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REVIEW 4 major objections 4 minor 37 references

Practically feasible robust quantum money with classical verification

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

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Pith's one-line read A private quantum money scheme based on Sampling Matching claims unconditional security against forging while tolerating noise up to 21.4% with a fixed passive linear-optical verification circuit.

desk verdict Plausible SM-based money scheme with a clean single-photon correctness analysis, but the advertised coherent-state implementation has no security proof and the single-photon proof leans on imported, partly conjectural ingredients. read the letter →

arxiv 1908.04114 v5 pith:XVR472KX submitted 2019-08-12 quant-ph

Niraj Kumar This is my paper · ORCID
classification quant-ph PACS 03.67.Dd
keywords quantummoneyclassicalverificationSamplingMatchinglinearopticsunconditionalsecuritynoisetoleranceweakcoherentstatessingle-photon
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 claims that the Sampling Matching problem, a one-way communication task, can be turned into the verification step of a private quantum money scheme. The scheme uses single-photon states that encode secret n-bit strings, and the verification requires only one round of classical communication between the local verifier and the Bank. The claimed payoff is unconditional security against any forging adversary while tolerating noise up to 21.4%, achieved with a fixed circuit made of passive 50/50 beam splitters; in the weak-coherent-state implementation the circuit shrinks to a single beam splitter and two single-photon threshold detectors. This matters because previous matching-based money schemes needed more optical components as the desired noise tolerance grew, which made high-robustness versions impractical.

What carries the argument

The load-bearing object is the Sampling Matching verification scheme: the noteholder's single-photon state is mixed mode-by-mode with a verifier-prepared equal-amplitude local state on 50/50 beam splitters, and the pattern of two-photon clicks—both in the same output family versus one photon in each family—yields the parity $x_k \oplus x_l$ for a sampled tuple $(k,l)\in T_n$. The quantitative engine is the conversion of forging into a fidelity-maximization SDP, whose bound $\bar F \le \frac12 + \frac1n$ directly lower-bounds the adversary's error probability; the fixed-hardware claim comes from the coherent-state version of Sampling Matching, where the whole interaction is a single beam splitter followed by two threshold detectors.

What would settle it

Send the verifier a single copy prepared as a two-photon state with one photon in mode $k$ and one in mode $l$ ($k\neq l$), then run the Sampling Matching test. It yields two single-photon clicks, so it passes the local two-click check; if such a two-photon state, or a mixture of two-photon states, can also make the Bank's parity comparison succeed often enough, the claimed unconditional security against all adversaries is false. This is directly checkable with the weak-coherent-state setup.

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Extended reading notes

Core claim

The paper's central claim is that Sampling Matching can serve as the verification primitive for private quantum money in a way that is both equipment-light and unconditionally secure. The Bank encodes each secret string $x\in\{0,1\}^n$ as a single-photon state $|x\rangle = \frac{1}{\sqrt n}\sum_{k=1}^n (-1)^{x_k}\hat a_k^\dagger|0\rangle$; the verifier interferes this state mode-by-mode with his own equal-superposition state through 50/50 beam splitters. Two single-photon clicks in distinct output modes reveal the parity $x_k \oplus x_l$ of a sampled tuple, while two photons in one mode give no information. The security proof reduces any forging strategy to maximizing the average fidelity with the honest state, imports the bound $\bar F \le \frac12 + \frac1n$, and shows that the forger's success probability decays exponentially whenever the honest success rate exceeds the forger's by the noise tolerance $\frac14 - \frac{1}{2n}$; at $n=14$ that tolerance is 21.4%. In the weak-coherent-state implementation, the same verification uses a single 50/50 beam splitter and two single-photon threshold detectors, independent of the note size.

Load-bearing premise

The proof assumes that any forger must send exactly one photon distributed over the $n$ modes for each copy, because it only analyzes single-photon mixed states; the verifier's local test, however, does not reject states with two photons in two different modes, which also produce two single-photon clicks.

