46,509 research outputs found
Almost separable matrices
An mĂn matrix A with column supports {Si} is k-separable if the disjunctions âiâKSi are all distinct over all sets K of cardinality k. While a simple counting bound shows that m>klog2n/k rows are required for a separable matrix to exist, in fact it is necessary for m to be about a factor of k more than this. In this paper, we consider a weaker definition of âalmost k-separabilityâ, which requires that the disjunctions are âmostly distinctâ. We show using a random construction that these matrices exist with m=O(klogn) rows, which is optimal for k=O(n1âÎČ) . Further, by calculating explicit constants, we show how almost separable matrices give new bounds on the rate of nonadaptive group testing
Tracy-Widom distribution for the edge eigenvalues of Gram type random matrices
Large dimensional Gram type matrices are common objects in high-dimensional
statistics and machine learning. In this paper, we study the limiting
distribution of the edge eigenvalues for a general class of high-dimensional
Gram type random matrices, including separable sample covariance matrices,
sparse sample covariance matrices, bipartite stochastic block model and random
Gram matrices with general variance profiles. Specifically, we prove that under
(almost) sharp moment conditions and certain tractable regularity assumptions,
the edge eigenvalues, i.e., the largest few eigenvalues of non-spiked Gram type
random matrices or the extremal bulk eigenvalues of spiked Gram type random
matrices, satisfy the Tracy-Widom distribution asymptotically. Our results can
be used to construct adaptive, accurate and powerful statistics for
high-dimensional statistical inference. In particular, we propose
data-dependent statistics to infer the number of signals under general noise
structure, test the one-sided sphericity of separable matrix, and test the
structure of bipartite stochastic block model. Numerical simulations show
strong support of our proposed statistics. The core of our proof is to
establish the edge universality and Tracy-Widom distribution for a rectangular
Dyson Brownian motion with regular initial data. This is a general strategy to
study the edge statistics for high-dimensional Gram type random matrices
without exploring the specific independence structure of the target matrices.
It has potential to be applied to more general random matrices that are beyond
the ones considered in this paper.Comment: 67 pages, 9 figure
Optimal Ensemble Length of Mixed Separable States
The optimal (pure state) ensemble length of a separable state, A, is the
minimum number of (pure) product states needed in convex combination to
construct A. We study the set of all separable states with optimal (pure state)
ensemble length equal to k or fewer. Lower bounds on k are found below which
these sets have measure 0 in the set of separable states. In the bipartite case
and the multiparticle case where one of the particles has significantly more
quantum numbers than the rest, the lower bound for non-pure state ensembles is
sharp. A consequence of our results is that for all two particle systems,
except possibly those with a qubit or those with a nine dimensional Hilbert
space, and for all systems with more than two particles the optimal pure state
ensemble length for a randomly picked separable state is with probability 1
greater than the state's rank. In bipartite systems with probability 1 it is
greater than 1/4 the rank raised to the 3/2 power and in a system with p qubits
with probability 1 it is greater than (2^2p)/(1+2p), which is almost the square
of the rank.Comment: 8 page
On the separability of unitarily invariant random quantum states - the unbalanced regime
We study entanglement-related properties of random quantum states which are
unitarily invariant, in the sense that their distribution is left unchanged by
conjugation with arbitrary unitary operators. In the large matrix size limit,
the distribution of these random quantum states is characterized by their
limiting spectrum, a compactly supported probability distribution. We prove
several results characterizing entanglement and the PPT property of random
bipartite unitarily invariant quantum states in terms of the limiting spectral
distribution, in the unbalanced asymptotical regime where one of the two
subsystems is fixed, while the other one grows in size.Comment: New section on PPT matrices with large Schmidt numbe
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