Loophole-Free Bell Inequality Violation Using Electron Spins Separated by 1.3 Kilometres
Nature, 2015 · paper · Physics
Reports a Bell inequality test using entangled electron spins 1.3 km apart that closes known loopholes. Across 245 trials the authors find S = 2.42 ± 0.20, violating the local-realist bound of 2, with p = 0.039.
Checked: PDF from arxiv.org, fetched 4 Oct 2026. Page numbers are those of that PDF.
2 findings
Substance Page 1 gives box B's directions as "(−Z + X)/√2 (for b = 0) or (−Z − X)/√2 (for b = 1)"; with |ψ−⟩ and Eq. (1) as printed these give S = 0, not the stated "S = 2√2", and they are the reverse of the page 6 angles "−3/4π − ϵ (for b = 0), 3/4π + ϵ (for b = 1)".
- p. 1 right column, paragraph after Eq. (1)
(−Z + X)/√2 (for b = 0) or (−Z − X)/√2 (for b = 1)
- p. 1 Eq. (1)
+ ⟨x · y⟩(1,0) − ⟨x · y⟩(1,1) ≤ 2
- p. 6 left column, readout bases
−3/4π − ϵ (for b = 0), 3/4π + ϵ (for b = 1)
- p. 1 Introduction, right column, last paragraph
quantum theory predicts a value of S = 2√2
Only the authors can say what is right here.
Judge's note: Read Eq. (1), the direction sentence and the state on the page 1 image, and the angle list on the page 6 image. Own reworking: for the singlet, E(u,v) = −u·v. With A = Z (a=0), X (a=1) and B as printed: E00 = −Z·(−Z+X)/√2 = +1/√2; E01 = −Z·(−Z−X)/√2 = +1/√2; E10 = −X·(−Z+X)/√2 = −1/√2; E11 = −X·(−Z−X)/√2 = +1/√2. Eq. (1) puts the minus sign on the (1,1) term, so S = (1+1−1−1)/√2 = 0. Exchanging the two B directions gives E00 = E01 = E10 = +1/√2, E11 = −1/√2 and S = 4/√2 = 2√2. Page 6 measures angles from Z with a=1 at +π/2 (towards +X), so b=0 at −3π/4 is (−Z−X)/√2 and b=1 at +3π/4 is (−Z+X)/√2, i.e. the exchanged assignment; Fig. 4a also shows the negative correlation at (1,1). No convention stated in the paper reconciles the page 1 sentence with Eq. (1). The stated value 2√2 stands for the exchanged labels; the printed setting is what is wrong. Either the b labels on page 1 or the sign placement would have to change, so left as an author query.
Second check: another model agreed
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Reference or label The text ("underlying the data (see Methods)") and the Fig. 4 caption ("number generators into account (Methods)") point to a Methods section that this 8-page document does not contain.
- p. 6 left column, 'complete' analysis
probability distributions underlying the data (see Methods)
- p. 6 Fig. 4 caption (b)
number generators into account (Methods)
- p. 6 Left column, paragraph on the 'complete' analysis
underlying the data (see Methods)
- p. 6 Fig. 4 caption (b)
taking the measured imperfections of the random number generators into account (Methods)
Only the authors can say what is right here.
Judge's note: Read all eight pages of the text: the article runs from the introduction through results and outlook into the reference list and acknowledgements, with no Methods section or heading. Every other detail pointer says 'Supplementary Information'. Both 'Methods' pointers are visible on the page 6 image and crop. The supplementary file is not among the files provided, so the statistical analysis may be described there under that or another name; in this version the pointer has no target.
Second check: another model agreed
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12 minor (typos and the like; most checked by the judge only)
Typo Affiliation 4 reads "ICREA-Institució Catalana de Recerca i Estudis Avanats"; the word is presumably "Avançats" with the "ç" dropped.
- p. 1 Affiliation 4
Estudis Avanats
- p. 1 Affiliation 4
ICREA-Institució Catalana de Recerca i Estudis Avanats
- p. 1 Affiliation 4
Recerca i Estudis Avanats
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: the institution's name is 'Institució Catalana de Recerca i Estudis Avançats'. The crop and the page 1 image both show 'Avanats' with no letter between 'n' and 'a', so this is in the PDF and not an extraction artefact.
Second check: none (judge only)
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Typo Affiliation 5 gives the postcode as "OX110QR" with no space.
- p. 1 Affiliation 5
OX110QR
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: UK postcodes are written with a space before the final three characters (here OX11 0QR). The page 1 image shows 'Didcot, Oxfordshire OX110QR, United Kingdom.' with no space. Purely cosmetic.
Second check: none (judge only)
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Typo "at the same time - commonly referred to as a loophole-free Bell test14,17,23 is of foundational importance": the aside opened with a dash is not closed before "is".
- p. 3 left column, third paragraph
at the same time - commonly referred to as a loophole-free Bell test14,17,23 is
Judge's note: Page 3 image, left column, paragraph beginning 'Because of unclosed loopholes': there is an opening dash after 'at the same time' and no dash or comma after the reference numbers 14,17,23.
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Typo Fig. 2 caption (a): "The event ready signal lies outside" is unhyphenated, where the paper otherwise writes "event-ready signal".
