Entanglement and Minimal Hilbert Space in the Classical Dual States of Quantum Theory

TL;DR

The paper investigates how quantum states reveal their classical dual content through the minimal Metaplectic representation Mp(2d), focusing on entanglement of dual states on circle and cylinder topologies. It computes Mp(2) projections of various entangled circle and cylinder states, deriving the entanglement probabilities P_{++}, P_{+-}, and P_{--}, and analyzes how topology, angular variables, and phase control (rho) govern classicalization. A key finding is that cylinder topology exhibits stronger classicalization (faster decay of entanglement with n) than the circle, with coset Mp(2) states further enhancing locality in the disk and reducing tails compared to Schrödinger-cat projections. The work also contrasts Mp(2) projections with non-Mp(2) projections, and with Schrödinger cat states, highlighting implications for quantum-classical information processing and foundational aspects of quantum-classical duality.

Abstract

A precise physical description and understanding of the classical dual content of quantum theory is necessary in many disciplines today: from concepts and interpretation to quantum technologies and computation. In this paper we investigate Quantum Entanglement with the new approach APL Quantum 2, 016104 (2025) on dual Classicalization. Thus, the results of this paper are twofold: Entanglement and Classicalization and the relationship between them. Classicalization truly occurs only under the action of the Metaplectic group Mp(n) (Minimal Representation group, double covering of the Symplectic group). Some of the results of this paper involves the computation and analysis of the entanglement for different types of coherent (coset and non coset) states and topologies: in the circle and the cylinder. We project the entangled wave functions onto the even (+) and odd (-) irreducible Hilbert Mp(n) subspaces, and compute their square norms: Entanglement Probabilities P++, P--, P+-, (eg in the same or in the different subspaces), and the Total sum of them, and more. These theoretical and conceptual results can be of experimental and practical real-world interest.

Figures (5)

• Figure 1: Some Main New Features of Entanglements and Classicalization of this Paper. Sections I and VI provide more Summary and explanation.
• Figure 2: Probabilities of the Entanglement of two circle (London) states with $\Delta = 0$ ($\varphi = \varphi^{\prime }$): coincident states projected onto the Metaplectic group (Minimal Representation Group): Classicalization: Left side with the control entanglement phase $\rho = 0$. Right side with $\rho =\pi$: e.g Antipodal Entanglement . In this case, the Antipodal condition (regulated by the control parameter $\rho = \pi$) does not modify essentially any of the Entanglement Mp (2) Classicalizations.
• Figure 3: Probabilities of the Entanglement of two circle London states $\Delta =\pi /2$ ($\varphi =\varphi ^{\prime }$ +$\pi /2$): orthogonal states projected onto the Metaplectic group (Minimal Representation Group): Classicalization. Left side entanglement with the control paramenter phase $\rho = 0$. Right side with $\rho = \pi$: Antipodal Entanglament. Here the antipodal condition is relevant, inverting the maxima of the Entanglement Probabilities.
• Figure 4: Schrodinger cat and circle Entanglement Probability of even and odd states: $P^{\,cat} {(+- )}$ (left side) and the Antipodal one $P_{A} ^{\,cat}{(+- )}$ (right side)of the circle entangled orthogonal states$\varphi = \varphi^{\prime} + \pi/2 : \Delta \rightarrow \pi /2$ projected onto the Schrodinger cat even and odd states. The parameters $\alpha$ and $\beta$ in the Figure represent the respective norms, being the range of the parameters $0\leq \left\vert \alpha \right\vert$ and $\left\vert \beta \right\vert \,< 2$ to appreciate the shape of the Entanglement Probability.
• Figure 5: Mp(2) and circle Entanglement Probability of even and odd states: $P{(+-)}$ (left side) and the Antipodal one $P_{A}(+ -)$ (right side) of the circle orthogonal states $\varphi = \varphi^{\prime} + \pi/2 , ( \Delta \rightarrow \pi /2 )$, projected on the basic Mp(2) states: Classicalization. The variables $\omega$ and $\sigma$ in the Figure representing the respective norms of the analytic functions in the unitary disk, have the range $0 \,\leq \left\vert \omega \right\vert$ and $\,\left\vert \sigma \right\vert \,< 1$.