Continuous-Variable Source-Independent Quantum Random Number Generator with a Single Phase-Insensitive Detector

Hongyi Zhou

REVIEW 5 major objections 5 minor 1 cited by

Continuous-Variable Source-Independent Quantum Random Number Generator with a Single Phase-Insensitive Detector

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A single phase-insensitive detector can certify randomness from an untrusted source.

desk verdict Promising protocol and the right dimension-reduction idea, but the central security proof as printed is not valid: the dual in Eq. (13) does not bound the primal, and Eq. (8) states an equality that is only an inequality. read the letter →

arxiv 2411.14817 v1 pith:E3I4RPDJ submitted 2024-11-22 quant-ph

Hongyi Zhou This is my paper · ORCID

classification quant-ph

keywords source-independentquantumrandomnumbergeneratorcontinuous-variablecryptographysemi-definiteprogrammingphase-insensitivedetectordimensionreductionmin-entropyFockbasistime-multiplexed

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 proposes a source-independent quantum random number generator that uses only a single phase-insensitive optical detector, such as a single-photon detector, and proves its security with semidefinite programming. The technical core is a dimension-reduction argument: because the detector's POVM is diagonal in the Fock basis, the infinite-dimensional guessing-probability optimization can be rigorously bounded by a finite-dimensional SDP whose optimum gives a guaranteed lower bound on the min-entropy. The bound holds for any source state compatible with the observed detection statistics and with the estimated mean photon number. Simulated time-multiplexed single-photon detection shows extraction rates above $10^{-2}$ bits per sample for typical weak coherent states. The scheme matters because it removes the need for conjugate measurements or a squashing model and simplifies practical implementation.

What carries the argument

The load-bearing object is a dimension-reduction bound on the probability weight of a source state living above $N$ photons. For a POVM diagonal in the Fock basis, the projection onto $N$ photons commutes with each $M_j$; Hölder's inequality gives $\operatorname{tr}(M_j \bar P \rho \bar P) \le \|\rho_{\bar N}\|_1 \|M_{j,\bar N}\|_\infty$, and the tail weight $\|\rho_{\bar N}\|_1$ is bounded by $\langle n\rangle/N$ via the dual program Eq. (11) with feasible point $x=0$, $y=1/N$. These two inequalities turn the infinite-dimensional SDP (5) into the finite-dimensional SDP (12), whose dual (13) is solved to obtain $d^*_N$.

What would settle it

Use a two-mode source engineered to give the same single-mode click probabilities and mean photon number while placing more than $\langle n\rangle/N$ weight above the cutoff; if its true guessing probability exceeds $d^*_N$, the finite-dimensional constraints have missed part of the feasible set.

Watch

Extended reading notes

Core claim

The central claim is that for any single-mode source state whose photon-number statistics match the observed outcomes and whose mean photon number is $\langle n\rangle$, the optimal value $d^*_N$ of the finite-dimensional SDP in Eq. (12), and its dual in Eq. (13), is an upper bound on the guessing probability of the infinite-dimensional problem, so $H_{\min}(A|E) \ge -\log_2 d^*_N$. The argument splits each optimal sub-state into an $N$-photon part and a tail; the tail contributes at most $\langle n\rangle/N$ through the feasible point $x=0$, $y=1/N$ of the dual tail-weight program, and that slack is folded into loosened constraints. The bound is therefore valid even though the optimization is truncated to photon numbers below $N$.

Load-bearing premise

The proof assumes the source is a single optical mode and that the mean photon number is known exactly, because the tail-weight bound $\langle n\rangle/N$ relies on that dual feasible point.

Editorial extensions

If this is right

Any source state compatible with the measured outcome probabilities and mean photon number is covered by the same bound, so the security statement is not tied to the coherent-state simulation.
Detectors whose POVM elements are diagonal in the Fock basis, including common single-photon detectors, can serve as the measurement, so no squashing model is needed.
The randomness yield per sample is at least $-\log_2 d^*_N$ in the asymptotic limit, and solving the dual problem avoids relying on the exact optimum of the truncated primal under finite precision.
The simulation indicates that with a time-multiplexed detector and weak coherent states the lower bound exceeds $10^{-2}$ bits per sample, pointing to practical rates.

