108 research outputs found
Generalizations of Fano's Inequality for Conditional Information Measures via Majorization Theory
Fano's inequality is one of the most elementary, ubiquitous, and important
tools in information theory. Using majorization theory, Fano's inequality is
generalized to a broad class of information measures, which contains those of
Shannon and R\'{e}nyi. When specialized to these measures, it recovers and
generalizes the classical inequalities. Key to the derivation is the
construction of an appropriate conditional distribution inducing a desired
marginal distribution on a countably infinite alphabet. The construction is
based on the infinite-dimensional version of Birkhoff's theorem proven by
R\'{e}v\'{e}sz [Acta Math. Hungar. 1962, 3, 188{\textendash}198], and the
constraint of maintaining a desired marginal distribution is similar to
coupling in probability theory. Using our Fano-type inequalities for Shannon's
and R\'{e}nyi's information measures, we also investigate the asymptotic
behavior of the sequence of Shannon's and R\'{e}nyi's equivocations when the
error probabilities vanish. This asymptotic behavior provides a novel
characterization of the asymptotic equipartition property (AEP) via Fano's
inequality.Comment: 44 pages, 3 figure
Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels
This thesis combines two parallel research directions: an exploration into the
continuity properties of certain entropic quantities, and an investigation
into a simple class of physical systems whose time evolution
is given by the repeated application of a quantum channel.
In the first part of the thesis, we present a general technique for
establishing local and uniform continuity bounds for Schur concave functions;
that is, for real-valued functions which are decreasing in the majorization
pre-order. Continuity bounds provide a quantitative measure of robustness,
addressing the following question: If there is some uncertainty or error in
the input, how much uncertainty is there in the output? Our technique uses a
particular relationship between majorization and the trace distance between
quantum states (or total variation distance, in the case of probability
distributions). Namely, the majorization pre-order attains a maximum and a
minimum over ε-balls in this distance. By tracing the path of the
majorization-minimizer as a function of the distance ε, we obtain the
path of ``majorization flow’’. An analysis of the derivatives of Schur
concave functions along this path immediately yields tight continuity bounds
for such functions.
In this way, we find a new proof of the Audenaert-Fannes continuity bound for
the von Neumann entropy, and the necessary and sufficient conditions for its
saturation, in a universal framework which extends to the other functions,
including the Rényi and Tsallis entropies. In particular, we prove a novel
uniform continuity bound for the α-Rényi entropy with α > 1 with
much improved dependence on the dimension of the underlying system and the
parameter α compared to previously known bounds. We show that this
framework can also be used to provide continuity bounds for other Schur
concave functions, such as the number of connected components of a certain
random graph model as a function of the underlying probability distribution,
and the number of distinct realizations of a random variable in some fixed
number of independent trials as a function of the underlying probability mass
function. The former has been used in modeling the spread of epidemics, while
the latter has been studied in the context of estimating measures of
biodiversity from observations; in these contexts, our continuity bounds
provide quantitative estimates of robustness to noise or data collection
errors.
In the second part, we consider repeated interaction systems, in which a
system of interest interacts with a sequence of probes, i.e. environmental
systems, one at a time. The state of the system after each interaction is
related to the state of the system before the interaction by the so-called
reduced dynamics, which is described by the action of a quantum channel. When
each probe and the way it interacts with the system is identical, the reduced
dynamics at each step is identical. In this scenario, under the additional
assumption that the reduced dynamics satisfies a faithfulness property, we
characterize which repeated interaction systems break any initially-present
entanglement between the system and an untouched reference, after finitely
many steps. In this case, the reduced dynamics is said to be eventually
entanglement-breaking. This investigation helps improve our
understanding of which kinds of noisy time evolution destroy entanglement.
