13,581 research outputs found
Entanglement and quantum combinatorial designs
We introduce several classes of quantum combinatorial designs, namely quantum
Latin squares, cubes, hypercubes and a notion of orthogonality between them. A
further introduced notion, quantum orthogonal arrays, generalizes all previous
classes of designs. We show that mutually orthogonal quantum Latin arrangements
can be entangled in the same way than quantum states are entangled.
Furthermore, we show that such designs naturally define a remarkable class of
genuinely multipartite highly entangled states called -uniform, i.e.
multipartite pure states such that every reduction to parties is maximally
mixed. We derive infinitely many classes of mutually orthogonal quantum Latin
arrangements and quantum orthogonal arrays having an arbitrary large number of
columns. The corresponding multipartite -uniform states exhibit a high
persistency of entanglement, which makes them ideal candidates to develop
multipartite quantum information protocols.Comment: 14 pages, 3 figures. Comments are very welcome
Non-adaptive probabilistic group testing with noisy measurements: Near-optimal bounds with efficient algorithms
We consider the problem of detecting a small subset of defective items from a
large set via non-adaptive "random pooling" group tests. We consider both the
case when the measurements are noiseless, and the case when the measurements
are noisy (the outcome of each group test may be independently faulty with
probability q). Order-optimal results for these scenarios are known in the
literature. We give information-theoretic lower bounds on the query complexity
of these problems, and provide corresponding computationally efficient
algorithms that match the lower bounds up to a constant factor. To the best of
our knowledge this work is the first to explicitly estimate such a constant
that characterizes the gap between the upper and lower bounds for these
problems
The capacity of non-identical adaptive group testing
We consider the group testing problem, in the case where the items are
defective independently but with non-constant probability. We introduce and
analyse an algorithm to solve this problem by grouping items together
appropriately. We give conditions under which the algorithm performs
essentially optimally in the sense of information-theoretic capacity. We use
concentration of measure results to bound the probability that this algorithm
requires many more tests than the expected number. This has applications to the
allocation of spectrum to cognitive radios, in the case where a database gives
prior information that a particular band will be occupied.Comment: To be presented at Allerton 201
The Capacity of Adaptive Group Testing
We define capacity for group testing problems and deduce bounds for the
capacity of a variety of noisy models, based on the capacity of equivalent
noisy communication channels. For noiseless adaptive group testing we prove an
information-theoretic lower bound which tightens a bound of Chan et al. This
can be combined with a performance analysis of a version of Hwang's adaptive
group testing algorithm, in order to deduce the capacity of noiseless and
erasure group testing models.Comment: 5 page
Summary Based Structures with Improved Sublinear Recovery for Compressed Sensing
We introduce a new class of measurement matrices for compressed sensing,
using low order summaries over binary sequences of a given length. We prove
recovery guarantees for three reconstruction algorithms using the proposed
measurements, including minimization and two combinatorial methods. In
particular, one of the algorithms recovers -sparse vectors of length in
sublinear time , and requires at most
measurements. The empirical oversampling constant
of the algorithm is significantly better than existing sublinear recovery
algorithms such as Chaining Pursuit and Sudocodes. In particular, for and , the oversampling factor is between 3 to 8. We provide
preliminary insight into how the proposed constructions, and the fast recovery
scheme can be used in a number of practical applications such as market basket
analysis, and real time compressed sensing implementation
Solution to the Mean King's problem with mutually unbiased bases for arbitrary levels
The Mean King's problem with mutually unbiased bases is reconsidered for
arbitrary d-level systems. Hayashi, Horibe and Hashimoto [Phys. Rev. A 71,
052331 (2005)] related the problem to the existence of a maximal set of d-1
mutually orthogonal Latin squares, in their restricted setting that allows only
measurements of projection-valued measures. However, we then cannot find a
solution to the problem when e.g., d=6 or d=10. In contrast to their result, we
show that the King's problem always has a solution for arbitrary levels if we
also allow positive operator-valued measures. In constructing the solution, we
use orthogonal arrays in combinatorial design theory.Comment: REVTeX4, 4 page
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