Kohei Kishida

Minimum quantum resources for strong non−locality

We analyse the minimum quantum resources needed to realise strong non-locality, as exemplified e.g. by the classical GHZ construction. It was already known that no two-qubit system, with any finite number of local measurements, can realise strong non-locality. For three-qubit systems, we show that strong non-locality can only be realised in the GHZ SLOCC class, and with equatorial measurements. However, we show that in this class there is an infinite family of states which are pairwise non LU-equivalent that realise strong non-locality with finitely many measurements. These states have decreasing entanglement between one qubit and the other two, necessitating an increasing number of local measurements on the latter.

Public announcements under sheaves

The goal of this article is to bring together the frameworks of model-update semantics for (propositional) public-announcement logic [9] and of sheaf semantics for first-order modal logic [2,10,14], and to thereby obtain a sheaf semantics for first-order public-announcement logic. The first attempt to extend dynamic epistemic logics to the first order was made by Kooi [15], who introduced terms to refer to epistemic agents, and an extension of public-announcement logic to the first order was briefly given by Ma [18]; both of these extensions used constant domains for interpreting first-order vocabulary. (A first-order extension of dynamic logic was given in [12,13], also with constant domains.) This article pushes ahead with these extensions by employing a sheaf structure, providing a progress toward a more flexible and useful treatment of first-order notions.

Possibilities determine the combinatorial structure of probability polytopes

Abstract We study the set of no-signalling empirical models on a measurement scenario, and show that the combinatorial structure of the no-signalling polytope is completely determined by the possibilistic information given by the support of the models. This is a special case of a general result which applies to all polytopes presented in a standard form, given by linear equations together with non-negativity constraints on the variables.

Stochastic Relational Presheaves and Dynamic Logic for Contextuality.

Presheaf models provide a formulation of labelled transition systems that is useful for, among other things, modelling concurrent computation. This paper aims to extend such models further to represent stochastic dynamics such as shown in quantum systems. After reviewing what presheaf models represent and what certain operations on them mean in terms of notions such as internal and external choices, composition of systems, and so on, I will show how to extend those models and ideas by combining them with ideas from other category-theoretic approaches to relational models and to stochastic processes. It turns out that my extension yields a transitional formulation of sheaf-theoretic structures that Abramsky and Brandenburger proposed to characterize non-locality and contextuality. An alternative characterization of contextuality will then be given in terms of a dynamic modal logic of the models I put forward.

Categories for Dynamic Epistemic Logic

The primary goal of this paper is to recast the semantics of modal logic, and dynamic epistemic logic (DEL) in particular, in category-theoretic terms. We first review the category of relations and categories of Kripke frames, with particular emphasis on the duality between relations and adjoint homomorphisms. Using these categories, we then reformulate the semantics of DEL in a more categorical and algebraic form. Several virtues of the new formulation will be demonstrated: The DEL idea of updating a model into another is captured naturally by the categorical perspective -- which emphasizes a family of objects and structural relationships among them, as opposed to a single object and structure on it. Also, the categorical semantics of DEL can be merged straightforwardly with a standard categorical semantics for first-order logic, providing a semantics for first-order DEL.