Continuous discretization of infinite dimensional Hilbert spaces

Abstract

In quantum theory, observables with a continuous spectrum are known to be fundamentally different from those with a discrete and finite spectrum. While some fundamental tests and applications of quantum mechanics originally formulated for discrete variables have been translated to continuous ones, this is not the case in general. For instance, despite their importance, no experimental demonstration of nonlocality exists in the continuous variables regime. Attempts to bridge this gap and put continuous variables on a closer footing to discrete ones used dichotomization. However, this approach considers only discrete properties of the continuum, and its infinitesimal properties are not fully exploited. Here we show that it is possible to manipulate, detect and classify continuous variable states using observables with a continuous spectrum revealing properties and symmetries which are analogous to finite discrete systems. Our approach leads to an operational way to define and adapt, to arbitrary continuous quantum systems, quantum protocols and algorithms typical to discrete systems.

5 pages + Supplementary Information

References in corpus (4)

• Continuous variable quantum computation with spatial degrees of freedom of photons
• Experimental Violation of Bell's Inequality in Spatial-Parity Space
• Detecting entanglement in spatial interference
• Experimental Observation of Quantum Correlations in Modular Variables

Cited by in corpus (16)

• Modular Bosonic Subsystem Codes
• Heisenberg-Weyl Observables: Bloch vectors in phase space
• Quantum information processing in phase space: A modular variables approach
• Toolbox for continuous variable entanglement production and measurement using spontaneous parametric down conversion
• Quantum Information Processing by Weaving Quantum Talbot Carpets
• Uncertainty Relations for coarse-grained measurements: an overview
• Coherence and realism in the Aharonov-Bohm effect
• Testing the Clauser-Horne-Shimony-Holt inequality using observables with arbitrary spectrum
• Proposal for a macroscopic test of local realism with phase-space measurements
• General conditions for maximal violation of non-contextuality in discrete and continuous variables
• Dynamical nonlocality in quantum time via modular operators
• Mutually Unbiased Coarse-Grained Measurements of Two or More Phase-Space Variables
• Entropic uncertainty relations for mutually unbiased periodic coarse-grained observables resemble their discrete counterparts
• Periodic discretized continuous observables are neither continuous nor discrete
• Hidden tensor structures
• Arbitrarily large violations of non-contextuality in single mode photon states with positive Wigner function