Experimental Randomness Amplification
Nature, 2026 · paper · Physics
Demonstrates device-independent randomness amplification, generating near-perfect randomness from an imperfect physical source, using a loophole-free Bell test with superconducting circuits and a recent theoretical protocol requiring high Bell violation and large data volume.
Checked: PDF from arxiv.org, fetched 5 Oct 2026. Page numbers are those of that PDF.
4 findings
Substance Eqs. (6)-(8) give "t_protocol,A→B = t_detection,B + t_ΔB + t_ΔAB + t_latch,A + t_RNG,A", but with the Table II entries this sums to 115.65 ns, not the printed "t_protocol,A→B 106.65". The B→A total (106.45) is not reproduced either. The printed totals come out only when t_latch is subtracted and each total uses a single node's segments.
- p. 13 Eqs. (6)-(8)
+t∆AB +tlatch,A +tRNG,A.
- p. 13 Table II, last row
106.65
- p. 13 Table II, last row
106.45
- p. 13 Eqs. (6)–(8)
+t∆AB +tlatch,A +tRNG,A
Only the authors can say what is right here.
Judge's note: I checked p.13 (Eqs. 6-8, Table II, and the text giving 106.7 ns with a 3.1 ns margin to t_d = 109.83 ns) and the upright Fig. 9 on p.14. Eq. (8) clearly prints +t_latch,A, and the table has no minus signs. Substituting the Table II values into the printed formula gives 115.65 ns for A→B and 115.85 ns for B→A, the latter computed with the mirror-image formula. Both exceed t_d = 109.83 ns. Fig. 9 draws t0,OsciA inside the t_RNG,A interval, with the latch after t0. Combined with the definitions in the text, this means t0,OsciA − t⋆A = t_RNG,A − t_latch,A, so t_latch should enter with a minus sign. Even with that sign, the cross-node terms named in Eq. (7) give 111.65 ns (or 107.65 ns if t_ΔB were negative), not 106.65. The printed 106.65 is reproduced only from node-A values with t_latch subtracted (17.1 − 2.0 + 5.0 + 0.55 + 86.0), and 106.45 only from node-B values. The equation (the sign of t_latch and which node's terms enter) and the table (column or label assignment) therefore disagree. Fig. 9 also draws t_latch,A longer than t_latch,B, the reverse of the table's 2.0 vs 7.2 ns, though the figure may be schematic. Only the authors can say which combination is correct. I found no erratum and no other definition in the files that reconciles them.
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Substance Theorem 5's output-length bound (18), "m ≤ n/6(η + 2 log 1/(1/2+µ) − 2) − log 6/(ε − 6εs)", carries a penalty of only 1·log(1/(ε−6εs)). Combining Theorems 2, 3 and 4 as printed, which is what the proof ('a direct calculation shows') says it does, certifies only about m ≤ K/6 − (5/3)·log(1/(ε−6εs)) − const. Eq. (21) inherits the same term.
- p. 15 Theorem 5, Eq. (18)
−log 6/(ε −6εs)
- p. 16 Proof of Theorem 5
a direct calculation shows that the condition in equation (18) implies that our extractor is secure
- p. 16 Theorem 6, Eq. (21)
−log 6/(ε−6εs)
Only the authors can say what is right here.
Judge's note: I reworked the step from pp.15-16. Write N = 2n, K = nη + 2n log(1/(1/2+µ)) − 2n, and ε' = ε − 6εs. Theorem 4 with δ1 = εs and δ2 = 0 requires a quantum-proof (k1 − L1, k2 − L2, εq) extractor and gives total error 6εs + 2ε1 + 2ε2 + 2εq. Theorem 3 requires a classical (k1c, k2c, εc) extractor with k_ic + log(1/εc) ≤ k_i − L_i, and gives 2εq = sqrt(3·εc·2^m). Theorem 2 gives log(1/εc) = (k1c + k2c − N − 2m)/2, so 4 log(1/εc) = K − 2m − L1 − L2. Setting ε1 = ε2 = a gives 2εq = sqrt(3)·2^{(6m−K)/8}·a^{−1/4}. The constraints 4a ≤ ε' and 2εq ≤ ε' then force 6m − K ≤ −10·log(1/ε') − const, that is, m ≤ K/6 − (5/3)·log(1/ε') − const. The K/6 term matches (18), but the ε-coefficient is 5/3, not 1. For small ε', Eq. (18) therefore allows larger m than the stated theorems certify: at ε = 1e-12 the gap is roughly 25 bits. The numerical effect against m ≈ 2×10^7 is negligible, but as printed the bound is not shown by the proof given. It could hold only if variants of the cited results are used that the paper does not state.
