Capacity Characterization of Pinching-Antenna Systems

Chongjun Ouyang; Yuanwei Liu; Zhaolin Wang; Zhiguo Ding

REVIEW 3 major objections 5 minor 44 references

Capacity Characterization of Pinching-Antenna Systems

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For two users sharing one pinched waveguide, the paper derives the exact capacity region in closed form for the single-pinch case and proves that as the number of pinches grows, the entire capacity region—uplink and downlink—collapses to…

desk verdict First real capacity-region result for pinching antennas; the single-pinch part is solid, but the asymptotic collapse to zero rests on a per-antenna noise model that the paper never justifies. read the letter →

arxiv 2506.14298 v1 pith:7G7MLBU7 submitted 2025-06-17 cs.IT eess.SPmath.IT

Chongjun Ouyang , Zhaolin Wang , Yuanwei Liu , Zhiguo Ding This is my paper

classification cs.ITeess.SPmath.IT MSC 94A1594A05

keywords pinchingantennascapacityregionrate-profilemethoduplink-downlinkdualityTDMAFDMAarraygainscalingline-of-sightchannels

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks what a pinching-antenna waveguide—a dielectric line with small radiating elements clipped at chosen positions—can do for two users sharing one base station, and answers at the level of exact capacity regions. For the single-pinch uplink it derives the capacity region in closed form: the optimal element position always lies on the interval between the two users' projections onto the waveguide, and every frontier rate pair comes from balancing the two users' rate profiles. That region provably contains the capacity region of a conventional fixed-antenna system, and the FDMA rate region contains the TDMA rate region. For multiple pinches the paper supplies an inner bound by element-wise position optimization, an outer bound by the Cauchy–Schwarz inequality, and a proof that as the number of pinches grows, the whole capacity region shrinks to the origin because per-element noise accumulates linearly while the best achievable array gain only grows like $\mathcal{O}((\ln N)^2/N)$. The same statements carry to the downlink through uplink-downlink duality.

What carries the argument

The machinery is the rate-profile method applied to a position-tunable channel. For each decoding order $\pi$ and profile factor $\alpha\in[0,1]$, one maximizes the sum rate $R$ subject to one user receiving $\alpha R$ and the other $(1-\alpha)R$; the union over $\alpha$ and $\pi$, convex-hulled, is the capacity region. For $N=1$ the effective channel satisfies $\lvert h_k(q_1)\rvert = \sqrt{\eta}/\sqrt{d_k^2+(x_k-q_1)^2}$, which reduces the optimization to one dimension and lets the optimum be found in closed form by locating where the two weighted rate functions cross. For $N>1$, the same rate-profile problem is attacked element-wise for the inner bound, while the outer bound uses the Cauchy–Schwarz inequality $\lvert h_k(q)\rvert^2 \le \frac{1}{N}\sum_{n=1}^N \eta/(d_k^2+(x_k-q_n)^2)$, whose best concentration over a fixed aperture gives the array-gain scaling $\mathcal{O}((\ln N)^2/N)$. The $1/\sqrt{N}$ in the effective channel and the $N$ independent noise terms together carry the asymptotic collapse.

What would settle it

A measurement campaign on one dielectric waveguide with $N$ active pinches, a fixed feed, and two fixed users would settle the claim: record per-user signal-to-noise ratio for $N=10,100,500$ at fixed transmit power. Under the paper's per-pinch noise model the SNR should decay at least as fast as $(\ln N)^2/N$; if it instead stabilizes or grows, as would happen with a single dominant noise source at the feed, then the predicted collapse in Eqs. (54)-(55) does not describe that hardware.

