Abstract
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.
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Acknowledgments
The support by SFB/TR12 Symmetries and Universality in Mesoscopic Systems program of the Deutsche Forschungsgemeischaft and Polish MNiSW grant DFG-SFB/38/2007 is gratefully acknowledged.
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Communicated by M.B. Ruskai
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Sawicki, A., Huckleberry, A. & Kuś, M. Symplectic Geometry of Entanglement. Commun. Math. Phys. 305, 441–468 (2011). https://doi.org/10.1007/s00220-011-1259-0
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DOI: https://doi.org/10.1007/s00220-011-1259-0