Optimal Statistical Analyses of Bell Experiments

[Richard Gill]

This topic is about bringing standard ideas and methodology from present day statistical science into the toolbox of experimentalists who need to analyse the data of Bell experiments. A paper by myself has just come out,

https://www.mdpi.com/2673-9909/3/2/23

Abstract: We show how both smaller and more reliable p-values can be computed in Bell-type experiments by using statistical deviations from no-signalling equalities to reduce statistical noise in the estimation of Bell’s S or Eberhard’s J. Further improvement was obtained by using the Wilks likelihood ratio test based on the four tetranomially distributed vectors of counts of the four different outcome combinations, one 4-vector for each of the four setting combinations. The methodology was illustrated by application to the loophole-free Bell experiments of 2015 and 2016 performed in Delft and Munich, at NIST, and in Vienna, respectively, and also to the earlier (1998) Innsbruck experiment of Weihs et al. and the recent (2022) Munich experiment of Zhang et al., which investigates the use of a loophole-free Bell experiment as part of a protocol for device-independent quantum key distribution (DIQKD).
Keywords: Bell experiment; quantum entanglement; Bell-CHSH inequality; maximum likelihood estimation; testing statistical hypotheses; Wilks generalized likelihood ratio test

Richard

Comments please!

[Richard Gill]

Here's the paper.

[Kupczynski, Marian]

Dear Richard

After reading the introduction, it seems to be an  excellent paper. Congratulations

Marian

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From: bell_quantum...@googlegroups.com <bell_quantum...@googlegroups.com> on behalf of Richard Gill <gill...@gmail.com>
Sent: Tuesday, May 16, 2023 6:55 AM
To: Bell inequalities and quantum foundations <Bell_quantum...@googlegroups.com>
Subject: [Bell_quantum_foundations] Re: Optimal Statistical Analyses of Bell Experiments

 

Attention: L’émetteur de ce courriel est externe à l’Université du Québec en Outaouais.

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[Richard Gill]

Thank you Marian! I worked this out for a Växjö talk four years ago, https://www.slideshare.net/gill1109/yet-another-statistical-analysis-of-the-data-of-the-loophole-free-experiments-of-2015

I gave the task of writing up the details to a new PhD student, but there was no progress, she got sick, had all kinds of problems; she came from Turkey on a grant from the Turkish science research council, but then there was the failed coup and the crackdown on Erdogan’s perceived enemies, and she became a refugee in the Netherlands. She should have stuck with number theory and not tried to move to quantum information.

Richard

[Richard Gill]

I'm preparing my slides for my talk at the upcoming Växjö conference. Here's a preview. 

https://www.slideshare.net/gill1109/vaxjo2023rdgpdf

[Austin Fearnley]

Hi Richard

Hope you have a very useful and enjoyable meeting.  Sounds attractive.

I have read your slides in a very condensed single page, all that was available without subscribing to your slide service.

Just one odd, low-brow thought about S=2.07.  I will be glad to be told that I am unnecessarily concerned about the following and I am prepared to accept being informed that I may be misunderstanding the statistical theory for this.

We have had a series of posts where Bryan adds and/or averages correlation coefficients.  I thought, and I think wrote, that correlations should not be averaged or added if it can be avoided.  Instead it is better to use one single calculation for the total. I think Bryan should do this and give up on averaging.

However, to calculate a CHSH S value one needs to 'add+add+add+subtract' four expectation values or their sample estimations. I wonder what effect this adding may have on the S statistic when correlation coefficient are not on a robust physical scale suitable for adding or averaging?  In a previous life I have routinely calculated hundreds of correlation coefficients (one per school or assessment set) in single trials and used Fisher's z to transform the correlations onto a more robust scale.  Then averaged them on the Fisher scale, and then find the average correlation by transforming back the transformed average using the inverse Fisher's z transformation.  I usually found that taking the median correlation, without any transformations, corresponded well with the transformed average.

I wonder what the effect is of adding the four sample estimates in the S calculation?
One could easily set up random data to test this.  One using sets of data without z transformation and one using sets of data with z transformations.  

To get S=2 one has four r's of about 0.5.  This is well away from the traditionally dangerous part of the correlation range.  The dangerous parts being near +1 and -1.  However, S=2.07 is equivalent to an average correlation of 0.50175  which is not far from 0.5.  S=2.07 strikes me as being very disappointing despite the high significance gained by one million data point.  If you are living dangerously a million times, it does not strike me as having much ... what is the word? [Note: You used the word twice in your recent paper but I have forgotten it.  It is the age of my brain letting me down.  It is a more attractive kind of significance.  More valid then merely statistical.]