Watch

Extended reading notes

Core claim

The central claim is a complete information-theoretic characterization of a two-user pinching-antenna channel. With $N=1$, the uplink capacity region is $C^U_S = \operatorname{Conv}\bigl(\bigcup_{\alpha,\pi} C^U_\pi(q^\alpha_\pi)\bigr)$ in (23), and the optimizing antenna position $q^\alpha_\pi$ is either one of the two users' projected coordinates or the point between them where $f^{(2)}_\pi(x)/\alpha = f^{(1)}_\pi(x)/(1-\alpha)$; no position outside $[x_1,x_2]$ is ever needed. The paper proves the inclusion chain $R^U_{S,T}\subseteq R^U_{S,F}\subseteq C^U_S$, that the fixed-antenna pentagon is contained in $C^U_S$, and that the same relations hold in the downlink via duality. For $N>1$ it establishes $C^U_{M,\mathrm{IB}}\subseteq C^U_M\subseteq C^U_{M,\mathrm{OB}}$ and, more boldly, that $\lim_{N\to\infty} C^U_M = \{(0,0)\}$ (and likewise for the TDMA/FDMA regions and the downlink), because the per-user array gain is bounded by $\mathcal{O}((\ln N)^2/N)$. Hence there is an optimal finite number of pinching antennas rather than a monotone benefit from adding elements.

Load-bearing premise

The collapse to zero as $N$ grows depends on the model in Eq. (3a) that each pinching antenna contributes its own independent Gaussian noise of power $\sigma^2$, so the combined noise power is $N\sigma^2$; if a single receiver noise at the waveguide feed dominated instead, the capacity region would not be forced to zero by the $1/N$ channel scaling.

Editorial extensions

If this is right

A single activated pinch placed on the segment between the two users' projections achieves the full uplink capacity region, so deployment rules can be stated without exhaustive search.
Because the fixed-antenna capacity region is nested inside the PASS region in both uplink and downlink, reconfigurable pinching does not sacrifice any rate pair that a conventional antenna achieves.
FDMA dominates TDMA, and in the multiple-pinch case both nearly fill the capacity region, meaning orthogonal access can approach the information-theoretic limit without successive interference cancellation or dirty-paper coding.
The capacity region collapses to $\{(0,0)\}$ as $N\to\infty$, so system design must stop at an optimal antenna count rather than deploy as many pinches as possible.
Downlink capacity and rate regions inherit the same inclusions and the same collapse through duality, so the conclusions transfer without separate optimization.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A consequence the paper does not draw: if the receiver noise were a single noise source at the waveguide feed rather than independent noise at every pinch, the $1/N$ normalization would not be offset by growing noise, and the capacity region could grow with $N$ instead of collapsing; the optimal-finite-$N$ conclusion is therefore tied to the per-element noise model.
The closed-form single-pinch solution relies on the crossing of two monotone rate functions, so extending the rate-profile argument to more than two users by enumerating decoding orders will likely need a multidimensional fixed-point or iterative step rather than a one-dimensional bracket.
The near-triangular multiple-pinch region suggests that the practical role of many pinches is to equalize the two users' effective channel gains; a testable design heuristic would be to maximize the minimum of the two per-user SNRs rather than the sum rate.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: this is a genuine first, not just another beamforming paper. It gives an actual capacity region for a two-user pinching-antenna MAC/BC, and the single-pinch part holds up. Lemma 2, Theorems 1 and 2, and the inclusion chain R_U_S,T ⊆ R_U_S,F ⊆ C_U_S are correct as far as I can tell. The result that PASS contains the fixed-antenna capacity region is clean, and the FDMA-over-TDMA conclusion is plausible and supported by the numerics.

Where I would push back: the multiple-pinch material is heavier. The inner bound is an alternating-optimization heuristic with no global optimality guarantee, which is fine as an inner bound, but the text sometimes treats it as the capacity region. The outer bound is loose by construction. The claims that the bounds are tight, that the region is nearly triangular, and that OMA approaches capacity are numerical observations, not theorems. Theorem 3's proof is also a sketch: Appendix C says it is 'straightforward to show' and points to a figure. I believe the claim, but a referee should ask for a real argument.

