Certifying the dimensionality of any quantum channel with minimal assumptions

Bivas Mallick; Pratik Ghosal; Saheli Mukherjee

REVIEW 2 major objections 4 minor 61 references

Certifying the dimensionality of any quantum channel with minimal assumptions

T0 review · 2 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper proves that any quantum channel capable of preserving entanglement beyond a fixed dimension can be certified with a game that assumes trust only in state preparation.

desk verdict A real proof gap in the soundness argument, but a clean repair exists; the result deserves serious peer review, not a desk reject. read the letter →

arxiv 2511.10758 v3 pith:GKEPLZ4O submitted 2025-11-13 quant-ph

Saheli Mukherjee , Bivas Mallick , Pratik Ghosal This is my paper · ORCID

classification quant-ph MSC 81P4581P40 PACS 03.67.-a03.65.Ud

keywords Schmidt-number-breakingchannelssemiquantumsignalinggamesentanglementdimensioncertificationmeasurement-device-independentChoi–JamiołkowskiisomorphismNPT-breakingchannel

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 claims that the property of a quantum channel being non-k-Schmidt-number-breaking—meaning it can preserve entanglement of dimension greater than k—can be faithfully certified using semiquantum signaling games, assuming only trusted preparation of input product states and no trust in measurement devices. The method is faithful: every non-k-SNB channel admits a game with negative average payoff, while every k-SNB channel gives nonnegative payoff for all measurement choices. This extends earlier measurement-device-independent certification of non-entanglement-breaking channels to the whole hierarchy of Schmidt-number-breaking channels. The same framework also certifies non-NPT-breaking channels.

What carries the argument

The Choi–Jamiołkowski isomorphism maps each channel to a bipartite state (its CJ operator), allowing resource classes to be seen as convex sets of states. The proof uses Lemma 2, which says that composing a k-SNB channel with any completely positive map yields a k-SNB CP map, to extend the witness inequality from channels to the broader class of CP maps. This extension is the load-bearing step in proving soundness.

What would settle it

Take any non-k-SNB channel N and its separating witness W from Eq. (7), then find a k-SNB channel E and a CP map F such that Tr[W J_{F*∘E}] < 0. Numerically, one could fix a qutrit depolarizing channel (non-2-SNB for appropriate λ), use the optimal witness from Example 1, and search over simple CP maps F (e.g., constant maps outputting a pure state) to see if the inequality in Eq. (21) fails.

Watch

Extended reading notes

Core claim

Theorem 1 states that for every channel N not in the set of k-Schmidt-number-breaking channels, there exists a semiquantum signaling game whose average payoff is negative on N and nonnegative on every k-SNB channel. The game is constructed from a witness operator W that separates N from the convex set of k-SNB channels; the payoff is defined so that its average equals Tr[W J_N] for a specific measurement, and the nonnegativity for all k-SNB channels is argued through a structural closure property.

Load-bearing premise

The proof of Theorem 1's soundness step assumes that the Hahn-Banach witness operator W, which is constructed to be nonnegative on all k-SNB channels, is also nonnegative on the Choi operators of all k-SNB completely positive maps—a strictly larger set that the separation theorem does not guarantee.

Editorial extensions

If this is right

Any non-k-SNB channel can be certified without false positives from k-SNB channels, regardless of Bob's measurement choices, provided the input states are prepared faithfully.
The method removes the need for entangled input states and side channels, making it practical for quantum memory and communication link verification.
The same construction works for non-NPT-breaking channels, and more generally for any resource-breaking class that satisfies analogues of Lemmas 1 and 2.
The proposed optical circuit (inverse controlled-shift plus inverse QFT) gives a concrete path to experimental implementation.

Reading between the lines

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

If the witness-extension gap is resolved, the theorem would show that semiquantum games are complete for a whole family of channel resource theories; if not, the soundness guarantee might only hold for a restricted subset of witnesses.
Because the soundness proof currently assumes the witness is nonnegative on all k-SNB CP maps, a counterexample would likely come from a k-SNB channel composed with a CP map that stretches the CJ operator outside the witness's positive cone.
The trusted-preparation assumption could be quantified by examining how tolerances in state preparation affect the payoff, parallel to how noise-robustness analyses are done for steering-based certificates.

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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

First thing to know: the paper is not ready as written, but the main theorem looks correct and the flaw is a single unjustified step with a straightforward fix. If you're deciding whether to engage, the answer is yes—with the expectation of a revised proof.

What's actually new here: this is the first faithful minimal-assumption certification of all non-k-SNB channels, going beyond the steering-based approach [41], which is not faithful. The move of using Lemma 2 (closure of k-SNB under composition with arbitrary CP maps) to lift Rosset et al.'s argument from EB channels to SNB channels is genuine, and the same trick extends cleanly to NPT-breaking channels. The explicit qutrit examples with finite-dimensional witnesses and the circuit in Fig. 2 are concrete and useful.

