Robust logarithmic lower bound on shared-resource cost for $f$-routing

Kevin Bogner

REVIEW 2 major objections 4 minor 21 references

Robust logarithmic lower bound on shared-resource cost for $f$-routing

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Even with 0.09 error allowed, one-round f-routing for inner product mod 2 forces shared states of rank Ω(n/log n).

desk verdict Genuine advance on f-routing lower bounds; the proof holds up and the stress-test worry about the information-disturbance constant is misplaced. read the letter →

arxiv 2608.05775 v1 pith:GCAMCO5V submitted 2026-08-06 quant-ph

Kevin Bogner This is my paper · ORCID

classification quant-ph

keywords f-routingnon-localquantumcomputationSchmidtrankshared-resourcecostsigninformation-disturbancetradeofftriangulardiscriminationinnerproductmod2

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

In one-round $f$-routing, Alice and Bob share a quantum state before receiving inputs; depending on the value of a public function $f$, either Alice or Bob must recover a qubit from their own final system. The paper proves that when $f$ is the inner product modulo 2, every protocol with worst-case error at most $0.09$ in both routing cases — even with unlimited message lengths and local operations — must pre-share a state whose smaller marginal support dimension $d$ is at least $c n/\log n$ for an absolute constant $c$. The equivalent cost statement is $E_{\dim}(\rho_{LR})=\log_2 d \ge \log_2 n - \log_2\log_2 n - O(1)$. The result matters because it is robust: the closest earlier growing lower bound for an explicit routing function required zero error in one routing case, while this one tolerates constant two-sided error. The proof works by converting correctness into a matrix with a constant gap between the two routing cases, approximating that matrix by a low-rank matrix whose rank depends only on $d$, and comparing with a sign-rank lower bound for the inner-product matrix.

What carries the argument

The central object is the symmetric-logarithmic-derivative (SLD) quadratic form $\chi_{SLD}(\rho,\sigma)=\operatorname{Tr}[Z L_\Sigma^+(Z)]$ with $\Sigma=\rho+\sigma$, $Z=\rho-\sigma$, and $L_\Sigma(X)=(\Sigma X+X\Sigma)/2$. This quantity — equal to twice the measured quantum triangular discrimination and to the Bures quantum $\chi^2$ divergence — is the statistic that is both controlled by the trace-norm routing gap and compatible with the tensor-product expansions of Bob's states. The other load-bearing devices are the cost-preserving purification to a pure state of Schmidt rank $d$, the mixing of each state with a small product state $\Omega_{x,y}=\alpha_x\otimes\beta_y$ so the average state lies between two multiples of that product state, and the sign-rank lower bound for the Walsh–Hadamard matrix. The inverse of $L_{\tau_{x,y}}$ is handled by a Fourier expansion whose complex powers separate into $x$-dependent and $y$-dependent factors precisely because $\Omega_{x,y}$ is a product state; this separation is what makes the approximate rank depend only on $d$ and not on message dimensions.

What would settle it

Exhibit an explicit one-round f-routing protocol for the inner product modulo 2 with worst-case diamond-norm error at most $0.09$ whose shared state has smaller marginal rank $d=o(n/\log n)$; this would directly contradict Theorem 1. Alternatively, since the paper's machine-checked formalization treats the information-disturbance bound as a hypothesis, a concrete counterexample to Lemma 3 at $\epsilon=0.09$ — an isometry and a recovery channel of error $0.09$ for which Bob's two reduced states differ in trace norm by more than $1.2$ — would close the constant gap and invalidate the proof.

