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Europhysics Letters (EPL)

There is more than one way to host a new guest in the quantum Hilbert hotel

Canio Noce

Published May 22, 2020• 1 citation
Publisher

Abstract

\n\nThe Hilbert hotel is an inn with infinite rooms where even when completely full, vacancies can be created by means of suitable room switches. Interestingly, the energy levels of a bounded quantum system can mimic the rooms in the Hilbert hotel if one shows how to perform the shift in the quantum spectrum with an unlimited number of energy levels. Here, we suggest a quantum realization of the Hilbert hotel and indicate how to experimentally improve the theoretical proposal. This goal may be achieved measuring the electric conductance of many quantum wells suitably arranged.\n

DOI
10.1209/0295-5075/130/20002

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\n\nThe Hilbert hotel is an inn with infinite rooms where even when completely full, vacancies can be created by means of suitable room switches. Interestingly, the energy levels of a bounded quantum system can mimic the rooms in the Hilbert hotel if one shows how to perform the shift in the quantum spectrum with an unlimited number of energy levels. Here, we suggest a quantum realization of the Hilbert hotel and indicate how to experimentally improve the theoretical proposal. This goal may be achieved measuring the electric conductance of many quantum wells suitably arranged.\n
In quantum mechanics a localized attractive potential typically supports a (possibly infinite) set of bound states, characterized by a discrete spectrum of allowed energies, together with a continuum of scattering states, characterized (in one dimension) by an energy-dependent phase shift. The 1∕x2 potential on 0<x<∞ confounds all of our intuitions and expectations. Resolving its paradoxes requires sophisticated theoretical machinery: regularization, renormalization, anomalous symmetry-breaking, and self-adjoint extensions. Our goal is to introduce the essential ideas at a level accessible to advanced undergraduates.
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 infinite state spaces: here we show that such a theoretical apparatus can accommodate an analog of Hilbert's hotel paradox. We devise a protocol that, mimicking what happens to the guests of the hotel, maps the amplitudes of an infinite eigenbasis to twice their original quantum number in a coherent and deterministic manner, producing infinitely many unoccupied levels in the process. We demonstrate the feasibility of the protocol by experimentally realizing it on the orbital angular momentum of a paraxial field. This new non-Gaussian operation may be exploited, for example, for enhancing the sensitivity of NOON states, for increasing the capacity of a channel, or for multiplexing multiple channels into a single one.
This paper addresses the paradox that arises from a naive synthesis of Hilbert's Grand Hotel and Schrödinger's Cat, where a quantum cat in a superposition of states would lead to a 'total occupancy' paradox, nullifying the hotel's properties. We introduce the KnoWellian Grand Hotel as a formal resolution, grounded in the KnoWellian Universe Theory (KUT). This model replaces the conventional Platonic ontology of completed, static infinite sets with a procedural ontology. The framework is governed by the Law of KnoWellian Conservation, which partitions reality into two mutually exclusive states: rendered actuality (m(t)) and unmanifested potentiality (w(t)). The paper demonstrates that 'rooms' in the hotel can only be occupied if they are first rendered from potential into actuality. Consequently, Schrödinger's Cat in its unobserved state does not occupy an infinite set of actual rooms. Instead, it exists as a single, coherent superposition within the unrendered potential field (w(t)). Through the 'Bernharda' thought experiment and a formal proof, we establish that any question regarding the properties of all potential guests (i.e., the unmanifested potential) is fundamentally un-renderable by any observer existing within the rendered hotel (m(t)). The paradox is thereby dissolved not by introducing a new mechanism of accommodation, but by revealing the ontological incompatibility of the question itself—a question formulated in a Platonic language that is invalid in a procedural universe.
We propose an efficient hyperentanglement source emitting photon pairs entangled in both energy and polarization. The compact electrically driven room-temperature source, based on intersubband two-photon emission from semiconductor quantum wells (QWs) exhibits pair generation rates several orders of magnitude higher than alternative conventional schemes. A theoretical formalism is derived for the calculation of photon pair generation spectra and rates. The results are presented for superlattice structures similar to quantum cascade lasers of GaAs/AlGaAs QWs emitting in the mid-IR and far-IR and for InN/AlN QW structures suitable for telecommunication wavelengths.