- p. 4 Fig. 2 caption (a)
The event ready signal lies outside
- p. 3 main text
event-ready signal
Judge's note: Page 4 image: the caption has 'event-ready signal' two sentences earlier and then 'The event ready signal lies outside the future light-cone', with the words on one line (no line break). Page 3 and the rest of the text use 'event-ready'.
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Typo Fig. 2 caption (c): "when the spin is prepared in ms = 0. (ms = ±1)." has a stray full stop before the parenthetical.
- p. 4 Fig. 2 caption, panel (c)
when the spin is prepared in ms = 0. (ms = ±1).
- p. 4 Fig. 2 caption, panel (c)
when the spin is prepared in ms = 0. (ms = ±1). Green
Judge's note: Page 4 image, Fig. 2 caption: 'Blue (orange) line: fidelity of outcome +1 (-1) when the spin is prepared in m_s = 0. (m_s = ±1). Green line: ...'. The parenthetical pairs with '(orange)' and '(-1)' and belongs to the same sentence, so the full stop after 'm_s = 0' is misplaced.
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Typo "heralded entanglement experiments over 3 meter27": singular unit; the paper elsewhere writes "metres" or "m".
- p. 5 main text, last paragraph
over 3 meter27
- p. 8 Reference 27
separated by three metres
Judge's note: Page 5 image, right column, last paragraph reads 'over 3 meter' followed by the superscript 27. Reference 27 on page 8 is titled '...separated by three metres' and distances elsewhere are given as 'm' or 'km'.
Second check: none (judge only)
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Typo Spelling varies within the paper: "single-mode optical fiber" and "1.7 km optical fibre" on the same page, "optical fiber connections" in the acknowledgements, "fibre" in the Fig. 1 caption.
- p. 5 main text
single-mode optical fiber
- p. 5 main text
1.7 km optical fibre
- p. 8 Acknowledgements
optical fiber connections
Judge's note: Page 5 image, right column: 'of 1.7 km of single-mode optical fiber' near the top of the text and '(8 dB/km) in the 1.7 km optical fibre' in the last paragraph. The Fig. 1 caption on page 2 uses 'fibre' throughout; the acknowledgements text has 'optical fiber connections'. Either spelling is legitimate; only the mixture is at issue.
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Typo Fig. 4 caption (b): "Statistical analysis for n = 245 trials We test the null-hypothesis" lacks a full stop after "trials".
- p. 6 Fig. 4 caption, panel (b)
n = 245 trials We test the null-hypothesis
Judge's note: Page 6 image and crop: '(b) Statistical analysis for n = 245 trials We test the null-hypothesis that...' with no punctuation between 'trials' and 'We'.
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Typo Fig. 4 caption (b): "formula (1) , for equal" has a space before the comma.
- p. 6 Fig. 4 caption (b)
formula (1) , for equal
Judge's note: The crop shows '(Note that I equals S from formula (1) , for equal n(a,b)).' with a clear gap between ')' and the comma. Cosmetic.
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Typo Reference 4 title is printed "Using Time- Varying Analyzers" with a stray space after the hyphen; it should read "Time-Varying Analyzers".
- p. 7 Reference 4
Using Time- Varying Analyzers
Judge's note: Viewed the crop of page 7. Reference 4 reads "of Bell's Inequalities Using Time- Varying Analyzers." with a clear word space between "Time-" and "Varying" in the middle of the line, so it is not a line-break hyphenation. Other hyphenated compounds in the same list ("Hidden-Variable", "Loophole-Free") are set without a space. The layout text shows the same space.
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Typo The acknowledgements print "Marissa Guistina", while the paper's own reference 10 spells the surname "Giustina, M. et al."; the two spellings disagree and the acknowledgement appears to transpose "iu".
- p. 8 Acknowledgements
Marissa Guistina
- p. 7 Reference 10
Giustina, M. et al.
- p. 8
Marissa Guistina, Laura Mancinska
Judge's note: Viewed the page 8 crop: the acknowledgements read "Peter Grünwald, Marissa Guistina, Laura Mancinska". Viewed the page 7 crop: reference 10 reads "Giustina, M. et al. Bell violation using entangled photons without the fair-sampling assumption. Nature 497, 227-230 (2013)." The inconsistency is internal to the paper (same initial M., same field, thanked for Bell-test discussions), so it does not rest on outside knowledge. That the reference spelling is the right one is the natural reading, since it is the cited author's published name.
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Typo The acknowledgements print "Thomas Vidic"; this is probably a misspelling of "Thomas Vidick" (final "k" missing).
- p. 8 Acknowledgements
Thomas Vidic
Only the authors can say what is right here.
Judge's note: Rests on outside knowledge: that the researcher in device-independent cryptography and Bell-test theory thanked here is named Thomas Vidick. Viewed the page 8 crop: the text reads "Hans Mooij, Thomas Vidic, Harald Weinfurter". The name appears nowhere else in the article (searched the layout text for "Vidic"; one hit only), so the paper itself offers no second spelling to compare with. The page says what the finding quotes, but only the authors can confirm who is meant.
Second check: none (judge only)