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Convexity and Operational Interpretation of the Quantum Information Bottleneck Function

Nilanjana Datta, Christoph Hirche, Andreas Winter

arXiv:1810.03644v3quant-phcs.ITmath-phmath.ITmath.MP

TL;DR

The paper addresses the conjectured convexity and entropic-task formulation underlying an earlier operational interpretation of the quantum IB function. It proves convexity and gives an interpretation through quantum source coding with side information at the decoder, while noting a limitation on a similar interpretation for the privacy funnel.

  • Problem

    An earlier operational interpretation of the quantum IB function relied on a convexity assumption and included a quantum mutual-information constraint in the task definition.

    Source: §II · p. 2

  • Method

    The paper proves convexity and relates the quantum IB function to the rate region of source coding with quantum side information at the decoder.

    Source: §VII · pp. 6–7

  • Results

    The paper establishes convexity of the quantum IB function and provides an alternative operational interpretation through source coding with side information at the decoder.

    Source: §VII · pp. 6–7

  • Takeaways & Limitations

    The paper concludes that the quantum privacy funnel is unlikely to characterize an optimal asymptotic rate because its classical version lacks an essential additivity property.

    Source: §V · p. 5

    Whether entanglement is necessary in the source-coding task remains open, as does removing the exponentially limited entanglement requirement in the converse.

    Source: §VII · p. 7

Abstract

from arXiv · show

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.

I. INTRODUCTION

The classical information bottleneck treats compression of X as preserving information relevant to Y. Its rate function describes the trade-off between compression and a lower bound on the information retained about Y.

I. INTRODUCTION

  • The information bottleneck compresses X into W while retaining as much information about Y as possible.The paper frames this as lossy data compression with side information Y.Source: §I · p. 1
  • The IB method generalizes rate-distortion theory by determining distortion between X and W from the joint distribution pXY.The paper notes applications including neural networks, video processing, clustering, and polar coding.Source: §I · p. 1
  • The IB constraint requires I(Y;W) to meet a lower bound, and the IB function gives the corresponding rate.I(Y;W) measures the information about Y contained in W.Source: §I · p. 1
  • A related function F(a), defined by minimizing H(Y|W) subject to H(X|W)≥a, is convex; this implies IY(R) is concave and R(IY) is convex.The paper states that IY(a)=H(Y)−F(H(X)−a).Source: §I · pp. 1–2

Operational interpretation of the classical IB function

The classical IB function has an operational interpretation through the Wyner-Ahlswede-Körner problem: source coding with side information at the decoder. The function gives the minimum rate for encoding Y under a rate constraint on a helper's encoding of X.

Operational interpretation of the classical IB function

II. QUANTUM INFORMATION BOTTLENECK

The quantum information bottleneck optimizes compression of X to a quantum memory while retaining information about Y under a constraint on the encoded information. Prior operational interpretation depended on a convexity conjecture and used an entropic constraint in the task definition; the paper addresses both concerns.

II. QUANTUM INFORMATION BOTTLENECK

  • The quantum IB compresses X into a memory M, maximizing I(M;Y) subject to an upper bound on I(M;R).R purifies the input state, and Y is produced by sending R through a quantum channel.Source: §II · p. 2
  • Salek et al. interpreted the quantum IB function as an optimal asymptotic rate, assuming its convexity.Their task was constrained entanglement-assisted lossy compression using a classical channel.Source: §II · p. 2
  • The paper proves the conjectured convexity and offers an alternative operational interpretation to avoid an entropic constraint in the task definition.The authors describe such a constraint as making the earlier task somewhat contrived.Source: §II · p. 2

III. CONVEXITY OF THE QIB FUNCTION

The quantum IB function is convex. The proof uses an equivalent representation and a flagged channel that combines optimizing channels, completing a convexity conjecture relevant to its operational interpretation.

