Symplectic Geometry of Entanglement

Adam Sawicki*, Alan Huckleberry, Marek Kus

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

35 Citations (Scopus)


We present a description of entanglement in composite quantum systems in terms of symplectic geometry. We provide a symplectic characterization of sets of equally entangled states as orbits of group actions in the space of states. In particular, using the Kostant-Sternberg theorem, we show that separable states form a unique symplectic orbit, whereas orbits of entangled states are characterized by different degrees of degeneracy of the canonical symplectic form on the complex projective space. The degree of degeneracy may be thus used as a new geometric measure of entanglement. The above statements remain true for systems with an arbitrary number of components, moreover the presented method is general and can be applied also under different additional symmetry conditions stemming, e. g., from the indistinguishability of particles. We show how to calculate the degeneracy for various multiparticle systems providing also simple criteria of separability.

Original languageEnglish
Pages (from-to)441-468
Number of pages28
JournalCommunications in Mathematical Physics
Issue number2
Publication statusPublished - Jul 2011



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