25,984 research outputs found
Bounds on Information Combining With Quantum Side Information
"Bounds on information combining" are entropic inequalities that determine
how the information (entropy) of a set of random variables can change when
these are combined in certain prescribed ways. Such bounds play an important
role in classical information theory, particularly in coding and Shannon
theory; entropy power inequalities are special instances of them. The arguably
most elementary kind of information combining is the addition of two binary
random variables (a CNOT gate), and the resulting quantities play an important
role in Belief propagation and Polar coding. We investigate this problem in the
setting where quantum side information is available, which has been recognized
as a hard setting for entropy power inequalities.
Our main technical result is a non-trivial, and close to optimal, lower bound
on the combined entropy, which can be seen as an almost optimal "quantum Mrs.
Gerber's Lemma". Our proof uses three main ingredients: (1) a new bound on the
concavity of von Neumann entropy, which is tight in the regime of low pairwise
state fidelities; (2) the quantitative improvement of strong subadditivity due
to Fawzi-Renner, in which we manage to handle the minimization over recovery
maps; (3) recent duality results on classical-quantum-channels due to Renes et
al. We furthermore present conjectures on the optimal lower and upper bounds
under quantum side information, supported by interesting analytical
observations and strong numerical evidence.
We finally apply our bounds to Polar coding for binary-input
classical-quantum channels, and show the following three results: (A) Even
non-stationary channels polarize under the polar transform. (B) The blocklength
required to approach the symmetric capacity scales at most sub-exponentially in
the gap to capacity. (C) Under the aforementioned lower bound conjecture, a
blocklength polynomial in the gap suffices.Comment: 23 pages, 6 figures; v2: small correction
A general approach to small deviation via concentration of measures
We provide a general approach to obtain upper bounds for small deviations in different norms, namely the supremum
and - H\"older norms. The large class of processes under
consideration takes the form , where and
are two possibly dependent stochastic processes. Our approach provides an upper
bound for small deviations whenever upper bounds for the \textit{concentration
of measures} of - norm of random vectors built from increments of the
process and \textit{large deviation} estimates for the process are
available. Using our method, among others, we obtain the optimal rates of small
deviations in supremum and - H\"older norms for fractional Brownian
motion with Hurst parameter . As an application, we discuss
the usefulness of our upper bounds for small deviations in pathwise stochastic
integral representation of random variables motivated by the hedging problem in
mathematical finance
Von Neumann Entropy Penalization and Low Rank Matrix Estimation
A problem of statistical estimation of a Hermitian nonnegatively definite
matrix of unit trace (for instance, a density matrix in quantum state
tomography) is studied. The approach is based on penalized least squares method
with a complexity penalty defined in terms of von Neumann entropy. A number of
oracle inequalities have been proved showing how the error of the estimator
depends on the rank and other characteristics of the oracles. The methods of
proofs are based on empirical processes theory and probabilistic inequalities
for random matrices, in particular, noncommutative versions of Bernstein
inequality
Rough differential equations driven by signals in Besov spaces
Rough differential equations are solved for signals in general Besov spaces
unifying in particular the known results in H\"older and p-variation topology.
To this end the paracontrolled distribution approach, which has been introduced
by Gubinelli, Imkeller and Perkowski ["Paracontrolled distribution and singular
PDEs", Forum of Mathematics, Pi (2015)] to analyze singular stochastic PDEs, is
extended from H\"older to Besov spaces. As an application we solve stochastic
differential equations driven by random functions in Besov spaces and Gaussian
processes in a pathwise sense.Comment: Former title: "Rough differential equations on Besov spaces", 37
page
Concentration for Coulomb gases and Coulomb transport inequalities
We study the non-asymptotic behavior of Coulomb gases in dimension two and
more. Such gases are modeled by an exchangeable Boltzmann-Gibbs measure with a
singular two-body interaction. We obtain concentration of measure inequalities
for the empirical distribution of such gases around their equilibrium measure,
with respect to bounded Lipschitz and Wasserstein distances. This implies
macroscopic as well as mesoscopic convergence in such distances. In particular,
we improve the concentration inequalities known for the empirical spectral
distribution of Ginibre random matrices. Our approach is remarkably simple and
bypasses the use of renormalized energy. It crucially relies on new
inequalities between probability metrics, including Coulomb transport
inequalities which can be of independent interest. Our work is inspired by the
one of Ma{\"i}da and Maurel-Segala, itself inspired by large deviations
techniques. Our approach allows to recover, extend, and simplify previous
results by Rougerie and Serfaty.Comment: Improvement on an assumption, and minor modification
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