Exploring the limitations of quantum networking through butterfly-based networks

Kieran N. Wilkinson; Stefano Pirandola; Thomas P. W. Cope

REVIEW 3 major objections 4 minor 48 references

Exploring the limitations of quantum networking through butterfly-based networks

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper shows that classical multicast communication can outpace quantum communication in butterfly-based networks, with the per-receiver gap reaching one bit when many butterfly blocks are wired in parallel.

desk verdict Concrete rate formulas for butterfly networks, but the quantum 'multicast' rate counts a shared GHZ qubit, not a qubit per receiver, so the headline gap compares different tasks. read the letter →

arxiv 1909.02342 v3 pith:2F3SJIKT submitted 2019-09-05 quant-ph cond-mat.otherphysics.app-ph

Kieran N. Wilkinson , Thomas P. W. Cope , Stefano Pirandola This is my paper · ORCID

classification quant-phcond-mat.otherphysics.app-ph

keywords butterflynetworkquantumcodingmulticastcommunicationrelativeentropyofentanglementdepolarizingchannelerasureteleportation-covariantcapacitybounds

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

The paper asks whether classical network topologies built from butterfly blocks are inherently worse at carrying quantum information than classical information. It compares achievable classical single-message multicast rates with upper bounds on quantum multicast rates for networks whose links are identity, depolarizing, or erasure channels. The central finding is that the classical rate per receiver exceeds the quantum upper bound over wide noise ranges, and for perfect identity channels the gap grows to one full bit per receiver as butterfly blocks are added in parallel. The paper's purpose is to identify these 'negative case' configurations so that quantum networks can be designed away from them.

What carries the argument

The load-bearing object is the butterfly network block and its rectangular grids. The quantum side rests on Eq. (10): the quantum multicast rate is bounded by (1/r) times the minimum-cut sum of relative entropies of entanglement of the Choi matrices of the channels crossing the cut, where senders and receivers are treated as super-users and r is the number of receivers. The classical side uses network coding: XOR at the bottleneck node lets each sender's message reach both receivers through a single channel use, and the achievable rates are computed by decomposing the network into effective point-to-point channels and optimizing input distributions. The comparison of these two quantities is what produces the gap.

What would settle it

Demonstrate a single-message multicast over a parallel identity-channel butterfly network (with r receivers) that delivers more than (2r - 1)/r logical qubits per use per receiver; this would refute the REE-based ceiling that the paper's gap relies on.

Watch

Extended reading notes

Core claim

For a single butterfly network with two senders and two receivers using identical teleportation-covariant channels, the achievable classical multicast rate per receiver can be strictly larger than the relative-entropy-of-entanglement upper bound on the quantum multicast rate. In the perfect-channel limit, one butterfly block transmits 2 classical bits per use per receiver but is bounded by 1.5 qubits. When N_x butterfly blocks are connected in parallel (with r = N_x + 1 receivers), the quantum bound becomes (2r - 1)/r qubits per receiver, approaching 2 qubits, while the achievable classical rate approaches 3 bits, so the gap tends to one bit per receiver. For depolarizing channels the classical rate exceeds the quantum bound over essentially the whole noise range, and for erasure channels it exceeds the bound below a critical erasure probability (about 0.159 without inter-node communication and 0.244 with it); adding blocks in parallel or allowing classical communication between nodes widens the range.

Load-bearing premise

The entire quantum-rate gap rests on the validity of the relative-entropy-of-entanglement network bound, including the super-user relaxation and the division by the number of receivers; if that bound does not actually limit single-message multiple multicasts, the claimed quantum rates are not ceilings and the gap could shrink or vanish.

Editorial extensions

If this is right

A quantum network built by simply replacing the wires of a butterfly-based classical network with quantum channels will carry fewer logical qubits than classical bits, by up to one bit per receiver in the parallel-block limit.
Inter-node classical communication helps classical erasure networks widen their advantage: the critical erasure probability at which classical beats quantum rises from about 0.159 to about 0.244 for a single block.
Adding butterfly blocks in parallel increases the classical-versus-quantum gap monotonically; adding blocks in series decreases both rates but the classical rate still beats the quantum bound for a range of erasure probabilities.
The results give a concrete design rule: to avoid quantum performance penalties, do not duplicate butterfly-block topologies that rely on network coding for classical gains.