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Reference or label Fig. 2 caption says "neutral atoms (squares), trapped ions (pentagon)", but the plot has one square (Nadlinger et al. (2022)) and two pentagons (Rosenfeld et al. (2017), Zhang et al. 2022); the two shapes appear to be assigned the wrong way round.
- p. 3 Fig. 2 caption
neutral atoms (squares), trapped ions (pentagon)
- p. 3 Fig. 2 plot markers
Rosenfeld et al. (2017)
- p. 3 Fig. 2 caption
atoms (squares), trapped ions (pentagon)
- p. 3 Fig. 2 plot markers
Nadlinger et al. (2022)
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that Nadlinger et al. 2022 is a trapped-ion experiment and that Rosenfeld et al. 2017 and Zhang et al. 2022 are neutral-atom experiments; the paper does not state the platforms. On the page image I see one square marker (Nadlinger) and two pentagon markers (Rosenfeld, Zhang 2022). The caption's plural "squares" and singular "pentagon" also fit the opposite assignment to the one printed.
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Reference or label The employed extractor is cited as "we choose the two-source extractor from [55]" in the main text but as "the employed two-source randomness extractor [63]" in Methods and "construction based on non-cyclic right shift matrices from [63]" (Theorem 2, "Lemma 4 in [63]") in the supplement.
- p. 5 Main text, right column
source extractor from [55]
- p. 7 Methods, Definition of randomness
ness extractor [63]
- p. 15 Supplement VI, before Theorem 2
struction based on non-cyclic right shift matrices from [63]
- p. 5 Main text, extractor paragraph (right column, top)
we choose the two-source extractor from [55]
Only the authors can say what is right here.
Judge's note: Page 5 image shows [55]; the reference list gives [55] as U. Vazirani, STOC '87. The text of pages 7 and 15 cites [63], listed as Dodis, Elbaz, Oliveira and Raz. The one extractor is attributed to two different references with no explanation in the paper. Only the authors can say whether [55] is meant as the historical origin or is a wrong pointer.
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16 minor (typos and the like; most checked by the judge only)
Wording "the excess predictability specified by the manufacturer of our QRNGs is below ε < 1·10−5 [43]" doubles the comparison ("below ... <") and uses ε, which the paper otherwise defines as the security parameter (ε = 10−12).
- p. 5 Main text, left column, choice of µ
our QRNGs is below ε < 1·10−5 [43]
Judge's note: Seen on the page 5 image, left column. On the same page ε is the security parameter, "chosen to be ε = 10^-12". Here the symbol stands for a different quantity (the manufacturer's excess predictability) and the phrasing is redundant. No number is affected; this is a notation and wording point.
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Wording Supplement II says the synchronisation "shows effectively no jitter on week-long time windows", but the same section says the QRNG drift is "limiting the system to be in a jitter-free configuration for about 2.5 days".
- p. 9 Supplement II, intro
effectively no jit- ter on week-long time windows
- p. 9 Phase drift measurements
jitter-free configuration for about 2.5 days
- p. 9 Supplement II, introduction
shows effectively no jitter on week-long time windows
- p. 9 Supplement II, Phase drift measurements
limiting the system to be in a jitter-free configuration for about 2.5 days
Only the authors can say what is right here.
Judge's note: Both sentences are on the page 9 image. Table I gives the QRNG granularity as 2 ns and its drift as 16.9 ps/hr, over a measurement of about 50 hours. Starting from the middle of the granularity window, 1 ns of margin at that drift lasts about 59 h, or about 2.5 days, which matches the second sentence and not "week-long". The paper does not say that week-long stability relies on periodic phase re-adjustment or excludes the QRNG, so the wording is left unreconciled.
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Wording Text says "The absolute drifts summarized in Table I are measured relative to the AWG5014" while the Table I column is headed "Relative drift in ps/hr".
- p. 9 Phase drift measurements
The absolute drifts summarized in Table I are measured relative to the AWG5014
- p. 11 Table I header
Relative drift
Judge's note: Both seen on the page images (page 9 text, page 11 table). "Absolute" may mean the magnitude of a drift measured relative to a reference, which would make the two compatible. It is a loose-terminology point and no value is affected.
