Space from entanglement: An information-geometric perspective

In this paper, we study the construction of classical geometry from the quantum entanglement structure by using information geometry. In the information geometry of classical spacetime, the Fisher information metric is related to a blurred metric of a classical physical space. We first show that a local information metric can be obtained from the entanglement contour in a local subregion. This local information metric measures the fine structure of entanglement spectra inside the subregion, which suggests a quantum origin of the information-geometric blurred space. We study both the continuous and the classical limits of the quantum-originated blurred space by using the techniques from the statistical sampling algorithms, the sampling theory of spacetime and the projective limit. A scheme for going from a blurred space with quantum features to a classical geometry is also explored.

Virtual Markov Chains

We introduce the space of virtual Markov chains (VMCs) as a projective limit of the spaces of all finite state space Markov chains (MCs), in the same way that the space of virtual permutations is the projective limit of the spaces of all permutations of finite sets.We introduce the notions of virtual initial distribution (VID) and a virtual transition matrix (VTM), and we show that the law of any VMC is uniquely characterized by a pair of a VID and VTM which have to satisfy a certain compatibility condition.Lastly, we study various properties of compact convex sets associated to the theory of VMCs, including that the Birkhoff-von Neumann theorem fails in the virtual setting.

Lifting Dual Connections with the Riemann Extension

Let (M,g) be a Riemannian manifold equipped with a pair of dual connections (∇,∇*). Such a structure is known as a statistical manifold since it was defined in the context of information geometry. This paper aims at defining the complete lift of such a structure to the cotangent bundle T*M using the Riemannian extension of the Levi-Civita connection of M. In the first section, common tensors are associated with pairs of dual connections, emphasizing the cyclic symmetry property of the so-called skewness tensor. In a second section, the complete lift of this tensor is obtained, allowing the definition of dual connections on TT*M with respect to the Riemannian extension. This work was motivated by the general problem of finding the projective limit of a sequence of a finite-dimensional statistical manifold.

On the domain of implicit functions in a projective limit setting without additional norm estimates

We examine how implicit functions on ILB-Fréchet spaces can be obtained without metric or norm estimates which are classically assumed. We obtain implicit functions defined on a domain D which is not necessarily open, but which contains the unit open ball of a Banach space. The corresponding inverse function theorem is obtained, and we finish with an open question on the adequate (generalized) notion of differentiation, needed for the corresponding version of the Fröbenius theorem.

ARITHMETIC STRUCTURES FOR DIFFERENTIAL OPERATORS ON FORMAL SCHEMES

Let $\mathfrak{o}$ be a complete discrete valuation ring of mixed characteristic $(0,p)$ and $\mathfrak{X}_{0}$ a smooth formal $\mathfrak{o}$ -scheme. Let $\mathfrak{X}\rightarrow \mathfrak{X}_{0}$ be an admissible blow-up. In the first part, we introduce sheaves of differential operators $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ on $\mathfrak{X}$ , for every sufficiently large positive integer $k$ , generalizing Berthelot’s arithmetic differential operators on the smooth formal scheme $\mathfrak{X}_{0}$ . The coherence of these sheaves and several other basic properties are proven. In the second part, we study the projective limit sheaf $\mathscr{D}_{\mathfrak{X},\infty }=\mathop{\varprojlim }\nolimits_{k}\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ and introduce its abelian category of coadmissible modules. The inductive limit of the sheaves $\mathscr{D}_{\mathfrak{X},\infty }$ , over all admissible blow-ups $\mathfrak{X}$ , is a sheaf $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$ on the Zariski–Riemann space of $\mathfrak{X}_{0}$ , which gives rise to an abelian category of coadmissible modules. Analogues of Theorems A and B are shown to hold in each of these settings, that is, for $\mathscr{D}_{\mathfrak{X},k}^{\dagger }$ , $\mathscr{D}_{\mathfrak{X},\infty }$ , and $\mathscr{D}_{\langle \mathfrak{X}_{0}\rangle }$ .

On the generalization of the Darboux theorem

Darboux theorem to more general context of Frechet manifolds we face an obstacle: in general vector fields do not have local flows. Recently, Fr\'{e}chet geometry has been developed in terms of projective limit of Banach manifolds. In this framework under an appropriate Lipchitz condition The Darboux theorem asserts that a symplectic manifold $(M^{2n},\omega)$ is locally symplectomorphic to $(R^{2n}, \omega_0)$, where $\omega_0$ is the standard symplectic form on $R^{2n}$. This theorem was proved by Moser in 1965, the idea of proof, known as the Moser’s trick, works in many situations. The Moser tricks is to construct an appropriate isotopy $ \ff_t $ generated by a time-dependent vector field $ X_t $ on $M$ such that $ \ff_1^{*} \omega = \omega_0$. Nevertheless, it was showed by Marsden that Darboux theorem is not valid for weak symplectic Banach manifolds. However, in 1999 Bambusi showed that if we associate to each point of a Banach manifold a suitable Banach space (classifying space) via a given symplectic form then the Moser trick can be applied to obtain the theorem if the classifying space does not depend on the point of the manifold and a suitable smoothness condition holds. If we want to try to generalize the local flows exist and with some restrictive conditions the Darboux theorem was proved by Kumar. In this paper we consider the category of so-called bounded Fr\'{e}chet manifolds and prove that in this category vector fields have local flows and following the idea of Bambusi we associate to each point of a manifold a Fr\'{e}chet space independent of the choice of the point and with the assumption of bounded smoothness on vector fields we prove the Darboux theorem.