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Extended reading notes

Core claim

The central claim is Theorem 1: for the inner product modulo 2, $f_n(x,y)=\bigoplus_{i=1}^n x_i y_i$, every protocol in the standard one-round $f$-routing model with worst-case error at most $0.09$ in the full, unhalved diamond norm satisfies $d\log_2(2d)\ge c n$ for an absolute constant $c>0$, where $d=\min\{\operatorname{rank}\rho_L,\operatorname{rank}\rho_R\}$ is the smaller marginal support dimension of the pre-shared state. Consequently the cost $E_{\dim}(\rho_{LR})=\log_2 d$ is at least $\log_2 n-\log_2\log_2 n-O(1)$. The proof first shows that any mixed shared state can be replaced, at no extra cost, by a pure state of Schmidt rank $d$. It then uses the information-disturbance tradeoff to show that Bob's two states $\rho^0_{x,y}$ and $\rho^1_{x,y}$ differ in trace norm by at least $2(1-\epsilon)$ when Bob must recover the qubit and by at most $4\sqrt{\epsilon}$ when Alice must; at $\epsilon=0.09$ these bounds are $1.82$ and $1.2$, a constant gap. The symmetric-logarithmic-derivative statistic $\chi(x,y)=\chi_{SLD}(\rho^0_{x,y},\rho^1_{x,y})$ inherits this gap, and after mixing with a small product state and applying a Chebyshev-plus-Fourier approximation, the matrix $[\chi(x,y)]$ is entrywise within $g/2$ of a real matrix of rank $\exp(O(d\log(2d)))\log(2N)$, with $N=2^n$. Because the sign-rank of the inner-product matrix is at least $\sqrt{N}$, comparing the two rank bounds forces $d\log(2d)=\Omega(n)$.

Load-bearing premise

The argument rests on a single borrowed bound: whenever one party can recover the qubit with error at most $\epsilon$, the other party's two possible states must be within distance $4\sqrt{\epsilon}$ of each other; if that bound were any weaker, the constant gap between the two routing cases would disappear and nothing in the paper would go through.

Editorial extensions

If this is right

Any one-round $f$-routing protocol for the inner product modulo 2 with worst-case diamond error at most $0.09$ must pre-share a state whose smaller marginal rank is $\Omega(n/\log n)$, so communication and local operations cannot substitute for this shared resource.
The cost lower bound $E_{\dim}(\rho_{LR})\ge \log_2 n-\log_2\log_2 n-O(1)$ holds even though the proof never bounds message lengths, local systems, or local operations.
Because the rank bound depends only on $d$, the same style of argument applies to any function whose sign matrix has sufficiently large sign rank, with the inner product mod 2 being one such function.
A polynomial lower bound on $E_{\dim}$ remains open; the paper's techniques yield only logarithmic growth, leaving a gap between $\log n$ and any $n^\delta$.

Reading between the lines

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

If the information-disturbance tradeoff constant were sharpened, the allowable error of $0.09$ could likely be increased and the same proof structure would survive; the paper does not explore this direction.
The final step's reliance on sign-rank suggests the result generalizes: any Boolean function with sign-rank $2^{\Omega(n)}$ should admit the same $E_{\dim}$ lower bound in this routing model, but the paper only proves this for inner product mod 2.
Because the paper's machine-checked formalization treats the information-disturbance tradeoff, the triangular-discrimination comparison, and the sign-rank bound as hypotheses, the theorem's correctness currently rests on those unverified external results; a rigorous attack would target them.
The cost measure charges shared randomness and separable correlations as well as entanglement, so the lower bound should be understood as a statement about all pre-shared correlations, not about entanglement specifically.

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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

The important thing to know about this paper: it delivers what it claims. It proves the first lower bound on the shared-resource cost E_dim for an explicit f-routing function that tolerates two-sided error, using the inner product modulo 2. The bound is only logarithmic, d = Ω(n/log n), but it improves on the zero-error Schmidt-rank bounds from ACM25, and the technique is genuinely new: converting correctness into a matrix with a constant gap, then controlling approximate rank via SLD quadratic form, mixing with a product state, Fourier expansions, and Chebyshev approximation.

The proof looks sound. I checked the constants that matter: the mixing step with s=0.05 leaves a gap around 0.167 between the two routing cases, and the approximation error g/2 preserves the sign pattern. The Lean 4 formalization machine-checks the internal argument; it treats the information-disturbance tradeoff, Liu's comparison, and Forster's sign-rank bound as hypotheses, which is honest and reasonable.

The stress-test concern about Lemma 3's constant does not survive reading the paper. Under the paper's explicit unhalved diamond norm, if ∥Φ - id∥⋄ = ε, then the normalized Choi states have trace distance ε; Uhlmann gives purifications with vector distance ≤ √ε, and reducing onto Bob's system loses a factor 2. So the correct bound is 2√ε, not 2√(2ε). The alternative constant would only appear under the halved norm. The gap does not close.

