General sending-or-not-sending twin field protocol for quantum key distribution with asymmetric source parameters

Cong Jiang; Xiang-Bin Wang; Xiao-Long Hu; Zong-Wen Yu

REVIEW 3 major objections 5 minor 96 references

General sending-or-not-sending twin field protocol for quantum key distribution with asymmetric source parameters

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that sending-or-not-sending twin-field QKD stays secure under asymmetric source parameters when a single ratio condition holds, and that this gives large key-rate gains on unbalanced channels.

desk verdict A clean generalization of SNS twin-field QKD to asymmetric source parameters with a simple ratio constraint; the main open question is whether the decoy-state step in the proof is fully justified in the post-selected setting. read the letter →

arxiv 1908.05073 v1 pith:FPGTL3FH submitted 2019-08-14 quant-ph

Xiao-Long Hu , Cong Jiang , Zong-Wen Yu , Xiang-Bin Wang This is my paper · ORCID

classification quant-ph PACS 03.67.Dd42.81.Gs03.67.Hk

keywords twin-fieldquantumkeydistributionsending-or-not-sendingprotocolasymmetricsourceparametersdecoy-statemethodsecurityproofphase-fliperrorratechannelsfinite-keyanalysis

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 extends the sending-or-not-sending (SNS) twin-field quantum key distribution protocol to let Alice and Bob use different source intensities and different sending probabilities. Its main claim is that the protocol stays secure provided the source parameters satisfy the single ratio condition $\mu_{Ak}/\mu_{Bk} = [\epsilon_A(1-\epsilon_B)\mu'_A e^{-\mu'_A}]/[\epsilon_B(1-\epsilon_A)\mu'_B e^{-\mu'_B}]$ for each decoy intensity $k>0$. The condition makes the single-photon part of the signal windows match the $|\chi^+\rangle$ state controlled by the decoy windows, so the decoy-state and tagged-model arguments from the original protocol carry over. The practical consequence is for channels whose two arms differ: numerical simulation with a 100 km length difference gives key rates tens to hundreds of times higher than the original SNS protocol applied to the same setup. The paper also supplies a four-intensity decoy-state parameter-estimation method and finite-key formulas, so the generalization is directly usable in experiments.

What carries the argument

The load-bearing mechanism is the ratio-matching condition of Eq. (1), placed on the source parameters so that, for every decoy intensity $k>0$, the ratio $\mu_{Ak}/\mu_{Bk}$ equals the ratio of the single-photon weights $\epsilon_A(1-\epsilon_B)\mu'_A e^{-\mu'_A}$ and $\epsilon_B(1-\epsilon_A)\mu'_B e^{-\mu'_B}$. This makes the single-photon component of the effective Z-window state coincide with the $|\chi^+\rangle$ state in the X-window decomposition, a two-mode superposition of exactly one photon on Alice's side or Bob's side. With that match, the tagged-model decomposition $\Omega=\sum_r q_r\Omega_r$ reduces security of the full mixed source state to security of its single-photon part, and the decoy-state method estimates the phase-flip error rate $e^{ph}_1$ from X-window data. The final key length is then $N_f = n_1[1-H(e^{ph}_1)] - f n_t H(E_Z)$, with $n_1$ the estimated number of single-photon effective events and $E_Z$ the bit-flip error rate in signal windows.

What would settle it

Prepare single-photon-level states directly in the $|\chi^+\rangle$ form and send them through a realistic asymmetric channel, then compare their observed error rate with the upper bound inferred from the four-intensity decoy analysis under Eq. (1); if the inferred phase-flip rate systematically underestimates the directly measured error rate of the $Z_1$ events, the equivalence at the center of the proof fails. A numerical search over channel models with intensity-dependent loss that preserves the decoy counting rates while changing the single-photon component would settle the same question.

