Richard's aventures in two entangled wonderlands

Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.

1. Richard Gill, Leiden University, Netherlands
31 May, 2024
Adventures in wonderland
Or: Richard down two rabbit holes
https://www.quantumsc.nl/Events/Upcoming-events/event/106/11th-QSC-General-
Assembly-in-Delft
11th Quantum Software Consortium General Assembly in Delft

2. •Deniers often exploit “loopholes”
•Understanding of how to avoid them led to “loophole-free Bell
tests” of 2020 and to Nobel prizes of 2022
Started way back: Accardi, Hess & Phillip, …
Rabbit hole 1: fighting Bell deniers

3. Open Peer Review on Qeios
Bell's theorem is an exercise in the statistical theory of
causality
Richard Gill
1
1 Leiden University
Funding: No specific funding was received for this work.
Potential competing interests: No potential competing interests to declare.
Abstract
In this short note, I derive the Bell-CHSH inequalities as an elementary result in the present-day theory of statistical
causality based on graphical models or Bayes' nets, defined in terms of DAGs (Directed Acyclic Graphs) representing
direct statistical causal influences between a number of observed and unobserved random variables. I show how
spatio-temporal constraints in loophole-free Bell experiments, and natural classical statistical causality considerations,
lead to Bell's notion of local hidden variables, and thence to the CHSH inequalities. The word “local” applies to the way
that the chosen settings influence the observed outcomes. The case of contextual setting-dependent hidden variables
(thought of as being located in the measurement devices and dependent on the measurement settings) is automatically
covered, despite recent claims that Bell's conclusions can be circumvented in this way.
Richard D. Gill
Mathematical Institute, Leiden University

In this short note I will derive the Bell-CHSH inequalities as an exercise in the modern theory of causality based on Bayes'
nets: causal graphs described by DAGs (directed acyclic graphs). The note is written in response to a series of papers by
M. Kupczynski
[1][2][3][4]
in which that author claims that Bell-CHSH inequalities cannot be derived (the author in fact writes
may not be derived) when one allows contextual setting-dependent hidden variables thought of as being located in the
measurement devices and with probability distributions dependent on the local setting. The result has of course been
known for a long time, but it seems worth writing out in full for the benefit of "the probabilistic opposition" as a vociferous
group of critics of Bell's theorem like to call themselves.
Figure 1 gives us the physical background and motivation for the causal model described in the DAG of Figure 2. How
that is arranged (and it can be arranged in different ways) depends on Alice and Bob's assistant, Charlie, at the
intermediate location in Figure 1. There is no need to discuss his or her role in this short note. Very different arrangements
can lead to quite different kinds of experiments, from the point of view of their realisation in terms of quantum mechanics.
Qeios, CC-BY 4.0 · Article, March 21, 2023
Qeios ID: K40UP4 · https://doi.org/10.32388/K40UP4 1/5
https://www.qeios.com/read/K40UP4
https://arxiv.org/abs/2211.05569

5. •Lucia de Berk 2001–2010
•Daniela Poggiali
•Ben Geen (!)
•Lucy Letby 2015– ?
•These different cases are horrifically similar. But the Lucy Letby case has
new levels of horror (due to social media, UK tabloid newspapers, …)
•I am now well known as a sick and deluded conspiracy theorist
Rabbit hole 2: fighting unjust
convictions of serial killer nurses

6. https://www.mirror.co.uk/news/uk-news/lucy-letby-activist-who-argues-32151666

7. EXCLUSIVE: Lucy Letby activist who argues baby killer is
innocent gives talk at university.
Academic Richard Gill, who claims prosecution evidence against baby
killer Lucy Letby was bogus, was allowed to give a talk to maths
students at Liverpool University.
A professor who campaigns to free  Lucy Letby was allowed to lecture
students near the hospital where she murdered babies.
Academic Richard Gill claims prosecution evidence was bogus and trolls
an NHS doctor who helped secure the serial killer’s conviction. But Gill’s
alarming web posts did not deter Liverpool University from inviting him to
lecture.
It is 17 miles from the Countess of Chester Hospital, where Letby was a
nurse. Gill, 72, a professor of statistics, travelled from his home in Holland for
the class at the department of mathematical scientists.
Although he did not mention the Letby case during his talk last Wednesday,
he later boasted online he had also visited Chester, and had given “an
informal talk on the Letby case”. Letby, 34, was jailed for life in August after a
10-month trial for murdering seven tots and attempting to kill six.