The bigger issue is the noise model. Equation (3a) places independent Gaussian noise at each passive pinching antenna, so the combined noise grows as Nσ² while the signal is normalized by 1/√N. Physically, pinching elements are dielectric particles; the dominant noise should be added at the waveguide feed. Under a single-feed noise model, the 1/N normalization disappears, and the O((ln N)^2/N) array gain from Lemma 3 would not drive rates to zero. The paper gives no physical justification for per-element noise, and the asymptotic collapse in Eq. (54), plus the optimal-N design rule in Remark 6, depends entirely on that choice. There is also an internal inconsistency: Eq. (12a) uses σ² in the denominator, but Eq. (3a) implies the combined noise variance is Nσ². This is load-bearing, not cosmetic.

The citation pattern looks fine. The authors rely on their own [21] and [24] for the array-gain scaling, but those are independently derived results, and the core single-pinch capacity derivation does not depend on them.

Verdict: this deserves a serious referee. The single-pinch results alone are publishable. The multi-pinch asymptotic claim should be revised or explicitly relabeled as an artifact of the per-antenna noise assumption, and the OMA near-capacity claim should be softened or proven. I would send it to review with a request to fix the noise model before publication.

Referee Report

3 major / 5 minor

Summary. The paper considers a two-user single-waveguide pinching-antenna system (PASS) and characterizes uplink and downlink capacity/rate regions. For a single activated pinching antenna, it derives closed-form expressions for the optimal antenna position, the exact uplink capacity region, and the TDMA/FDMA achievable rate regions, and proves the chain C_U_f ⊆ C_U_S and R_U_S,T ⊆ R_U_S,F ⊆ C_U_S. For multiple pinching antennas, it gives an alternating-optimization inner bound and a Cauchy-Schwarz outer bound on the capacity region, analogous TDMA/FDMA regions, and an asymptotic analysis concluding that the capacity region shrinks to {(0,0)} as N → ∞, implying an optimal finite number of antennas. The downlink results are obtained by applying uplink-downlink duality.

Significance. The single-pinch capacity-region characterization is a genuine and useful contribution: the rate-profile method is applied carefully, the optimal-position structure q1 ∈ [x1,x2] is plausible and proved, and the inclusion results are the first of their kind for PASS. The paper also provides machine-checkable formulas and clear numerical validation for those parts. The multiple-pinch inner bound is valid as an inner bound, and the duality extension is standard. However, the asymptotic collapse and the 'optimal N' design rule depend on a specific per-antenna independent noise model and on an N→∞ limit that is not consistent with the finite-waveguide constraint as stated; these issues must be resolved before the broader claims can be accepted.

major comments (3)

[II-A, Eq. (3a); III-D4, Eqs. (54)-(55), Remark 6] The model in Eq. (3a) places independent Gaussian noise z_n^U ∼ CN(0, σ²) at every pinching antenna, so the combined noise power is Nσ². Since pinching antennas are passive dielectric particles and the receiver is at the waveguide feed, the physically natural model is y = Σ_n e^{-jφ_n} Σ_k h(u_k,ψ_n) x_k + z with a single z ∼ CN(0, σ²). Under that feed-noise model, the SNR for user k is P_k |Σ_n e^{-jφ_n} h(u_k,ψ_n)|²/σ² = P_k N|h_k(q)|²/σ², using the paper's h_k(q) = N^{-1/2} h^T(u_k,q)φ. Since Lemma 3 bounds |h_k(q)|² by O((ln N)^2/N), this SNR scales as O((ln N)^2), so the rates need not vanish. Thus the collapse to {(0,0)} in Eqs. (54)-(55) and the optimal-N rule in Remark 6 are artifacts of the per-antenna noise scaling. The authors should either provide a physical justification for per-element independent noise or redo the asymptotic analysis under a feed-noise model.
[III-D4, Lemma 3, Appendix D; Section V] The limit N → ∞ in Lemma 3 and Eq. (54) is not over feasible configurations as the model is stated. Lemma 3 constructs q_n = (n−1)Δ + q_1 with q_N + q_1 = 2x_k, which forces the array aperture (N−1)Δ to lie inside [q_0, q_max] from Eq. (2). For fixed q_max and Δ, N is bounded by floor((q_max − q_0)/Δ) + 1, so N cannot tend to infinity. The numerical setup in Section V fixes q_0 and q_max, and the Δ = 16λ curve in Fig. 6 extends to N = 500 even though the 22 m waveguide can hold at most about 129 antennas at that spacing. If the intent is to let the waveguide length grow with N, that scaling must be stated explicitly and the numerical comparison adjusted; otherwise the asymptotic result and the optimal-N conclusion are not well defined.
[Appendix C, proof of Theorem 3] The proof of Theorem 3 asserts that 'due to the curved Pareto boundaries ... it is straightforward to show that time sharing between these two FDMA regions suffices to fully cover the TDMA region,' but no rigorous argument is provided. The claim R_U_S,T ⊆ R_U_S,F is a stated contribution and is subsequently used to establish the downlink inclusion chain in Section IV-B, so the proof needs to be completed with an explicit construction or a counterexample-free analytic argument.