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Extended reading notes

Core claim

The central claim is that symmetric source parameters are not needed for secure sending-or-not-sending twin-field QKD; what is needed is one ratio condition on the two users' intensities and sending probabilities. Under that condition, the single-photon component of the signal-window state is proportional to $\mu_{A1}|10\rangle\langle10|\otimes|10\rangle\langle10| + \mu_{B1}|01\rangle\langle01|\otimes|01\rangle\langle01|$, which is exactly the $|\chi^+\rangle$ component controlled by the decoy-window analysis. The security proof proceeds through virtual protocols and reductions that connect the effective Z-window state to the X-window decoy states, so the phase-flip error rate of the key bits can be bounded from observed decoy data. Finite-key formulas are then given, and numerical simulation shows large key-rate gains for asymmetric channel losses.

Load-bearing premise

The proof depends on the assumption that the single-photon component of the signal windows is statistically interchangeable with the $|\chi^+\rangle$ state used in the decoy windows, so that the phase error rate measured on decoy data genuinely bounds the phase error of the key bits.

Editorial extensions

If this is right

Alice and Bob can choose source intensities matched to their own channel losses instead of being forced to use identical settings.
On a 100 km difference between the two channel lengths, the optimized key rate is tens to hundreds of times higher than the original SNS protocol in the paper's simulations.
The four-intensity decoy-state formulas give explicit lower and upper bounds for the single-photon counting rate and phase-flip error rate, so the protocol is implementable with finite data.
Setting Alice's and Bob's parameters equal recovers the original SNS protocol as a special case.
Because the security reduction does not depend on the untrusted third party's honesty, the measurement-device-independent character of SNS twin-field QKD is preserved in the asymmetric-source version.

Reading between the lines

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

My inference: the ratio condition can be read as an impedance-matching condition between the two arms, and the same construction could be carried over to related twin-field variants that currently assume identical sources by identifying their analogous single-photon component.
My inference: an adaptive implementation could recompute source intensities in real time to keep Eq. (1) satisfied as channel losses drift, which would make the protocol more robust in field deployment than fixed symmetric settings.
My inference: the numerical gap over the approach of adding compensating loss to the shorter channel grows with channel asymmetry, so the protocol is most valuable precisely when the two arms are very different.

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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

Xu — quick take on Hu et al. 1908.05073. The real content is a single constraint, Eq (1), that lets Alice and Bob use different intensities and sending probabilities in SNS twin-field QKD while keeping the single-photon components of the Z and X windows aligned. The extension is non-obvious, and it pays off: for a 100 km channel imbalance the key rate jumps by tens to hundreds of times compared to the original or loss-modulated variant. The numerical comparisons are honest and finite-size formulas are included.

What's genuinely new is the security proof for asymmetric parameters. It follows the standard virtual-protocol chain — tagged model, quasi-purification, decoy-state estimation — and the appendix gives the four-intensity formulas. The mathematics looks plausible, a solid within-subfield contribution. The connection to ref [78] is mishandled: the authors cite a 2019 PRA that likely covers the same asymmetric scenario and never say what is added. That needs to be fixed; if ref [78] truly subsumes the result, the novelty claim collapses.

The soft spot that matters is the decoy-state step. The paper says in Step 5 that the method applies 'as if the phases δA and δB were not announced' and points to ref [68], but it does not reproduce the argument or adapt it to the asymmetric ratio. The stress-test note raises the worry that Charlie could use a phase-dependent POVM after the phases are announced. I don't think that attack works as stated, because the protocol has Charlie announce his measurement result before the phases are revealed, so his POVM is fixed and cannot depend on the private phases. But the deeper concern is not timing; it's that the post-selected X-window ensemble (the tilde-X windows) is not a phase-randomized mixture of Fock states. Observed error counts in that set are not automatically Poisson-weighted sums of photon-number yields. The proof needs an explicit statement of why the decoy-state linearity survives post-selection. The original ref [68] may cover it, but the paper only cites it. A referee should ask for a self-contained argument or a precise quotation of the relevant theorem.

Overall, the central idea is clean and the numbers make it worth reading. The gaps are presentational and verification-level, not a clear fatal flaw. I'd bring it to a reading group and I'd want it peer-reviewed, with the authors pushed on the ref [78] overlap and the decoy-state justification.

