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Invariant measures on multimode quantum Gaussian states

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Published copy (DOI)


  • C. Lupo
  • S. Mancini
  • A. De Pasquale
  • P. Facchi
  • G. Florio
  • S. Pascazio


Publication details

JournalJournal of Mathematical Physics
DatePublished - 19 Dec 2012
Issue number12
Original languageEnglish


We derive the invariant measure on the manifold of multimode quantum Gaussian states, induced by the Haar measure on the group of Gaussian unitary transformations. To this end, by introducing a bipartition of the system in two disjoint subsystems, we use a parameterization highlighting the role of nonlocal degrees of freedom-the symplectic eigenvalues-which characterize quantum entanglement across the given bipartition. A finite measure is then obtained by imposing a physically motivated energy constraint. By averaging over the local degrees of freedom we finally derive the invariant distribution of the symplectic eigenvalues in some cases of particular interest for applications in quantum optics and quantum information.

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