Related papers: Quantum metrics from the trace on full matrix alge…
The paper is devoted to geometrical investigation of the Gromov-Hausdorff distance on the classes of all metric spaces and of all bounded metric spaces. The main attention is paid to pass connectivity questions. The pass connected…
The Gromov--Hausdorff distance (hereinafter referred to as the GH-distance) is a measure of non-isometricity of metric spaces. In this paper, we study a modification of this distance that also takes topological differences into account. The…
We show how quantum mechanics can be understood as a space-time theory provided that its spatial continuum is modelled by a variable real number (qrumber) continuum. Such a continuum can be constructed using only standard Hilbert space…
We propose a new method to define theories of random geometries, using an explicit and simple map between metrics and large hermitian matrices. We outline some of the many possible applications of the formalism. For example, a…
Given a quantum state in the finite-dimensional Hilbert space $ \C^n $, the range of possible values of a quantum observable is usually identified with the discrete spectrum of eigenvalues of a corresponding Hermitian matrix. Here any such…
Under the assumption that the approximating function $\psi$ is monotonic, the classical Khintchine-Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of the set of $\psi$-approximable matrices in $\R^{mn}$.…
Inspired by regularization in quantum field theory, we study topological and metric properties of spaces in which a cut-off is introduced. We work in the framework of noncommutative geometry, and focus on Connes distance associated to a…
We present an extension of the Gromov-Hausdorff metric on the set of compact metric spaces: the Gromov-Hausdorff-Prokhorov metric on the set of compact metric spaces endowed with a finite measure. We then extend it to the non-compact case…
The notion of quantum matrix pairs is defined. These are pairs of matrices with non-commuting entries, which have the same pattern of internal relations, q-commute with each other under matrix multiplication, and are such that products of…
We introduce a logic modelling some aspects of the behaviour of the measurement process, in such a way that no direct mention of quantum states is made, thus avoiding the problems associated to this rather evasive notion. We then study some…
A complete set of mutually unbiased bases for a Hilbert space of dimension N is analogous in some respects to a certain finite geometric structure, namely, an affine plane. Another kind of quantum measurement, known as a symmetric…
We investigate the behaviour of the Tolman metrics within the formalism of the trace-free (or unimodular) gravity. While this approach is similar to the standard Einstein field equations, some subtlety arises. The effective number of…
With the help of a given distance matrix of size $n$, we construct an infinite family of distances $d_p$ (where $p \geq 2$) on the complex projective space $\mathbb{P}(\mathbb{C}^n)$ modelling the space of pure states of an $n$-level…
Let $V$ be an $n$-dimensional vector space over the finite field consisting of $q$ elements and let $\Gamma_{k}(V)$ be the Grassmann graph formed by $k$-dimensional subspaces of $V$, $1<k<n-1$. Denote by $\Gamma(n,k)_{q}$ the restriction of…
We show that Bunce-Deddens algebras, which are AT-algebras, are also limits of circle algebras for Rieffel's quantum Gromov-Hausdorff distance, and moreover, form a continuous family indexed by the Baire space. To this end, we endow…
We define (non-Einsteinian) universal metrics as the metrics that solve the source-free covariant field equations of generic gravity theories. Here, extending the rather scarce family of universal metrics known in the literature, we show…
We show that there exists a natural counterpart of the Gromov-Hausdorff metric in the class of ultrametric spaces. It is proved, in particular, that the space of all ultrametric spaces whose metric take values in a fixed countable set is…
When it comes to discriminating between two quantum states, trace distance is one of the well-known metrics used in quantum computation and quantum information theory. While there are several quantum algorithms for calculating the trace…
We introduce the notion of timed-Gromov--Hausdorff distance for timed-metric spaces. We prove that this distance is bi-Lipschitz equivalent to the intrinsic timed-Hausdorff distance of Sakovich--Sormani, and therefore induces the same…
We establish a previously unexplored conservation law for the Quantum Fisher Information Matrix (QFIM) expressed as follows; when the QFIM is constructed from a set of observables closed under commutation, i.e., a Lie algebra, the spectrum…