459 research outputs found
The quantum capacity with symmetric side channels
We present an upper bound for the quantum channel capacity that is both
additive and convex. Our bound can be interpreted as the capacity of a channel
for high-fidelity quantum communication when assisted by a family of channels
that have no capacity on their own. This family of assistance channels, which
we call symmetric side channels, consists of all channels mapping symmetrically
to their output and environment. The bound seems to be quite tight, and for
degradable quantum channels it coincides with the unassisted channel capacity.
Using this symmetric side channel capacity, we find new upper bounds on the
capacity of the depolarizing channel. We also briefly indicate an analogous
notion for distilling entanglement using the same class of (one-way) channels,
yielding one of the few entanglement measures that is monotonic under local
operations with one-way classical communication (1-LOCC), but not under the
more general class of local operations with classical communication (LOCC).Comment: 10 pages, 4 figure
A Quantized Johnson Lindenstrauss Lemma: The Finding of Buffon's Needle
In 1733, Georges-Louis Leclerc, Comte de Buffon in France, set the ground of
geometric probability theory by defining an enlightening problem: What is the
probability that a needle thrown randomly on a ground made of equispaced
parallel strips lies on two of them? In this work, we show that the solution to
this problem, and its generalization to dimensions, allows us to discover a
quantized form of the Johnson-Lindenstrauss (JL) Lemma, i.e., one that combines
a linear dimensionality reduction procedure with a uniform quantization of
precision . In particular, given a finite set of points and a distortion level , as soon as , we can (randomly) construct a mapping from
to that approximately
preserves the pairwise distances between the points of .
Interestingly, compared to the common JL Lemma, the mapping is quasi-isometric
and we observe both an additive and a multiplicative distortions on the
embedded distances. These two distortions, however, decay as when increases. Moreover, for coarse quantization, i.e., for high
compared to the set radius, the distortion is mainly additive, while
for small we tend to a Lipschitz isometric embedding. Finally, we
prove the existence of a "nearly" quasi-isometric embedding of into . This one involves a non-linear
distortion of the -distance in that vanishes for distant
points in this set. Noticeably, the additive distortion in this case is slower,
and decays as .Comment: 27 pages, 2 figures (note: this version corrects a few typos in the
abstract
A family of quantum protocols
We introduce two dual, purely quantum protocols: for entanglement
distillation assisted by quantum communication (``mother'' protocol) and for
entanglement assisted quantum communication (``father'' protocol). We show how
a large class of ``children'' protocols (including many previously known ones)
can be derived from the two by direct application of teleportation or
super-dense coding. Furthermore, the parent may be recovered from most of the
children protocols by making them ``coherent''. We also summarize the various
resource trade-offs these protocols give rise to.Comment: 5 pages, 1 figur
Small Width, Low Distortions: Quantized Random Embeddings of Low-complexity Sets
Under which conditions and with which distortions can we preserve the
pairwise-distances of low-complexity vectors, e.g., for structured sets such as
the set of sparse vectors or the one of low-rank matrices, when these are
mapped in a finite set of vectors? This work addresses this general question
through the specific use of a quantized and dithered random linear mapping
which combines, in the following order, a sub-Gaussian random projection in
of vectors in , a random translation, or "dither",
of the projected vectors and a uniform scalar quantizer of resolution
applied componentwise. Thanks to this quantized mapping we are first
able to show that, with high probability, an embedding of a bounded set
in can be achieved when
distances in the quantized and in the original domains are measured with the
- and -norm, respectively, and provided the number of quantized
observations is large before the square of the "Gaussian mean width" of
. In this case, we show that the embedding is actually
"quasi-isometric" and only suffers of both multiplicative and additive
distortions whose magnitudes decrease as for general sets, and as
for structured set, when increases. Second, when one is only
interested in characterizing the maximal distance separating two elements of
mapped to the same quantized vector, i.e., the "consistency width"
of the mapping, we show that for a similar number of measurements and with high
probability this width decays as for general sets and as for
structured ones when increases. Finally, as an important aspect of our
work, we also establish how the non-Gaussianity of the mapping impacts the
class of vectors that can be embedded or whose consistency width provably
decays when increases.Comment: Keywords: quantization, restricted isometry property, compressed
sensing, dimensionality reduction. 31 pages, 1 figur
Relating quantum privacy and quantum coherence: an operational approach
We describe how to achieve optimal entanglement generation and one-way
entanglement distillation rates by coherent implementation of a class of secret
key generation and secret key distillation protocols, respectively.