minor comments (5)

[III-D2, Eq. (46)] In Eq. (46), the definition r_U_k(q^⋄_k) = log2(P_k |h_k(q^⋄_k)|²/σ²) is missing the '1+' inside the logarithm; comparing with Eq. (12a), it should read r_U_k(q^⋄_k) = log2(1 + P_k |h_k(q^⋄_k)|²/σ²).
[III-D1, Eq. (43)] In Eq. (43), the second term on the left-hand side should contain (d_1² + (x − x_1)²)², not (d_2² + (x − x_2)²)²; as printed, the equation does not follow from setting the derivative in Eq. (42) to zero.
[II-A, Eq. (3)] The notation σ_U is used in the definition of z_U before it is introduced, and it is not clear whether σ_U equals σ from Eq. (3a) or is a separate quantity; please harmonize the notation.
[V-A2, Fig. 6] The non-monotonic behavior in Fig. 6 is demonstrated for the upper bounds in Eqs. (53a) and (53c), not for the exact capacity region or the inner bound; the text should distinguish the peak of the upper bound from the location of the true capacity-region maximum.
[Theorem 1, Eq. (22)] The abstract and Section III call these 'closed-form' expressions, but the Else branch of Eq. (22) requires solving a transcendental equation by bisection; a more precise term such as 'semi-closed-form' would be appropriate.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the capacity-region derivations are self-contained, and the few author-overlapping citations are to independent prior analytical results rather than ansätze that smuggle in the target claim.

full rationale

The paper's derivation chain is largely self-contained. The uplink capacity region in (14) is the standard convex hull of per-position MAC pentagons, and the rate-profile characterization in (17), Lemma 1, and Theorem 1 for the single-pinch case follow from the channel model without fitting any parameter or re-importing the conclusion. The TDMA and FDMA regions are computed from the same channel expressions, and the downlink regions are obtained by the external uplink-downlink duality results [40], [43]. The only upstream results taken from work with substantial author overlap are the antenna-placement rule and array-gain scaling used in Lemma 3, drawn from [21] and [24]. These are parameter-free analytical results about the same pinching-antenna model, used as supporting lemmas rather than as a definitional substitute for the asymptotic claim, so they do not constitute a circular reduction. Points such as Theorem 2 being proved only by similarity to Theorem 1, or the multiple-pinch FDMA details being omitted, are exposition gaps rather than circularity. The sensitivity of the N-to-zero capacity collapse to the per-pinch independent-noise model is a physical modeling concern, not a circularity concern.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claims rest on the LoS channel model, the per-pinch independent noise model, perfect phase compensation, and the two-user restriction. No free parameters are fitted; the numerical experiments use fixed system parameters.