For general correlation the barriers at +1 and -1 cause non-linear effects on the correlation scale making it no longer a ratio scale.  However, the Bell correlation when not exceeded is a barrier at 0.5. There may be barrier effects at r = 0.5 for this type of correlation?  I wonder if the S calculation is skewed by barrier effects?  S = 2.07 is rather disappointing, and it is maybe worth someone running some random trials to see whether one can get 2.07 by random data exceeding 2.0 because of adding on a non-linear un-physical scale?  The Fisher's z transformation would not be appropriate for a boundary effect at r=0.5 so another transformation would need to be used.

Austin

On Wednesday, May 31, 2023 at 11:45:03 AM UTC+1 Richard Gill wrote:

I'm preparing my slides for my talk at the upcoming Växjö conference. Here's a preview. 

https://www.slideshare.net/gill1109/vaxjo2023rdgpdf

etc.

[Richard Gill]

Dear Austin

You are completely misunderstanding the theory behind this.

The significance of the experiment in question is that it used superconducting qubits of the type used in the quantum computers which have so far been engineered by Google, IBM and others.

I am going to see them at the lab at ETH Zürich in two hours from now.

Richard

Sent from my iPhone

On 31 May 2023, at 18:25, Austin Fearnley <ben...@hotmail.com> wrote:

Hi Richard

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[Richard Gill]

PS did you actually read my paper on this topic yet? It is exactly about the issues you raise.

Why don’t you do the simulations you’d like to see? 

Sent from my iPhone

On 31 May 2023, at 18:25, Austin Fearnley <ben...@hotmail.com> wrote:

Hi Richard

[ben smith]



Hi Richard 

I will do a simulation.  I can run a basic Bell simulation four times with small Ns which have a barrier at r=0.5 .  Next, compare adding the four outcomes against a direct computation of the overall correlation.  If the experiment outcome had been S= 2.4 instead of 2.07, I would have had no concerns about boundary effects.

Ok, I will have a better read of your paper …

I am working on quantum computer calculations of probabilities but will comment about that in a different thread/conversation.

Best wishes 

Austin 

On 1 Jun 2023, at 06:21, Richard Gill <gill...@gmail.com> wrote:

 PS did you actually read my paper on this topic yet? It is exactly about the issues you raise.

[Richard Gill]

I don’t understand what you mean by doing a direct calculation of the overall correlation. What overall correlation? Why the hell would you? The experiment consists of four sub experiments with different experimental conditions. The question is whether three of the correlations are large and positive and one is large and negative.

Sent from my iPhone

On 1 Jun 2023, at 08:04, ben smith <ben...@hotmail.com> wrote:



[Austin Fearnley]

I am not being deliberately awkward.  Merely naive.

If people are only interested in the four separate experiments' individual outcomes, then why do they take the step of adding their results to get an S value for the outcome of the four experiments combined?  E.g. S=2.07.  It is the validity of this combination of the four into one S value which is concerning me.  It is back to basics for me to accept the validity of the S calculation and not especially connected with the 2022 experiment.  Of course you can measure the worth of each of the four individual experiments separately.  Anyway, I believe know how to simulate some results to test my concern.  

All the best

Austin

[Richard Gill]

Austin, Why don’t you read Bell’s paper? Or Wikipedia? The rationale is clearly explained. I’m at a loss to know what else to say to you. 

But here I will try yet again. The step is taken because people are interested in testing the hypothesis of local hidden variables. If you are not interested in LHV theories then you are not interested in Bell experiments.

By the way, concerning the interest of the new ETH experiment, read this paper:

One is interested in the four correlations found in the four subexperiments. You could put confidence intervals around each correlation, using the Bonferroni method in order to have a simultaneous confidence region. Would also be fine.

S is just a convenient way to see if the vector of 16 probabilities corresponding to the four subexperiments each with four joint outcomes lies inside the region covered by LHV theories. Did you never read my Statistical Science article? Read Tsirelson? Read Bell?

In fact, S is the optimal way to do this, if the experiment actually exhibits the symmetries which (according to QM) one expects from the optimal experiment.

