308,163 research outputs found
Separable approximations of density matrices of composite quantum systems
We investigate optimal separable approximations (decompositions) of states
rho of bipartite quantum systems A and B of arbitrary dimensions MxN following
the lines of Ref. [M. Lewenstein and A. Sanpera, Phys. Rev. Lett. 80, 2261
(1998)]. Such approximations allow to represent in an optimal way any density
operator as a sum of a separable state and an entangled state of a certain
form. For two qubit systems (M=N=2) the best separable approximation has a form
of a mixture of a separable state and a projector onto a pure entangled state.
We formulate a necessary condition that the pure state in the best separable
approximation is not maximally entangled. We demonstrate that the weight of the
entangled state in the best separable approximation in arbitrary dimensions
provides a good entanglement measure. We prove in general for arbitrary M and N
that the best separable approximation corresponds to a mixture of a separable
and an entangled state which are both unique. We develop also a theory of
optimal separable approximations for states with positive partial transpose
(PPT states). Such approximations allow to decompose any density operator with
positive partial transpose as a sum of a separable state and an entangled PPT
state. We discuss procedures of constructing such decompositions.Comment: 12 pages, 2 figure
Non-separable AF-algebras
We give two pathological phenomena for non-separable AF-algebras which do not
occur for separable AF-algebras. One is that non-separable AF-algebras are not
determined by their Bratteli diagrams, and the other is that there exists a
non-separable AF-algebra which is prime but not primitive.Comment: 8 page
Some strongly bounded classes of Banach spaces
We show that the classes of separable reflexive Banach spaces and of spaces
with separable dual are strongly bounded. This gives a new proof of a recent
result of E. Odell and Th. Schlumprecht, asserting that there exists a
separable reflexive Banach space containing isomorphic copies of every
separable uniformly convex Banach spaces.Comment: 10 page
Genericity of Fr\'echet smooth spaces
If a separable Banach space contains an isometric copy of every separable
reflexive Fr\'echet smooth Banach space, then it contains an isometric copy of
every separable Banach space. The same conclusion holds if we consider
separable Banach spaces with Fr\'echet smooth dual space. This improves a
result of G. Godefroy and N. J. Kalton.Comment: 34 page
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