13 research outputs found
Proof of an entropy conjecture for Bloch coherent spin states and its generalizations
Wehrl used Glauber coherent states to define a map from quantum density
matrices to classical phase space densities and conjectured that for Glauber
coherent states the mininimum classical entropy would occur for density
matrices equal to projectors onto coherent states. This was proved by Lieb in
1978 who also extended the conjecture to Bloch SU(2) spin-coherent states for
every angular momentum . This conjecture is proved here. We also recall our
1991 extension of the Wehrl map to a quantum channel from to , with corresponding to the Wehrl map to classical densities.
For each and we show that the minimal output entropy for
these channels occurs for a coherent state. We also show that coherent
states both Glauber and Bloch minimize any concave functional, not just
entropy.Comment: Version 2 only minor change
Quantization and Compressive Sensing
Quantization is an essential step in digitizing signals, and, therefore, an
indispensable component of any modern acquisition system. This book chapter
explores the interaction of quantization and compressive sensing and examines
practical quantization strategies for compressive acquisition systems.
Specifically, we first provide a brief overview of quantization and examine
fundamental performance bounds applicable to any quantization approach. Next,
we consider several forms of scalar quantizers, namely uniform, non-uniform,
and 1-bit. We provide performance bounds and fundamental analysis, as well as
practical quantizer designs and reconstruction algorithms that account for
quantization. Furthermore, we provide an overview of Sigma-Delta
() quantization in the compressed sensing context, and also
discuss implementation issues, recovery algorithms and performance bounds. As
we demonstrate, proper accounting for quantization and careful quantizer design
has significant impact in the performance of a compressive acquisition system.Comment: 35 pages, 20 figures, to appear in Springer book "Compressed Sensing
and Its Applications", 201
Probabilistic frames: An overview
Finite frames can be viewed as mass points distributed in -dimensional
Euclidean space. As such they form a subclass of a larger and rich class of
probability measures that we call probabilistic frames. We derive the basic
properties of probabilistic frames, and we characterize one of their subclasses
in terms of minimizers of some appropriate potential function. In addition, we
survey a range of areas where probabilistic frames, albeit, under different
names, appear. These areas include directional statistics, the geometry of
convex bodies, and the theory of t-designs