Read my. Paper https://arxiv.org/abs/quant-ph/0307125 for more on how to design optimal Bell experiments. It’s nearly 20 years old!

Published as IEEE-Transactions on Information Theory 51 (2005), 2812-2835

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[Austin Fearnley]

Hi Richard

We have had a few posts offline since the post above.  I will try not to repeat myself too much.

I have already read Bell, CHSH and Tsirelson years ago. My history is relevant to my current ideas.  When I did read Bell it became trivially easy to agree with the idea that r=0.5 was a real barrier in classical physics (and as you say, not just in physics).  This agrees with my 'essence of Bell' idea that 0.707 is the correlation obtained using real numbers of projection lengths between a unit vector at 0 degrees and a unit vector at 45 degrees.  But when you restrict the projections to integers -1 and +1 you get attenuation or reduction of the correlation to 0.5.  Next, you cannot disattenuate the correlation from 0.5 back to 0.707 while still using integers. Not in classical calculations. Full stop.  End of story.

This is where I became disenchanted with Bell discussions of the kind years ago on Joy's/Fred's forums.  I did not want to get lost in maths in following in detail quite a few peoples' efforts at trying to prove mathematically that there was a flaw in Bell's Theorem.  So I switched off from reading flawed, lost in maths, reasoning and was content with my 'essence of Bell' idea.

At this stage, long ago, I grew weary of hearing about action at a distance to explain the disattenuated correlation.  I will not complain too much as I have brought back the idea of action at a distance via retrocausality.  I note that it is OK for Richard to discuss action at a distance, which classically is an unreal effect [ignoring Newton's laws], and yet retrocausality is strongly dismissed by him and most others.  Richard says action at a distance in the lab while I say backwards in time antiparticles in their own compactified 4D cause action at a distance in the lab. I say that I have explained the reason behind action at a distance while, if I am correct, Richard says he is not a physicist and does not need to explain action at a distance?  Someone else's job.

So the Nobel prize has been won, is it all over now? Certainly not.  IMO Einstein will eventually be seen as also being correct (everyone is a winner) because LHV (using retrocausality) will apply to the Bell experiments.  The oddity arises because at the particle level, there is locality and realism (not the realism that we currently accept) but at the level of Alice and Bob in the lab there is action at a distance.  Both perspectives are correct descriptions which exist simultaneously.  Also, Bell experiments live on into the future in quantum computing/communication.

I now move on, thanks to Bell experiments, by accepting that QM calculations are correct.  I still need to check that all quantum gates and circuits are compatible with entanglement for 'retrocausal' particles.  I have doubts of course.  It may be that this work will test my retrocausal model to destruction, but it may show that not all supposed Bell states support entanglement.  Or I might have worried unnecessarily and they all agree.  This also reduces QM calculations by analogy to Monty Hall calculations about which box holds the prize.  In my model the single particle LHV spin is the prize in the box.  But you also need the QM calculations to get the probability of finding the single particle spin in that box were it to be measured/opened.

------------

As I said, I agree more with four sub-experiment correlations having a joint confidence region using Bonferroni correction than with using the S statistic, which I have never liked.  In the early 1980s a colleague published in BERJ a paper:"Think before you square correlations, or do anything with them".  [People were squaring correlations to find the amount of shared variance.] One extra point: it would be easier for Bryan to work with his four separate sub-experiment correlations rather than use S.  Aim for four r=0.707 results, and avoid r=0.5.  Also, as a first step, ignore the pol and coh subsets and combine their data before calculating the four full-experiment correlations.  Avoids adding correlations altogether.  If you get a significant overall result then you can find correlations for pol and coh separately, again without needing to add correlations.  The S statistic may be optimal but in my opinion it is flawed merely because of adding four untransformed correlations together. Saying its scale is unphysical and only used for statistical significance does not seem adequate.  But Bryan should calculate the final S statistic, as is the norm, but only when he is happy that it agrees with the correlation results.  I still suspect that he will get correlation = approx 0.35.

[Richard Gill]

Dear Austin, 

dear all,

In my opinion, retrocausality = action at a distance. If you can go back in the past and then move to the future, while moving in space all the time, you can get somewhere else apparently instantaneously. So: retrocausality is as crazy as non-locality. Bell believed in realism, hence was forced to accept nonlocality. Austin: you are the same. You believe in Bell's reasoning, and you believe in QM. You also believe in realism, hence (because you are ruled by logic, unlike for instance Bryan Sanctuary) you are obliged to accept non-locality. You simply dress it up as retrocausality.