This short paper is a high-level descrioption of our detailed papers [8] and
[10].Comment: 4 pages, revtex
A Resource Framework for Quantum Shannon Theory
Quantum Shannon theory is loosely defined as a collection of coding theorems,
such as classical and quantum source compression, noisy channel coding
theorems, entanglement distillation, etc., which characterize asymptotic
properties of quantum and classical channels and states. In this paper we
advocate a unified approach to an important class of problems in quantum
Shannon theory, consisting of those that are bipartite, unidirectional and
memoryless.
We formalize two principles that have long been tacitly understood. First, we
describe how the Church of the larger Hilbert space allows us to move flexibly
between states, channels, ensembles and their purifications. Second, we
introduce finite and asymptotic (quantum) information processing resources as
the basic objects of quantum Shannon theory and recast the protocols used in
direct coding theorems as inequalities between resources. We develop the rules
of a resource calculus which allows us to manipulate and combine resource
inequalities. This framework simplifies many coding theorem proofs and provides
structural insights into the logical dependencies among coding theorems.
We review the above-mentioned basic coding results and show how a subset of
them can be unified into a family of related resource inequalities. Finally, we
use this family to find optimal trade-off curves for all protocols involving
one noisy quantum resource and two noiseless ones.Comment: 60 page
Shape Analysis Using Spectral Geometry
Shape analysis is a fundamental research topic in computer graphics and computer vision. To date, more and more 3D data is produced by those advanced acquisition capture devices, e.g., laser scanners, depth cameras, and CT/MRI scanners. The increasing data demands advanced analysis tools including shape matching, retrieval, deformation, etc. Nevertheless, 3D Shapes are represented with Euclidean transformations such as translation, scaling, and rotation and digital mesh representations are irregularly sampled. The shape can also deform non-linearly and the sampling may vary. In order to address these challenging problems, we investigate Laplace-Beltrami shape spectra from the differential geometry perspective, focusing more on the intrinsic properties. In this dissertation, the shapes are represented with 2 manifolds, which are differentiable.
First, we discuss in detail about the salient geometric feature points in the Laplace-Beltrami spectral domain instead of traditional spatial domains. Simultaneously, the local shape descriptor of a feature point is the Laplace-Beltrami spectrum of the spatial region associated to the point, which are stable and distinctive. The salient spectral geometric features are invariant to spatial Euclidean transforms, isometric deformations and mesh triangulations. Both global and partial matching can be achieved with these salient feature points. Next, we introduce a novel method to analyze a set of poses, i.e., near-isometric deformations, of 3D models that are unregistered. Different shapes of poses are transformed from the 3D spatial domain to a geometry spectral one where all near isometric deformations, mesh triangulations and Euclidean transformations are filtered away. Semantic parts of that model are then determined based on the computed geometric properties of all the mapped vertices in the geometry spectral domain while semantic skeleton can be automatically built with joints detected. Finally we prove the shape spectrum is a continuous function to a scale function on the conformal factor of the manifold. The derivatives of the eigenvalues are analytically expressed with those of the scale function. The property applies to both continuous domain and discrete triangle meshes. On the triangle meshes, a spectrum alignment algorithm is developed. Given two closed triangle meshes, the eigenvalues can be aligned from one to the other and the eigenfunction distributions are aligned as well. This extends the shape spectra across non-isometric deformations, supporting a registration-free analysis of general motion data
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