The successful loophole-free Bell experiments of 2015 and 2016 in Delft, Munich, NIST and Vienna were milestone achievements. They pushed quantum technology to the limit and paved the way for DIQKD and quantum internet. They were the first successful implementations of the ideal experimental protocol described by John Bell in his 1981 masterpiece "Bertlmann's socks and the nature of reality" (https://cds.cern.ch/record/142461/files/198009299.pdf). Still, there is a vociferous but small community of proponents of local realism, who continue to grasp at the straws offered by some shortcomings of the 2015+ experiments. I'll discuss the loopholes in the loophole-free experiments, explain how the experimenters could have claimed much smaller p-values "for free" by using standard likelihood-based inference, but also relativise the meaning of a 25 standard deviation violation of local realism. I suggest that particle physicists have wisely settled on 5 standard deviations because of Chebyshev's inequality and the fact that 1 / 5 squared = 0.04, just significant at the 5% level. https://arxiv.org/abs/2209.00702 "Optimal statistical analyses of Bell experiments" https://arxiv.org/abs/2208.09930 "Kupczynski's contextual setting-dependent parameters offer no escape from Bell-CHSH"

1. Richard Gill, Mathematical Institute, Leiden University
Statistical evaluation of
the 2015 loophole-free
Bell experiments
aQa seminar, 21 September 2022

2. • Setting choices 𝑎, 𝑏 ∈ {1,2} for Alice and Bob
• Measurement outcomes 𝑥, 𝑦 ∈ {−1, +1} for Alice and Bob
• Think of (𝑎, 𝑏, 𝑥, 𝑦) as the outcome of r.v’s (𝐴, 𝐵, 𝑋, 𝑌)
• Define 𝜌𝑎𝑏 = 𝐸(𝑋𝑌 ∣ 𝐴 = 𝑎, 𝐵 = 𝑏)
• Then under Local Realism, 𝜌11 + 𝜌12 + 𝜌21 − 𝜌22 ≤ 2
• This is one of the 8 one-sided Bell-CHSH inequalities
• However, under QM we can have 𝜌11 + 𝜌12 + 𝜌21 − 𝜌22 =
2 2
Each trial of a Bell experiment has …
Bell-CHSH inequalities

3. • Suppose 𝑢, 𝑣, 𝑥, 𝑦 ∈ {−1, +1}
• 𝑢𝑥 + 𝑢𝑦 + 𝑣𝑥 − 𝑣𝑦 = 𝑢(𝑥 + 𝑦) + 𝑣(𝑥 − 𝑦)
• 𝑥 = ±𝑦 = ±1, hence one bracketed term equals 0, the other
± 2
• The coefficients 𝑎, 𝑏 = ±1
• Hence 𝑢𝑥 + 𝑢𝑦 + 𝑣𝑥 − 𝑣𝑦 = ±2
• Hence, for four r.v’s taking the values ±1,
𝐸(𝑋1𝑌1) + 𝐸(𝑋1𝑌2) + 𝐸(𝑋2𝑌1) − 𝐸(𝑋2𝑌2) ∈ [−2,2]
Basic algebraic fact

4. • Bell experiment: assume local hidden variables (aka local realism)
• Denote Alice and Bob’s settings by 𝑎, 𝑏
• LHV: There exist functions 𝐴, 𝐵 taking values in {–1, +1} and a variable 𝜆 with
probability distribution 𝜌 not depending on 𝑎, 𝑏, such that Alice’s outcome 𝑥 =
𝐴(𝑎, 𝜆), Bob’s outcome 𝑦 = 𝐵(𝑏, 𝜆)
• Thus
𝐸(𝑋𝑌 ∣ 𝐴 = 𝑎, 𝐵 = 𝑏) = ∫{𝜆∈Λ}𝐴(𝑎, 𝜆)𝐵(𝑏, 𝜆)𝜌(𝑑𝜆)
𝐸(𝑋) = ∫{𝜆∈Λ}𝐴(𝑎, 𝜆)𝜌(𝑑𝜆), 𝐸(𝑌) = ∫{𝜆∈Λ}𝐵(𝑏, 𝜆)𝜌(𝑑𝜆)
aka Local Realism
Local hidden variables