8. •Proof using statistical causality theory (DAG’s etc)
Bell’s theorem
Back to sanity (for a while)

9. •Nodes stand for random variables
•Arrows stand for direct “causal” dependence: they specify the
causal structure of the following statistical simulation:
REPEAT: Till there are none left, pick a node whose parents
have all been instantiated, and simulate its value using a
conditional probability distribution for chosen node, given
values of its parent nodes
•Corollary: root nodes stand for some variables which
are all statistically independent of one another
Encapsulate assumptions about statistical dependencies and
independencies between a bunch of random variables (also conditional …)
Causal models based on DAGs
[Regularity conditions?]

11. •Summary: Theorem, the model is equivalent to a deterministic
model where each node equals a deterministic function of parent
nodes and an independent random “innovation”
•So we can define the counterfactual outcomes “what each node
would have been had the parent nodes been different”
•Now just follow Bell’s CHSH proof … but please rewrite it in the
modern notation of probability theory
•Basic fact: for X1, X2, Y1, Y2 = +/–1
X1 Y1 – X1 Y2 – X2 Y2 – X2 Y1 = –2 or +2
Please read the paper
The proof
[Regularity conditions? Measurability?]

12. Really, the less said about it the better, but …
Marian Kupczynski’s model
https://www.mdpi.com/2624-960X/5/2/32

13. •Let settings A, B be outcomes of measurements of spin in x
direction of two ancillary spin-half particles prepared in spin-up (z
direction) state
•So these models are allowed inside conventional quantum
information theory, do not require “free will” or “sentient agents
outside of quantum theory”
•As Sabine Hossenfelder©™ likes to say (and I agree), “super
determinism” is just a derogatory term for “determinism”
•Tim Palmer “thins” conventional QIT to dense subsets of states and
measurements, making some triples (bipartite state, measurement
1, measurement 2) unavailable and a Bell experiment impossible
(one of the four measurement combinations is not available)
“How can we be sure these models are
consistent with the laws of quantum physics?”
Tim Palmer’s model

14. •This looks to me like a conspiracy theory, hidden in a smokescreen
of sophisticated maths which no one can quickly understand
•Bell’s theorem does not depend on any conception that
measurement settings are set exactly or reproduced exactly
•Palmer asks us to believe that what happened in the deep past of a
classical universe ensures that quantum correlations arise naturally
in certain contexts
•He does not give us any physical mechanism whereby this should
be expected
•Even if you believe in determinism, you do not believe this
“How can we be sure these models are
consistent with the laws of quantum physics?”
Tim Palmer’s model

16. A causal model of one trial in a Bell experiment
NB a *classical* causal model, not a quantum causal model!
X Y
A B
(Hidden)
Experimenter
Settings A,B∈{1,2};outcomes X,Y∈{−1,+1}
The deep past
Tim Palmer’s model

17. “We explain how a simple probabilistic locally causal model is able to reproduce quantum
correlations in Bell tests.”
Marian Kupczynski’s model
settings (x, y) as they wish: randomly or in a systematic way. In our model it does not matter.
The choice of (x, y) does not depend on (λ1, λ2) describing incoming signals.
Simple probabilistic model incorporating contextual hidden variables allows reproducing
the long range correlations between clicks observed by Alice and Bob in different settings [52, 53]:

(1)
where Λxy= Λ1 x Λ2 x Λx x Λy , Ax(λ1,λx ) and By(λ2,λy ) are equal 0, ±1 . Random experiments
performed in different settings are described using different parameter spaces Λxy in agreement
with QM and with Kolmogorov theory of probability [10, 34, 45, 51]. The oversimplification
made by Bell was the assumption that Λxy= Λ =Λ1 x Λ2 what is equivalent to assuming that the
correlations in different incompatible experiments may be deduced from a joint probability
distribution on some unique probability space Λ [25, 26]. If the model (1) is used Bell-type
inequalities may not be proven and quantum predictions may be reproduced. In spite of the fact
that it has been explained by several authors the speculations about nonlocality of Nature and the
quantum magic continue.
Using the model (1) one obtains explicitly local expectation values of single clicks observed
by Alice


12
112
,
( | x) A ( , ) ( )P( , )
x
xxxx
EA P

  
(2)
and Bob:

12
212
,
(B | y) B ( , ) ( )P( , )
y
yyyy
EP

  
(3)
The model (1-3) allows explaining why single counts deduced from the estimated
correlations between Alice’s and Bob’s may depend on the settings and it does not mean that
Einsteinian no-signaling is violated [53].
Several authors argue, evoking the Bayes theorem, that setting dependent parameters
compromise experimenters’ freedom of choice. It has been proven recently in [52] that such
conclusion is incorrect.
Therefore the violation of Bell-type inequalities neither proves the nonlocality of Nature nor
the completeness of QM.
In the next section we discuss a general problem whether a probabilistic description of a
random experiments may be considered complete. We conclude that we still don’t know whether
QM is predictably complete and that it should be tested.
4. Can a statistical description of phenomena be considered complete?
In this section we reproduce again after some editing several paragraphs form [46].
A statistical description is not a description of individual objects but it is a description of
regularities observed in large populations or in the outcomes of a series of repeated random
experiments.
Let us examine a series of coin flipping experiments. Instead of coins having head and tails
we have coins with one side ”blue”(B) and one side ”red”(R). If we want to provide a complete
description of a coin using concepts of classical physics and mechanics we may say that a coin is
12 12
( , | x, y) A ( , )B ( , ) ( ) ( )P( , )
xy
xxy yxxyy
EAB P P

    



18. settings (x, y) as they wish: randomly or in a systematic way. In our model it does not matter.
The choice of (x, y) does not depend on (λ1, λ2) describing incoming signals.
Simple probabilistic model incorporating contextual hidden variables allows reproducing
the long range correlations between clicks observed by Alice and Bob in different settings [52, 53]:

(1)
where Λxy= Λ1 x Λ2 x Λx x Λy , Ax(λ1,λx ) and By(λ2,λy ) are equal 0, ±1 . Random experiments
performed in different settings are described using different parameter spaces Λxy in agreement
with QM and with Kolmogorov theory of probability [10, 34, 45, 51]. The oversimplification
made by Bell was the assumption that Λxy= Λ =Λ1 x Λ2 what is equivalent to assuming that the
correlations in different incompatible experiments may be deduced from a joint probability
distribution on some unique probability space Λ [25, 26]. If the model (1) is used Bell-type
inequalities may not be proven and quantum predictions may be reproduced. In spite of the fact
that it has been explained by several authors the speculations about nonlocality of Nature and the
quantum magic continue.
Using the model (1) one obtains explicitly local expectation values of single clicks observed
by Alice


12
112
,
( | x) A ( , ) ( )P( , )
x
xxxx
EA P

  
(2)
and Bob:

12
212
,
(B | y) B ( , ) ( )P( , )
y
yyyy
EP

  
(3)
The model (1-3) allows explaining why single counts deduced from the estimated
correlations between Alice’s and Bob’s may depend on the settings and it does not mean that
Einsteinian no-signaling is violated [53].
Several authors argue, evoking the Bayes theorem, that setting dependent parameters
compromise experimenters’ freedom of choice. It has been proven recently in [52] that such
conclusion is incorrect.
Therefore the violation of Bell-type inequalities neither proves the nonlocality of Nature nor
the completeness of QM.
In the next section we discuss a general problem whether a probabilistic description of a
random experiments may be considered complete. We conclude that we still don’t know whether
QM is predictably complete and that it should be tested.
4. Can a statistical description of phenomena be considered complete?
In this section we reproduce again after some editing several paragraphs form [46].
A statistical description is not a description of individual objects but it is a description of
regularities observed in large populations or in the outcomes of a series of repeated random
experiments.
Let us examine a series of coin flipping experiments. Instead of coins having head and tails
we have coins with one side ”blue”(B) and one side ”red”(R). If we want to provide a complete
description of a coin using concepts of classical physics and mechanics we may say that a coin is
12 12
( , | x, y) A ( , )B ( , ) ( ) ( )P( , )
xy
xxy yxxyy
EAB P P

