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Noisy channel coding via privacy amplification and information reconciliation

Joseph M. Renes, Renato Renner

arXiv:1012.4814v2quant-phcs.ITmath.IT

TL;DR

The paper constructs noisy-channel coding protocols from privacy amplification and information reconciliation, with optimal primitives yielding optimal channel-coding protocols. It frames the constructions through smooth min- and max-entropies, whose duality in quantum information means one primitive can suffice.

  • Problem

    The paper addresses constructing channel-coding protocols from simpler information-theoretic primitives and characterizing their capacities in the one-shot setting.

    Source: §V · p. 6

  • Method

    The construction combines privacy amplification or randomness extraction with information reconciliation, using distribution shaping to reproduce the desired input distribution.

    Source: C. Distribution Shaping · p. 4, §V · p. 6

  • Results

    When the underlying primitives are optimal, the constructed channel-coding protocols are optimal; in quantum information, duality of the two primitives means one primitive suffices.

    Source: §V · p. 6

  • Takeaways & Limitations

    The paper frames two-terminal information-theoretic protocols through smooth min- and max-entropies, and quantum duality permits construction from a single primitive.

    Source: §V · p. 6

    The uncertainty-principle variant in cannot be used for this purpose because it subtly relies on the decoder construction the proof seeks to avoid.

    Source: §V · p. 6

Abstract

from arXiv · show

We show that optimal protocols for noisy channel coding of public or private information over either classical or quantum channels can be directly constructed from two more primitive information-theoretic tools: privacy amplification and information reconciliation, also known as data compression with side information. We do this in the one-shot scenario of structureless resources, and formulate our results in terms of the smooth min- and max-entropy. In the context of classical information theory, this shows that essentially all two-terminal protocols can be reduced to these two primitives, which are in turn governed by the smooth min- and max-entropies, respectively. In the context of quantum information theory, the recently-established duality of these two protocols means essentially all two-terminal protocols can be constructed using just a single primitive.

I. DEFINITIONS AND RESULTS

The section defines public and private coding for classical-to-quantum channels and states capacity bounds in terms of smooth min- and max-entropies.

I. DEFINITIONS AND RESULTS

  • The channel model accepts classical inputs and quantum output states; requiring those states to be simultaneously diagonalizable recovers a classical channel.For a physical channel accepting quantum inputs, fixing the input state for each classical value can be treated as part of the channel or encoder.Source: §I · p. 2
  • A coding scheme maps classical messages to channel inputs and maps channel outputs to guesses of the input messages.The message transmission process is represented as a Markov chain, with the primed output intended to match the input message.Source: §I · p. 2
  • The public and private capacities are defined as the largest message sizes achievable at a chosen error level, with private coding adding a secrecy requirement.The private scheme requires each message to be approximately unknown to the eavesdropper.Source: §I · p. 2
  • Capacity upper and lower bounds are expressed using smooth conditional min- and max-entropies.The passage states these bounds for both public and private communication.Source: §I · p. 2

II. ACHIEVABILITY

The achievability proof builds channel coding from information reconciliation and distribution shaping, converting a uniform message into a channel input distribution suited to communication.

II. ACHIEVABILITY

  • The proof constructs a channel encoder and decoder by adapting information reconciliation to uniformly distributed channel inputs, then shaping those inputs to mimic a desired distribution.Privacy amplification can be used in the shaping step so the eavesdropper learns nothing about the message.Source: §II · p. 3, Figure 1 · p. 3

A. Average Case Coding Implies Worst Case Coding

A scheme with low average error and secrecy can be restricted to a subset of messages that meets both guarantees in the worst case.

A. Average Case Coding Implies Worst Case Coding

  • A subset containing at least half the messages has error and secrecy parameters at most ε for every retained message.Markov's inequality bounds the fraction exceeding either threshold by one-quarter, leaving at least half satisfying both.Source: A. Average Case Coding Implies Worst… · p. 3

B. Channel Coding From Information Reconciliation For Uniformly-Distributed Inputs

For uniformly distributed inputs, an information reconciliation compressor/decompressor yields a channel code; the construction uses a linear compressor and arbitrary decompressor.