Referee Report

3 major / 5 minor

Summary. The paper proposes a general version of the sending-or-not-sending (SNS) twin-field QKD protocol in which Alice and Bob are allowed to use different source intensities and sending probabilities, subject to the ratio constraint Eq. (1). The authors give a security proof via a sequence of virtual protocols, a four-intensity decoy-state parameter estimation, and a finite-key analysis using the Chernoff bound. Numerical simulations for asymmetric channel lengths show a large key-rate improvement over the original SNS protocol and over a 'modified SNS' protocol that adds extra loss to balance the channels.

Significance. If the security proof is correct, the protocol is a practically valuable extension: it removes the equal-source-parameter restriction of the original SNS protocol and substantially improves key rates in asymmetric channel deployments, which are common in real QKD networks. The design constraint Eq. (1) is simple and the finite-key formulas are provided. The paper is clearly written and follows the standard virtual-protocol structure that the authors and others have used for SNS/TFQKD proofs. The main unresolved issue is the rigor of the decoy-state reduction in the asymmetric-parameter setting, which is load-bearing for the central security claim.

major comments (3)

[Sec. III.C (Virtual Protocol 3, Reduction 3) and Step 5 Note, Sec. II; Appendix Eq. (40)] The paper asserts that the decoy-state method can be applied 'as if the phases δA and δB were not announced' (Step 5 Note), but it does not prove this for the asymmetric-parameter case. The bound on the phase-flip error rate in Eq. (40) is obtained by subtracting the vacuum contribution from the observed error counting rate T_Δ; this is a valid one-sided bound only if the observed error rate is a Poisson-weighted sum over total-photon-number components, i.e., if Charlie's measurement is block-diagonal in total photon number. The protocol does not explicitly prevent Charlie from storing the pulses and performing a phase-dependent measurement after the phases δA, δB are announced. Under such a strategy, the POVM can have coherences between different total photon numbers and Eq. (40) may not hold. Because Eq. (1) changes the weights of the photon-number components relative to the symmetric SNS proof, the citation to Ref. [68] is not sufficient. The authors need to provide an explicit derivation of Eq. (40) under arbitrary adversarial channels, or modify the protocol/analysis to close this gap.
[Sec. III.A, Eqs. (18)-(21)] In the asymmetric case, the real-photon states |χ0⟩ and |χ1⟩ are not orthogonal: their inner product is (μA1−μB1)/(μA1+μB1). The paper uses these states to define the phase-flip error rate and to relate the X-window data to the Z-window phase error, but it does not discuss how the non-orthogonality affects the estimates. The original SNS proof relies on the symmetric case where these states are orthogonal (μA1=μB1). The authors should justify that the formula for E(a,d) and the reductions leading to Eq. (31) remain valid when the two states are not orthogonal, or show explicitly why the non-orthogonality is irrelevant for the security argument.
[Sec. III.C, Eq. (32)-(33)] The decomposition of the phase-randomized coherent-state density matrix into the mixture {|ψ_l⟩} is stated without derivation and appears to be incorrect as written: the phase-randomized state is diagonal in the Fock basis, whereas the states |ψ_l⟩ in Eq. (33) contain coherences within each total-photon-number subspace. The decoy-state method requires well-defined yields for the components of the mixture. This decomposition is the basis for the claim that the single-photon component |ψ_1⟩ is exactly |χ+⟩ under Eq. (1), which is central to the security proof. The authors should provide a careful derivation of the decomposition and of the resulting decoy-state estimates, rather than citing Ref. [68] without addressing the asymmetric weights.

minor comments (5)