5. • 𝜆 is created at the source and transmitted to Alice and Bob
• No. 𝜆 may have many components located throughout the experimental system
• a and b are angles hence not determined precisely hence …
• No. Alice and Bob flick switches to “up” or “down” positions, the intended angle
settings are irrelevant
• The word “realism” in the expression “local realism” is linked to the philosophical position
called “realism”. The word “local” in the phrase “local hidden variables” refers to locality of
the variables
• No. Locality is expressed through the function A not depending on b. “Local realism”
means “classical” or “locally deterministic”
• We study particle pairs so must condition on events when two particles are detected
• No. We study time-slots. We don’t assume anything about “particles”
What does “LHV” and “LR” actually mean?
Common misunderstandings

6. • 𝑃(𝑋 = 𝑥 ∣ 𝐴 = 𝑎, 𝐵 = 𝑏) cannot depend on 𝑏
• 𝑃(𝑌 = 𝑦 ∣ 𝐴 = 𝑎, 𝐵 = 𝑏) cannot depend on 𝑎
• Problem: the “loophole free” Bell experiments of 2015 actually exhibit
possible signalling, in this sense
• An explanation: during the duration of the experiment, physical
parameters change and may even jump. This also affects the physical
random number generators used to select settings. This must be a
spurious correlation: correlation due to a hidden confounder.
• Since the likely confounder is known (time!) we can control for it!
• Remedy: martingale based tests
Under both QM and LR
No-signalling

7. • Detection loophole: outcomes in {–1, 0, +1} where 0 means “no
detection”
• Locality loophole: Alice’s outcome fixed before Bob’s setting could
have reached her location
• Delft and Munich use entanglement swapping to achieve
detection efficiency almost 100%
• NIST and Vienna merge “–1” and “0” and use Eberhard’s
inequality to enable QM violation of local realism even at detector
efficiency 66.67% with much less than maximally entangled states
Delft, Munich, NIST, Vienna
Loophole-free Bell experiments

8. Delft raw counts
+ – + –
+ 23 3 26 33 11 44
– 4 23 27 5 30 35
27 26 53 38 41 79
+ 22 10 32 4 20 24
– 6 24 30 21 6 27
28 34 62 25 26 51

9. 196 245
+ – + –
+ 0,43 0,06 1,49 0,42 0,14 0,56
– 0,08 0,43 0,51 0,06 0,38 0,44
0,51 0,49 1,00 0,48 0,52 1,00
+ 0,35 0,16 0,52 0,08 0,39 0,47
– 0,10 0,39 0,48 0,41 0,12 0,53
0,45 0,55 1,00 0,49 0,51 1,00
P(X= + | 11) = 0.49, P(X = + | 12) = 0.56
P(X= + | 21) = 0.52, P(X = + | 22) = 0.47
P(Y= + | 11) = 0.51, P(Y = + | 21) = 0.35
P(Y= + | 21) = 0.56, P(Y = + | 22) = 0.47
r11 = 0.74, r12 = 0.59, r21 = 0.48, r22 = –0.61 S = 2.42
Delft relative frequencies

10. • In each trial
Win if either setting =“1” and outcomes are equal, and also
win if both settings = “2” and outcomes are opposite.
Otherwise lose
• Suppose for each trial, local realism holds, conditional on past
• Suppose for each trial, the two settings are fair coin tosses
• Then the total number of wins in n trials has a probability distribution with smaller right
tail probabilities than the binomial(n, 3/4)
(Roughly speaking, there is no way “locally, deterministically” to achieve a success
probability for any trial better than 3/4, even allowing for arbitrary capitalisation on all data
from either wing of the experiment for all past trials)
• Still true with “3/4” replaced by something a bit larger if we allow some given small
amount of bias in the settings (on average)
Bell game
How to discount signalling

