Communicating Properties of Quantum States over Classical Noisy Channels

Frank H. P. Fitzek; Jiechen Chen; Nikhitha Nunavath; Osvaldo Simeone; Riccardo Bassoli

REVIEW 2 major objections 3 minor 12 references

This paper claims that a receiver can estimate the expected values of arbitrary local Pauli observables of an unknown quantum state sent over a noisy classical channel using a number of transmitted bits that grows logarithmically with the n

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-03 18:29 UTC pith:KSPSYXZ7

load-bearing objection Useful UEP-for-classical-shadows idea, but the n-independence claim is wrong and the main guarantee assumes independent known residual errors the implemented system doesn't deliver. the 2 major comments →

arxiv 2512.04913 v4 pith:KSPSYXZ7 submitted 2025-12-04 quant-ph eess.SP

Communicating Properties of Quantum States over Classical Noisy Channels

Nikhitha Nunavath , Jiechen Chen , Osvaldo Simeone , Riccardo Bassoli , Frank H. P. Fitzek This is my paper

classification quant-ph eess.SP MSC 81P4581P47 PACS 03.67.-a03.67.Hk

keywords shadow tomographyclassical shadowsunequal error protectionquantum communicationPauli observablessample complexitydebiasingnoisy classical channel

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper introduces STT-UEP, a protocol for sending properties of an unknown quantum state over a classical noisy channel without requiring the receiver to reconstruct the full state. It measures copies of the state in random Pauli bases, then transmits the basis descriptions and the measurement outcomes separately, giving the basis bits a stronger error-correcting code. A debiasing factor derived from the bit-error probability exactly removes the statistical bias that outcome bit flips introduce, so the number of state copies needed grows only logarithmically with the number of requested observables and exponentially only with the largest observable weight, independent of system size. If correct, this removes the exponential-in-qubits communication bottleneck for distributed quantum computing and sensing scenarios where the receiver only needs specific properties.

Core claim

STT-UEP combines classical shadow tomography with unequal error protection: the encoder measures N copies of the state in randomly chosen single-qubit Clifford bases (Z, X, or Y), and transmits the basis descriptions under a low-rate code and the outcome bits under a higher-rate code. The decoder keeps only measurements whose basis is compatible with each target observable, forms a product estimate on the support, and multiplies by the debiasing factor a_m = 3^|S_m| / (1 - 2 p_err)^|S_m|, which exactly cancels the selection bias (probability 3^{-|S_m|}) and the bit-flip bias ((1 - 2 p_err)^|S_m|). Proposition 1 shows that with probability 1 - P_outage, the accuracy requirement Pr(|hat o_m -

What carries the argument

The central mechanism is the pairing of basis-compatibility filtering with a multiplicative debiasing factor. The decoder selects only the approximately 3^{-|S_m|} fraction of measurements whose random Pauli basis matches the target observable on its support, and then scales the averaged product outcome by a_m = 3^{|S_m|}/(1 - 2 p_err)^{|S_m|}. This factor simultaneously corrects for the chance that a basis is compatible and for the expected attenuation of the product observable under independent bit flips. Unequal error protection with code rates R_u < R_b ensures the basis-identifying bits survive the channel, since an error there is catastrophic, while outcome bit flips enter only as a kn

Load-bearing premise

The proof treats bit errors on the transmitted measurement outcomes as independent across all Nn bits with a known, fixed probability p_err, and the debiasing factor collapses if residual errors are correlated after decoding or if p_err is unknown.

What would settle it

Transmit through a channel with correlated (bursty) errors without interleaving, use the claimed debiasing factor with a fixed p_err, and measure the empirical bias of the estimator over many trials. If the bias exceeds epsilon, or if the required N from (21) fails to satisfy the accuracy requirement (4), the central claim is refuted.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

The total number of transmitted bits is B = (n/R_b + ceil(log2 S)/R_u) N, with N given by (21), so communication cost scales logarithmically with M and exponentially only in the maximum observable weight w, not the system size n.
Outcome bit errors can be both corrected and debiased without retransmission: as long as the bit-flip probability p_err is below 1/2 and known, the estimator \(\hat o_m = a_m \hat o^{biased}_m\) is unbiased and satisfies the Hoeffding-based guarantee.
In the numerical experiments, giving the basis bits a stronger code than the outcomes (R_u < R_b, including uncoded outcomes with R_b = 1) achieves the highest success probability for a fixed total bit budget, outperforming equal error protection and conventional state-vector quantization.
The protocol is agnostic to the receiver's chosen observables: the encoder need not know which M Pauli observables the decoder will request, only the maximum weight w.
A channel outage, caused by failure to decode the basis bits despite the CRC check, sets the overall failure probability to P_outage = BLER(ceil(log2 S) N, R_u).

