Dependence Balance and Capacity Bounds for Multiterminal Communication and Wiretap Channels

Amin Gohari; Gerhard Kramer

arxiv: 2009.14814 · v4 · submitted 2020-09-30 · 💻 cs.IT · math.IT

Dependence Balance and Capacity Bounds for Multiterminal Communication and Wiretap Channels

Amin Gohari , Gerhard Kramer This is my paper

Pith reviewed 2026-05-24 14:55 UTC · model grok-4.3

classification 💻 cs.IT math.IT

keywords dependence balancefractional partitionmultiterminal channelsrelay channelwiretap channelshared randomnesscut-set boundGaussian channels

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The pith

A dependence balance inequality derived from fractional partitions tightens upper bounds on shared randomness and secret rates in multiterminal channels.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a general dependence balance inequality using an information measure based on fractional partitions of a set. This inequality produces new upper bounds on the rates of reliable communication and secret communication over multiterminal channels, including a bound on shared randomness generation that serves as a counterpart to the cut-set bound. The authors introduce auxiliary receivers to obtain these bounds and prove that Gaussian distributions optimize them for Gaussian channels. The resulting bounds are applied to multiaccess channels with generalized feedback and to relay channels, where they improve upon the cut-set bound for scalar Gaussian cases.

Core claim

An information measure based on fractional partitions yields a dependence balance inequality that upper-bounds the rate of shared randomness among terminals; when combined with auxiliary receivers, the same inequality supplies improved capacity upper bounds for reliable and wiretap multiterminal channels, and these bounds are achieved by Gaussian inputs on Gaussian channels.

What carries the argument

The fractional-partition information measure, which produces a dependence balance inequality that constrains mutual information terms across partitions of the terminal set and is tightened by the introduction of auxiliary receivers.

If this is right

The cut-set bound is strictly improved for scalar Gaussian relay channels.
Gaussian inputs optimize the new bounds on Gaussian multiaccess channels with feedback and on Gaussian relay channels.
The same dependence balance inequality supplies upper bounds on secret rates for wiretap versions of the same multiterminal channels.
Auxiliary receivers can be chosen to recover known single-letter bounds as special cases while sometimes yielding stricter results.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may extend to continuous-time or non-Gaussian channels if the fractional-partition measure can be defined for those alphabets.
If the bound is tight for certain relay networks, it would imply that shared randomness cannot exceed a computable fraction of the cut-set value.
The auxiliary-receiver technique could be combined with other outer-bound methods such as genie-aided arguments to obtain further improvements.

Load-bearing premise

The inequality obtained from the fractional-partition information measure remains valid and produces nontrivial bounds once auxiliary receivers are added to the model.

What would settle it

An explicit achievable rate tuple for a scalar Gaussian relay channel that exceeds the new upper bound derived from the dependence balance inequality.

read the original abstract

An information measure based on fractional partitions of a set is used to derive a general dependence balance inequality for communication. This inequality is used to obtain new upper bounds on reliable and secret rates for multiterminal channels. For example, we obtain a new upper bound on the rate of shared randomness generated among terminals, a counterpart of the cut-set bound for reliable communication. The bounds for reliable communication use the concept of auxiliary receivers, and we show that they are optimized by Gaussian distributions for Gaussian channels. The bounds are applied to multiaccess channels with generalized feedback and relay channels, and improve the cut-set bound for scalar Gaussian channels. The improvement for Gaussian relay channels complements results obtained with other methods.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper introduces a fractional-partition dependence balance inequality that bounds shared randomness and improves cut-set bounds on Gaussian relay channels via auxiliary receivers, though the extension step needs explicit verification.

read the letter

The main takeaway is that this paper derives a dependence balance inequality from fractional partitions of the terminal set and uses it to bound the rate of shared randomness among terminals, then applies auxiliary receivers to obtain tighter upper bounds on reliable rates for multiaccess channels with generalized feedback and relay channels. It shows these bounds are optimized by Gaussian distributions on Gaussian channels and reports an improvement over the cut-set bound for scalar Gaussian relay channels. This is presented as distinct from prior cut-set and other bounding methods. The fractional-partition approach and the auxiliary-receiver construction are the concrete new elements, and the Gaussian optimality result for the Gaussian cases is a clear positive. The work does well in giving a general inequality that acts as a counterpart to cut-set for shared randomness and in applying it to specific models where it yields measurable improvement. The applications complement results from other techniques on relay channels, which adds usefulness. The soft spot is the extension of the inequality to auxiliary receivers. The derivation starts from the original channel outputs and partitions of terminals, but it is not automatic that the same fractional functional remains valid and produces a strictly tighter bound once auxiliary outputs are added to the joint distribution. The abstract gives no explicit conditions for this step, so the claimed improvements rest on whether the proof carries over without extra restrictions or looseness. If that part does not hold tightly, the new bounds may collapse to existing ones. The paper is aimed at specialists in multiterminal information theory who work on capacity bounds for networks with feedback or relays. Readers who track new bounding techniques and cut-set refinements will get value from the inequality and the Gaussian results. It shows clear engagement with the literature by building on cut-set ideas rather than ignoring them. I recommend sending it for peer review because the technique is original enough and the applications are concrete enough to justify referee time, even if the auxiliary extension requires careful checking.