[Sec. II, Eq. (4)] Equation (4) contains a typographical issue: 'fn tH(EZ)' should read 'f n_t H(E_Z)' (i.e., f times n_t times H(E_Z)). Similar spacing issues appear in Eqs. (35) and (36).
[Sec. II, Step 1] The description of the bit values is confusing: 'she (he) decides to send ... and puts down a bit value 1 (0)' followed by the opposite for not sending. It would be clearer to define Alice's and Bob's bit values explicitly in a table or a short formula.
[Appendix A, Eqs. (37)-(39)] The notation for lower bounds is ambiguous: in Eq. (37), the same symbol ⟨s^Z_1⟩ appears on both sides of the inequality, which is confusing. The authors should use distinct notations for the quantity and its lower bound (e.g., an underline or superscript L), as is common in the decoy-state literature.
[Sec. IV, Table I] The table header 'Nt ed d ηd fe ξ α' is hard to parse; it would be clearer to separate the two dark-count-related parameters as 'e_d' and 'd' explicitly in the header, matching the caption.
[Sec. V] The conclusion states that 'the intensities and the probabilities for sending' should satisfy Eq. (1), but Eq. (1) is a constraint on the intensities only; the sending probabilities appear in the ratio but are not independently constrained. The sentence should be phrased more precisely.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Eq. (1) is a designed constraint, not a fitted prediction, and the security proof rests on a new reduction plus standard decoy-state/GLLP lemmas.

full rationale

The central claim is the sufficiency of the parameter-ratio constraint Eq. (1) for security of the asymmetric SNS protocol. That constraint is not fitted to the target key rate or to the security conclusion; it is imposed as a mathematical condition and used in Reduction 3 (Sec. III.C) to make the single-photon component |ψ_1> of the phase-randomized X-window state coincide with the |χ+> component of the Z-window entangled state. The subsequent phase-error bound Eq. (40) is a standard decoy-state estimate; it does not rename an input or reuse the output. The only load-bearing self-citation is the Step 5 Note ('As proved in ref. [68], decoy state method can applied to our protocol as if the phases δA and δB were not announced'), together with the use of refs. [84,85] for finite-size formulas. These are parameter-free lemmas from the authors' prior SNS work, applied to the same phase-randomized-coherent-state structure; the novel asymmetric step (Eq. (1) and the Reduction 3 identification) is supplied in this paper, so the argument does not reduce to the cited results. The skeptical concern about phase-dependent adversarial POVMs is a correctness/security-model question, not a circularity: nothing in the proof is defined in terms of the conclusion or fitted to the predicted key rates. The numerical improvement over the original SNS protocol is a simulation result, not a fitted prediction, and therefore does not create circularity.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The protocol introduces no new physical entities. The free parameters listed are optimization variables in the numerical simulation; the central security proof does not depend on their specific values. The axioms are standard assumptions in decoy-state QKD, with the notable new restriction being the ratio constraint Eq (1).

free parameters (5)

Signal intensities µ'A, µ'B = Not stated
Optimized in the numerical simulation for maximum key rate; values not tabulated in the paper.
Sending probabilities ǫA, ǫB = Not stated
Optimized in the simulation; constrained by Eq (1) to relate to decoy intensities.
Decoy intensities µA1, µB1, µA2, µB2 = Not stated
Used in the four-intensity decoy-state method; the ratio µA1/µB1 must satisfy Eq (1).
Window probabilities pZ_A, pZ_B, pX_Ak, pX_Bk = Not stated
Probability of choosing signal or decoy windows; optimized in the simulation.
Postselection parameter λ and phase offset Δϕ = Not stated
Chosen by Alice and Bob based on channel calibration to balance key rate and error tolerance; the paper sets Δϕ=0 for simplicity.

assumptions (4)

domain assumption Private phases δA and δB are uniformly random and secret, so phase-randomized coherent states are classical mixtures of Fock states.
Stated in Section II: the private phases are random and kept secret; this is essential for the density-matrix decomposition in Eq (6) and for the decoy-state analysis.
domain assumption Channel behavior depends only on the input intensities and is independent of the users' basis choices and intensity settings (decoy-state stationarity).
Used implicitly throughout Section IV and the Appendix to estimate counting rates and error rates from observed data; standard in decoy-state QKD but not proven in this paper.
standard math The GLLP tagging model applies to the decomposition of the Z-window state Ω into tagged and untagged parts.
The security proof uses the tagged model (refs [10,11]) to handle the mixture Ω = Σ q_r Ω_r in Eq (8), assuming the untagged part Ω_1 can be distilled securely.
standard math Universally composable security framework and Chernoff bound for finite-size analysis.
The finite-key analysis in Appendix C relies on the universally composable definition (ref [87]) and Chernoff bounds (ref [86]) with given failure probabilities.

