相关论文: Simple experimental realization of optical Hilbert…

The Hilbert hotel is an old mathematical paradox about sets of infinite numbers. This paradox deals with the accommodation of a new guest in a hotel with an infinite number of occupied rooms. Over the past decade, there have been many…

In 1924 David Hilbert conceived a paradoxical tale involving a hotel with an infinite number of rooms to illustrate some aspects of the mathematical notion of "infinity". In continuous-variable quantum mechanics we routinely make use of…

What is known as "Hilbert's hotel" is a story of an imaginary hotel with infinitely many rooms that illustrates the bizarre consequences of assuming an actual infinity of objects or events. Since the 1970s it has been used in a variety of…

We demonstrate how the unusual mathematics of transfinite numbers, in particular a nearly perfect realization of Hilbert's famous hotel paradox, manifests in the propagation of light through fractional vortex plates. It is shown how a…

In this article, we explore the notion of infinity by studying Cantor's contribution to this field. A brief history of set theory is given. As an example of infinity, we consider Hilbert's famous hotel. A graphical construction is used to…

In this article, some classical paradoxes of infinity such as Galileo's paradox, Hilbert's paradox of the Grand Hotel, Thomson's lamp paradox, and the rectangle paradox of Torricelli are considered. In addition, three paradoxes regarding…

This text tries to give an elementary introduction to the mathematical properties of infinite sets. The aim is to keep the approach as simple as possible. Advanced knowledge of mathematics is not necessary for a proper understanding, and…

The concepts of independence and totalness of subspaces are introduced in the context of quasi-probability distributions in phase space, for quantum systems with finite-dimensional Hilbert space. It is shown that due to the…

Light beams with an azimuthal phase dependency of $e^{i\ell\phi}$ have helical phase fronts and thus carry orbital angular momentum (OAM), a strictly conserved quantity with propagation. Here, we engineer quasi three-dimensional (3D)…

We consider the problem of singular beams in optics as a part of the general questions of interactions, shaping and transformations of vortex states with fractional topological charges in physics, in particular, in hydrodynamic and quantum…

Integrated optics allow the generation and control of increasingly complex photonic states on chip based architectures. Here, we implement two entangled qutrits - a 9-dimensional quantum system - and demonstrate an exceptionally high degree…

We develop a Hilbert space framework for a number of general multi-scale problems from dynamics. The aim is to identify a spectral theory for a class of systems based on iterations of a non-invertible endomorphism. We are motivated by the…

Hilbert's machine is a supertask machine inspired by Hilbert's Hotel whose functioning leads to a contradictory result involving w-ordering.

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as…

The full structuration of light in the transverse plane, including intensity, phase and polarization, holds the promise of unprecedented capabilities for applications in classical optics as well as in quantum optics and information…

The primary aim of Hilbert's proof theory was to establish the consistency of classical mathematics using finitary means only. Hilbert's strategy for doing this was to eliminate the infinite (in the form of unbounded quantifiers) from…

Within the framework of test-experiments, an original pointing set-up based on speed-induced deflection of a light-beam and using a high-resolution opto-electronic array as a position detector, is proposed. The device would provide a new…

The gravitational path integral suggests a striking result: the Hilbert space of closed universes in each superselection sector, a so-called $\alpha$-sector, is one-dimensional. We develop an abstract formalism encapsulating recent…

Given a countable metric space, we can consider its end. Then a basis of a Hilbert space indexed by the metric space defines an end of the Hilbert space, which is a new notion and different from an end as a metric space. Such an indexed…

The ultimate miniaturization of electronic devices will likely require local and coherent control of single electronic wavefunctions. Wavefunctions exist within both physical real space and an abstract state space with a simple geometric…