Symplectic Geometry of Entanglement

Abstract

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.

Static correlated functionals for reduced density matrix functional theory

“…It is worth mentioning that the fully polarized case can be easily tackled for the Borland-Dennis state by separating the natural orbitals in two sets, namely {1, 2, 4}, for which we use latin letters (a, b...), and {3, 5, 6}, for which we use greek letters (µ, ν, ...). Similar expressions to (31) and (33) can be easiiy obtained.…”

Implications of pinned occupation numbers for natural orbital expansions. II: rigorous derivation and extension to non-fermionic systems

“…The solution of one-body quantum marginal problem is obtained mutatis mutandis to the case of fermions as all the essential mathematical structures are also present for distinguishable particles, i.e. ( )   is a Kähler manifold and μ arises as the momentum map of the local unitary action on the space of states (see [5,6,[11][12][13][14][15][16][17][18][19][20][21][22][23] for more examples of the usage of geometric techniques in quantum information). A similar observation applies to bosons.…”

Stratified manifold of quantum states, actions of the complex special linear group

“…Composite quantum systems and quantum entanglement manipulation are of fundamental importance in the context of quantum information theory and the geometry of quantum entanglement is a fascinating and very complex subject [3,6,11,29,30,32,33,34,37,46,47,48,49,50,51,55]. Here we want to present some possible connections between the geometry of quantum entanglement and the geometrical tools introduced in the previous sections.…”

“…Our understanding of the geometry of the space of quantum states is in constant evolution and there are different fields of application in which it is possible to use the knowledge we gain. For instance, geometrical ideas have been successfully exploited when addressing the foundations of quantum mechanics [4,7,8,10,13,20,22,23,25,31,35,40], quantum information theory [5,15,19,27,36,39,43,45,54], quantum dynamics [9,12,14,16,17,18,24], entanglement theory [3,6,11,29,30,34,48,49].…”