    



19. •Problem (raised by many): when we flick a switch to choose
setting 1 or 2, the exact time this happens, the pressure we put
on the switch, the trembling of our finger, … are also physical
inputs to the system we are studying
•We must redraw our DAG with new “hidden” nodes, and with
settings = outputs, not inputs
All models are wrong, some are useful!
But, still, has Bell sent us down a rabbit hole?

20. A causal model of one trial in a Bell experiment
NB a *classical* causal model, not a quantum causal model!
X Y
A B
(Hidden)
Experimenter
Settings A,B∈{1,2};outcomes X,Y∈{−1,+1}
Recent past in the labs
A
B

21. •My final questions to Tim Palmer on queios.com:
•Dear Tim, Thanks for your comments. As you know, I fundamentally
disagree with you. I planned for a long time to write up a careful analysis
and refutation of your arguments, but right now I’m rather tied up with
fighting the biggest miscarriage of justice in UK history ever: the Lucy
Letby case. In particular, we are developing causal models for the events at
Countess of Chester Hospital to show that beyond reasonable doubt, Lucy
is innocent. You might find your claim that these models are inconsistent
with the laws of physics soon being adopted by the hysterical mobs incited
by the UK tabloids, so: maybe we will be reading about it in the Daily Mail
or The Telegraph, any day soon! I hope soon to be visiting Oxford so we
can discuss your theory of quantum physics (not my theory of Lucy Letby)
at the blackboard. Would you even like to invite me to give a seminar? That
is a causal intervention which you could make, if (FAPP) you have free will,
that will force me to write up my ideas properly. Yours, Richard.
How many angels can dance on the tip of a pin?
End of quantum talk

22. •Gerard ’t Hooft anecdote concerning the case of Lucia de Berk
•Without intervention of Gerhard ’t Hooft, Lucia de Berk might still
be in jail
•Watch “Lola runs”. You can make a difference
•Your UK colleagues can more easily make a difference, than you
can, and you can influence them
•Oh yes, you can! Butterfly effect
•Or do you really believe in superdeterminism? In predestination?
Does that absolve you from making moral choices?
Yes, everything is connected
So, did those rabbit holes meet?

23. •Trial lasted 10 months (2022–23)
•Police investigation lasted 2017–2022, 60 police inspectors full time,
cost: a million pounds per year
•Lucy’s defence team has applied for an appeal
•The application has been turned down
•The grounds of the application and the reasons it was rejected are secret
•The prosecution appealed against one “no verdict” (baby K) and Lucy
will be retried starting June 10
•An enquiry will start in September to determine why management tried
to prevent senior doctors from going to the police, in order to prevent
cases like this in the future
In jail for life, key is thrown away
Lucy Letby, current situation

24. •Article by Rachel Aviv in “New Yorker” (geoblocked in UK)
•Public opinion is shifting (not yet far enough)
•Next step should be an application to the CCRC …
Or just the lights of the approaching train
Light at the end of the tunnel?
https://www.newyorker.com/magazine/2024/05/20/lucy-letby-was-found-
guilty-of-killing-seven-babies-did-she-do-it

25. •Read a bit about the case and talk to your friends in the UK
about it
•Starting points could be: my blog https://www.gill1109.com;
New Yorker article …
What can you do?

26. •Read a bit about the case and talk to your friends in the UK
about it
•Help me finish my paper with more material on Tim Palmer’s
approach
•I need helpers to convert existing written material into WordPress
web pages
What can you do?