When the probes and their interactions with the system are slowly-varying
(i.e. adiabatic), we analyze the saturation of Landauer's bound, an inequality
between the entropy change of the system and the energy change of the probes,
in the limit in which the number of steps tends to infinity and both the
difference between consecutive probes and the difference between their
interactions vanishes. This analysis proceeds at a fine-grained level by means
of a two-time measurement protocol, in which the energy of the probes is
measured before and after each interaction. The quantities of interest are
then studied as random variables on the space of outcomes of the energy
measurements of the probes, providing a deeper insight into the interrelations
between energy and entropy in this setting.Cantab Capital Institute for the Mathematics of Informatio
Scaling of disorder operator at deconfined quantum criticality
We study scaling behavior of the disorder parameter, defined as the
expectation value of a symmetry transformation applied to a finite region, at
the deconfined quantum critical point in (2+1) in the - model via
large-scale quantum Monte Carlo simulations. We show that the disorder
parameter for U(1) spin rotation symmetry exhibits perimeter scaling with a
logarithmic correction associated with sharp corners of the region, as
generally expected for a conformally-invariant critical point. However, for
large rotation angle the universal coefficient of the logarithmic corner
correction becomes negative, which is not allowed in any unitary conformal
field theory. We also extract the current central charge from the small
rotation angle scaling, whose value is much smaller than that of the free
theory.Comment: 8 pages, 6 figures; v2 improved measurement on disorder operato
From Blackwell Dominance in Large Samples to Rényi Divergences and Back Again
We study repeated independent Blackwell experiments; standard examples include drawing multiple samples from a population, or performing a measurement in different locations. In the baseline setting of a binary state of nature, we compare experiments in terms of their informativeness in large samples. Addressing a question due to Blackwell (1951), we show that generically an experiment is more informative than another in large samples if and only if it has higher Rényi divergences
Probing and detecting entanglement in synthetic quantum matter
In the past decades, rapid advances in the experimental control of quantum systems have opened up unparalleled capabilities of engineering exotic quantum states. Nowadays, we may consider ourselves witnesses of a second quantum revolution as strongly correlated quantum matter is regularly created, on a daily basis, within the most diverse platforms, e.g. arrays of Rydberg atoms, ultra-cold atoms in optical lattices, superconducting qubits, trapped ions, and quantum dots.
In this era, dubbed Noisy-Intermediate Scale Quantum (NISQ) era, the effort in pursuing research for realizing quantum technologies with practical purposes, such as quantum computing, simulations, communication and metrology, has greatly accelerated, and we are just starting to experience the immense progress that could be achieved.
The remarkable breakthrough we are facing builds upon seminal theoretical and experimental advances in quantum physics. The key achievements in atomic, molecular, and optical (AMO) physics have opened up the possibility of controlling, trapping, and measuring single atoms, one by one, with high accuracy and reliability. In parallel, from a theoretical point of view, the study of quantum entanglement and correlations has bridged AMO physics and quantum information with crucial proposals for the realization of universal quantum computers.
In this context, entanglement has emerged as one of the key tools to characterize and to exploit quantum many-body systems for quantum information purposes.
We will study quantum entanglement in several scenarios to investigate and probe complex quantum many-body systems. We will consider examples ranging from generic mixed states in equilibrium to out-of-equilibrium dynamics, with and without dissipation, and topologically non-trivial systems.
The leitmotif of this work will be how entanglement and correlations can be exploited to characterize the many-body quantum state describing a physical system and how entanglement can be detected in an experimentally efficient manner
Divergence Measures
Data science, information theory, probability theory, statistical learning and other related disciplines greatly benefit from non-negative measures of dissimilarity between pairs of probability measures. These are known as divergence measures, and exploring their mathematical foundations and diverse applications is of significant interest. The present Special Issue, entitled “Divergence Measures: Mathematical Foundations and Applications in Information-Theoretic and Statistical Problems”, includes eight original contributions, and it is focused on the study of the mathematical properties and applications of classical and generalized divergence measures from an information-theoretic perspective. It mainly deals with two key generalizations of the relative entropy: namely, the R_ényi divergence and the important class of f -divergences. It is our hope that the readers will find interest in this Special Issue, which will stimulate further research in the study of the mathematical foundations and applications of divergence measures
The Statistical Foundations of Entropy
In the last two decades, the understanding of complex dynamical systems underwent important conceptual shifts. The catalyst was the infusion of new ideas from the theory of critical phenomena (scaling laws, renormalization group, etc.), (multi)fractals and trees, random matrix theory, network theory, and non-Shannonian information theory. The usual Boltzmann–Gibbs statistics were proven to be grossly inadequate in this context. While successful in describing stationary systems characterized by ergodicity or metric transitivity, Boltzmann–Gibbs statistics fail to reproduce the complex statistical behavior of many real-world systems in biology, astrophysics, geology, and the economic and social sciences.The aim of this Special Issue was to extend the state of the art by original contributions that could contribute to an ongoing discussion on the statistical foundations of entropy, with a particular emphasis on non-conventional entropies that go significantly beyond Boltzmann, Gibbs, and Shannon paradigms. The accepted contributions addressed various aspects including information theoretic, thermodynamic and quantum aspects of complex systems and found several important applications of generalized entropies in various systems
- …