I have no objection to all that. I too tend to believe that QM does correctly describe nature, and moreover, that what the EPR-B model predicts is not only allowed by QM, but also allowed by physical reality. I think the Zurich experiment represents a great breakthrough since it essentially consists of a 30 meter long quantum computer of just two qubits. The experiment proves that quantum. entanglement was engineered in the lab and that 2 one-qubit quantum computers could do something which no network of two classical computers ever could have done.

Slides of my talk for QIP 2023, Växjö: https://www.slideshare.net/gill1109/vaxjo2023rdgpdf-258203493

Richard

[David Marcus]

> If you can go back in the past and then move to the future, while moving
> in space all the time, you can get somewhere else apparently
> instantaneously.

Indeed. How does calling it "retrocausality" simplify anything?

David

[ben smith]

I was going to start a thread/conversation here on Retrocausality but after this comment, and no doubt many more like it, will wait until I have written a paper on retrocausality, TSVF, quantum gates & bell states.  That may take a year. 

My retrocausal model does not allow David or any other person to go back in time and neither does it allow any person to perform zigzags in time.  Not unless anybody knows someone who is entirely made of antiparticles and existing within a four dimensional spacetime which appears to be compactified in our spacetime.  And even they could not return to our time via a zigzag as they would forever travel backwards in time.  

Austin 

On 2 Jun 2023, at 21:05, David Marcus <david.ma...@gmail.com> wrote:

> If you can go back in the past and then move to the future, while moving

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[Richard Gill]

Dear all

Getting back to the topic of this conversation (though I'm looking forward to Austin's paper on retrocausality, TSVF, quantum gates & Bell states - in particular, what does TSVF stand for?), I have just been down to Zürich and visited Simon Storz, PhD student in Andreas Wallraff's group. They have been working for 10 years on a loophole free Bell experiment and finally it got done and published in Nature, a month or so ago. I've been trying to understand it. The qubits are physically realised on semi-conducting quantum interference devices. They are in a vacuum at a temperature of several milliKelvin and connected by a 30 metre long tube similarly empty and cold. Actually the qubits are qutrits. The basis states are called g, e, f where g stands for ground state and where f is used for transmission of superposition from A to B by microwave.

They start by creating the state (e + f) / sqrt 2 in Qutrit A, and state g in Qutrit B. Here, I am writing e, f, g as shorthand for kets |e>, |f>, |g>

Then they do something called emission/absorption which moves something from A to B
Then they do a rotation on B
They end up with a Bell state ( e, g + g, f ) / sqrt 2.   Here I am writing  e , g for |e> \otimes |g>. Thus the comma in e , g stands for tensor product of ket "e" in A with a ket "g" in B.

In my simplified notation, the initial bipartite state was ( e, g  +  f, g ) / sqrt 2

I would like to know what is the state after the emission/absorption, but before the rotation

Does anyone know where this is written up the language of quantum information processing / quantum computing?  Basically I am asking for two 9 x 9 matrices with 0's and 1's implementing two permutations of the 9 basis states of the product system. The first one creates entanglement, ie superposition of product states; the second one only changes the state of the second qutrit (but it stays entangled with the other).

Richard

The experiment:

https://www.nature.com/articles/s41586-023-05885-0

I analyse the data in

https://www.slideshare.net/gill1109/vaxjo2023rdgpdf-258203493

The beautiful thing is that the sample size is so large and the symmetry of the data so perfect that the Bell game test statistic, the test based on CHSH/standard error, and the test based on optimised CHSH/optimised standard error, and the test based on Wilks likelihood ratio test, all have essentially the same p-value. 

[Richard Gill]

David, mathematically, calling it "retrocausality" might be very appropriate. It might help mathematical intuition and hence it might be very useful for creative exploration of the physics. I don't think it helps *explain* quantum weirdness. But if you like, you can say "quantum weirdness is not actually weird, it is just elementary particles going backwards and forward in time and interfereing as waves with themselves" or something like that. I have no objection to people using language like that. It isn't helpful for me, but if it is helpful for others, why not? Fundamental particles are fictions, anyway. They correspond to solutions of equations and they seem also to correspond to some phenomena which can be measured in a lab and those phenomena also have to do with things we experience in daily life like a mutations of our DNA caused by a single photon from outer space giving us cancer...

