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Topological Net-Field Representation of Squeezed Quantum States

Shin Tamaki

Published Sep 21, 2026• 0 citations
Publisher

Abstract

We propose a topological Net Field that geometrically represents squeezed states, photon-number structure, and interference. Anisotropic tension, twist, and pivot nuclei reveal nonclassical features including negative Wigner regions, quiet bands, and knot-like entanglement. This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.

DOI
10.5281/zenodo.22862446

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We propose a topological Net Field that geometrically represents squeezed states, photon-number structure, and interference. Anisotropic tension, twist, and pivot nuclei reveal nonclassical features including negative Wigner regions, quiet bands, and knot-like entanglement. This work is licensed under the Creative Commons Attribution 4.0 International (CC BY 4.0) license.
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Much attention has been given in the past to two classes of squeezed states: the squeezed vacuum and coherent states. Here we study the effects of squeezing on number states and on thermal field states. The statistical properties of these various squeezed states are discussed using the second-order correlation functions. The quasiprobabilities of the Wigner, Q, and positive P representations are calculated and compared for the squeezed states. The Glauber P representation for the squeezed thermal state explicitly shows the limit of its applicability. The photon number distributions of the squeezed number and squeezed thermal states are extensively discussed and new interference effects in phase space are shown to lead to highly structured number distributions.
Phase-space-interference approaches employ interference on the phase plane to explain a quantum state's photon-number distribution. This paper generalizes the approach so that knowledge of the photon-number distribution explains the state's quantum phase-plane distribution. The required generalization is as follows. The Bohr-Sommerfeld ``band'' or trajectory of the mixture of number states \ensuremath{\sim}(\ensuremath{\Vert}N〉〈N\ensuremath{\Vert}+\ensuremath{\Vert}M〉〈M\ensuremath{\Vert}) consists of two annuli centered on the origin. In contrast, the trajectory of the pure number state superposition \ensuremath{\sim}(\ensuremath{\Vert}N〉+\ensuremath{\Vert}M〉) (with M>N) contains additional interference structures, namely, (M-N) symmetrically positioned radial spokes connecting the annuli. These spokes intersect the circular band attributed to a photon-number state at (M-N) locations, with the resulting destructive interference eliminating states \ensuremath{\Vert}N+1〉 to \ensuremath{\Vert}M-1〉. These structures build state phase-plane quantum probability distributions, as we illustrate for the squeezed vacuum. This state is missing every odd-numbered photon number state, and thus its trajectory consists of a repeated pattern of two heavily emphasized bars eliminating every odd-numbered state. These bars sum to generate the traditional ``cigar'' shape of the squeezed state.
This paper explores the intersection of non-linear geometry topology and quantum entanglement, proposing a novel framework for describing quantum states and investigating the interplay between quantum entanglement and spacetime geometry. We introduce a new quantum state representation based on topological structures, leveraging their inherent properties to design a computational framework specifically tailored for simulating quantum entanglement. The core claim is that this approach, when applied to quantum entanglement, will yield a more precise understanding of the underlying correlations and potentially reveal critical insights into the nature of spacetime geometry and its influence on quantum information. This research offers a potential pathway toward advancing quantum computation and quantum information security by providing a fundamentally different approach to managing and analyzing complex quantum states.