58,564 research outputs found
Achieving Marton's Region for Broadcast Channels Using Polar Codes
This paper presents polar coding schemes for the 2-user discrete memoryless
broadcast channel (DM-BC) which achieve Marton's region with both common and
private messages. This is the best achievable rate region known to date, and it
is tight for all classes of 2-user DM-BCs whose capacity regions are known. To
accomplish this task, we first construct polar codes for both the superposition
as well as the binning strategy. By combining these two schemes, we obtain
Marton's region with private messages only. Finally, we show how to handle the
case of common information. The proposed coding schemes possess the usual
advantages of polar codes, i.e., they have low encoding and decoding complexity
and a super-polynomial decay rate of the error probability.
We follow the lead of Goela, Abbe, and Gastpar, who recently introduced polar
codes emulating the superposition and binning schemes. In order to align the
polar indices, for both schemes, their solution involves some degradedness
constraints that are assumed to hold between the auxiliary random variables and
the channel outputs. To remove these constraints, we consider the transmission
of blocks and employ a chaining construction that guarantees the proper
alignment of the polarized indices. The techniques described in this work are
quite general, and they can be adopted to many other multi-terminal scenarios
whenever there polar indices need to be aligned.Comment: 26 pages, 11 figures, accepted to IEEE Trans. Inform. Theory and
presented in part at ISIT'1
Polar Codes for Quantum Reading
Quantum readout provides a general framework
for formulating statistical discrimination of quantum channels.
Several paths have been taken for such this problem. However,
there is much to be done in the avenue of optimizing channel
discrimination using classical codes. At least two open questions
can be pointed out: how to construct low complexity encoding
schemes that are interesting for channel discrimination and,
more importantly, how to develop capacity-achieving protocols.
This paper aims at presenting a solution to these questions
using polar codes. Firstly, we characterize the information rate
and reliability parameter of the channels under polar encoding.
We also show that the error probability of the scheme proposed
decays exponentially with the square root of the code length.
Secondly, an analysis of the optimal quantum states to be used
as probes is given
Polar codes for degradable quantum channels
Channel polarization is a phenomenon in which a particular recursive encoding
induces a set of synthesized channels from many instances of a memoryless
channel, such that a fraction of the synthesized channels becomes near perfect
for data transmission and the other fraction becomes near useless for this
task. Mahdavifar and Vardy have recently exploited this phenomenon to construct
codes that achieve the symmetric private capacity for private data transmission
over a degraded wiretap channel. In the current paper, we build on their work
and demonstrate how to construct quantum wiretap polar codes that achieve the
symmetric private capacity of a degraded quantum wiretap channel with a
classical eavesdropper. Due to the Schumacher-Westmoreland correspondence
between quantum privacy and quantum coherence, we can construct quantum polar
codes by operating these quantum wiretap polar codes in superposition, much
like Devetak's technique for demonstrating the achievability of the coherent
information rate for quantum data transmission. Our scheme achieves the
symmetric coherent information rate for quantum channels that are degradable
with a classical environment. This condition on the environment may seem
restrictive, but we show that many quantum channels satisfy this criterion,
including amplitude damping channels, photon-detected jump channels, dephasing
channels, erasure channels, and cloning channels. Our quantum polar coding
scheme has the desirable properties of being channel-adapted and symmetric
capacity-achieving along with having an efficient encoder, but we have not
demonstrated that the decoding is efficient. Also, the scheme may require
entanglement assistance, but we show that the rate of entanglement consumption
vanishes in the limit of large blocklength if the channel is degradable with
classical environment.Comment: 12 pages, 1 figure; v2: IEEE format, minor changes including new
figure; v3: minor changes, accepted for publication in IEEE Transactions on
Information Theor
Magic state distillation with punctured polar codes
We present a scheme for magic state distillation using punctured polar codes.
Our results build on some recent work by Bardet et al. (ISIT, 2016) who
discovered that polar codes can be described algebraically as decreasing
monomial codes. Using this powerful framework, we construct tri-orthogonal
quantum codes (Bravyi et al., PRA, 2012) that can be used to distill magic
states for the gate. An advantage of these codes is that they permit the
use of the successive cancellation decoder whose time complexity scales as
. We supplement this with numerical simulations for the erasure
channel and dephasing channel. We obtain estimates for the dimensions and error
rates for the resulting codes for block sizes up to for the erasure
channel and for the dephasing channel. The dimension of the
triply-even codes we obtain is shown to scale like for the binary
erasure channel at noise rate and for the dephasing
channel at noise rate . The corresponding bit error rates drop to
roughly for the erasure channel and for
the dephasing channel respectively.Comment: 18 pages, 4 figure
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