Reading between the lines

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

A possible extension is to replace the single-mode tail bound with a multi-mode bound, tracking the photon-number operator per mode; the single-mode assumption is load-bearing in the present proof.
The asymptotic argument could be turned into a finite-size bound by estimating the mean photon number and outcome probabilities from a finite block and propagating confidence intervals through the loosened constraint, which the paper does not carry out.
The same projection-and-tail idea may extend to POVMs that are not Fock-diagonal if off-diagonal terms are bounded separately, which would broaden the detector classes beyond phase-insensitive ones.

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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 core idea here is sensible and worth pursuing: combine the dimension-reduction trick from Upadhyaya et al. with a single phase-insensitive detector to get a CV-SI-QRNG that avoids squashing models. The identification of diagonal-Fock POVMs as the compatible class is a genuine, practically useful observation, and the time-multiplexed detector simulation is a reasonable illustration of the framework. That part I like.

The problem is that the security proof, as printed, does not work. The reader's stress-test is correct: Eq. (13) is not a valid dual of Eq. (12). Take m=2, M1=0.9I, M2=0.1I, N=20, nbar=0, and vacuum statistics p1=0.9, p2=0.1. The primal has value 0.9 (rho1=|0><0|), but lambda=eta=xi=0 is feasible in Eq. (13) with objective 0. So the claimed certified bound d*_N >= p* fails even in the idealized single-mode, exactly-estimated-nbar setting. The sign errors in the Lagrange derivation in Appendix A are the source; Eq. (A3) has both the eta-lambda sign and the xi term wrong. This is a load-bearing flaw, not a typo, because Eq. (15) uses d*_N as the min-entropy bound.

There are also two smaller proof issues that point the same direction. Eq. (8) states an equality between the tail contribution and 1 - sum tr(rho*_{k,N}), but the correct relation is an inequality (each M_{k,Nbar} <= I). And Eq. (11) has a sign error in the dual objective: the correct dual is min x + y, not -x - y. The feasible point x=0, y=1/N still gives the intended n/N bound, so that one is cosmetic. Eq. (8) is more serious but fixable by changing the equality to <= and adjusting the subsequent optimization accordingly.

The good news is that the high-level approach is sound. The dimension reduction for diagonal POVMs is the right tool, and the errors are localized enough that a careful rewrite of Section IV could repair the proof. The novelty claim is a bit inflated—the technique itself is from [20]—but the specific assembly with a phase-insensitive detector is new.

This paper deserves a serious referee. The idea is good, the field would benefit from a correct version, and the current flaws are identifiable and correctable. I would not cite the current version or trust its bounds, but I would send it to review with a clear request to fix the dual formulation and the inequality in Eq. (8).

Referee Report

5 major / 5 minor

Summary. The manuscript proposes a continuous-variable source-independent quantum random number generator (CV-SI-QRNG) that uses a single phase-insensitive detector whose POVM elements are diagonal in the Fock basis. To avoid squashing models, the authors introduce a dimension-reduction technique: they project the infinite-dimensional optimization for the guessing probability onto the subspace of photon numbers below a cutoff N, bound the tail weight outside this subspace by the mean photon number divided by N, and formulate a finite-dimensional SDP (Eq. (12)) with constraints derived from the observed probabilities. They then give a claimed dual SDP (Eq. (13)) and assert that its optimal value d*_N provides a rigorous upper bound on the original infinite-dimensional guessing probability, leading to the min-entropy bound H_min >= -log2 d*_N (Eq. (15)). The protocol is simulated for a time-multiplexed single-photon detector with weak coherent states and parameters N=20, m=10, Nmode=2^5, yielding min-entropy lower bounds above 10^-2 bits per sample for mean photon numbers below 1. The central claim is that the finite-dimensional SDP and its dual certify security against arbitrary source states compatible with the observed statistics and mean photon number.