Referee Report

2 major / 1 minor

Summary. The paper introduces an information measure based on fractional partitions of a terminal set to derive a general dependence-balance inequality. This inequality yields new upper bounds on reliable communication rates and secret rates for multiterminal channels and wiretap channels. A new upper bound is obtained on the rate of shared randomness generated among terminals. Reliable-communication bounds are derived via auxiliary receivers and shown to be optimized by Gaussian distributions on Gaussian channels. The bounds are applied to multiple-access channels with generalized feedback and relay channels, improving the cut-set bound for scalar Gaussian channels.

Significance. If the dependence-balance inequality extends rigorously to auxiliary-receiver constructions and the Gaussian-optimality claim holds, the work supplies strictly tighter outer bounds than the cut-set bound for Gaussian relay and MAC-with-feedback channels. Such improvements are load-bearing for capacity results in multiterminal information theory and complement existing techniques for Gaussian networks.

major comments (2)

[Sections deriving the auxiliary-receiver bounds (likely §4–5)] The central dependence-balance inequality is stated for the original terminal outputs. Its extension to the joint distribution that includes auxiliary-receiver outputs (used to obtain the new bounds) is not automatically guaranteed by the same proof steps; an explicit verification that the fractional-partition functional remains valid under this augmentation is required, otherwise the claimed improvement over the cut-set bound may not hold.
[Gaussian-channel optimality argument] Gaussian optimality is asserted for the auxiliary-receiver bounds on Gaussian channels, but the proof must confirm that the fractional-partition functional preserves the extremal property under the same covariance constraints used for the cut-set bound; otherwise the improvement may be an artifact of the auxiliary construction rather than a genuine tightening.

minor comments (1)

[Abstract] The abstract states that the bounds 'improve the cut-set bound for scalar Gaussian channels' but does not quantify the improvement or identify the specific parameter regime (e.g., SNR range) where the gain appears.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments on our manuscript. We address each major comment below and will make the necessary revisions to strengthen the presentation.

read point-by-point responses

Referee: [Sections deriving the auxiliary-receiver bounds (likely §4–5)] The central dependence-balance inequality is stated for the original terminal outputs. Its extension to the joint distribution that includes auxiliary-receiver outputs (used to obtain the new bounds) is not automatically guaranteed by the same proof steps; an explicit verification that the fractional-partition functional remains valid under this augmentation is required, otherwise the claimed improvement over the cut-set bound may not hold.

Authors: The dependence balance inequality is derived for an arbitrary joint distribution over the terminal outputs, and the auxiliary receivers are incorporated by augmenting the terminal set with these additional outputs. The fractional partition is defined on this extended set. Nevertheless, to address the referee's concern and ensure the argument is self-contained, we will include an explicit verification of the inequality under this augmentation in the revised version of the manuscript. revision: yes
Referee: [Gaussian-channel optimality argument] Gaussian optimality is asserted for the auxiliary-receiver bounds on Gaussian channels, but the proof must confirm that the fractional-partition functional preserves the extremal property under the same covariance constraints used for the cut-set bound; otherwise the improvement may be an artifact of the auxiliary construction rather than a genuine tightening.

Authors: In the manuscript, the Gaussian optimality follows from the fact that the mutual information terms and the dependence measure are optimized by jointly Gaussian distributions under the given covariance constraints, as is standard for Gaussian channels. The fractional-partition functional is a linear combination of such terms, hence inherits the same optimality. We will add a more detailed argument in the revision to explicitly confirm that the extremal property is preserved for the augmented functional. revision: yes

Circularity Check

0 steps flagged

No circularity: new fractional-partition measure yields independent dependence-balance inequality

full rationale

The derivation begins with a newly defined information measure on fractional partitions of the terminal set and produces a dependence-balance inequality that is then applied to obtain upper bounds. This chain is presented as a forward derivation from the new measure rather than a fit, renaming, or self-citation reduction. Auxiliary-receiver constructions are introduced after the inequality and the Gaussian optimality claim is shown by direct substitution into the resulting expressions for Gaussian channels. No equation is shown to equal its own input by construction, and no load-bearing step collapses to a prior self-citation or fitted parameter. The paper is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.0 · 5635 in / 1069 out tokens · 19000 ms · 2026-05-24T14:55:13.502068+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 17 canonical work pages

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Secrecy capacities for multiterminal channel models,

I. Csisz´ ar and P. Narayan, “Secrecy capacities for multiterminal channel models,” IEEE Trans- actions on Information Theory , vol. 54, no. 6, pp. 2437–2452, 2008

work page 2008

[2] [2]