Where Pith is reading between the lines

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

The UEP principle likely extends beyond Pauli observables: in any shadow-tomography pipeline where the transmitted classical description of the random basis is more safety-critical than the outcome bits, protecting the basis-identifying bits with a stronger code should yield the same kind of logarithmic communication advantage.
If the decoder does not know p_err exactly, the debiasing factor becomes mismatched and the estimator develops a bias proportional to the error in p_err; the bound suggests this could be handled by estimating p_err from pilot bits and paying a multiplicative penalty of (1 - 2 p_err)^{-w}.
A testable extension: on a channel with bursty errors and no interleaving, the independence assumption for bit flips fails, so the debiasing factor should no longer cancel the bias; one would expect the outage-free accuracy guarantee to degrade exactly in proportion to the residual error correlation.
The rate-splitting idea generalizes to other semantic communication settings: bits that determine 'which question is being asked' (the basis) are worth more than bits that answer the question, so giving them unequal protection should be near-optimal whenever the question identity is discrete and the answer is a real-valued statistic.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Useful UEP-for-classical-shadows idea, but the n-independence claim is wrong and the main guarantee assumes independent known residual errors the implemented system doesn't deliver.

read the letter

Read it. The core idea is worth taking seriously: when you send classical shadow data over a noisy bit channel, protect the basis bits more than the outcome bits. The paper makes that case cleanly, and the numerical work backs it up.

What is new: the observation that basis-selection errors are catastrophic while outcome bit flips are statistically debiasable, and the UEP protocol that codes bases at a lower rate. The appendix proof is correct under its stated assumptions: the debiasing factor 3^w/(1-2p)^w is the right fix if residual bit flips are independent and the decoder knows p_err. The Hoeffding step is standard, and (21) follows honestly from those assumptions.

Two soft spots matter.

First, the "independent of n" claim is false for transmitted bits. There are 3^n possible random bases, so specifying the bases costs n log2(3) bits per copy. Total bits are B = O(n N) = O(n 9^w log M). Sample complexity is n-independent; bit complexity is not. The authors conflate the two in the abstract and Section I-C. This should be corrected.

Second, the guarantee (21) relies on independent, known p_err after LDPC decoding. LDPC residual errors are neither independent nor known to the decoder, and interleaving before transmission does not decorrelate errors after decoding. The paper only says "approximately" and moves on. If the decoder's assumed p_err is off, the debiasing factor amplifies bias by 3^w/(1-2p)^w. This is a genuine gap between the theorem and the system simulated. The authors need either to specify how p_err is estimated and show robustness to mismatch, or to prove a bound that holds under correlated residual errors.

Minor things: a few typos (ST-UEP, STT-UER) and the abstract oversells "arbitrary observables" when the results are for local Pauli observables.

Proportionate verdict: the protocol is plausible, the math is correct under stated assumptions, and the experiments are honest. These are fixable issues, not a dead end. The paper deserves a serious referee, but the referee should demand a corrected n-scaling claim and an honest treatment of the residual-error model before acceptance.

Referee Report

2 major / 3 minor

Summary. The paper considers an encoder with access to N copies of an n-qubit state. The encoder measures each copy in a random Pauli basis from {I,H,HS^dagger}^n, then transmits the basis choices and measurement outcome bits over a classical noisy channel using unequal error protection: basis bits are encoded with a stronger code (rate R_u) than outcome bits (rate R_b > R_u). The decoder selects measurements compatible with each of M Pauli observables of weight at most w, applies a debiasing factor that depends on the bit-error probability p_err, and forms estimates of the observable expectation values. Proposition 1 gives a lower bound on N that is logarithmic in M and exponential in w, and numerical simulations compare the scheme with shadow tomography with equal error protection and with quantized full state transmission.

Significance. The underlying idea is attractive and potentially useful: it connects classical shadow tomography with classical error-control coding, and it isolates the observation that basis errors are catastrophic while outcome errors can be partially corrected by debiasing. The derivation of the debiasing factor is clean and parameter-free, and the bound in Proposition 1 is a genuine sample-complexity statement for the idealized Bernoulli-error model assumed in the proof. The numerical comparison with relevant baselines is useful and shows a consistent advantage for the UEP strategy in the tested regime. However, the paper overstates the communication-complexity result and does not fully bridge the gap between the theorem's assumptions and the LDPC-coded channel used in simulations.

major comments (2)