Reading between the lines

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

The same REE-cut technique could be applied to continuous-variable teleportation-covariant channels (e.g., Gaussian channels) to check whether the classical-over-quantum gap persists in that regime.
Because the quantum bound uses the super-user relaxation, a tighter multi-sender upper bound might reduce the gap; finding such a bound would be a direct test of how much of the reported gap is real versus an artifact of the relaxation.
The classical rates assume XOR network coding and simple routing; more sophisticated classical coding could push the classical rate higher, making the gap even larger.
One could experimentally realize a small erasure-butterfly network and measure the multicast throughput to see whether the predicted crossing at erasure probability about 0.159 is observed.

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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

This paper gives explicit rate formulas for classical and quantum communication over butterfly networks and shows that the classical rate exceeds an upper bound on the quantum rate, with the gap growing when you add parallel blocks. The erasure-channel derivations are clean counting arguments, and the identity-channel limit is a neat observation. The paper is worth reading for those formulas alone.

The soft spots are real. First, Eq. (12) has a sign error: the REE for the depolarizing channel should be 1 - H2(3p/4), not 1 - H2((1-3p)/4). The p=0 limit gives H2(1/4) ≈ 0.811, which would imply an entanglement of about 0.189 for a maximally entangled state. That's clearly wrong. The correct expression appears later in Eq. (19), so this is likely a typo, but it's in a load-bearing position.

Second, the depolarizing classical rate is said to come from a numerical optimization, but no algorithm or code is given. A reader cannot verify the curve in Fig. 2. That's a reproducibility gap.

The bigger issue is conceptual. The quantum rate RQ in Eq. (10) is obtained by dividing a physical-qubit bound by the number of receivers r. The quantum message is a GHZ-like logical qubit shared by the receivers. Each receiver does not locally hold the qubit; they hold a share. Classical multicast gives each receiver an independent bit. So RC and RQ are rates for different tasks. In the identity case, saying '2 bits per receiver vs 1.5 qubits per receiver' compares a locally accessible bit with a collective quantum state. The abstract's 'one extra bit per receiver' is not a gap in transmission of the same kind of information. If the quantum task were to send a qubit that each receiver can decode, no-cloning makes that impossible for two receivers, so the REE bound would not be the relevant upper bound.

This does not mean the erasure formulas are wrong; they are correct for the task the authors define. But the paper's framing oversells the comparison. A referee should ask the authors to either redefine the quantum task to match the classical one or to state clearly that they are comparing a shared quantum state with independent classical messages.

The citation pattern is fine; the REE bound is taken from prior work and used as a theorem.

My verdict: the paper deserves a serious referee but needs major revision. If the authors fix the typo, provide the depolarizing calculation, and reframe the quantum task honestly, it could be a useful contribution to the network-coding literature. As it stands, the central claim is not as strong as advertised.

Referee Report

3 major / 4 minor

Summary. The manuscript studies butterfly networks and networks assembled from butterfly blocks, comparing achievable classical multicast rates with upper bounds on quantum multicast rates. For identity, depolarizing, and erasure channels, the authors derive explicit per-receiver rates: the classical rate RC is computed from counting arguments or numerical optimization, while the quantum upper bound RQ comes from the relative entropy of entanglement (REE) network bound of Refs. [32,35,36], divided by the number of receivers. The central claim is that RC exceeds RQ in all cases considered, with the gap growing with the number of parallel butterfly blocks and reaching up to one extra bit per receiver in the erasure/identity limit. The paper also treats series and grid arrangements, showing that the parameter region where classical communication wins remains nonempty.