My present opinion is that quantum weirdness cannot be explained, it just has to be accepted. The maths is basically simple, but how reality could actually be like that is a matter for wonder, I think. Enjoy it, take advantage of it, while you can.

[Mark Hadley]

There is a lot for an explanation of QM to achieve.

The fine structure constant relates the quantized unit of electric charge with planks constant ( the relation between energy and frequency)

We don't know what particles are. The question that drove centuries of scientific exploration is no longer asked, but until we can answer it we won't be able to explain the particle mass spectrum.

A theory that explains these things will be accepted and the insight it gives to QM will then become the orthodoxy. All we know now is that it will be wierd and that, by whatever name, it will have a fundamental nonlocal character. 

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[Austin Fearnley]

Hi Richard

TSVF is Two State Vector Formalism.  Not originated by Y Aharonov, Tel Aviv university, but he is up and running with it.  I am just an amateur and my own interpretation of TSVF is that a particle, say the photon, has two parts, not one.  One forward in time and one backwards in time. (In my preon model the photon has at least 96 states, half are forwards in time and half are backwards.)

On a Time Symmetric Formulation of Quantum Mechanics, 1995
https://arxiv.org/abs/quant-ph/9501011
(see page 6 for advanced and retarded states)

This paper is interesting, though I do not like weak measurements myself.
https://arxiv.org/abs/1610.09025
The Case of the Disappearing (and Re-Appearing) Particle

Yakir Aharonov: Finally making sense of the double-slit experiment
https://www.youtube.com/watch?v=Dm_Tpt7GWwY

TSVF from 28 mins onwards.  I find it difficult to follow the soundtrack.

IMO there is an absolute time in one sense.  It is given by the trivector of the global space. But the trivector is not a true vector and does not point in one static vector direction.  It is a little like my polarisation vector (in my gyroscope model of the electron) which for an electron points in one hemisphere.  To that extent it is absolutely giving the particle spin hemisphere, which is why repeated measurements in one direction give the same outcomes.

Austin

[Austin Fearnley]

Hi Richard

I am having trouble following the entanglements.

In the Susskind online lectures on entanglement, the singlet state was
(|ud> - |du>)/sqrt 2   where up and down have opposite polarisations.  

In this paper there are e(excited1), g(ground) and f(excited2) states.  I speculate that in absorbing the microwave energy something (neutral atom?) gets excited.  Or does not, for the ground state?  So excited/ground may be equated to say up/down?

What do they mean by  "time-reversal-symmetric photon" ( I know this cannot be retrocausality!)

I looked up transmon qbits. They are an improvement on Cooper-pair boxes.   Cooper pairs (of low temperature electrons) can have unusual fractional  charges of electricity which  may cause aproblem?   Fascinating stuff.

I suspect that entanglement before rotation is (|eg> - |ge>)/sqrt 2 whereas
after rotation it is (|gg> + |ee>)/sqrt 2, but how?
[The latter entanglement looks to me to be something one could only get after entanglement swapping?]
Is the f state something that appears and then disappears, like an enzyme that is only there to aid a  process?

I get confused by the measurement bases.  In my own simulations the particle and measuring device polarisation angles are easy to separate out in ones mind.  Here they seem to be confounded.  I am sure they are correct but it confuses me.
For example in Table 1 they write: "measurement bases of the two qubits".  I suppose the qubit is the measuring device!?

The experimental details are awesome. Awesome as in frightening.

NB        9x9 matrices.  I have looked online but only found 8x8. Sounds like you are talking about a two-gate process with a Hadamard plus a controlled Not.
https://en.wikipedia.org/wiki/Controlled_NOT_gate

But I know very little at this stage.  Yours seems a more complicated arrangement.


Austin

[Richard Gill]

I think I figured it out now.

The "9" I mentioned was 3x3. A two qutrit gate. Three qubit gates are 8x8

[Richard Gill]

One must see this as a tripartite system. A and B are qutrits. In between is the (near) vacuum which has basis |0>, |1>, |2>, ...

First of all the qutrit at A gets "half excited" and gets into a Schrödinger cat state using two of its three basis states. The live cat emits a photon into the vacuum and returns to death. As the photon propagates towards B it gets spread out and only part of it hits the qutrit at B at the best moment. This is why they only get S = 2.05. The photon emission/absorption just involves |0> and |1> of the vacuum and two of the three basis elements of the qutrit. It just a swap of two of the four basis states of the composite effectively 2x2 system. It's reversible. A emits the photon, B absorbs it.