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Cite this review

Pith. "Pith review of General sending-or-not-sending twin field protocol for quantum key distribution with asymmetric source parameters." pith.science (2026). https://pith.science/paper/FPGTL3FH

@misc{pith2026190805073,
  author       = {Pith},
  title        = {Pith review of: General sending-or-not-sending twin field protocol for quantum key distribution with asymmetric source parameters},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FPGTL3FH}},
  note         = {Machine review of arXiv:1908.05073}
}

read the original abstract

The sending-or-not-sending (SNS) protocol of the twin-field quantum key distribution (TFQKD) can tolerant large misalignment error and its key rate can exceed the bound of repeaterless QKD. But the original SNS protocol requires the two users to use the same source parameters. Here we propose a general protocol with asymmetric source parameters and give the security proof of this protocol. Our general protocol has a much better performance than that of the original SNS protocol when the channel of the system is asymmetric.

Figures

Figures reproduced from arXiv: 1908.05073 by the authors.

**Figure 1.** FIG. 1: A schematic of the setup for the general SNS pro [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2: (Color online)The optimized key rates (per pulse pai [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

Figures from the paper (1 more)

**Figure 3.** Figure 3: FIG. 3: (Color online)The optimized key rates (per pulse pai [PITH_FULL_IMAGE:figures/full_fig_p008_3.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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[3] [1]

They preshare an extended state Ω 0 = |Ψ ⟩⟨Ψ | (14) where |Ψ ⟩ =C(√ µA1eiγ A|10⟩ ⊗ |10⟩ + √µB1eiγ B |01⟩ ⊗ |01⟩) (15) and γA and γB are announced publicly

Preparation stage For each time window i, they preshare the classical information about whether this time window is an X- window or Z-window. They preshare an extended state Ω 0 = |Ψ ⟩⟨Ψ | (14) where |Ψ ⟩ =C(√ µA1eiγ A|10⟩ ⊗ |10⟩ + √µB1eiγ B |01⟩ ⊗ |01⟩) (15) and γA and γB are announced publicly. (Remind that Ω 0 and |Ψ ⟩ vary in diﬀerent time windows.) I...

[4] [2]

(16) to Charlie and keep the ancillary photons locally

Protocol V1-1: At any Z-window (X-window), they send the real photons of Ω Z (Ω X ) deﬁned in Eq. (16) to Charlie and keep the ancillary photons locally. V1-2: Charlie measures the real photons from Alice and Bob with a beam splitter after taking phase com- pensation according to the strong reference lights with phases γA and γB. He announces his measurem...

[5] [3]

They preshare an extended state Ω Z in Eq.(16) for Z- windows, an real-photon state |χ+⟩ for X0-windows and an real state |χ−⟩ for X1-windows

Preparation stage For each time window i, they preshare the classical information about whether this time window is an X0- window, an X1-window or a Z-window. They preshare an extended state Ω Z in Eq.(16) for Z- windows, an real-photon state |χ+⟩ for X0-windows and an real state |χ−⟩ for X1-windows

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At any X0-window (X1-window), they send |χ+⟩ (|χ−⟩) to Charlie

Protocol V2-1: At any Z-window, they send out the real pho- tons of Ω Z to Charlie and keep the ancillary photons locally. At any X0-window (X1-window), they send |χ+⟩ (|χ−⟩) to Charlie. V2-2: Charlie measures the real photons from Alice and Bob with a beam splitter after taking phase com- pensation according to the strong reference lights with phases γA ...

[7] [5]

the left detector clicks and cos( δA −δB)< 0

[8] [6]

Therefore, we can deﬁne these two kinds of eﬀective events as error events and the corresponding time windows are deﬁned as error windows

the right detector clicks and cos( δA −δB) ≥ 0. Therefore, we can deﬁne these two kinds of eﬀective events as error events and the corresponding time windows are deﬁned as error windows . If they set the value of λ small enough and Charlie perform the compensation honestly, they may get few error events so that the phase-ﬂip error rate will be quite low a...