117 research outputs found
R\'enyi Bounds on Information Combining
Bounds on information combining are entropic inequalities that determine how
the information, or entropy, of a set of random variables can change when they
are combined in certain prescribed ways. Such bounds play an important role in
information theory, particularly in coding and Shannon theory. The arguably
most elementary kind of information combining is the addition of two binary
random variables, i.e. a CNOT gate, and the resulting quantities are
fundamental when investigating belief propagation and polar coding. In this
work we will generalize the concept to R\'enyi entropies. We give optimal
bounds on the conditional R\'enyi entropy after combination, based on a certain
convexity or concavity property and discuss when this property indeed holds.
Since there is no generally agreed upon definition of the conditional R\'enyi
entropy, we consider four different versions from the literature. Finally, we
discuss the application of these bounds to the polarization of R\'enyi
entropies under polar codes.Comment: 14 pages, accepted for presentation at ISIT 202
An improved rate region for the classical-quantum broadcast channel
We present a new achievable rate region for the two-user binary-input
classical-quantum broadcast channel. The result is a generalization of the
classical Marton-Gelfand-Pinsker region and is provably larger than the best
previously known rate region for classical-quantum broadcast channels. The
proof of achievability is based on the recently introduced polar coding scheme
and its generalization to quantum network information theory.Comment: 5 pages, double column, 1 figure, based on a result presented in the
Master's thesis arXiv:1501.0373
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
Efficient achievability for quantum protocols using decoupling theorems
Proving achievability of protocols in quantum Shannon theory usually does not
consider the efficiency at which the goal of the protocol can be achieved.
Nevertheless it is known that protocols such as coherent state merging are
efficiently achievable at optimal rate. We aim to investigate this fact further
in a general one-shot setting, by considering certain classes of decoupling
theorems and give exact rates for these classes. Moreover we compare results of
general decoupling theorems using Haar distributed unitaries with those using
smaller sets of operators, in particular -approximate 2-designs. We
also observe the behavior of our rates in special cases such as
approaching zero and the asymptotic limit.Comment: 5 pages, double column, v2: added referenc
Convexity and Operational Interpretation of the Quantum Information Bottleneck Function
In classical information theory, the information bottleneck method (IBM) can
be regarded as a method of lossy data compression which focusses on preserving
meaningful (or relevant) information. As such it has recently gained a lot of
attention, primarily for its applications in machine learning and neural
networks. A quantum analogue of the IBM has recently been defined, and an
attempt at providing an operational interpretation of the so-called quantum IB
function as an optimal rate of an information-theoretic task, has recently been
made by Salek et al. However, the interpretation given in that paper has a
couple of drawbacks; firstly its proof is based on a conjecture that the
quantum IB function is convex, and secondly, the expression for the rate
function involves certain entropic quantities which occur explicitly in the
very definition of the underlying information-theoretic task, thus making the
latter somewhat contrived. We overcome both of these drawbacks by first proving
the convexity of the quantum IB function, and then giving an alternative
operational interpretation of it as the optimal rate of a bona fide
information-theoretic task, namely that of quantum source coding with quantum
side information at the decoder, and relate the quantum IB function to the rate
region of this task. We similarly show that the related privacy funnel function
is convex (both in the classical and quantum case). However, we comment that it
is unlikely that the quantum privacy funnel function can characterize the
optimal asymptotic rate of an information theoretic task, since even its
classical version lacks a certain additivity property which turns out to be
essential.Comment: 17 pages, 7 figures; v2: improved presentation and explanations, one
new figure; v3: Restructured manuscript. Theorem 2 has been found previously
in work by Hsieh and Watanabe; it is now correctly attribute
Quantum Network Discrimination
Discrimination between objects, in particular quantum states, is one of the
most fundamental tasks in (quantum) information theory. Recent years have seen
significant progress towards extending the framework to point-to-point quantum
channels. However, with technological progress the focus of the field is
shifting to more complex structures: Quantum networks. In contrast to channels,
networks allow for intermediate access points where information can be
received, processed and reintroduced into the network. In this work we study
the discrimination of quantum networks and its fundamental limitations. In
particular when multiple uses of the network are at hand, the rooster of
available strategies becomes increasingly complex. The simplest quantum network
that capturers the structure of the problem is given by a quantum superchannel.
We discuss the available classes of strategies when considering copies of a
superchannel and give fundamental bounds on the asymptotically achievable rates
in an asymmetric discrimination setting. Furthermore, we discuss achievability,
symmetric network discrimination, the strong converse exponent, generalization
to arbitrary quantum networks and finally an application to an active version
of the quantum illumination problem.Comment: 39 pages, 1 Table, 9 Figures incl. 1 Animatio
Polar codes in network quantum information theory
Polar coding is a method for communication over noisy classical channels
which is provably capacity-achieving and has an efficient encoding and
decoding. Recently, this method has been generalized to the realm of quantum
information processing, for tasks such as classical communication, private
classical communication, and quantum communication. In the present work, we
apply the polar coding method to network quantum information theory, by making
use of recent advances for related classical tasks. In particular, we consider
problems such as the compound multiple access channel and the quantum
interference channel. The main result of our work is that it is possible to
achieve the best known inner bounds on the achievable rate regions for these
tasks, without requiring a so-called quantum simultaneous decoder. Thus, our
work paves the way for developing network quantum information theory further
without requiring a quantum simultaneous decoder.Comment: 18 pages, 2 figures, v2: 10 pages, double column, version accepted
for publicatio
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