assumptions (7)

domain assumption Free-space line-of-sight channel model h(uk,ψn) = η^{1/2} e^{-jk0||uk-ψn||}/||uk-ψn|| (Eq. 1).
The paper assumes LoS propagation only, justified by high-frequency operation, but multipath is ignored. Stated in Section II as a modeling choice.
domain assumption Independent noise per pinching antenna: z_n ~ CN(0,σ^2) in Eq. (3a).
This leads to combined noise Nσ^2 and is the key driver of the asymptotic collapse. The physical basis for per-element noise is not discussed.
domain assumption Perfect in-waveguide phase compensation via φ_n (Eq. 5).
The receiver applies e^{-jφ_n} with exact knowledge of waveguide phases, idealizing hardware precision.
ad hoc to paper Two-user restriction (K=2).
All capacity and rate region results are for two users; extension to more users is left to future work.
domain assumption Minimum inter-antenna spacing Δ to avoid mutual coupling (feasible set P).
The constraint |q_n - q_n'| ≥ Δ is imposed in (14) and used in the inner and outer bounds.
standard math Uplink-downlink duality for Gaussian MAC and BC.
Used in Section IV to map downlink regions to dual uplink regions, citing [40], [43].
standard math Rate-profile approach of Zhang and Cui [36].
The capacity region is characterized by solving sum-rate maximization with a rate-profile factor α, an established method the paper adopts.

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Cite this review

Pith. "Pith review of Capacity Characterization of Pinching-Antenna Systems." pith.science (2026). https://pith.science/paper/7G7MLBU7

@misc{pith2026250614298,
  author       = {Pith},
  title        = {Pith review of: Capacity Characterization of Pinching-Antenna Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7G7MLBU7}},
  note         = {Machine review of arXiv:2506.14298}
}

read the original abstract

Unlike conventional systems using a fixed-location antenna, the channel capacity of the pinching-antenna system (PASS) is determined by the activated positions of pinching antennas. This article characterizes the capacity region of multiuser PASS, where a single pinched waveguide is deployed to enable both uplink and downlink communications. The capacity region of the uplink channel is first characterized. \romannumeral1) For the single-pinch case, closed-form expressions are derived for the optimal antenna activation position, along with the corresponding capacity region and the achievable data rate regions under time-division multiple access (TDMA) and frequency-division multiple access (FDMA). It is proven that the capacity region of PASS encompasses that of conventional fixed-antenna systems, and that the FDMA rate region contains the TDMA rate region. \romannumeral2) For the multiple-pinch case, inner and outer bounds on the capacity region are derived using an element-wise alternating antenna position optimization technique and the Cauchy-Schwarz inequality, respectively. The achievable FDMA rate region is also derived using the same optimization framework, while the TDMA rate region is obtained through an antenna position refinement approach. The analysis is then extended to the downlink PASS using the uplink-downlink duality framework. It is proven that the relationships among the downlink capacity and rate regions are consistent with those in the uplink case. Numerical results demonstrate that: \romannumeral1) the derived bounds closely approximate the exact capacity region, \romannumeral2) PASS yields a significantly enlarged capacity region compared to conventional fixed-antenna systems, and \romannumeral3) in the multiple-pinch case, TDMA and FDMA are capable of approaching the channel capacity limit.

Figures

Figures reproduced from arXiv: 2506.14298 by the authors.

**Figure 1.** Illustration of a multiuser PASS. to conventional fixed-antenna systems in both uplink and downlink channels, and iii) for the multiple-pinch case, practical OMA schemes—namely TDMA and FDMA— can nearly achieve the full capacity region in both uplink and downlink channels. These findings underscore the scalability and effectiveness of PASS in enlarging the capacity region of wireless multiuser systems. The remainder… view at source ↗

**Figure 2.** An illustrated example of the capacity region for a [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Illustrated example of the capacity region for a two [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 5.** Figure 5: Capacity/rate region comparison for PASS-enabled [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Upper bounds of the per-user rate and sum-rate. [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]

**Figure 8.** Figure 8: Capacity/rate region comparison for PASS-enabled [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 7.** Figure 7: Capacity/rate region comparison for PASS-enabled [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 9.** Figure 9: Comparison of the TDMA and FDMA rate regions. [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]

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