Entanglement and minimal Hilbert space in the classical dual states of quantum theory

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 (2025) 016104, https://doi.org/10.1063/5.0247698 on dual Classicalization. Thus, the results of this paper are two-fold: entanglement and classicalization and the relationship between them. Classicalization truly occurs only under the action of the metaplectic group Mp[Formula: see text] (minimal representation group, double covering of the symplectic group). The results of this paper are: (1) We compute and analyze 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[Formula: see text] subspaces, and compute their square norms: entanglement probabilities [Formula: see text], [Formula: see text], [Formula: see text] (e.g. in the same or in the different subspaces), and the total sum of them. (2) Entanglements in the circle: (i) of orthogonal states are [Formula: see text] even for the control phase [Formula: see text] and are preserved for any phase [Formula: see text]. [Formula: see text] is separable (entanglement is broken) only if [Formula: see text]. (ii) For coincident states, entanglements are [Formula: see text] and separable for any control parameter [Formula: see text]. If [Formula: see text], all three [Formula: see text], [Formula: see text], [Formula: see text] vanish. (iii) The entanglements of coset circle states depend on the angles and the coherent complex displacement parameter [Formula: see text] through a single variable [Formula: see text] which condition on the disk [Formula: see text] guarantees both analyticity and normalization. (iv) Entanglement classicalization in the circle is stronger for coset states than for non-coset ones: a tail decreasing with increasing [Formula: see text] is absent in the non-coset case. (3) Entanglement in the cylinder depends weakly on the angles ([Formula: see text]) and strongly classicalizes: rapid exponential decay suppression [Formula: see text] for large [Formula: see text], stronger than in the circle (be coset or not). It expresses in terms of Theta functions [Formula: see text] and [Formula: see text] in all classicalization projections in the degeneracy (equal states) limit. (4) Comparison with other (non-Mp(2)) entanglements, as the Schrödinger cat states, show very clear differences (e.g. Figs. [4] and [5]). (5) The antipodal and non-antipodal entanglements (as regulated by the phase control parameter [Formula: see text] and [Formula: see text] respectively), are clearly different in all types (circle, cylinder, coset or not) of orthogonal states studied here. The antipodal entanglement is the minimal. These theoretical and conceptual results can be of experimental and practical real-world interest.