Significance. If the proof can be corrected, the result would be a useful step: it would extend numerical security analysis to infinite-dimensional CV-SI-QRNGs without restrictive squashing assumptions, for a broad class of phase-insensitive detectors, using only standard SDP duality and no fitted parameters. The proposed implementation with a single time-multiplexed detector is experimentally simple, and the simulation gives a concrete, falsifiable prediction of the achievable min-entropy rate. However, the manuscript as printed contains load-bearing mathematical errors in Eqs. (8), (11), and (13), so the security claim is not currently established; the significance is therefore conditional on a successful correction.

major comments (5)

[IV, Eq. (13) and Appendix A] Eq. (13) is not a valid dual of Eq. (12). The point λ=η=ξ=0 is always feasible in Eq. (13) because its constraint reduces to M_{k,N} − I ⪯ 0, and the objective at that point is 0. However, the primal Eq. (12) can have positive optimal value; for example, with m=2, M_1=0.9 I, M_2=0.1 I, N=20, ⟨n⟩=0, and p=(0.9,0.1), the feasible choice ρ_1=|0⟩⟨0|, ρ_2=0 gives objective 0.9. Hence the claimed inequality d*_N ≥ p* cannot be true, and Eq. (15) is unsupported as printed. The Lagrange derivation in Appendix A contains algebraic errors: starting from Eq. (A1), the constant term and the signs in the coefficient of ρ_{k,N} lead to a different dual, namely min_{λ,η,ξ≥0} 1 + Σ_j λ_j p_j − Σ_j η_j p^L_j + ξ subject to M_{k,N} + Σ_j(η_j − λ_j)M_{j,N} − (1 + ξ)I ⪯ 0 (with λ for the upper constraints and η for the lower constraints). The printed Eqs. (A4), (A5), and (13) have the wrong signs on the probability terms and are missing the constant 1.
[IV, Eq. (8)] The second equality in Eq. (8) is not generally valid. It asserts Σ_k tr(ρ*_{k,bar N} M_{k,bar N}) = 1 − Σ_k tr(ρ*_{k,N}), which would require each M_{k,bar N} to be the identity on the tail subspace. In general only the inequality Σ_k tr(ρ*_{k,bar N} M_{k,bar N}) ≤ Σ_k tr(ρ*_{k,bar N}) = 1 − Σ_k tr(ρ*_{k,N}) holds, since Σ_k M_{k,bar N} = I_{bar N} and each M_{k,bar N} ⪯ I_{bar N}. The desired upper bound can be repaired by replacing the equality with this inequality, but as written the derivation is mathematically incorrect.
[IV, Eqs. (10)-(11)] The dual problem stated in Eq. (11) has the wrong objective sign. For the maximization max tr(ρ bar P) subject to tr(ρ a†a) = ⟨n⟩ and trρ = 1, the standard Lagrange dual is min_{x,y∈R} x + y⟨n⟩ subject to bar P − xI − y a†a ⪯ 0. The feasible point x=0, y=1/N then gives the upper bound ⟨n⟩/N on the tail weight, as used in the text. With the printed objective −x−y⟨n⟩, the same feasible point gives −⟨n⟩/N, so the claimed tail bound does not follow from the displayed dual.
[IV, Eq. (12) and protocol step 3] The tail bound relies on an exact value of the mean photon number ⟨n⟩. The protocol (step 3) only says that ⟨n⟩ is "well estimated" by a phase-insensitive detector. If ⟨n⟩ is merely estimated from data, an underestimation makes the constraint p^L_j too large, so the finite SDP Eq. (12) need not contain the projection of the true feasible set and d*_N can fail to be an upper bound. The manuscript should either state explicitly that ⟨n⟩ is a known a priori upper bound on the source's mean photon number, or incorporate finite-sample or uncertainty bounds on ⟨n⟩ into p^L_j. This assumption is load-bearing, not merely a practical detail.
[IV, Eq. (10)] The optimization in Eq. (10) and the POVM model in Eq. (6) assume the source is a single bosonic mode. The security claim in Eq. (15) is phrased for "any source state compatible with observed statistics"; if the physical source emits light in several modes and the detector collects all of them, the operator a†a in Eq. (10) must be replaced by the total photon-number operator of the relevant modes and the tail bound may no longer scale as ⟨n⟩/N. The single-mode assumption should be stated as an explicit protocol assumption, and the multi-mode case should be either analyzed or explicitly excluded.