Editorial extensions

If this is right

  • If the central claim is correct, a fixed passive linear-optical circuit—one 50/50 beam splitter and two threshold detectors in the coherent-state version—suffices to verify quantum money at any noise tolerance up to the threshold, whereas earlier matching-based schemes needed more optical elements as the tolerance grew.
  • Because verification uses a single round of classical communication, the Bank does not need to keep an active memory of the interaction, so many local verifiers can authenticate notes concurrently.
  • Each note can be reused for a number of verifications linear in its size, with the Bank capping the total number of attempts before the note is retired.
  • The security guarantee is information-theoretic rather than computational, so it would survive an adversary with unbounded quantum computing power.
  • If the fidelity bound $\bar F \le \frac12 + \frac1n$ holds for all $n$, the scheme's asymptotic noise tolerance reaches 25%, the conjectured ceiling for matching-based money schemes.

Reading between the lines

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

  • Inference: Since the coherent-state Sampling Matching circuit has already been demonstrated with weak pulses, running the verification at $n=14$ and measuring the honest-holder error rate would turn the 21.4% tolerance from a theoretical number into a directly testable one.
  • Inference: The same parity-sampling game could plausibly be reused for other linear-optics verification tasks—checking that a prover knows an encoded string while revealing only one parity at a time—so the fixed-circuit simplification may outlive this particular money scheme.
  • Inference: If the fidelity bound $\bar F \le \frac12 + \frac1n$ is ever proven for all $n$, the fixed circuit would achieve the conjectured matching-scheme ceiling of 25% noise tolerance without any increase in optical hardware.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript introduces a private quantum money mini-scheme whose verification is based on the Sampling Matching problem. The Bank prepares q copies of single-photon superposition states encoding random n-bit strings; a verifier interferes each selected copy with a local reference state through 50/50 beam splitters, checks the click statistics, and forwards parity outcomes to the Bank. The paper claims information-theoretic unforgeability with a noise tolerance of 21.4%, a fixed passive linear-optical verification circuit, and re-usability that grows linearly with note size. A practical implementation using weak coherent states, a single 50/50 beam splitter, and two threshold detectors is also described. Correctness is analyzed with Chernoff bounds; unforgeability is reduced to a fidelity bound imported from [AA17], with a short argument that coherent attacks reduce to collective attacks. Section 3.4 explicitly states that no full security proof is provided for the coherent-state scheme.

Significance. If the security proof were complete, the scheme would be a genuinely useful simplification of matching-based quantum money: a fixed passive linear-optical verification circuit, a single round of classical communication with the Bank, and a claimed noise tolerance comparable to or better than previous schemes. The correctness calculation is simple, and the circuit simplification for the coherent-state implementation is attractive. However, the advertised unconditional security is not established by the manuscript. The central adversary bound is imported from another paper and is only numerically verified and partly conjectured; the proof for the practical threshold-detector implementation explicitly disclaims a full analysis; and the reduction from coherent to collective attacks is asserted rather than proved. These are load-bearing gaps in the central claim, not presentation issues.