Significance. If the comparison is meaningful, the paper provides a quantitative demonstration that standard classical network-coding architectures can outperform quantum communication, and it gives explicit, parameter-free counting formulas for the erasure case. The erasure formulas (17)-(18) and (21)-(22) are transparent and checkable, and the use of closed-form REE bounds avoids fitted parameters. However, the significance is undermined by a conceptual mismatch between the tasks counted by RC and RQ: the classical rate counts locally decodable bits per receiver, while the quantum rate counts shares of a GHZ-like logical qubit that no individual receiver can decode. This issue, together with a sign error in Eq. (12), means the quantitative gap claimed in the abstract is not yet established for the stated comparison.

major comments (3)

[Section II, Eq. (10); Section IV A, Eqs. (19)-(21); Abstract] The comparison between RC and RQ conflates two different tasks. RQ is defined for a single-message multicast in which each sender transmits a GHZ-like logical qubit α|0...0>+β|1...1> encoded in r physical qubits. No individual receiver can locally decode this logical qubit: the reduced state of any single receiver is diagonal and carries no coherence between |0> and |1>. In contrast, each classical bit counted in RC is delivered to and locally decodable by every receiver. The claimed gap of up to one extra bit per receiver (Eqs. (19)-(21)) therefore compares locally accessible classical bits with per-receiver shares of a globally distributed quantum state. If 'multicast' is understood in the standard sense that every receiver obtains the message, then a perfect single-qubit multicast to r>=2 receivers is forbidden by no-cloning, so RQ is not an upper bound for that task. The manuscript should either define the quantum task as entanglement distribution and compare with an appropriate classical analogue, or restrict the claims accordingly.
[Section III, Eq. (12) and Eq. (13); Fig. 2] The argument of the binary entropy in Eq. (12) is incorrect. For the qubit depolarizing channel defined in Eq. (11), the Choi state has fidelity F = 1 - 3p/4, so the REE bound is 1 - H2(1 - 3p/4), equivalently 1 - H2(3p/4) by symmetry of H2. Equation (12) instead writes 1 - H2((1 - 3p)/4), which gives 1 - H2(1/4) ≈ 0.189 at p=0 instead of 1, and becomes ill-defined for p > 1/3. This error propagates into Eq. (13) and the depolarizing curves in Fig. 2, and the numerical results in those panels should be recalculated after the fix.
[Section III, depolarizing case; Fig. 2] The achievable classical rate RC for the depolarizing butterfly is not specified. The text states that the network is deconstructed into equivalent channels and that the overall rate is 'found numerically', but no transition matrix, optimization problem, or code is provided. Since the depolarizing panels of Fig. 2 and the associated claim of a gap over the whole range of p rely on this rate, the authors should give the explicit channel decomposition and the optimization procedure (or a derivation) so that the lower bound is verifiable.

minor comments (4)

[Eq. (19)] Equation (19) writes H2(3p/4) while Eq. (12) writes H2((1-3p)/4); after correcting Eq. (12), use the same expression in both places to avoid apparent inconsistency.
[Section II, text before Eq. (10)] The sentence 'the total number of logical qubits correctly received by the destination set is equal to the total number of physical qubits correctly received by each individual receiver' is confusing; since each logical qubit uses r physical qubits, the division in Eq. (10) deserves a clearer one-sentence justification.
[Section III, Eqs. (17)-(18)] The exponent 5 in the network-coding term is not derived in the text; a short explanation of which five edges must succeed for a network-coded bit (e.g., both senders' edges to R1, R1->R2, R2->Bi, and the direct edge Ai->Bi for decoding) would make the counting transparent.
[Fig. 3 and Fig. 6] The crossing points eta and eta' are quoted to three decimal places, but the method used to compute them (root finding of which curve difference?) is not described; please state how these values are obtained.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the claimed quantum vs classical rate gap compares an internally consistent classical achievable-rate calculation with an externally imported REE upper bound; no part of the central claim reduces to its own input by construction.