Secrecy generation for multiaccess channel models ,

——, “Secrecy generation for multiaccess channel models ,” IEEE transactions on information theory, vol. 59, no. 1, pp. 17–31, 2012

work page 2012

[3] [3]

Secret key capacity for multip leaccess channel with public feed- back,

H. Tyagi and S. Watanabe, “Secret key capacity for multip leaccess channel with public feed- back,” in 2013 51st Annual Allerton Conference on Communication, Contro l, and Computing (Allerton). IEEE, 2013, pp. 1–7. 8

work page 2013

[4] [4]

Wiretap channel with secure rate-limited feedback,

E. Ardestanizadeh, M. Franceschetti, T. Javidi, and Y.- H. Kim, “Wiretap channel with secure rate-limited feedback,” IEEE Transactions on Information Theory , vol. 55, no. 12, pp. 5353– 5361, 2009

work page 2009

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The general gaussian multiple-ac cess and two-way wiretap chan- nels: Achievable rates and cooperative jamming,

E. Tekin and A. Yener, “The general gaussian multiple-ac cess and two-way wiretap chan- nels: Achievable rates and cooperative jamming,” IEEE Transactions on Information Theory , vol. 54, no. 6, pp. 2735–2751, 2008

work page 2008

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The wiretap channel wi th feedback: Encryption over the channel,

L. Lai, H. El Gamal, and H. V. Poor, “The wiretap channel wi th feedback: Encryption over the channel,” IEEE Transactions on Information Theory , vol. 54, no. 11, pp. 5059–5067, 2008

work page 2008

[7] [7]

Strongly secure communica tions over the two-way wiretap channel,

A. J. Pierrot and M. R. Bloch, “Strongly secure communica tions over the two-way wiretap channel,” IEEE Transactions on Information Forensics and Security , vol. 6, no. 3, pp. 595– 605, 2011

work page 2011

[8] [8]

Achievable secrecy rate regions for the two-way wiretap channel,

A. El Gamal, O. O. Koyluoglu, M. Youssef, and H. El Gamal, “ Achievable secrecy rate regions for the two-way wiretap channel,” IEEE Transactions on Information Theory , vol. 59, no. 12, pp. 8099–8114, 2013

work page 2013

[9] [9]

The role of feedback in two-way secure communications,

X. He and A. Yener, “The role of feedback in two-way secure communications,” IEEE Trans- actions on Information Theory , vol. 59, no. 12, pp. 8115–8130, 2013

work page 2013

[10] [10]

A channel model of transceivers for multiterminal secret key agreement,

A. Poostindouz and R. Safavi-Naini, “A channel model of transceivers for multiterminal secret key agreement,” arXiv:2008.02977

work page arXiv 2008

[11] [11]

Information-theoretic k ey agreement of multiple terminalspart i,

A. Gohari and V. Anantharam, “Information-theoretic k ey agreement of multiple terminalspart i,” IEEE Transactions on Information Theory , vol. 56, no. 8, pp. 3973–3996, 2010

work page 2010

[12] [12]

Information-theoretic key agreement of multiple terminalspart ii: Channel model,

——, “Information-theoretic key agreement of multiple terminalspart ii: Channel model,” IEEE Transactions on Information Theory , vol. 56, no. 8, pp. 3997–4010, 2010

work page 2010

[13] [13]

Multiterminal secret key agreeme nt,

C. Chan and L. Zheng, “Multiterminal secret key agreeme nt,” IEEE transactions on informa- tion theory, vol. 60, no. 6, pp. 3379–3412, 2014

work page 2014

[14] [14]

Outer bounds for multiuser setti ngs: the auxiliary receiver approach

A. Gohari and C. Nair, “Outer bounds for multiuser setti ngs: the auxiliary receiver approach.” [Online]. Available: http://chandra.ie.cuhk.edu.hk/pu b/papers/NIT/Auxiliary-Receiver.pdf

work page

[15] [15]

Multivariate mutual information inspired by secret-key agreement,

C. Chan, A. Al-Bashabsheh, J. B. Ebrahimi, T. Kaced, and T. Liu, “Multivariate mutual information inspired by secret-key agreement,” Proceedings of the IEEE , vol. 103, no. 10, pp. 1883–1913, 2015

work page 1913

[16] [16]

Dependence balance and the ga ussian multiaccess channel with feedback,

G. Kramer and M. Gastpar, “Dependence balance and the ga ussian multiaccess channel with feedback,” in 2006 IEEE Information Theory Workshop-ITW’06 Punta del Este . IEEE, 2006, pp. 198–202

work page 2006

[17] [17]

Dependence balance based oute r bounds for gaussian networks with cooperation and feedback,

R. Tandon and S. Ulukus, “Dependence balance based oute r bounds for gaussian networks with cooperation and feedback,” IEEE Transactions on Information Theory , vol. 57, no. 7, pp. 4063–4086, 2011. 9

work page 2011