[Sec. III-A, Eq. (11); Abstract] The claimed communication complexity is not independent of the number of qubits n. Since S = 3^n, log_2(S) = n log_2(3), so Eq. (11) gives B = (n/R_b + n log_2(3)/R_u) N. With N from Eq. (21), B is linear in n: B = Theta(n * 9^w * log(M/delta) / ((1-2p_err)^{2w} epsilon^2)). The result is a sample-complexity bound independent of n, not a bit-complexity bound independent of n. The Abstract and the 'Main Contributions' section should be revised to avoid this conflation, since the stated contribution is communication complexity in bits.
[Sec. III-B, Eqs. (13)-(19); Appendix (22)-(25)] Proposition 1 rests on two assumptions that are not satisfied by the simulated system: (i) the post-decoding bit flips are independent, and (ii) the decoder knows p_err exactly. The debiasing factor in Eq. (19) and the Hoeffding bound in Eq. (25) both require independent flips. Interleaving before transmission does not make LDPC residual errors independent after decoding, and the paper does not explain how the decoder obtains p_err. If the assumed and true p_err differ, every estimate is biased, with the bias amplified by 3^w/(1-2p)^{2w}. Thus Eq. (21) is a theorem for an idealized Bernoulli-error model, not for the LDPC-coded AWGN channel used in Section IV. The authors should either state this model explicitly as an assumption and align the simulations with it, or provide a correlation- and mismatch-robust version of the guarantee.

minor comments (3)

[Sec. II, Eq. (4)] The formal requirement should read Pr(max_m |o_m_hat - <O_m>| > epsilon) < delta, i.e. the union bound over m is needed. As written, 'for all m' inside the probability states a different, weaker event that does not match the union-bound proof in the Appendix.
[Sec. IV, Fig. 4] The text refers to 'Proposition 2', but the paper contains only Proposition 1. The reference should be corrected.
[Sec. IV, Fig. 3 caption] The caption uses 'STT-UER' in one place; this should be 'STT-UEP'.

Circularity Check

0 steps flagged

No significant circularity: the sample-complexity bound is derived from classical shadows and the stated bit-error model; the few self-citations are contextual, not load-bearing.

full rationale

The paper's derivation chain starts from established classical-shadow results (refs. [5], [6]) and applies a debiasing factor a_m = 3^{|S_m|}/(1-2p_err)^{|S_m|}. The unbiasedness proof in the Appendix uses only the compatibility probability 3^{-|S_m|} and the independent-bit-flip model with probability p_err; neither quantity is fitted to the quantities being predicted. Proposition 1 is a Hoeffding/union-bound consequence of the estimator's bounded range and unbiasedness; it is not equivalent to its inputs by construction. The treatment of p_err as known and the approximation that post-decoding bit errors are independent are assumptions/limitations, not circular steps: no parameter is fitted to the target observables and then renamed a prediction. Self-citations (e.g., ref. [7] to the authors' earlier quantum-semantic-communication paper) appear only in Related Work and are not load-bearing for the main theorem. The central bound therefore has independent content: it predicts sample complexity in terms of w, M, epsilon, delta, and p_err, and is benchmarked against CQCR and STT-CC without reverse-engineering those benchmarks. Score 1 reflects only the presence of non-load-bearing self-citations in the related-work section, not any circularity in the derivation.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central bound rests on standard shadow-tomography facts plus a channel model (independent known p_err). The only hand-chosen parameters are the code rates; the paper does not fit any constants to data.

free parameters (2)

R_b (outcome code rate) = 1 (uncoded) in main experiments; 0.4 in comparison
Chosen by hand; sets outcome bit-error probability p_err = BER(Nn, R_b) and contributes n/R_b bits per copy.
R_u (basis code rate) = 0.4 in main experiments; adjusted to match total B in Fig. 3
Chosen by hand; sets outage probability BLER(log2(S)*N, R_u) and contributes log2(S)/R_u bits per copy.

axioms (5)

standard math Random local Pauli shadow property: a basis matches a weight-|S| Pauli observable with probability 3^-|S|, and matched noiseless outcome products have expectation equal to the observable
Used in the Appendix to get E[o_biased] = 3^-|S| (1-2p)^|S| <O_m>; imported from classical shadows literature [6].
domain assumption After decoding and interleaving, bit flips on the Nn outcome bits are independent with common probability p_err = BER(Nn, R_b)
Stated in Section III-B after (13); needed for unbiasedness (22)-(24) and for Hoeffding (25).
domain assumption CRC detects all residual basis errors except with probability exponentially small in CRC size
Used to equate outage probability with BLER (12).
domain assumption BLER and BER are increasing functions of code rate R
Used to justify UEP: lower R_u gives stronger protection for bases.
domain assumption p_err <= 0.5
Required so 1 - 2p_err >= 0 and the debiasing factor is well-defined.

pith-pipeline@v1.3.0-alltime-deepseek · 7233 in / 16514 out tokens · 146092 ms · 2026-08-03T18:29:01.946246+00:00 · methodology

0 comments

read the original abstract

Transmitting information about quantum states over classical noisy channels is an important problem with applications to science, computing, and sensing. This task, however, poses fundamental challenges due to the exponential scaling of state space with system size. We introduce shadow tomography-based transmission with unequal error protection (STT-UEP), a novel communication protocol that enables efficient transmission of properties of quantum states, allowing decoder-side estimation of arbitrary local Pauli observables. Unlike conventional approaches requiring the transmission of a number of bits that is exponential in the number of qubits, STT-UEP achieves communication complexity that scales logarithmically with the number of observables, depending on the observable weight. The protocol exploits classical shadow tomography for measurement efficiency, and applies unequal error protection by encoding measurement bases with stronger channel codes than measurement outcomes. We provide theoretical guarantees on estimation accuracy as a function of the bit error probability of the classical channel, and validate the approach against several benchmarks via numerical results.