What's soft: the proof of Theorem 1's soundness condition. Eq. (21) asserts Tr[W J_{F*∘E}] ≥ 0 because W is a k-SNB witness, but the witness in Eq. (7) was only separated against the set of k-SNB channels, not against all k-SNB CP maps. J_{F*∘E} need not be a channel, so its CJ operator does not have the fixed marginal that channel CJ operators have. The step does not follow as written. This is not a trivial footnote; it's the part that ensures no false positives.

However, the gap is repairable. Since N ∉ k-SNBC, Lemma 1 gives SN(J_N) > k, so J_N can be separated from the compact convex set of states with Schmidt number at most k by a strong Schmidt-number witness W' that is nonnegative on that whole state set, which by scaling extends to all positive semidefinite operators with SN ≤ k. With that W' in place, the contested inequality goes through. The same fix applies to the NPT result using PPT witnesses. The examples in the paper already use strong witnesses (e.g., W_opt^2), so the experimental demonstrations are unaffected.

So the result likely stands. The paper deserves a serious referee, not a desk reject. The referee should ask for the proof to be recast using the strong witness, and for a brief note on why the strong witness can be expressed in the required separable form (it can, since it's Hermitian).

Bottom line: a solid advance with one repairable hole. Worth citing once the fix is in.

Referee Report

2 major / 4 minor

Summary. The paper introduces a measurement-device-independent method for certifying that a quantum channel is not k-Schmidt-number-breaking (k-SNB). The proposed semiquantum signaling game uses trusted quantum input states and an untrusted joint measurement on the output of the channel and a second prepared state. The central result, Theorem 1, claims that for every channel outside k-SNBC there is such a game whose average payoff is negative, while every k-SNB channel yields nonnegative payoff for all measurements. The proof constructs a witness operator by Hahn-Banach separation from the set of k-SNB channels and decomposes it into local density matrices to define the game. The method is also extended to non-NPT-breaking channels in Appendix A, and experimental implementations are discussed with explicit examples.

Significance. If the central claim is correct, the paper provides a substantial advance: a faithful (i.e., applicable to all channels) certification of effective entanglement dimension under minimal assumptions, avoiding the need for trusted measurements, entangled inputs, or auxiliary side channels. The paper also demonstrates the reach of the method with explicit examples and a proposed optical implementation, and it extends the framework to NPT-breaking channels. The proof strategy, based on Lemma 2 and the Choi-Jamiołkowski isomorphism, is elegant. However, the soundness proof of Theorem 1 contains a significant gap: the witness from Eq. (7) is used beyond its proven domain. The gap is repairable by using a Schmidt-number witness constructed instead from state-space separation, and with that amendment the central claim is likely sound.

major comments (2)

[Sec. III, Theorem 1, Eq. (21)] The soundness condition (ii) of Theorem 1 is not proven. The witness W is obtained by Hahn-Banach separation from the convex set k-SNBC, so Eq. (7) guarantees Tr[W J_E] ≥ 0 only when J_E is the Choi operator of a k-SNB channel, i.e., a PSD operator with the fixed marginal Tr_B[J_E]=I/d (up to normalization). In the soundness argument, Π0^AB is identified with the Choi operator of an arbitrary CP map F, and Eq. (21) applies the witness to J_{F*∘E}. Lemma 2 only shows that this Choi operator has Schmidt number at most k; it does not put it on the channel slice, and Eq. (7) gives no control there. The sentence 'the final inequality follows from the properties of W as a k-SNB witness' is therefore unsupported. This is load-bearing because condition (ii) must hold for every measurement, including those whose associated F is not trace-preserving. The gap is repairable: choose W as a Schmidt-nu
[Appendix A, Theorem 2] The same gap appears in the NPT-breaking extension. The witness W̃ from Eq. (A3) is only known to be nonnegative on Choi operators of NPT-breaking channels. The proof applies it to J_{F*∘S}, which by Lemma 4 is PPT but is not necessarily the Choi operator of a channel. Nonnegativity of W̃ on all PPT operators does not follow from separation from the set of NPT-breaking channels. The proof should either construct W̃ as a PPT witness, nonnegative on all PPT operators, or explicitly justify that the separation in (A2)-(A3) can be chosen to have this stronger property. Without this, the soundness of Theorem 2 is also incomplete.

minor comments (4)

[Sec. III, Eq. (13)] The notation for the inverse Choi isomorphism is hard to follow, especially the double transpose ([ξ^x_{A'}]^⊺)^⊺. Please clarify the convention and the subsystem labels so the identity N(ρ)=Tr_{A'}[J^N_{A'A}(ρ^T⊗I)] is unambiguous.
[Sec. IV, Example 1] The matrix in Eq. (27) is poorly formatted and difficult to read; a properly typeset matrix would help. Also, the relation between the states ξ^x, the unitary gates U_x, and the input states ψ^x, φ^y should be stated before Eq. (28) to avoid confusion.
[Sec. II, Lemma 2] The term 'k-SNB CP map' is used before being defined. Please define it explicitly, e.g., as a CP map whose Choi operator has Schmidt number at most k, and state how the CJ isomorphism is normalized for non-trace-preserving maps.
[General] The proof of Eq. (16) is somewhat terse; a few extra lines showing the substitution of Eq. (13) and the contraction with W would improve readability. This is a presentation issue only.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the witness-based game construction is self-contained; the Eq. (21) gap is a correctness issue, not a circular reduction.