The soft spots are minor. The three external lemmas are load-bearing, and the paper doesn't reproduce KSW's theorem state—so a referee should verify that one constant, but it is right. The Lean artifact has no commit hash, so the formalization isn't pinned. And the lower bound remains logarithmic; the polynomial gap is still open, which the paper says plainly.

Who is this for? People working on f-routing, non-local quantum computation, or quantum communication complexity more generally. It deserves a serious referee; the result is real, the proof is intricate, and the formalization is a plus.

Recommendation: accept with the standard nudges—ask the author to pin the Lean version and state the exact lemma from KSW in the text. But don't send this back for major surgery; the center holds.

Referee Report

2 major / 4 minor

Summary. The paper studies one-round f-routing with a shared resource charged by E_dim(ρ_LR)=log_2 min{rank ρ_L, rank ρ_R}, with no limits on message lengths, local systems, or local operations. For f(x,y)=IP_n(x,y) (inner product mod 2), it claims that every protocol with worst-case error at most 0.09 in the full, unhalved diamond norm satisfies d log_2(2d)=Ω(n) for d=min{rank ρ_L, rank ρ_R}, and consequently E_dim(ρ_LR)≥log_2 n − log_2 log_2 n − O(1). The proof purifies the shared state to Schmidt rank d, applies an information-disturbance tradeoff to establish a constant trace-norm gap between the two routing cases, defines an SLD quadratic-form statistic χ(x,y) that inherits this gap, then shows (after mixing and Fourier/product-expansion arguments with explicit constant bookkeeping) that the matrix [χ^s(x,y)] is entrywise g/2-close to a real matrix of rank exp(O(d log(2d))) log(2N). Comparing this with the √N sign rank of the inner-product matrix yields the lower bound. A Lean 4 formalization machine-checks the internal proof, treating the information-disturbance tradeoff, the quantum-triangular-discrimination comparison, and the sign-rank bound as external hypotheses.

Significance. The result, if correct, is a meaningful advance: it gives the first growing lower bound on the charged shared-resource cost for an explicit routing function that allows two-sided error in both routing cases, in a model with unrestricted messages and local operations, and with arbitrary mixed shared states. The proof technique—converting routing correctness into an approximate-rank bound via an SLD quadratic form and then invoking a sign-rank lower bound—is novel and likely to be reusable. The paper is unusually explicit about its constants (s=0.05, θ=1.306695, g=0.166695), and the Lean 4 formalization of the internal argument is a valuable complement, even though the three external lemmas are not machine-checked. The main caveat is that the numerical threshold 0.09 depends on the exact constant in the external information-disturbance tradeoff, so the headline numerical claim carries a correctness risk that is not covered by the formalization.

major comments (2)

[Section 3, Lemma 3 and Proposition 4; numerical gap in Equation (5)] The proof of Theorem 1 is critically sensitive to the exact constant 2√ε in Lemma 3. If the information-disturbance tradeoff from [KSW08, ABM+24] yields instead ∥N_B−R_ζ∥_⋄ ≤ 2√(2ε), then the f=0 bound in Proposition 4 becomes 4√(2ε) ≈ 1.70 at ε=0.09, and after the mixing step the upper value in Equation (5) becomes roughly 1.61, which exceeds the f=1 lower bound 1.47; the θ±g separation used in the sign-rank comparison of Section 5 would disappear. The manuscript does not prove Lemma 3 but cites [ABM+24] and asserts, without derivation, that the continuity theorem gives √ε isometry closeness. Please provide a complete derivation of Lemma 3, including the exact statement of the external theorem and the diamond-norm convention, or adjust the error parameter and the advertised 0.09 threshold to a value for which the separation holds.
[Lean certification paragraph (after the Introduction) and Section 3, Lemma 3] The paper explicitly states that the information-disturbance tradeoff, the comparison for measured quantum triangular discrimination, and the sign-rank bound are treated as hypotheses in the Lean formalization, and no commit hash is given. This makes the machine-checked claim conditional on the exact constants in these external results, in particular on the constant in Lemma 3 that drives the 0.09 threshold. The repository should record the precise statement of each axiom, with constants and norm conventions, and the paper should pin the repository version used. As it stands, the formalization cannot independently certify the robustness claim that is the paper's headline.