Figures

Figures reproduced from arXiv: 2512.04913 by Frank H. P. Fitzek, Jiechen Chen, Nikhitha Nunavath, Osvaldo Simeone, Riccardo Bassoli.

**Figure 2.** Figure 2: To estimate the expected value ⟨Om⟩, the decoder retains only measurements corresponding to compatible bases Pi. A measurement basis Pi is compatible with an observable Om only if the condition P j∈Sm 1(Pi,j = Om,j ) = |Sm| in (15) holds, so that all non-trivial Pauli matrices in Om are present in the corresponding position in Pi. For each index i ∈ Cm, we have a basis Pi and a received measurement outcome… view at source ↗

**Figure 4.** Figure 4: Success probability Psucc versus the number of copies N for the STT-UEP, with Ru = 0.4 and Rb = 1. for STT-UEP with code rates Rb = 0.4 and Rb = 1, STT-CC, and CQCR with quantization resolutions b = 2 bits, b = 4 bits and b = 8 bits. For a fair comparison, all schemes are constrained to use the same number of transmitted bits B, which is achieved by appropriately adjusting the code rate Ru for STT-UER, and… view at source ↗

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 2 linked inside Pith

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Pith/arXiv arXiv 2022

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Simeone,Classical and Quantum Information Theory: Uncertainty, Information, and Correlation

O. Simeone,Classical and Quantum Information Theory: Uncertainty, Information, and Correlation. Cambridge University Press, 2025

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J. R. Busemeyer and P. D. Bruza,Quantum Models of Cognition and Decision. Cambridge University Press, 2012

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2025

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Quantum semantic communi- cations for resource-efficient quantum networking,

M. Chehimi, C. Chaccour, and W. Saad, “Quantum semantic communi- cations for resource-efficient quantum networking,”IEEE Communica- tions Letters, vol. 28, no. 4, pp. 891–895, 2024

2024

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Shadow tomography of quantum states,

S. Aaronson, “Shadow tomography of quantum states,” inProceedings of the 50th annual ACM SIGACT Symposium on Theory of Computing, 2018, pp. 325–338

2018

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Predicting many properties of a quantum system from very few measurements,

H.-Y . Huang, R. Kueng, and J. Preskill, “Predicting many properties of a quantum system from very few measurements,”Nature Physics, vol. 16, no. 10, pp. 1050–1057, 2020

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N. Nunavath, M. I. Habibie, E. C. Strinati, R. Bassoli, and F. H. Fitzek, “Quantum semantic communications for graph-based models,” inProc. IEEE SPA WC, pp. 871–875, 2024

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Unequal error protection: An information-theoretic perspective,

S. Borade, B. Nakibo ˘glu, and L. Zheng, “Unequal error protection: An information-theoretic perspective,”IEEE Transactions on Information Theory, vol. 55, no. 12, pp. 5511–5539, 2009

2009

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Shadow tomography with noisy readouts,

H.-C. Nguyen, “Shadow tomography with noisy readouts,”arXiv preprint arXiv:2310.17328, 2023

Pith/arXiv arXiv 2023

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The randomized measurement toolbox,

A. Elben, S. T. Flammia, H.-Y . Huang, R. Kueng, J. Preskill, B. Ver- mersch, and P. Zoller, “The randomized measurement toolbox,”Nature Reviews Physics, vol. 5, no. 1, pp. 9–24, 2023

2023

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Interleaving for combating bursts of errors,

Y . Q. Shi, X. M. Zhang, Z.-C. Ni, and N. Ansari, “Interleaving for combating bursts of errors,”IEEE Circuits and Systems Magazine, vol. 4, no. 1, pp. 29–42, 2004

2004

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Sionna: An open-source library for next-generation physical layer research,

J. Hoydis, S. Cammerer, F. A. Aoudia, A. Vem, N. Binder, G. Marcus, and A. Keller, “Sionna: An open-source library for next-generation physical layer research,”arXiv preprint arXiv:2203.11854, 2022. APPENDIX To prove the unbiasedness of (18), note that thei-th basis Pi is compatible with the observableO m with probability 3−|Sm|. Furthermore, since the ch...

Pith/arXiv arXiv 2022