III. CONVEXITY OF THE QIB FUNCTION

  • An equivalent representation expresses the objective and constraint using information quantities of the same tripartite state.The proof uses a purification of ρXY and invariance of mutual information under isometries.Source: §III · p. 3
  • The quantum IB function Rq(a) is convex for all λ∈[0,1] and a0,a1≥0.The theorem establishes the convexity conjectured in.Source: §III · pp. 2–3
  • A flagged channel combines the optimizing channels for a0 and a1 to establish the convexity bound.The flag is a qubit, and the resulting state is block diagonal.Source: §III · p. 3
  • The convexity result completes the proof of the operational interpretation proposed in and is also of independent interest.The paper states both consequences of the result.Source: §III · pp. 2–3

IV. OPERATIONAL INTERPRETATION

The quantum IB function characterizes the achievable rate region for quantum source coding with quantum side information at the decoder, a bona fide information-theoretic task.

  • The quantum IB function precisely characterizes the achievable rate region for quantum source coding with quantum side information at the decoder.Source: §IV · p. 3
  • The operational interpretation uses a bona fide information-theoretic task: quantum source coding with quantum side information at the decoder.Source: §IV · p. 3
  • The paper relates the quantum IB function to the rate region of this task.Source: §IV · p. 3

The task: quantum Wyner-Ahlswede-Körner problem

The task generalizes the WAK problem to quantum source coding with side information at the decoder: a helper and an encoder send separate qubit messages to a decoder, who reconstructs the target state with high fidelity. Its achievable rate pairs form a closed, convex region characterized by Theorem 2.

The task: quantum Wyner-Ahlswede-Körner problem

  • Alice, the helper, and Bob, the encoder, send separate qubit messages to Charlie, the decoder, who aims to reconstruct the target state with high fidelity.Alice and Charlie may share prior entanglement; the rates are QX and QY for their respective messages.Source: The task: quantum… · p. 3
  • The protocol encodes the X and Y systems separately and has Charlie decode the transmitted systems together with the shared entanglement.Source: The task: quantum… · p. 3
  • Achievability requires encoding and decoding maps whose reconstruction error vanishes in the asymptotic limit.Source: The task: quantum… · p. 4
  • The achievable rate pairs form a closed, convex region extending above and to the right, described by a convex, monotonically non-increasing lower boundary.Source: The task: quantum… · p. 4, Figure 3 · p. 4
  • Theorem 2 gives an if-and-only-if characterization of which rate pairs are achievable for a source state and its purification.Source: The task: quantum… · p. 4
  • The achievability proof combines the Quantum Reverse Shannon Theorem, or state splitting, with Fully quantum Slepian-Wolf, or coherent state merging.Source: The task: quantum… · p. 4

V. PRIVACY FUNNEL

The paper defines classical and quantum privacy funnel functions and proves both are convex. It notes that an asymptotic operational interpretation is doubtful because the classical function lacks the relevant behavior under block extension, and conjectures a similar issue for the quantum function.

V. PRIVACY FUNNEL

  • The privacy funnel minimizes private information leaked while preserving enough public information for an analyst.Source: §V · p. 4
  • Both the classical and quantum privacy funnel functions are convex.The quantum proof follows the approach used for Theorem 1; the classical case was previously proven in.Source: §V · p. 4
  • The authors consider an asymptotic operational interpretation unlikely because the classical privacy funnel's behavior differs from that of its n-copy counterpart.The finite classical function is typically positive near t = 0, whereas the n-copy function does not share this behavior as n tends to infinity.Source: §V · p. 5
  • The proof that G(∞)(H(X∣Y)) = 0 uses privacy amplification by random hashing to extract a key from X^n with negligible information about Y^n.The extractable rate converges to H(X∣Y).Source: §V · p. 5
  • The authors suspect that the quantum privacy funnel has the same non-additivity issue, while leaving its analysis for another occasion.Source: §V · p. 5

VI. NUMERICS AND EXAMPLES

The examples examine how output dimension affects the information bottleneck, compare normalized bottleneck and privacy funnel functions, and use a classical case to check the numerical algorithm. They also show that dimension effects vary by example, while a general quantum bound on the required output dimension remains unknown.