B. Channel Coding From Information Reconciliation For Uniformly-Distributed Inputs

  • Information reconciliation compresses X and uses side information Y to reconstruct X with error probability at most ε.When X forms a linear space, the compression map may be linear.Source: B. Channel Coding From Information… · p. 3
  • An ε-good reconciliation protocol for the joint input-output pair of a uniform-input channel yields a code with message size |U|/|C| and error below ε.The stated construction assumes a linear compressor and allows an arbitrary decompressor.Source: B. Channel Coding From Information… · p. 3
  • The construction selects the compressor output class with the lowest average decoding error and encodes messages as inputs in that class.Linearity makes each class the same size, |U|/|C|; decoding applies the decompressor and inverts the encoding when its output lies in the selected class.Source: B. Channel Coding From Information… · pp. 3–4
  • If reconciliation has error ε2 on an input within total variation distance ε1 of uniform, its channel-code error on uniform input is at most ε1 + ε2.The bound follows from the triangle inequality.Source: B. Channel Coding From Information… · p. 4
  • Removing the linear-compression restriction costs a factor ε in message count and adds failure probability ε.Source: B. Channel Coding From Information… · p. 4

C. Distribution Shaping

A distribution shaper removes the uniform-input restriction by mapping uniform randomness to a chosen channel-input distribution. It is built from a randomness extractor, with additional randomness used to reverse the mapping.

C. Distribution Shaping

  • An extractor-derived shaper maps a uniform input to a chosen distribution by reconstructing X from U′ and auxiliary randomness R.The shaper samples X from the conditional distribution PX|U′=u′.Source: C. Distribution Shaping · p. 4
  • If the extractor performs privacy amplification against an eavesdropper’s Z, the shaper replicates X while hiding U′ from that eavesdropper.The passage attributes this to the eavesdropper’s relevant conditional states being the same in both cases.Source: C. Distribution Shaping · p. 4
  • The auxiliary randomness is needed because the extractor may map multiple X values to the same U′, making the mapping non-invertible.R lets the shaper select an X consistent with the observed U′.Source: C. Distribution Shaping · p. 4

D. Putting it all together

The construction combines distribution shaping with information reconciliation to build channel codes, then adjusts the resulting protocols for private or public communication. The bounds account for the shaping and reconciliation errors and allow nonuniform inputs.

D. Putting it all together

  • When the channel does not leak to an adversary, the construction specializes to classical capacity, with the additional modifications improving the bound’s constants.The passage also notes fixing a particular output value as an alternative to properly simulating X.Source: D. Putting it all together · pp. 4–5
  • Concatenating a privacy-amplification-based shaper with the channel creates a virtual channel that accepts roughly uniform inputs, to which an information-reconciliation code is applied.The shaper may be included in the encoder rather than the channel.Source: D. Putting it all together · p. 4
  • For private coding, at least half of the code choices meet both average error and leakage bounds after combining shaping and compression.The average error is at most ε1+ε2 and average leakage at most 2ε1 before the Markov-inequality selection.Source: D. Putting it all together · p. 4
  • The average guarantees can be converted to worst-case error and leakage of 8ε each by treating shaping as part of the encoder and applying Lemma 1.The passage sets ε ≡ 4 max(ε1, ε2).Source: D. Putting it all together · p. 4
  • Extractor output does not increase conditional max-entropy, and optimizing the input distribution yields the stated private-capacity lower bound.The construction also allows Alice to optimize a preceding mapping into X.Source: D. Putting it all together · p. 4
  • For public coding, the proof needs only an average-error guarantee, yielding worst-case error 2(ε1+ε2) without the private-code selection step.The encoder can also dispense with the randomness needed for shaping in this setting.Source: D. Putting it all together · p. 4

III. CONVERSE

The converse bounds private communication by applying smooth conditional entropy constraints implied by reliable decoding and secrecy. The classical converse additionally uses the fact that an encoder can be taken deterministic and injective.

III. CONVERSE

  • Reliability and secrecy of a private code imply smooth conditional entropy constraints on the message relative to the receiver’s output and the eavesdropper’s output.The proof uses trace-distance closeness to the ideal message and secrecy outputs, with a square-root conversion to purification distance.Source: §III · p. 5
  • Data processing bounds the message’s conditional max-entropy given the receiver’s output because the decoder generates its guess from that output.The passage cites the data processing inequality.Source: §III · p. 5
  • For classical capacity, the encoder may be taken deterministic and injective, allowing the message distribution to be represented on its image in the channel-input alphabet.A collision would necessarily cause an error; uniform messages have Hmin(M)=n.Source: §III · p. 5

IV. ASYMPTOTIC ANALYSIS

The asymptotic analysis recovers the established public and private communication rates from the one-shot bounds using smooth-entropy AEP arguments. It also identifies when regularization is needed and why private-rate optimization over input maps matters.