minor comments (4)

[Section 3, proof of Lemma 3] Please cite the specific theorem numbers in [KSW08] and [ABM+24] that yield the √ε isometry closeness, and state which normalization of the diamond norm they use.
[Section 4, proof of Lemma 12] The proof of the bound |F_ζ| ≤ M_j using noncommutative Hölder is only sketched; in particular, the case h_i=0 (where one uses the operator norm of a unitary) should be spelled out, since it is essential for the dimension independence.
[Title and Abstract] The term 'robust' is not formally defined; the introduction should state explicitly that it refers to allowing two-sided error in both routing cases.
[Bibliography entry [ACM25]] The arXiv identifier or DOI for [ACM25] would help readers verify the comparison with the closest earlier result.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation reduces to published external theorems and its own internal rank bounds, with no step equivalent to the conclusion by construction.

full rationale

The proof's load-bearing imports are external published results: Lemma 3 is cited to [KSW08] and [ABM+24], Lemma 5 to [Liu25], Lemma 16 to [For02], and Lemma 14's Chebyshev construction is standard. None of these sources is authored by the present author, and none is stated in a form that assumes the target lower bound. The internal chain (purification with Schmidt rank d, the trace-norm gap from correctness, mixing with a product state, the polynomial/Fourier approximate-rank argument, and the final sign-rank comparison) uses d only as the resource parameter and derives the bound d log(2d) = Omega(n) from comparing an exp(O(d log(2d))) log(2N) rank upper bound with the 2^{n/2} sign-rank lower bound. No parameter is fitted to the conclusion: the constants 0.09, s=0.05, and theta=1.306695 are chosen after the inequalities are derived and are not statistically calibrated to any data. The paper explicitly flags the information-disturbance tradeoff and the other external lemmas as Lean hypotheses rather than machine-checked theorems; that is a verification caveat, and the sensitivity of the 0.09 threshold to the external constant is a robustness risk, but neither makes the derivation circular. There are no self-citations by this author and no uniqueness claim imported from prior work to force the choice of construction.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central claim rests on three published external theorems (information-disturbance tradeoff, Liu's comparison, Forster's sign-rank bound) plus standard mathematical tools. The numbers s, θ, g are hand-chosen proof constants, not fitted data parameters. No new physical entities are introduced.

free parameters (3)

s (mixing parameter) = 0.05
Fraction of product state mixed into the routed states in Section 4 (after Lemma 6). Chosen so that the bounds in Equation (3) and Lemma 7 hold and the gap remains positive.
θ (threshold) = 1.306695
Midpoint of the post-mixing bounds in Equation (5), set as (1.14 + 1.47339)/2 in Equation (6).
g (gap) = 0.166695
Half of the post-mixing gap (1.47339 - 1.14)/2, set in Equation (6) and used as the approximation error budget in Proposition 15.

assumptions (5)

domain assumption Information-disturbance tradeoff: if D∘N_A is ε-close to the identity in diamond norm, then N_B is 2√ε-close to a replacer channel (Lemma 3).
External theorem from [KSW08] and [ABM+24], treated as a hypothesis in the Lean formalization. It provides the f=0 upper bound in Proposition 4.
standard math Quantum triangular discrimination comparison: T^2 ≤ χ_SLD/2 ≤ T for T = ||ρ-σ||_1/2 (Lemma 5).
From [Liu25]; treated as a hypothesis in the Lean formalization. Converts trace-norm gaps into gaps for the SLD quadratic form.
standard math Forster's sign-rank bound: for any N×N sign matrix S, signrank(S) ≥ N/||S||_op.
From [For02]; used in Lemma 16 to lower-bound the sign rank of the inner product matrix.
standard math Max-flow min-cut theorem (Ford-Fulkerson).
Used in Lemma 10 to construct edge weights on a tree.
standard math Beta integral and Fourier transform identities, including the reflection formula for the gamma function.
Used in Lemma 11 to bound the Fourier transform of g_θ.