full rationale

The central derivation constructs the semiquantum signaling game directly from a Hahn-Banach witness W that separates the tested channel N from the convex set k-SNBC. The payoff inequality J_avg(N)<0 is obtained by translating Tr[W J_N]<0 into game terms, and J_avg(E)>=0 by translating Tr[W J_E]>=0; no parameter is fitted from the quantity being certified and no prediction is renamed as an input. Lemma 2 is attributed to the authors' prior work [29], but the paper gives a self-contained proof using the monotonicity of Schmidt number under local CP maps, a standard external result [17], so the self-citation is not load-bearing. The other self-citations ([34] in the introduction) are background motivation, not used in the proof of Theorem 1 or Theorem 2. The only significant issue is Eq. (21): the final inequality applies the channel-slice witness W to J_{F*∘E}, whereas Eq. (7) guarantees nonnegativity only on CJ operators of k-SNB channels. This is an unproved extension and a genuine proof gap, but it is not an equivalence-by-definition or a fitted-input-called-prediction; it is a repairable correctness issue. Therefore no circular step is established.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central proof relies on standard convex-geometry and quantum-information tools. The critical unproven assumption is the extension of the witness property to all k-SNB CP maps (the cone generated by the set of k-SNB channels is not the full cone of k-SNB CP maps). This assumption is not stated in the paper and is the source of the soundness gap.

assumptions (5)

ad hoc to paper The witness W from Hahn-Banach separation of the set of k-SNB channels is nonnegative on all CJ operators of k-SNB CP maps, not just channels.
Used in Eq. (21) to conclude Tr[W J_{F*∘E}] ≥ 0. This property is not proven and is not implied by Eq. (7). It is load-bearing for the soundness condition.
standard math Hahn-Banach separation theorem provides a witness W satisfying Eq. (7).
Used to guarantee existence of a witness for every non-k-SNB channel.
standard math Choi-Jamiołkowski isomorphism and the inverse action formula (5).
Used throughout the proof to translate between channels and bipartite operators.
standard math Local CP maps cannot increase the Schmidt number of a bipartite state.
Used in the proof of Lemma 2 and to justify SN(J_{F*∘E}) ≤ k.
domain assumption Every Hermitian operator can be decomposed as a real linear combination of density operators.
Used in Eq. (11) to express the witness W as a sum of tensor products of density operators.

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Cite this review

Pith. "Pith review of Certifying the dimensionality of any quantum channel with minimal assumptions." pith.science (2026). https://pith.science/paper/GKEPLZ4O

@misc{pith2026251110758,
  author       = {Pith},
  title        = {Pith review of: Certifying the dimensionality of any quantum channel with minimal assumptions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GKEPLZ4O}},
  note         = {Machine review of arXiv:2511.10758}
}

read the original abstract

High-dimensional entanglement offers significant advantages over its low-dimensional counterpart in various information-processing tasks. However, to harness these advantages, it is crucial that the quantum channels used to store or transmit the subsystems of an entangled system not only preserve entanglement but also maintain its dimensionality above a certain threshold. The maximum entanglement dimension that a channel can preserve is referred to as its effective dimensionality, since the channel cannot be used to transmit information of dimension greater than that in a single use. In this work, we present a method to certify whether a quantum channel can preserve entanglement dimension above a given threshold. Unlike existing approaches, our method is faithful, i.e., it can be applied to any channel, and avoids common assumptions such as preparation of entangled states, auxiliary side channels, or perfect measurement devices. Moreover, the method can be extended to faithfully certify other classes of non-resource-breaking channels, such as non-nonpositive-partial-transpose-breaking channels (non-NPT-breaking channels). Finally, we discuss possible experimental realizations of our certification scheme through explicit examples.

Figures

Figures reproduced from arXiv: 2511.10758 by the authors.

**Figure 2.** Circuit for experimental realization of the certification scheme for an unknown channel [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Variation of the average payoff Javg with respect to the noise parameter (λ) for depolarizing and dephasing channels, illustrating the transition to non-EB and non-2-SNB behaviors. The blue (red) line corresponds to the dephasing (depolarizing) channel. A negative payoff in the solid (dotted) blue line denotes the range of noise parameters for which the dephasing channel is non-2-SNB (non–entanglement breaking). As … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Variation of the average payoff J˜ avg with respect to the parameter (α) for the channel given by Eq. (A5). A negative payoff denotes the range of parameters for which the channel is non-NPT-breaking [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

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