Sakurai's proof of Wigner's inequality

[Richard Gill]

Dear Alexandre, dear Bell group

Alexandre sent us a document “a_1.pdf” , which I attach right here:

[Richard Gill]

Here is an executive summary of that last email.

Alexandre'’s mistake was to use the same symbol “P” for four different experiments, each with a different outcome space and probability measure. One LHV experiment. Three QM experiments.

The LHV experiment is: pick a ball from a vase, duplicate it, send it to Alice and Bob. There are 8 different kinds of balls. Each ball contains instructions for Alice and for Bob’s assistants: “if your boss tells you to  measure a, b or c, then this is the outcome +/- you must give him or her.

Alice and Bob are the directors of two labs. They think they are doing a Bell experiment. But their assistants (together with Carol at a third location) are lazy. They pocketed the money for the apparatus for the very expensive experiment and figured out a cheap way to keep the directors happy.

However, the directors will be disappointed. No Bell violation.

Richard

On 29 Sep 2026, at 08:15, Richard Gill <gill...@gmail.com> wrote:

Dear Alexandre, dear Bell group

Alexandre sent us a document “a_1.pdf” , which I attach right here:

<a_1.pdf>

He wrote

On 28 Sep 2026, at 07:28, Alexandre de Castro <alx...@gmail.com> wrote:

Richard, 
I need to confine myself strictly to the calculations. You can see in "a_1.pdf" that quantum mechanical predictions for the singlet state (spin-1/2) can be derived from the violation. However, these same predictions can also be derived from a framework of uniformly distributed local hidden variables

He later added:

On 29 Sep 2026, at 03:22, Alexandre de Castro <alx...@gmail.com> wrote:

Richard, 

try to point out any error in the mathematical development of the document I shared with you. Just a single error, no matter how small, would strengthen your point. But please do this in the group so everyone else can see it too.

In “a_1.pdf”  he reproduces part of Sakurai’s proof of Wigner’s inequality, equation (3.435) in that book:

<Wigner Inequality (Sakurai equation 3.435).pdf>

The relevant part of the book is pages 227, 228 and 229, attached here:

<Sakurai p 227.pdf>

<Sakurai p 228.pdf>

<Sakurai p 229.pdf>

Please look at Table 3.2 on page 228, and let me define N = N_1 + N_2 + … + N_8. 

<Sakurai Table 3.2.pdf>

The LHV framework framework presented in Table 3.2 defines a single Kolmogorov probability space (Omega, F, P) with 8 elements omega_i, with elementary probabilities N_i / N. 

On that probability space, Alexandre’s "a_1.pdf”  Equation (2) is true. The three events which I will denote in shorthand as {a+, b+}, {a+, c+} and {c+, b+} are events in that probability space. 

By the event {a+, b+} I mean the event: Observer A measures “a", gets “+" and Observer B measures “b", gets “+" as well. For instance, the event {a+,b+} is {omega_3, omega_4}.

The probabilities in his Equation (1) are not probabilities of events in one Kolmogorov probability space.

Those three events are events in three different probability spaces  (Omega_i, F_i, P_i) , corresponding to three different experiments.

Observer A measures “a” and Observer B measures “b” in QM Experiment 1.

It has four outcomes and they have four probabilities which, assuming the singlet state, we can calculate with quantum mechanics.

Observer A measures “a” and Observer B measures “c” in QM Experiment 2.

It too has four outcomes and they have four probabilities which, assuming the singlet state, we can calculate with quantum mechanics.

Observer A measures “c” and Observer B measures “b”  in QM Experiment 3.

It too has four outcomes and they have four probabilities which, assuming the singlet state, we can calculate with quantum mechanics.

QM gives us 4 + 4 + 4 probabilities and they satisfy Alexandre’s "a_1.pdf"  Equation (1).

In Equation (1), it would have been useful to give the three instances of the symbol “P” different subscripts. They are probability measures on three different probability spaces (three different experiments), computed following QM.

In Equation (2), “P” is a fourth probability measure on a fourth probability space. A fourth, different, classical experiment: pick one ball out of a vase containing N balls. The LHV is the ball type: 1, 2, …, 8. Nature picks a ball, duplicates it, sends it to Alice and Bob. The ball they receive tells them what outcome they will get for each of the three measurements, each of them can make.

Yours

Richard

[Justo Pastor Lambare]

Richard

I don't understand. Alexander final his explanation reads:

"Thus, the issue being raised here is the following apparent tension:
How can the same LHV framework both reproduce the singlet-state probabilities and imply
a Bell–Wigner constraint that those same probabilities violate?"

It can't, isn't that what Bell proved and the meaning of the Bell theorem?i.e. that no local hidden variable can reproduce the quantum probabilities.

Regards,

      Justo Pastor Lambaré

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[Richard Gill]

Justo, you exactly right.

It can’t, it doesn’t.

Alexandre repeatedly writes P(some event) but he should write P(some event, in some experiment). His problem is purely a problem of notation.

Different experiments include events which, superficially, can be described with the same few words, but whose probabilities are computed according to different theoretical frameworks, and which moreover actually refer to different experiments. 

In QM the observer is active. In HV (a hidden variables model) the observer is inactive. Moreover, in a LHV model the observer is restricted by locality.

This, I believe, is also the root cause of apparently Bryan’s self-contradictory writings.

Richard

Sent from my iPad

On 29 Sep 2026, at 09:07, Justo Pastor Lambare <jup...@gmail.com> wrote:



[Alexandre de Castro]

Richard, 

(1) follows directly from Kolmogorov's third axiom. I had asked you to point out at least one mistake...

Em ter., 29 de set. de 2026, 03:15, Richard Gill <gill...@gmail.com> escreveu:

Dear Alexandre, dear Bell group

Alexandre sent us a document “a_1.pdf” , which I attach right here:

He wrote

On 28 Sep 2026, at 07:28, Alexandre de Castro <alx...@gmail.com> wrote:

Richard, 
I need to confine myself strictly to the calculations. You can see in "a_1.pdf" that quantum mechanical predictions for the singlet state (spin-1/2) can be derived from the violation. However, these same predictions can also be derived from a framework of uniformly distributed local hidden variables

He later added:

On 29 Sep 2026, at 03:22, Alexandre de Castro <alx...@gmail.com> wrote:

Richard, 

try to point out any error in the mathematical development of the document I shared with you. Just a single error, no matter how small, would strengthen your point. But please do this in the group so everyone else can see it too.

In “a_1.pdf”  he reproduces part of Sakurai’s proof of Wigner’s inequality, equation (3.435) in that book:

The relevant part of the book is pages 227, 228 and 229, attached here:

Please look at Table 3.2 on page 228, and let me define N = N_1 + N_2 + … + N_8. 

[Richard Gill]

I pointed out one mistake of omission, repeated throughout your note. You wrote P( … ) many times, but each time it was a different probability measure on a different sample space.

Sent from my iPad

[Alexandre de Castro]

So, you are reading a different document from the one I sent. 

The only probabilities mentioned are those of Wigner's inequality. I never defined probabilities — Sakurai was the one who did that.

[Richard Gill]

I read Sakurai and I read your document.

I read both, very carefully.

Sent from my iPad

On 29 Sep 2026, at 12:11, Alexandre de Castro <alx...@gmail.com> wrote: