Symplectic Structures in Quantum Entanglement

In this work, we explore the implications of applying the formalism of symplectic geometry to quantum mechanics, particularly focusing on many-particle systems. We extend the concept of a symplectic indicator of entanglement, originally introduced by Sawicki et al. \cite{sawicki2011}, to these complex systems. Specifically, we demonstrate that the restriction of the symplectic structure to manifolds comprising all states characterized by isospectral reduced one-particle density matrices, $M_{\mu(\psi)}^0$, exhibits degeneracy for non-separable states. We prove that the degree of degeneracy at any given state $\ket{\varphi} \in M_{\mu(\psi)}^0$ corresponds to the degree of degeneracy of the symplectic form $\omega$ when restricted to the manifold of states that are locally unitary equivalent with $\ket{\varphi}$. Additionally, we provide a physical interpretation of this symplectic indicator of entanglement, articulating it as an inherent ambiguity within the associated classical dynamical framework. Our findings underscore the pivotal role of symplectic geometry in elucidating entanglement properties in quantum mechanics and suggest avenues for further exploration into the geometric structures underlying quantum state spaces.