IV. ASYMPTOTIC ANALYSIS

  • The AEP recovers the usual memoryless-channel capacity expressions from smooth conditional min- and max-entropy bounds.The converse uses upper bounds and conditional-entropy bounds in terms of conditional von Neumann entropy.Source: §IV · pp. 5–6
  • The paper relates its classical rate to Shannon’s theorem [7] and its private rate to the wiretap-channel result and later extensions,.For quantum channels, the public rate is the HSW theorem [11],, and the private rate was proven by Devetak.Source: §IV · p. 5
  • The asymptotic private-capacity lower bound is max over maps T→X of H(T|Z)−H(T|Y).For the direct proof, the analysis chooses an i.i.d. sequence of T variables and applies the conditional smooth-entropy AEP.Source: §IV · pp. 5–6
  • Typical sequences and type classes are unnecessary in the protocol itself; the proof needs only the AEP statement about the capacity-formula entropies.The passage notes that such methods could establish the AEP, but are not necessary; it cites.Source: §IV · p. 6
  • For classical-output channels, the regularization limit can be removed for private communication, whereas quantum-output channels require regularization for both private and public rates.The public classical-output result is referenced to Theorem 4.2.1 in; the quantum-output results are attributed to and.Source: §IV · p. 6
  • A randomized encoding achieves a 1-bit private rate in the example, while the rate without random encoding is less than 1 bit.The example shows why optimizing the map T→X is generally necessary for the optimal private rate.Source: §IV · p. 6

V. CONCLUSIONS

The paper builds channel-coding protocols from privacy amplification and information reconciliation, rather than reusing only their proof techniques. In quantum information, duality between the primitives allows a decoupling-based construction that recovers familiar asymptotic capacity results.

V. CONCLUSIONS

  • The construction turns privacy amplification and information reconciliation into channel-coding protocols, with optimal primitives yielding optimal channel-coding protocols.The authors present this as a framework for building more complicated two-terminal protocols from simpler ones.Source: §V · p. 6
  • In quantum information, duality between the two primitives means that one primitive can suffice to construct more complicated protocols.The paper links this duality to the duality of smooth min- and max-entropies.Source: §V · p. 6
  • A privacy-amplification construction can provide the linear compressor/decompressor needed in the channel-coding proof, using the uncertainty principle and the duality result.The compressed output size is given in terms of the smooth min-entropy of the conjugate basis and the error parameters.Source: §V · p. 6
  • The resulting public- and private-capacity proofs for quantum channels can use decoupling of the purifying system instead of directly constructing a receiver decoder.The authors state that the resulting capacity is no smaller than the expression in Eq. 9 up to O(log 1/ε) terms.Source: §V · p. 6
  • In the asymptotic limit of many independent channel uses, the approach recovers the HSW, and private-capacity results.The paper describes this as the classical-quantum analogue of deriving quantum capacity using decoupling.Source: §V · p. 7

APPENDIX

The appendix reviews smooth-entropy tools and results for information reconciliation, then uses entropy bounds and conditional Markov structure in its private-capacity argument.

APPENDIX

  • Information reconciliation concerns compressing a classical variable using side information at the decoder while keeping recovery error below a target.The appendix identifies an optimal one-shot result for this task and introduces a classical-quantum information-reconciliation theorem.Source: APPENDIX · p. 7
  • For a linear function, every output has exactly |X|/|Y| preimages; for an arbitrary function, a random output has at least ε|X|/|Y| preimages with probability at least 1−ε.The arbitrary-function bound is derived using min-entropy and Markov's inequality.Source: APPENDIX · p. 7
  • Smooth conditional min- and max-entropies are bounded using continuity of conditional von Neumann entropy, with a correction depending on ε and dim(A).The min-entropy argument uses a nearby state attaining the smoothed value and the bound between purification and trace distance.Source: APPENDIX · pp. 7–8
  • The private-capacity proof reduces a sum of terms to a single-letter optimization using the Markov condition (T,V) ↔ X ↔ (Y,Z).The terms use V_i = Z_1…Z_{i−1}Y_{i+1}…Y_ℓ, and conditioning on V preserves the Markov condition.Source: APPENDIX · p. 8
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