major comments (4)
  1. [Section 3.4 / Abstract] The abstract promises 'unconditional security' for a practically feasible scheme using weak coherent states, but Section 3.4 states: 'Here we are not providing full security proof of our quantum money scheme using coherent states.' The security analysis in Section 3.3 is formulated for single-photon states of the form Eq. (22) and does not analyze the vacuum or multi-photon components of the coherent states used in Eq. (43). Since the coherent-state implementation is exactly the one advertised as practical, the central practical claim is unsupported by the proof given in the manuscript.
  2. [Section 3.3.1, Eq. (33)] The main adversary bound, Eq. (33), is taken directly from [AA17] as a numerically verified inequality for n <= 14 and as a conjecture for larger n. The paper provides no proof or rigorous numerical certification of this SDP bound and no detailed demonstration that the SDP optimization for the Sampling Matching scheme is identical to the one solved in [AA17]. Since Eq. (33) is the step that converts the fidelity calculation into an upper bound on the forging probability, the claimed 'unconditional' security is conditional on an unproven external bound. This is a substantive gap for an information-theoretic security claim.
  3. [Section 3.3.1, before Eq. (21)] The proof restricts the adversary to single-photon states of the form Eq. (22), justified by the assertion that any other state fails the step 5 check of the Verification phase with certainty. This assertion is valid only for ideal photon-number-resolving detection. The practical implementation in Section 3.4 uses threshold detectors, which do not resolve photon number, so a multi-photon component distributed over distinct modes can in principle produce the required click pattern. The reduction to Eq. (22) therefore does not cover the threshold-detector implementation, and the manuscript needs either a security analysis of multi-photon adversarial states in that setting or an explicit statement that the proof applies only to the PNR implementation.
  4. [Section 3.3.2] The reduction from coherent attacks to collective attacks is handled in one paragraph asserting that the Croke-Kent teleportation argument applies because the setting is 'very similar.' No formal mapping between the money-forging game and the bit-commitment setting is given, and the role of two independent verifiers together with authenticated classical communication to the Bank is not addressed. Since this reduction is needed to rule out arbitrary joint operations across the q' copies, the proof of unconditional security is incomplete at this step as well.
minor comments (4)
  1. [Definition 2.2, Eq. (4)] Equation (4) writes the second verification event as VerB_H($1) = 1 twice; the second event should be VerB_H($2) = 1.
  2. [Section 4, Eq. (41)] In the measure-and-resend example for n = 4, the adversary state of Eq. (40) provides correct parity information only for the measured tuple. Since the verifier can obtain any of the six tuples, the error probability is 5/12, not 1/3 as stated in Eq. (41).
  3. [Section 3.4, Fig. 7] The text repeatedly refers to 'single-photon clicks' in the coherent-state implementation, but threshold detectors are not photon-number-resolving and produce only binary click/no-click outcomes. The wording should be adjusted to avoid implying number resolution.
  4. [Throughout] There are several typographical errors, including 'upto' in the abstract, 'mdoe' in Section 2.2.2, 'prarity' in Section 3.3.2, and some inconsistent pronoun use. A careful proofreading pass is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the security proof is derived from the paper's own Sampling Matching analysis plus the external AA17 SDP bound, and KKD19 self-citations are background/implementation support rather than load-bearing.

full rationale

The derivation chain is not circular. The unforgeability proof in Section 3.3 does not assume the theorem it is trying to prove: it computes the adversary's per-copy error probability directly from the Sampling Matching output operator in Section 2.2.2 (Eqs. (9)-(14), (24)-(31)), then imports only the numerical fidelity bound F-bar <= 1/2 + 1/n for n <= 14 from Amiri and Arrazola [AA17], an external source; the paper explicitly says 'Here we directly use the SDP result of [AA17]'. The 21.4% noise tolerance is a consequence of that external bound for n = 14, not a fitted parameter. Citations to the author's own KKD19 work supply the Sampling Matching primitive and the coherent-state implementation, but the paper restates the SM scheme in Section 2.2.2 and describes the coherent-state circuit in Section 3.4 with equations; KKD19 is an experimentally demonstrated, published result, so it counts as independent support under the review rules rather than a load-bearing self-citation. The verification thresholds (l_min and delta) are defined from honest-holder expectations and are standard correctness checks, not predictions of security. The only notable gaps: the absence of a full security proof for the coherent-state scheme (Section 3.4: 'Here we are not providing full security proof of our quantum money scheme using coherent states') and the proof's restriction to single-photon states of the form Eq. (22) are completeness and correctness limitations, not circular reductions. Therefore the circularity score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The main burden is the unproven SDP bound from AA17 and the unjustified single-photon restriction in the security proof. Protocol parameters n, delta, and epsilon are hand-picked, not fitted to data.