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Wording Fig. 5 caption says "The only two trusted components of the RT electronics are a randomness source and an ADC", but Supplement I also counts as trusted "the field-programmable gate array (FPGA) used to perform state discrimination and post-processing".
- p. 10 Fig. 5 caption
The only two trusted components of the RT electronics are a randomness source and an ADC
- p. 9 Section I
the field-programmable gate array (FPGA) used to perform state discrimination
Only the authors can say what is right here.
Judge's note: Page 9 image: "we consider trusted the analog-to-digital converter (ADC) ... and the field-programmable gate array (FPGA) used to perform state discrimination and post-processing". Page 10 caption: only an RNG and an ADC among the RT electronics, plus timing verification equipment and classical memory. The FPGA is therefore not covered by the caption's "only two". The ADC may sit on the FPGA board, which would reconcile the two, but the paper does not say so; the main text and Supplement V list the ADC and do not mention the FPGA separately. This is a wording ambiguity for the authors to clarify.
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Wording Fig. 9, which carries a "10 ns" scale bar, draws tlatch,A (about 7 ns) much longer than tlatch,B (about 2.5 ns), whereas Table II gives tlatch,A = "2.0" and tlatch,B = "7.2"; the drawn t∆A (about 12 ns) also differs from the tabulated "5.0".
- p. 14 Fig. 9, Oscilloscope A / Oscilloscope B rows (with 10 ns scale bar)
tlatch, A
- p. 13 Table II
7.2
Only the authors can say what is right here.
Judge's note: Measured on the upright page 14 image. The scale bar is about 128 px per 10 ns. tlatch,A spans about 88 px (≈6.9 ns), tlatch,B about 32 px (≈2.5 ns), and t∆A about 152 px (≈11.9 ns). The latch lengths are the reverse of Table II on page 13. The figure may be schematic, but it has an explicit scale bar and its overall protocol span (about 106.5 ns) is drawn to scale. Pixel estimates are approximate, so this is a question for the authors. I did not check the reviewer's Fig. 8 claim.
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Wording Theorems 5 and 6 state "ε,εEA,εs ∈(0,1) be arbitrary" / "Let ε,εs ∈(0,1) be arbitrary.", yet the bound contains log 6/(ε − 6εs). That term is undefined for ε < 6εs, and the proof (via Theorem 4's 6δ1 = 6εs error term) cannot reach security ε when ε ≤ 6εs. The hypothesis 6εs < ε is implicit but not stated.
- p. 15 Theorem 5 statement and Eq. (18)
ε,εEA,εs ∈(0,1) be arbitrary
- p. 16 Theorem 6 statement and Eq. (21)
Let ε,εs ∈(0,1) be arbitrary.
Only the authors can say what is right here.
Judge's note: I checked the statements of Theorem 5 (p.15) and Theorem 6 (p.16) on the page images. Both say ε and εs are arbitrary in (0,1). Eqs. (18) and (21) contain log 6/(ε−6εs). Theorem 4's bound (Eq. 17) includes 6δ1 with δ1 = εs, so security parameter ε is unattainable once 6εs ≥ ε. The missing hypothesis does not affect the experiment, where εs is presumably far below ε/6, so this is a statement-level imprecision, not a false result.
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Typo Fig. 2 point label "Zhang et al. 2022" lacks the parentheses around the year that every other label uses (e.g. "Rosenfeld et al. (2017)").
- p. 3 Fig. 2, point label
Zhang et al. 2022
Judge's note: Seen in the figure crop: all other labels have the form "Author et al. (year)"; this one alone prints the year without parentheses. Cosmetic.
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Typo Fig. 3 caption has stray commas: "(xi, (red) for Alice, and yi, (blue) for Bob)".
- p. 4 Fig. 3 caption b
(xi, (red) for Alice, and yi, (blue) for Bob)
Judge's note: The crop shows "(x_i, (red) for Alice, and y_i, (blue) for Bob)" with a comma after each variable before the colour parenthesis. Punctuation slip only.
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Typo Block size is written "each block contains 4^13 (∼ 67 million) trials" in Section III but "20 blocks of 2^26 (∼ 67 million) trials each" in the main text; the two are numerically equal.