minor comments (5)

[II.B, Eq. (5)] In the first constraint, the summation index j is reused for the measurement outcome j; it should be Σ_k ρ_k rather than Σ_j ρ_j.
[III, Eq. (6)] The non-projective condition is stated as "the spectrum norm of an arbitrary POVM element M_j should be less than 1"; since the M_j are positive, this should be written as ||M_j||∞ < 1, and the completeness relation Σ_j M_j = I should be stated explicitly.
[IV, Eq. (8)] The symbols ρ*_{k,N}, M_{k,N}, and bar N are used in Eq. (8) before their definition in the following sentence; the notation should be introduced before the equation. The word "POP" in the explanation after Eq. (8) appears to be a typo.
[V, Fig. 2] The figure reports only a single curve; no solver, SDP tolerance, or verification of the dual bound is given, and the label "lg(R)" should state the base of the logarithm and clarify that the quantity is the min-entropy per sample.
[Title and throughout] There are several typographical errors, including "N umber" in the title, "aﬀact" in Section III, and "Arbitraty" in Section III; these should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No circularity: the SDP security bound is computed from measured inputs; the minor self-citation to the author's numerical framework is not load-bearing, while the apparent Eq. (13) weak-duality sign issue is a correctness concern, not a circularity.

full rationale

The paper's derivation chain is: Eqs. (2)-(5) reduce the guessing probability to an SDP; Eqs. (8)-(12) construct a finite-dimensional relaxation using the tail-weight bound; Eq. (13) is the claimed dual; Eq. (15) converts the SDP value into a min-entropy lower bound. No step defines the target quantity (the randomness lower bound or the guessing probability) in terms of itself. The probabilities p_j and mean photon number <n> are protocol inputs, not fitted parameters, and the SDP computes a certified upper bound from them. The tail-weight bound in Eq. (11) is a derived bound via a feasible dual point, not an assumed conclusion. The dimension-reduction technique is attributed to external Ref. [20] and is re-derived in Section IV. Ref. [17] is a self-citation to the author's earlier numerical framework, but Eq. (5) is independently justified in the text from the pure-state decomposition and POVM grouping, so this self-citation is not load-bearing. The apparent sign issue in the printed dual Eq. (13) and Appendix A, where the always-feasible point lambda=eta=xi=0 gives objective 0 while the primal can be positive, is a mathematical correctness or weak-duality concern rather than a circularity: it does not make the output equal to an input by construction. Hence there is no significant circularity; the score reflects only the presence of one minor, non-load-bearing self-citation.

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

The framework contributes no new physical entities. The main ledger items are the truncation parameter N and the idealized detector model, which are simulation choices, and the input mean photon number, which is a protocol statistic. The core proof borrows the dimension-reduction technique from Ref. [20] and the SDP formulation from prior QRNG security work.

free parameters (4)

photon number cutoff N = 20
Truncation parameter chosen for the simulation; the security bound depends on N through the tail correction <n>/N.
number of temporal modes N_mode = 25
Detector model parameter in the TMD simulation; affects the POVM elements and the computed bound.
number of POVM elements m = 10
Simulation parameter; in practice it is the number of detector outcomes.
mean photon number <n> = <= 1, swept in simulation
Protocol input estimated by Alice; the bound in Eq. (12) uses <n>/N as the tail correction.

assumptions (5)

domain assumption The SDP framework of Eqs. (1)-(5) correctly characterizes the SI-QRNG guessing probability.
Taken from prior numerical security analysis frameworks [8,17]; the paper does not re-derive this.
domain assumption The detector POVM elements M_j are exactly known and diagonal in the Fock basis.
Required for [M_j,P]=0 in Section IV; realistic detectors have losses and phases that break exact diagonality.
domain assumption The mean photon number <n> is exactly known from measurements.
Used to bound the tail weight via Eq. (11); estimation errors are not modeled.
standard math The dimension-reduction technique of Ref. [20] applies and is valid.
The paper relies on the projection-plus-tail-bound method from [20] without proving it in full.
domain assumption The time-multiplexed detector can be modeled as n balls into N_mode bins with equal probability.
Used in Eq. (17) to construct the POVM; assumes no loss, no dark counts, and ideal detection.

