29,003 research outputs found
Physical properties of the Schur complement of local covariance matrices
General properties of global covariance matrices representing bipartite
Gaussian states can be decomposed into properties of local covariance matrices
and their Schur complements. We demonstrate that given a bipartite Gaussian
state described by a covariance matrix \textbf{V}, the
Schur complement of a local covariance submatrix of it can be
interpreted as a new covariance matrix representing a Gaussian operator of
party 1 conditioned to local parity measurements on party 2. The connection
with a partial parity measurement over a bipartite quantum state and the
determination of the reduced Wigner function is given and an operational
process of parity measurement is developed. Generalization of this procedure to
a -partite Gaussian state is given and it is demonstrated that the
system state conditioned to a partial parity projection is given by a
covariance matrix such as its block elements are Schur complements
of special local matrices.Comment: 10 pages. Replaced with final published versio
On the propagation of semiclassical Wigner functions
We establish the difference between the propagation of semiclassical Wigner
functions and classical Liouville propagation. First we re-discuss the
semiclassical limit for the propagator of Wigner functions, which on its own
leads to their classical propagation. Then, via stationary phase evaluation of
the full integral evolution equation, using the semiclassical expressions of
Wigner functions, we provide the correct geometrical prescription for their
semiclassical propagation. This is determined by the classical trajectories of
the tips of the chords defined by the initial semiclassical Wigner function and
centered on their arguments, in contrast to the Liouville propagation which is
determined by the classical trajectories of the arguments themselves.Comment: 9 pages, 1 figure. To appear in J. Phys. A. This version matches the
one set to print and differs from the previous one (07 Nov 2001) by the
addition of two references, a few extra words of explanation and an augmented
figure captio
Uniform approximation for the overlap caustic of a quantum state with its translations
The semiclassical Wigner function for a Bohr-quantized energy eigenstate is
known to have a caustic along the corresponding classical closed phase space
curve in the case of a single degree of freedom. Its Fourier transform, the
semiclassical chord function, also has a caustic along the conjugate curve
defined as the locus of diameters, i.e. the maximal chords of the original
curve. If the latter is convex, so is its conjugate, resulting in a simple fold
caustic. The uniform approximation through this caustic, that is here derived,
describes the transition undergone by the overlap of the state with its
translation, from an oscillatory regime for small chords, to evanescent
overlaps, rising to a maximum near the caustic. The diameter-caustic for the
Wigner function is also treated.Comment: 14 pages, 9 figure
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