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arXiv:1506.00675 · Physical Review Letters

Quantum Hilbert Hotel

Václav Potoček, Filippo M. Miatto, Mohammad Mirhosseini, Omar S. Magaña‐Loaiza, Andreas C. Liapis, Daniel K. L. Oi, Robert W. Boyd, John Jeffers

Published Oct 15, 2015• 44 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevlett.115.160505

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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.
In Chapter 4 we saw that the canonical commutation relations force us to use a space of states of infinite dimension, in which rigor would require the use of advanced mathematical tools. Fortunately, physicists generally need only to carry the results for finite dimension over to infinite dimension with some simple modifications which we shall indicate here, without embarking on sophisticated mathematics. Nevertheless, it is useful to be aware of the lapses in rigor which are customarily made in physics in order to avoid possible unpleasant surprises. The objective of this chapter is, on the one hand, to present some concrete examples illustrating the new features which arise in infinite dimension and, on the other, to give the rules for practical calculations, in particular to write down the spectral decomposition of Hermitian and unitary operators. The mathematics we use is a bit more detailed than commonly found in most quantum mechanics textbooks. The reader interested purely in the practical aspects can proceed directly to Section 7.3, where the results essential for later on are summarized. Hilbert spaces Definitions The space of states of quantum mechanics is a Hilbert space ℌ, which in general is of infinite dimension. The axiomatic definition of a Hilbert space is the following. It is a vector space which, for the needs of quantum mechanics, is defined on complex numbers. The vectors of this space are denoted |ϕ〉. […]
In recent years decoherence has been widely discussed and accepted as a mechanism responsible for the emergence of classicality in quantum measurements and the absence in the real world of Schrödinger-cat-like states [ 2 , 6 , 7 , 11 , 12 ]. Let me start with recalling the Schrödinger cat paradox. It was a hypothetical experiment proposed by Schrödinger in 1935 which can be briefly described as follows. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
\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
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 a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.
Orbital angular momentum (OAM) multiplexing provides an efficient method to improve data-carrying capacity in various quantum communication protocols. It is a precondition to distribute OAM multiplexed quantum resources in quantum channels for implementing quantum communication. However, quantum steering of OAM multiplexed optical fields and the effect of channel noise on OAM multiplexed quantum resources remain unclear. Here, we generate OAM multiplexed continuous-variable (CV) entangled states and distribute them in lossy or noisy channels. We show that the decoherence property of entanglement and quantum steering of the OAM multiplexed states carrying topological charges l = 1 and l = 2 are the same as that of the Gaussian mode with l = 0 in lossy and noisy channels. The sudden death of entanglement and quantum steering of high-order OAM multiplexed states is observed in the presence of excess noise. Our results demonstrate the feasibility to realize high data-carrying capacity quantum information processing by utilizing OAM multiplexed CV entangled states.