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Cite this review

Pith. "Pith review of Continuous-Variable Source-Independent Quantum Random Number Generator with a Single Phase-Insensitive Detector." pith.science (2026). https://pith.science/paper/E3I4RPDJ

@misc{pith2026241114817,
  author       = {Pith},
  title        = {Pith review of: Continuous-Variable Source-Independent Quantum Random Number Generator with a Single Phase-Insensitive Detector},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E3I4RPDJ}},
  note         = {Machine review of arXiv:2411.14817}
}

read the original abstract

Quantum random number generators (QRNGs) harness quantum mechanical unpredictability to produce true randomness, which is crucial for cryptography and secure communications. Among various QRNGs, source-independent QRNGs (SI-QRNGs) relax the trust on the quantum source, allowing for flexible use of advanced detectors to achieve high randomness generation rates. Continuous-variable (CV) SI-QRNGs, in particular, hold promise for practical deployment due to their simplicity and randomness generation rates comparable to trusted-device QRNGs. In this work, we propose a novel CV-SI-QRNG scheme with a single phase-insensitive detector, and provide security proof based on semi-definite programming (SDP). We introduce a dimension reduction technique, which rigorously reduces an infinite-dimensional SDP problem to a finite-dimensional one, enabling efficient computation while maintaining valid randomness lower bound. We further validate our method through simulations. These results demonstrate the feasibility of our framework, paving the way for practical and simple SI-QRNG implementations.

Figures

Figures reproduced from arXiv: 2411.14817 by the authors.

**Figure 2.** FIG. 2. Simulation of the randomness lower bound versus the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [20]

ˇSupi´ c, P

I. ˇSupi´ c, P. Skrzypczyk, and D. Cavalcanti, Phys. Rev. A 95, 042340 (2017)

work page 2017

[2] [1]

The untrusted source sends an unknown quantum state to a phase-insensitive detector

work page

[3] [2]

The detector outputs a measurement result j ∈ {1, 2,...,m }, characterized by a set of POVM {Mj}m j=1

work page

[4] [3]

After repeating steps 1-2 for a large number of rounds, Alice records the probability of each out- comepj and estimate the mean photon number ⟨n⟩

work page

[5] [4]

Then she performs the post- processing to extract the ﬁnal random numbers

Alice calculates the upper bound of the probability of successful guessing. Then she performs the post- processing to extract the ﬁnal random numbers. The protocol description is general for all SI-QRNGs. For a CV-SI-QRNG, the key diﬀerence is that the dimension of Mj is inﬁnity, which make the security analysis dif- ﬁcult. One possible method to reduce t...

work page

[6] [5]

and the nu- merical security analysis becomes feasible. In this work, we will use another simpler approach, a novel dimension reduction technique based on the diago- nal form of POVM of phase-insensitive detectors to deal with the inﬁnite dimensional optimization problem. IV. SECURITY ANALYSIS Recalling Eq. ( 5), we can see that the security analysis of s...

work page

[7] [6]

To make it com- putable, we deﬁne a projection on the subspace where the photon number is less than N , i.e., P = ∑ N −1 n=0 |n⟩ ⟨n|, and its complement ¯P = I −P

is an inﬁnite dimensional SDP problem. To make it com- putable, we deﬁne a projection on the subspace where the photon number is less than N , i.e., P = ∑ N −1 n=0 |n⟩ ⟨n|, and its complement ¯P = I −P . Then after the pro- jection P , all operators in Eq. (

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[8] [7]

We try to ﬁnd an upper bound of the target function

are represented by N -dimensional matrices in Fock basis. We try to ﬁnd an upper bound of the target function. Suppose the solution to Eq. ( 5) is ρ∗ k, we have the follow- ing relation of the maximum guessing probability, m∑ k=1 tr(ρ∗ kMk) = m∑ k=1 tr(ρ∗ k,NMk,N ) + m∑ k=1 tr(ρ∗ k, ¯NMk, ¯N ) = m∑ k=1 tr(ρ∗ k,NMk,N ) + 1 − m∑ k=1 tr(ρ∗ k,N ) ≤ max ρk,N ∈...

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