free parameters (3)
  • n (string length / note size) = 14 for the claimed 21.4% tolerance
    The noise tolerance is computed as 1/4 - 1/(2n); the paper sets n=14 to obtain 21.4%, and relies on a conjecture for larger n.
  • delta (Bank cutoff) = (c - c_adv)/2
    The cutoff in Eq. (38) is chosen based on the gap between honest and adversarial success probabilities, which itself depends on the conjectured SDP bound.
  • epsilon (local test security factor) = not specified numerically
    Appears in the Chernoff bound for the local test and controls the honest-failure probability; its value is left to the implementer.
assumptions (6)
  • standard math Chernoff-Hoeffding bound
    Used in correctness and unforgeability calculations (Section 3.2, Eq. 19, Eq. 38).
  • standard math Quantum mechanics and no-cloning
    Underpins the entire quantum money setting and the Holevo bound used in Section 3.3.4.
  • domain assumption Ben-David and Sattath reduction from mini-scheme to full scheme requires collision-resistant hash functions secure against quantum adversaries
    The paper relies on BDS16 to lift security of the mini-scheme to a full money scheme, and acknowledges the hash function assumption (Section 2.1).
  • domain assumption The SDP bound Eq. (33): average fidelity <= 1/2 + 1/n, numerically verified for n<=14 and conjectured for all n
    The security bound and the 21.4% noise tolerance directly inherit this result from AA17; it is not proven in this paper.
  • domain assumption Croke and Kent proof that coherent attacks are no more powerful than collective attacks
    The paper states that 'our setting is very similar to their construction' and uses CK12 without a detailed reduction (Section 3.3.2).
  • ad hoc to paper Adversarial states can be restricted to single-photon states
    Section 3.3.1 asserts that an adversary must send a single-photon state or fail step 5 with certainty; this is not proven and is in fact false for multi-photon states in distinct modes.

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Cite this review

Pith. "Pith review of Practically feasible robust quantum money with classical verification." pith.science (2026). https://pith.science/paper/XVR472KX

@misc{pith2026190804114,
  author       = {Pith},
  title        = {Pith review of: Practically feasible robust quantum money with classical verification},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XVR472KX}},
  note         = {Machine review of arXiv:1908.04114}
}
read the original abstract

We introduce a private quantum money scheme with the note verification procedure based on Sampling Matching, a problem in the one-way communication complexity model introduced by Kumar et al.[Nature Communications 10, Article number: 4152]. Our scheme involves a Bank that produces and distributes quantum notes, noteholders who are untrusted and trusted local verifiers of the Bank to whom the holders send their notes in order to carry out transactions. The key aspects of our money scheme include: note verification procedure requiring a single round classical interaction between the local verifier and Bank; fixed verification circuit that uses only passive linear optical components; re-usability of each note in our scheme which grows linearly with the size of note, and an unconditional security against any adversary trying to forge the banknote while tolerating the noise of up to 21.4%. We further describe a practical implementation technique of our money scheme using weak coherent states of light and the verification circuit involving a single 50/50beam splitter and 2 single-photon threshold detectors. Previous best-known matching based money scheme proposal [AA17] involves a verification circuit where the number of optical components increase proportionally to the increase in desired noise tolerance (robustness). In contrast, we achieve any desired noise tolerance (up to a maximal threshold value) with only a fixed number of optical components. This considerable reduction of components in our scheme enables us to reach the robustness values that are not feasible for any existing money scheme with the current technology.

Figures

Figures reproduced from arXiv: 1908.04114 by the authors.

Figure 1
Figure 1. Sampling matching problem. Alice receives an input [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. An example of all possible distinct tuples for the size [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Superposition circuit of Alice to create a single-photon state in equal superposition over [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Sampling Matching circuit model in single-photon encoding. Alice encodes a secret string [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Illustration of a 50/50 beam splitter transforming input modes [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: Illustration of our private quantum money scheme based on the verification protocol using the [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: Sampling Matching (SM) circuit implementation using weak coherent states, 50/50 beam splitter [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: Illustration of our quantum money scheme based on the verification protocol using the SM [PITH_FULL_IMAGE:figures/full_fig_p022_8.png]

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Reference graph

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