full rationale

Walking the derivation chain, the paper does not fit any parameter to the gap it reports. The quantum side is imported as a theorem: Eq. (4) is the single-channel REE bound Q2(E) ≤ ER(E) from Ref. [32], Eq. (8) is the network min-cut REE bound from Ref. [35], and Eqs. (9)-(10) apply the super-user relaxation from Ref. [36] and then divide by the number of receivers. These are cited results with stated assumptions (teleportation-covariant channels, adaptive LOCC, super-user ensembles) that do not include the butterfly-specific gap. Under the review rules, a parameter-free external theorem with independent stated assumptions counts as real evidence even when the citation overlaps with the present authors, so the self-citation does not by itself make the derivation circular. The classical rates are obtained independently from standard capacities: Eq. (14) for the depolarizing channel's binary symmetric channel equivalent, Eq. (17) from erasure-channel combinatorics of the butterfly network, and Eqs. (21)-(22) from straightforward path counting. The reported 'gap' is an upper-bound-versus-lower-bound comparison, which is valid evidence of a separation and is not a prediction forced by a fitted input. The reader-level objection that RQ counts a GHZ-like logical qubit shared by all receivers while RC counts locally decodable bits is a substantive concern about whether the two metrics describe the same multicast task, but it is a correctness/task-definition issue, not a circularity. No equation is defined in terms of the conclusion it is used to establish, so the correct circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the REE network bound from the authors' prior work, the super-user relaxation, and standard classical information theory results. No free parameters are introduced and no new physical entities are postulated.

assumptions (4)

domain assumption REE network upper bound: for teleportation-covariant channels, the two-way assisted quantum capacity of a network is bounded by the min-cut sum of REEs of edge Choi states (Eqs. (4)-(10)).
This is the foundational bound from Refs. [32,35,36] on which all quantum rate bounds rest; it is not rederived here.
domain assumption Super-user relaxation: replacing each ensemble of senders/receivers by a single super-user with non-local operations yields a valid upper bound on the total qubit flow.
Invoked in the passage around Eq. (9); if non-local operations among users are not allowed, the bound may be loose, but it remains an upper bound.
standard math Classical network coding achievability: the butterfly network achieves a 2-bit multicast per receiver via XOR coding over noiseless links, with erasure generalizations.
Standard result from Ahlswede et al. [1], used to establish RC formulas.
standard math Depolarizing channel classical capacity: C(p) = 1 - H2((1-p)/2) for the qubit depolarizing channel.
Established result (Refs. [40-42]), used to construct the classical achievable rate.

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Cite this review

Pith. "Pith review of Exploring the limitations of quantum networking through butterfly-based networks." pith.science (2026). https://pith.science/paper/2F3SJIKT

@misc{pith2026190902342,
  author       = {Pith},
  title        = {Pith review of: Exploring the limitations of quantum networking through butterfly-based networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2F3SJIKT}},
  note         = {Machine review of arXiv:1909.02342}
}

read the original abstract

We investigate the classical and quantum networking regimes of the butterfly network and a group of larger networks constructed with butterfly network blocks. By considering simultaneous multicasts from a set of senders to a set of receivers, we analyze the corresponding rates for transmitting classical and quantum information through the networks. More precisely, we compare achievable rates (i.e., lower bounds) for classical communication with upper bounds for quantum communication, quantifying the performance gap between the rates for networks connected by identity, depolarizing and erasure channels. For each network considered, we observe a range over which the classical rate non-trivially exceeds the quantum capacity. We find that, by adding butterfly blocks in parallel, the difference between transmitted bits and qubits can be increased up to one extra bit per receiver in the case of perfect transmission (identity channels). Our aim is to provide a quantitative analysis of those network configurations which are particularly disadvantageous for quantum networking, when compared to classical communication. By clarifying the performance of these 'negative cases', we also provide some guidance on how quantum networks should be built.

Figures

Figures reproduced from arXiv: 1909.02342 by the authors.

**Figure 2.** FIG. 2: Rates (bits/qubits) per use and receiver for the depo [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** FIG. 4: Diagram of the construction of larger networks from [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (2 more)

**Figure 5.** Figure 5: FIG. 5: Classical rate (lower lines) and inter-node CC assis [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6: Upper panel: relative increase in the critical point [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Consider the two-pair communication problem in which the pair of senders A1 and A2 perform single-message multicasts to the pair of receivers B1 and B2 under the ﬂooding condi- tion that data can only be sent through unused connec- tions. We encounter a bottleneck at node R1 as data is waiting to be sent from both senders through the channel (R1, R2). In ...

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