11. • Delft has 196 successes in 245 trials
Delft (continued)
> 1 - pbinom(196, 245, 0.75)
[1] 0.02763698

12. • 𝑝(𝑥, 𝑦 ∣ 𝑎, 𝑏) =
1
4
+
1
2
(𝑝𝑎(𝑥) −
1
2
) +
1
2
(𝑞𝑏(𝑦) −
1
2
) ±
𝜌𝑎𝑏
4
• Local realism: 𝜌11 + 𝜌12 + 𝜌21 − 𝜌22 ≤ 2 (plus 7 others)
• General situation: 8 free parameters
• Null hypothesis: 7 free parameters
• Simply maximise the log likelihood under H0 and in general and
compare difference in 2 log likelihood to 50-50 mixture of chi-
square(0) and chi-square(1) distribution
Data = 16 counts = 4 tetranomially distributed vectors
Assume i.i.d. trials per setting
pair

13. • Conventional analysis:
S = 2.42 (0.21)
p = 0.02
• Improved analysis
S = 2.46 (0.20)
p = 0.01
Delft, multinomial

14. • Conventional analysis:
S = 2.000092 (0.000016)
z = 5.86
p = 2.0 x 10–9
• Improved analysis
S = 2.000051 (0.000007)
z = 7.64
p = 1.1 x 10–14
NIST, multinomial

15. • Consider the four tetranomial case
• Given the four no-signalling equalities, the 8 one-sided CHSH
inequalities are NASC for LHV
• Consider the 16-vectors (p(x, y | a, b))
• Due to non-negativity, normalisation, and no-signalling, they lie in an
8 dimensional polytope inside an 8 dimensional affine subspace of
R16
• The CHSH inequalities reduce this to a smaller polytope. Its vertices
are the 16 deterministic LHV models. The whole LR polytope is the
probability mixture of these 16 deterministic “deterministic, local
instruction sets”.
(Precursor in Boole 1854)
Fine’s theorem (1981)

16. • 7.5 standard errors is great but p = 1.1 x 10–14 is an illusion
• A more honest p-value is 1 / 7.52 = 1% (Chebyshev)
P-values based on asymptotic normality of a z-statistic
Final remarks (1)

17. Final remarks (2a)
DIQKD
Zhang, W., van Leent, T., Redeker, K. et al. A device-independent quantum key distribution system for
distant users. Nature 607, 687–691 (2022). https://doi.org/10.1038/s41586-022-04891-y

18. Final remarks (2b)
DIQKD
Zhang, W., van Leent, T., Redeker, K. et al. A device-independent quantum key distribution system for
distant users. Nature 607, 687–691 (2022). https://doi.org/10.1038/s41586-022-04891-y
BSM = Bell State Measurement

19. • DIKQD uses an embedded loophole-free Bell test. But this test is
a three party Bell experiment: Alice, Bob and Carol.
• Munich and Delft experiments: Alice and Bob have two settings
and two outcomes. Carol has no setting and four outcomes. One
studies p(x, y | z; a, b). All. three parties have to be well
separated: Carol’s outcome mustn’t influence Alice and Bob’s
outcomes. But current DIQKD experiments have two locations
only!
• Zhang et al. have merged Carol and Alice. Carol/Alice have *four*
inputs and *four outputs*
DIQKD
Final remarks (2c)
Zhang, W., van Leent, T., Redeker, K. et al. A device-independent quantum key distribution system for
distant users. Nature 607, 687–691 (2022). https://doi.org/10.1038/s41586-022-04891-y

20. S = 2.578 (0.075)
z = 7.71
Final remarks
(2d)

21. The trick
Schwonnek, R., Goh, K.T., Primaatmaja, I.W. et al. Device-independent quantum key distribution with
random key basis. Nat Commun 12, 2880 (2021). https://doi.org/10.1038/s41467-021-23147-3
Final remarks
(2e)