- p. 11 Section III
each block contains 413 (∼67 million) trials
- p. 4 main text, data acquisition
20 blocks of 226
- p. 4 main text
20 blocks of 2
- p. 11 Section III
each block contains 4
Judge's note: The page 11 image shows 4^13 and the page 4 crop shows 2^26. Both equal 67,108,864, so no value is wrong; this is only a notation inconsistency, and either form is legitimate.
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Typo Redundant repeated word: "we generate a private ε-random random bit-string K".
- p. 5 Main text, right column
erate a private ε-random random bit-string K
- p. 5 main text, 'As a final result'
private ε-random random bit-string K
- p. 5
or, we gen- erate a private ε-random random bit-string K consis
Judge's note: Seen on the page 5 image, right column. Later in the same paragraph the paper writes simply "guaranteed to be ε-random". Cosmetic.
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Typo The AWG5014 granularity is printed as "about 1.67 ns" in Supplement II but as "1.66 ns" in Table I, Section III and the Fig. 7 caption.
- p. 9 Supplement II, first paragraph
about 1.67 ns
- p. 11 Table I; Supplement III; Fig. 7 caption
1.66 ns
- p. 11 Table I and Section III text
smallest granularity of the control devices, which is 1.66 ns
- p. 9 Supplement II, first paragraph
comes to about 1.67 ns
Judge's note: Page 9 image shows "about 1.67 ns". Page 11 image shows 1.66 in Table I, in the Section III text ("which is 1.66 ns in our case") and in the Fig. 7 caption ("device granularity, 1.66 ns"). The Fig. 6 legend shows a 600 MHz clock, whose period is 1.667 ns, so 1.67 is the correctly rounded value and 1.66 is a truncation. The difference has no effect on any result.
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Typo Misspellings and copy slips: Fig. 5 label "Memory & extraxctor"; Fig. 6 caption "ascerain"; Fig. 7 caption "within our uncertainly calculations"; Sec. IV "the exchange of classical information exchange between the two nodes"; Table I caption "taken about a course of 50 hours".
- p. 10 Fig. 5, box label
Memory & extraxctor
- p. 11 Fig. 7 caption
within our uncertainly calculations
- p. 10 Fig. 6 caption
ascerain
- p. 11 Fig. 7 caption
uncertainly calculations
Judge's note: All five were seen on the images: "extraxctor" in the Fig. 5 box and "ascerain" in the Fig. 6 caption (page 10); "uncertainly" in the Fig. 7 caption and "about a course of 50 hours" in the Table I caption (page 11); the doubled "exchange" on page 12 (crop). All are cosmetic.
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Typo Fig. 5 caption says "Supplemental Information section II" whereas the rest of the paper uses "Supplementary Information".
- p. 10 Fig. 5 caption
Supplemental Information section II
Judge's note: Seen on the page 10 image. A text search finds this single use of "Supplemental" against several uses of "Supplementary Information section ...". It is a style inconsistency only.
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Typo Fig. 7 caption prints "which is within our uncertainly calculations"; "uncertainly" should be "uncertainty".
- p. 11 Fig. 7 caption
which is within our uncertainly
- p. 11 Fig. 7 caption
which is within our uncertainly calculations
Judge's note: Read the Fig. 7 caption on the page 11 image; the word is printed as "uncertainly". It is a plain misspelling with an evident fix.
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Typo Section IV prints "the exchange of classical information exchange between the two nodes is forbidden" (duplicated "exchange"), and the Table I caption prints "taken about a course of 50 hours" (should read "over the course of 50 hours").
- p. 12 Supplement IV, right column top
the exchange of classical information exchange between the two nodes
- p. 19 Ref. [24]
RAND Coorporation
- p. 11 Table I caption
taken about a course of 50 hours
- p. 12 Section IV
exchange of classical information exchange between the two nodes
Judge's note: Both are visible in the images: the duplicated "exchange" in the page 12 crop (right column, top) and "about a course of 50 hours" in the Table I caption on page 11. The cluster's third item, "RAND Coorporation", is also real but is adjudicated under C024 to avoid double counting.
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Typo Reference [24] prints "RAND Coorporation"; it should read "RAND Corporation".
- p. 19 Reference [24]
RAND Coorporation
Judge's note: The page 19 crop shows "RAND Coorporation, A Million Random Digits with 100,000 Normal Deviates". It is a plain misspelling of the word "Corporation", visible in the image and not an extraction artefact.
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