27 research outputs found
Secrecy Results for Compound Wiretap Channels
We derive a lower bound on the secrecy capacity of the compound wiretap
channel with channel state information at the transmitter which matches the
general upper bound on the secrecy capacity of general compound wiretap
channels given by Liang et al. and thus establishing a full coding theorem in
this case. We achieve this with a stronger secrecy criterion and the maximum
error probability criterion, and with a decoder that is robust against the
effect of randomisation in the encoding. This relieves us from the need of
decoding the randomisation parameter which is in general not possible within
this model. Moreover we prove a lower bound on the secrecy capacity of the
compound wiretap channel without channel state information and derive a
multi-letter expression for the capacity in this communication scenario.Comment: 25 pages, 1 figure. Accepted for publication in the journal "Problems
of Information Transmission". Some of the results were presented at the ITW
2011 Paraty [arXiv:1103.0135] and published in the conference paper available
at the IEEE Xplor
Entanglement transmission and generation under channel uncertainty: Universal quantum channel coding
We determine the optimal rates of universal quantum codes for entanglement
transmission and generation under channel uncertainty. In the simplest scenario
the sender and receiver are provided merely with the information that the
channel they use belongs to a given set of channels, so that they are forced to
use quantum codes that are reliable for the whole set of channels. This is
precisely the quantum analog of the compound channel coding problem. We
determine the entanglement transmission and entanglement-generating capacities
of compound quantum channels and show that they are equal. Moreover, we
investigate two variants of that basic scenario, namely the cases of informed
decoder or informed encoder, and derive corresponding capacity results.Comment: 45 pages, no figures. Section 6.2 rewritten due to an error in
equation (72) of the old version. Added table of contents, added section
'Conclusions and further remarks'. Accepted for publication in
'Communications in Mathematical Physics
Typical support and Sanov large deviations of correlated states
Discrete stationary classical processes as well as quantum lattice states are
asymptotically confined to their respective typical support, the exponential
growth rate of which is given by the (maximal ergodic) entropy. In the iid case
the distinguishability of typical supports can be asymptotically specified by
means of the relative entropy, according to Sanov's theorem. We give an
extension to the correlated case, referring to the newly introduced class of
HP-states.Comment: 29 pages, no figures, references adde
The invalidity of a strong capacity for a quantum channel with memory
The strong capacity of a particular channel can be interpreted as a sharp
limit on the amount of information which can be transmitted reliably over that
channel. To evaluate the strong capacity of a particular channel one must prove
both the direct part of the channel coding theorem and the strong converse for
the channel. Here we consider the strong converse theorem for the periodic
quantum channel and show some rather surprising results. We first show that the
strong converse does not hold in general for this channel and therefore the
channel does not have a strong capacity. Instead, we find that there is a scale
of capacities corresponding to error probabilities between integer multiples of
the inverse of the periodicity of the channel. A similar scale also exists for
the random channel.Comment: 7 pages, double column. Comments welcome. Repeated equation removed
and one reference adde
Quantum capacity under adversarial quantum noise: arbitrarily varying quantum channels
We investigate entanglement transmission over an unknown channel in the
presence of a third party (called the adversary), which is enabled to choose
the channel from a given set of memoryless but non-stationary channels without
informing the legitimate sender and receiver about the particular choice that
he made. This channel model is called arbitrarily varying quantum channel
(AVQC). We derive a quantum version of Ahlswede's dichotomy for classical
arbitrarily varying channels. This includes a regularized formula for the
common randomness-assisted capacity for entanglement transmission of an AVQC.
Quite surprisingly and in contrast to the classical analog of the problem
involving the maximal and average error probability, we find that the capacity
for entanglement transmission of an AVQC always equals its strong subspace
transmission capacity. These results are accompanied by different notions of
symmetrizability (zero-capacity conditions) as well as by conditions for an
AVQC to have a capacity described by a single-letter formula. In he final part
of the paper the capacity of the erasure-AVQC is computed and some light shed
on the connection between AVQCs and zero-error capacities. Additionally, we
show by entirely elementary and operational arguments motivated by the theory
of AVQCs that the quantum, classical, and entanglement-assisted zero-error
capacities of quantum channels are generically zero and are discontinuous at
every positivity point.Comment: 49 pages, no figures, final version of our papers arXiv:1010.0418v2
and arXiv:1010.0418. Published "Online First" in Communications in
Mathematical Physics, 201
Private information via the Unruh effect
In a relativistic theory of quantum information, the possible presence of
horizons is a complicating feature placing restrictions on the transmission and
retrieval of information. We consider two inertial participants communicating
via a noiseless qubit channel in the presence of a uniformly accelerated
eavesdropper. Owing to the Unruh effect, the eavesdropper's view of any encoded
information is noisy, a feature the two inertial participants can exploit to
achieve perfectly secure quantum communication. We show that the associated
private quantum capacity is equal to the entanglement-assisted quantum capacity
for the channel to the eavesdropper's environment, which we evaluate for all
accelerations.Comment: 5 pages. v2: footnote deleted and typos corrected. v3: major
revision. New capacity (single-letter!) theorem and implicit assumption
lifte
Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno's Theorem
In classical information theory, entropy rate and Kolmogorov complexity per
symbol are related by a theorem of Brudno. In this paper, we prove a quantum
version of this theorem, connecting the von Neumann entropy rate and two
notions of quantum Kolmogorov complexity, both based on the shortest qubit
descriptions of qubit strings that, run by a universal quantum Turing machine,
reproduce them as outputs.Comment: 26 pages, no figures. Reference to publication added: published in
the Communications in Mathematical Physics
(http://www.springerlink.com/content/1432-0916/
Coding Theorem for a Class of Quantum Channels with Long-Term Memory
In this paper we consider the transmission of classical information through a
class of quantum channels with long-term memory, which are given by convex
combinations of product channels. Hence, the memory of such channels is given
by a Markov chain which is aperiodic but not irreducible. We prove the coding
theorem and weak converse for this class of channels. The main techniques that
we employ, are a quantum version of Feinstein's Fundamental Lemma and a
generalization of Helstrom's Theorem.Comment: Some typos correcte
Partitioned trace distances
New quantum distance is introduced as a half-sum of several singular values
of difference between two density operators. This is, up to factor, the metric
induced by so-called Ky Fan norm. The partitioned trace distances enjoy similar
properties to the standard trace distance, including the unitary invariance,
the strong convexity and the close relations to the classical distances. The
partitioned distances cannot increase under quantum operations of certain kind
including bistochastic maps. All the basic properties are re-formulated as
majorization relations. Possible applications to quantum information processing
are briefly discussed.Comment: 8 pages, no figures. Significant changes are made. New section on
majorization is added. Theorem 4.1 is extended. The bibliography is enlarged
On Quantum Capacity of Compound Channels
In this paper we address the issue of universal or robust communication over
quantum channels. Specifically, we consider memoryless communication scenario
with channel uncertainty which is an analog of compound channel in classical
information theory. We determine the quantum capacity of finite compound
channels and arbitrary compound channels with informed decoder. Our approach in
the finite case is based on the observation that perfect channel knowledge at
the decoder does not increase the capacity of finite quantum compound channels.
As a consequence we obtain coding theorem for finite quantum averaged channels,
the simplest class of channels with long-term memory. The extension of these
results to quantum compound channels with uninformed encoder and decoder, and
infinitely many constituents remains an open problem.Comment: 16 pages, no figure