VI. NUMERICS AND EXAMPLES

  • A general quantum bound on the output dimension needed to reach the optimum is unknown, limiting evaluation and practical application.Classically, |W| ≤ |X| + 2 suffices; finding a quantum analogue is an open problem.Source: §VI · p. 5
  • For ρ(3), larger W dimensions achieve lower values than |W|=2, but the example suggests no gain beyond |W|=3.The |W|=3 and |W|=4 curves appear identical for p=0.2 and p=0.4.Source: §VI · pp. 5–6
  • For ρ(2), no advantage appears through |W|=16, and the nondifferentiable point remains; larger dimensions may still matter.The authors suggest the output dimension might need to be larger still.Source: §VI · p. 6
  • For a pure state, the normalized IB and privacy funnel functions coincide: Rq(a) = Pq(a) = a.The section also plots normalized IB and privacy funnel functions for ρ(3) with p=0.4, using solid |W|=3 and dashed |W|=2 curves.Source: §VI · p. 6
  • The classical binary example provides an analytical IB function against which the numerically computed quantum function verifies the algorithm.The algorithm was adapted from, with the fixed |W|=2 restriction removed and privacy-funnel evaluation added.Source: §VI · pp. 5–6
  • For classical states, Rq(a) ≤ 1/2; the observed Rq(1) = 1/2 reflects the normalization and the availability of classical W, not teleportation.The same bound applies to quantum inputs when W is restricted to be classical.Source: §VI · p. 6

VII. CONCLUSIONS

The paper establishes convexity of the quantum information bottleneck function and gives it an operational interpretation through quantum source coding with side information at the decoder. It closes by identifying open questions about entanglement, channel output dimension, and applications.

VII. CONCLUSIONS

  • The quantum information bottleneck function is shown to be convex and interpreted through source coding with side information at the decoder.The authors also give an alternative formulation for further investigation.Source: §VII · pp. 6–7
  • Figure 7 plots normalized classical and quantum IB functions for a binary symmetric channel with δ = 0.9, using yellow and black, respectively.Compare the curve shapes; the caption does not state an outcome.Source: §VII · p. 7
  • Whether entanglement is necessary for the source-coding task, and whether the converse can avoid exponentially limited entanglement, remain open questions.Source: §VII · p. 7
  • A quantum analogue of the classical output-dimension bound |W|≤|X|+2 remains to be found, which would aid evaluation and practical application.The bound stated for the classical case is attributed to.Source: §VII · p. 7
  • The authors identify translating classical IB applications to quantum settings and understanding quantum generalizations of information-combining bounds as future directions.Source: §VII · p. 7

APPENDIX

The appendix proves the rate-region characterization by establishing achievability and converse bounds, then single-letterizing them using additivity. In particular, the achievable rate region for a product of independent sources is the Minkowski sum of their individual regions.

Achievability

  • The achievability proof simulates Alice’s encoding channel with QRST, then uses FQSW to transmit Bob’s side information to Charlie; minimizing over encoders gives the stated achievable rate.The respective transmission rates are 1/2 I(W;RY)ω and 1/2 I(Y;RV)σ.Source: APPENDIX · p. 8

Converse/Optimality

  • The converse assumes noiseless quantum channels to Charlie and decomposes Alice’s operation into sharing entanglement and applying an encoding map.The protocol must deliver the specified systems to Charlie at rates QX and QY.Source: APPENDIX · p. 8
  • The converse uses decoupling of Bob’s discarded system from RnV, inferred from the fidelity criterion, to derive rate bounds with vanishing error.The continuity bound is made arbitrarily small for sufficiently large n.Source: APPENDIX · p. 9

Additivity and single-letterization

  • The proof establishes additivity of the rate-region set T(ψ), yielding single-letter rate bounds for n copies of the source.The additivity argument bounds the rates for the two component sources and combines the bounds.Source: APPENDIX · pp. 9–10
  • For two independent sources, the rate region of their product is the Minkowski sum of their individual rate regions.This is the stated implication of the additivity relation for T(ψ).Source: APPENDIX · p. 10

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