On Monday, June 5, 2023 at 12:06:49 AM UTC+2 ben...@hotmail.com wrote:

[ben smith]

Hi Richard 

Your 9x9 solution looks very good.

I have found it mentioned in the following website which appears to be software that lets you play with quantum gates?

Austin

On 5 Jun 2023, at 05:50, Richard Gill <gill...@gmail.com> wrote:

I think I figured it out now.

The "9" I mentioned was 3x3. A two qutrit gate. Three qubit gates are 8x8

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<the QM.jpg>

[Richard Gill]

How they did it in Zürich (corrected)

[ben smith]

Hi Richard 

I think I follow that, though not sure exactly.  I would like to follow it through using matrices.

The Hadamard gate creates an entangled state and its input is |0,0,0> which is standard zeros for an Hadamard input.

The first swap is when fa drops down to ga whilst channel 0 goes up to channel 1.

The second swap is when channel 1 drops to channel 0 which allows gb to go up to fb.

Not sure what is going on in the channel’s vacuum energy levels.

I am still not sure what is being measured, and where, and what by.

Austin 

On 5 Jun 2023, at 15:56, Richard Gill <gill...@gmail.com> wrote:



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<the QM.jpg>

[ben smith]

Or to simplify terms,

Start with 000 and apply entanglement gate to get

100 + 100

Swap to get 100 + 010

Swap again to get 100 + 001

This last entanglement entangles an energetic state for A with an energetic state for B.

Austin 

[Richard Gill]

Austin, at ETH Zürich they work with qutrits, not qbits.

The first step at ETH Zürich was to convert |g_A> to (1/sqrt 2) (|e_A> + |f_A>)

I did not yet investigate how that is done.

g = ground state, e and f = first two excited states.

Sent from my iPad

On 5 Jun 2023, at 18:55, ben smith <ben...@hotmail.com> wrote:



[ben smith]



Hi Richard 

Yes, I have lots of questions still.

My simplified terms matched yours exactly except I replaced ground states g by zero.  And replaced both excited states f and e by unity.  So I am unclear why you think I have reverted to qubits while you haven’t?

I note that the channel is used to swap states round and, like an enzyme, drops out of the end entanglement state, except  defaulting to zero states.   e,0,0 + 0,0,f

There are extra sections of information in the Nature paper which are inaccessible to me.  Presumably you can see them.

I will look at trying to convert  |g_A> to (1/sqrt 2) (|e_A> + |f_A>) using gates.

Austin 

On 5 Jun 2023, at 19:08, Richard Gill <gill...@gmail.com> wrote:

 Austin, at ETH Zürich they work with qutrits, not qbits.

[ben smith]

Hi Richard 

Yes, I have lots of questions still.

My simplified terms matched yours exactly except I replaced ground states g by zero.  And replaced both excited states f and e by unity.  So I am unclear why you think I have reverted to qubits while you haven’t?

I note that the channel is used to swap states round and, like an enzyme, drops out of the end entanglement state, except  defaulting to zero states.   e,0,0 + 0,0,f

There are extra sections of information in the Nature paper which are inaccessible to me.  Presumably you can see them.

I will look at trying to convert  |g_A> to (1/sqrt 2) (|e_A> + |f_A>) using gates.

Austin 

On 5 Jun 2023, at 19:08, Richard Gill <gill...@gmail.com> wrote:

 Austin, at ETH Zürich they work with qutrits, not qbits.

[Richard Gill]

The transition from g to (e + f) / sqrt 2 might be done by (g, e) <-> (e, g) and then (e,f) <-> (e +f, e - f) / sqrt 2

[ben smith]

Hi Richard 

I  have had other duties today.  Life cannot be all fun physics, unfortunately.

The following site looks relevant 

It starts and finishes with qutrits and in between engages in swapping pairs of states.

I have not been to the university bookshop since covid but I may go tomorrow.  I have looked online at the bookshop abstracts and can’t find anything very relevant.  But I can browse better in the actual shop.  Probably find something very basic as  primer for me.

Austin 

On 6 Jun 2023, at 08:02, Richard Gill <gill...@gmail.com> wrote:

The transition from g to (e + f) / sqrt 2 might be done by (g, e) <-> (e, g) and then (e,f) <-> (e +f, e - f) / sqrt 2

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