where \Si_\phi and $\R_\phi$ are two special objects (the `state-object' and `quantity-value object') in the appropriate topos, $\tau_\phi$. We discuss two different types of language that can be attached to a system, $S$. The first, \PL{S}, is a propositional language; the second, \L{S}, is a higher-order, typed language. Both languages provide deductive systems with an intuitionistic logic. With the aid of \PL{S} we expand and develop some of the earlier work (By CJI and collaborators.) on topos theory and quantum physics. A key step is a process we term `daseinisation' by which a projection operator is mapped to a sub-object of the spectral presheaf \Sig--the topos quantum analogue of a classical state space. The topos concerned is \SetH{}: the category of contravariant set-valued functors on the category (partially ordered set) \V{} of commutative sub-algebras of the algebra of bounded operators on the quantum Hilbert space \Hi.Comment: To appear in ``New Structures in Physics'' ed R. Coeck