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Cite this review

Pith. "Pith review of Robust logarithmic lower bound on shared-resource cost for $f$-routing." pith.science (2026). https://pith.science/paper/GCAMCO5V

@misc{pith2026260805775,
  author       = {Pith},
  title        = {Pith review of: Robust logarithmic lower bound on shared-resource cost for $f$-routing},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GCAMCO5V}},
  note         = {Machine review of arXiv:2608.05775}
}

abstract

In one-round $f$-routing, Alice receives an $n$-bit string and an unknown qubit, while Bob receives another $n$-bit string. They exchange one simultaneous message each, and the value of $f$ determines which party must recover the qubit. Message lengths, local systems, and local operations are unrestricted. We charge only $E_{\dim}(\rho_{LR})=\log_2\min\{\operatorname{rank}\rho_L,\operatorname{rank}\rho_R\}$, the logarithm of the smaller marginal support dimension of the shared state prepared before the inputs arrive. The state may be arbitrary and mixed. For the inner product modulo $2$, we prove that every protocol with worst-case error at most $0.09$ in both routing cases, measured in the full, unhalved diamond norm, satisfies $d\log_2(2d)=\Omega(n)$ for $d=\min\{\operatorname{rank}\rho_L,\operatorname{rank}\rho_R\}$. Thus $d=\Omega(n/\log n)$ and $E_{\dim}(\rho_{LR})\ge\log_2 n-\log_2\log_2 n-O(1)$. The closest earlier growing Schmidt-rank lower bound for an explicit routing function assumes zero error in one routing case. Our proof converts correctness into a matrix whose entries have a constant gap between the two routing cases. It approximates this matrix by one whose rank depends on $d$, but not on the dimensions of messages or local systems. A sign-rank lower bound for the matrix of the inner product modulo $2$ completes the argument. A lower bound polynomial in $n$ on $E_{\dim}$ remains open.

Figures

Figures reproduced from arXiv: 2608.05775 by the authors.

**Figure 1.** One round of f-routing. Time runs downward. Alice receives x and the qubit Q, Bob receives y, and they share the state ρLR prepared before the inputs arrive. Double lines carry the classical inputs. Alice applies Ax to QL, keeps A0, and sends the message CA→B. Bob applies By to R, keeps B0, and sends CB→A. The messages cross simultaneously. If f(x, y) = 0, Alice applies D x,y A to MA = A0CB→A and outputs the qubit. … view at source ↗

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Allerstorfer, H

R. Allerstorfer, H. Buhrman, A. May, F. Speelman, and P. Verduyn Lunel, Relating non-local quantum computation to information theoretic cryptography, Quantum 8 (2024), 1387

work page 2024

[2] [2]

V. R. Asadi, R. Cleve, E. Culf, and A. May, Linear gate bounds against natural functions for position-verification, arXiv:2402.18648v4, 2026

work page Pith review arXiv 2026

[3] [3]

V. R. Asadi, E. Culf, and A. May, Rank lower bounds on non-local quantum computation, in 16th Innovations in Theoretical Computer Science Conference, LIPIcs 325 (2025), 11:1--11:18

work page 2025

[4] [4]

V. R. Asadi, K. Kuroiwa, D. Leung, A. May, S. Pasterski, and C. Waddell, Conditional disclosure of secrets with quantum resources, Quantum 9 (2025), 1885

work page 2025

[5] [5]

Bluhm, M

A. Bluhm, M. Christandl, and F. Speelman, A single-qubit position verification protocol that is secure against multi-qubit attacks, Nature Physics 18 (2022), 623--626

work page 2022

[6] [6]

Bluhm, S

A. Bluhm, S. H\"ofer, A. May, M. Stasiuk, P. Verduyn Lunel, and H. Yuen, A complexity theory for non-local quantum computation, Quantum 10 (2026), 2152

work page 2026

[7] [7]

Boyd and L

S. Boyd and L. Vandenberghe, Convex optimization, Cambridge University Press, 2004

work page 2004

[8] [8]

Cleve and A

R. Cleve and A. May, Lower bounds on non-local computation from controllable correlation, arXiv:2602.00255, 2026

arXiv 2026