11,196 research outputs found
Polar codes for classical-quantum channels
Holevo, Schumacher, and Westmoreland's coding theorem guarantees the
existence of codes that are capacity-achieving for the task of sending
classical data over a channel with classical inputs and quantum outputs.
Although they demonstrated the existence of such codes, their proof does not
provide an explicit construction of codes for this task. The aim of the present
paper is to fill this gap by constructing near-explicit "polar" codes that are
capacity-achieving. The codes exploit the channel polarization phenomenon
observed by Arikan for the case of classical channels. Channel polarization is
an effect in which one can synthesize a set of channels, by "channel combining"
and "channel splitting," in which a fraction of the synthesized channels are
perfect for data transmission while the other fraction are completely useless
for data transmission, with the good fraction equal to the capacity of the
channel. The channel polarization effect then leads to a simple scheme for data
transmission: send the information bits through the perfect channels and
"frozen" bits through the useless ones. The main technical contributions of the
present paper are threefold. First, we leverage several known results from the
quantum information literature to demonstrate that the channel polarization
effect occurs for channels with classical inputs and quantum outputs. We then
construct linear polar codes based on this effect, and the encoding complexity
is O(N log N), where N is the blocklength of the code. We also demonstrate that
a quantum successive cancellation decoder works well, in the sense that the
word error rate decays exponentially with the blocklength of the code. For this
last result, we exploit Sen's recent "non-commutative union bound" that holds
for a sequence of projectors applied to a quantum state.Comment: 12 pages, 3 figures; v2 in IEEE format with minor changes; v3 final
version accepted for publication in the IEEE Transactions on Information
Theor
Performance of polar codes for quantum and private classical communication
We analyze the practical performance of quantum polar codes, by computing
rigorous bounds on block error probability and by numerically simulating them.
We evaluate our bounds for quantum erasure channels with coding block lengths
between 2^10 and 2^20, and we report the results of simulations for quantum
erasure channels, quantum depolarizing channels, and "BB84" channels with
coding block lengths up to N = 1024. For quantum erasure channels, we observe
that high quantum data rates can be achieved for block error rates less than
10^(-4) and that somewhat lower quantum data rates can be achieved for quantum
depolarizing and BB84 channels. Our results here also serve as bounds for and
simulations of private classical data transmission over these channels,
essentially due to Renes' duality bounds for privacy amplification and
classical data transmission of complementary observables. Future work might be
able to improve upon our numerical results for quantum depolarizing and BB84
channels by employing a polar coding rule other than the heuristic used here.Comment: 8 pages, 6 figures, submission to the 50th Annual Allerton Conference
on Communication, Control, and Computing 201
Efficient Quantum Polar Coding
Polar coding, introduced 2008 by Arikan, is the first (very) efficiently
encodable and decodable coding scheme whose information transmission rate
provably achieves the Shannon bound for classical discrete memoryless channels
in the asymptotic limit of large block sizes. Here we study the use of polar
codes for the transmission of quantum information. Focusing on the case of
qubit Pauli channels and qubit erasure channels, we use classical polar codes
to construct a coding scheme which, using some pre-shared entanglement,
asymptotically achieves a net transmission rate equal to the coherent
information using efficient encoding and decoding operations and code
construction. Furthermore, for channels with sufficiently low noise level, we
demonstrate that the rate of preshared entanglement required is zero.Comment: v1: 15 pages, 4 figures. v2: 5+3 pages, 3 figures; argumentation
simplified and improve
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
Polar Codes for Arbitrary Classical-Quantum Channels and Arbitrary cq-MACs
We prove polarization theorems for arbitrary classical-quantum (cq) channels.
The input alphabet is endowed with an arbitrary Abelian group operation and an
Ar{\i}kan-style transformation is applied using this operation. It is shown
that as the number of polarization steps becomes large, the synthetic
cq-channels polarize to deterministic homomorphism channels which project their
input to a quotient group of the input alphabet. This result is used to
construct polar codes for arbitrary cq-channels and arbitrary classical-quantum
multiple access channels (cq-MAC). The encoder can be implemented in operations, where is the blocklength of the code. A quantum successive
cancellation decoder for the constructed codes is proposed. It is shown that
the probability of error of this decoder decays faster than
for any .Comment: 30 pages. Submitted to IEEE Trans. Inform. Theory and in part to
ISIT201
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