“Simulating quantum ‘time travel’ disproves butterfly effect in quantum realm”—not!

A Special note for the Potential Employers from the Data Science field:

Recently, in April 2020, I achieved a World Rank # 5 on the MNIST problem. The initial announcement can be found here [^], and a further status update, here [^].

All my data science-related posts can always be found here [^].


This post is based on a series of tweets I made today. The original Twitter thread is here [^]. I have made quite a few changes in posting the same thought here. Further, I am also noting some addenda here (which are not there in the original thread).

Anyway, here we go!


1. The butterfly effect and QM: a new paper that (somehow) caught my fancy:

1.1. Why this news item interested me in the first place:

Nonlinearity in the wavefunction \Psi, as proposed by me, forms the crucial ingredient in my new approach to solving the QM measurement problem. So, when I spotted this news item [^] today, it engaged my attention immediately.

The opening line of the news item says:

Using a quantum computer to simulate time travel, researchers have demonstrated that, in the quantum realm, there is no “butterfly effect.”

[Emphasis in bold added by me.]

The press release by LANL itself is much better worded (PDF) [^]. In the meanwhile, I also tried to go through the arXiv version of the paper, here [^].

I don’t think I understand the paper in its entirety. (QC and all is not a topic of my main interests.) However, I do think that the following analogy applies.:

1.2. A (way simpler) analogy to understand the situation described in the paper:

The whole thing is to do with your passport-size photo, called “P”.

Alice begins with “P”, which is given in the PNG/BMP format. [Should the usage be the Alice? I do tend to think so! Anyway…]

She first applies a 2D FFT to it, and saves the result, called “FFT-P”, in a folder called “QC” on her PC. Aside: FFT’ed photos look like dots that show a “+”-like visual structure. Note, Alice saves both the real and the imaginary parts of the FFT-ed image. This assumption is important.

She then applies a further sequence of linear, lossless, image transformations to “FFT-P”. Let’s call this ordered set of transformations “T”. Note, “T” is applied to “FFT-P”, not to “P” itself.

As a result of applying the “T” transformations, she obtains an image which she saves to a file called “SCR-FFT-P”. This image totally looks like random dots to the rest of us, because the “T” transformations are such that they scramble whatever image is fed to them. Hence the prefix “SCR”, short for “scrambled”, in the file-name.

But Alice knows! She can always apply the same sequence of transformation but in the reverse direction. Let’s call this reverse transformation “T-inverse”.

Each step of “T” is reversible—that’s what “linear” and “lossless” transformation means! (To contrast, algorithms like “hi-pass” or “low-pass” filtering, or operators like the gradient or Laplacian are not loss-less.)

Since “T” is reversible, starting with “SCR-FFT-P”, Alice can always apply “T-inverse”, and get back to the original 2D FFT representation, i.e., to “FFT-P”.

All this is the normal processing—whether in the forward direction or in the reverse direction.

1.3. Enter [the [?]] Bob:

As is customary in the literature on the QC/entanglement, Bob enters the scene now! Alice and Bob work together.

Bob hates you. That’s because he believes in every claim made about QC, but you don’t. That’s why, he experiences an inner irrepressible desire to do some damage to your photograph, during its processing.

So, to, err…, “express” himself, Bob comes early to office, gains access to Alice’s “QC” folder, and completely unknown to her, he modifies a single pixel of the “FFT-P” image stored there, and even saves it. Remember, this is the FFT-ed version of your original photo “P”.

Let’s call the tampered version: “B-FFT-P”. On the hard-disk, it still carries the name “FFT-P”. But its contents are modified, and so, we need another name to denote this change of the state of the image.

1.4. What happens during Alice’s further processing?

Alice comes to the office a bit later, and soon begins her planned work for the day, which consists of applying the “T” transformation to the “FFT-P” image. But since the image has been tampered with by Bob, what she ends up manipulating is actually the “B-FFT-P” image. As a result of applying the (reversible) scrambling operations of “T”, she obtains a new image, and saves it to the hard-disk as “SCR-B-FFT-P”.

But something is odd, she feels. So, just to be sure, she decides to check that everything is OK, before going further.

So, she applies “T-inverse” operation to the “SCR-B-FFT-P” file, and obtains the “B-FFT-P” image back, which she saves to a file of name “Recovered FFT-P”. Observe, contents-wise, it is exactly the same as “B-FFT-P”, though Alice still believes it is identical to “FFT-P”.

Now, on a spur of the moment, she decides also to apply the reverse-FFT operation to “Recovered FFT-P”, i.e., to the Bob-tampered version of the FFT-ed version of your original photo. She saves the fully reversed image as “Recovered P”.

Just to be sure, she then runs some command that does a binary bit-wise comparison between “Recovered P” and the original “P”.

We know that they are not the same. Alice discovers this fact, but only at this point of time.

1.5. The question that the paper looks into:

If I understand it right, what the paper now ponders over is this question:

How big or small is the difference between the two images: “Recovered P” and the original “P”?

The expected answer, of course, is:

Very little.

The reason to keep such an expectation is this: FFT distributes the original information of any locality over the entire domain in the FFT-ed image. Hence, during reverse processing, each single pixel in the FFT-ed image maps back to all the pixels in the original image. [Think “holographic” whatever.] Therefore, tampering with just one pixel of the FFT-ed representation does not have too much effect in the recovered original image.

Hence, Alice is going to recover most of the look and feel of your utterly lovely, Official, passport-size visage! That is what is going to happen even if she in reality starts only from the scrambled tampered state “SCR-B-FFT-P”, and not the scrambled un-tampered state “SCR-FFT-P”. You would still be very much recognizable.

In fact, due to the way FFT works, the difference between the original photo and the recovered photo goes on reducing as the sheer pixel size of the original image goes on increasing. That’s because, regardless of the image size, Bob always tampers only one pixel at a time. So, the percentage tampering goes on reducing with an increase in the resolution of the original image.

1.6. The conclusion that the paper draws from the above:

Let’s collect the undisputable facts together:

  • There is very little difference between the recovered image, and the original image.
  • Whatever be the difference, it goes on reducing with the size of the original image.

The paper now says, IMO quite properly, that Bob’s tampering of the single pixel is analogous to his making a QM measurement, and thereby causing a permanent change to the concerned (“central”) qubit.

But then, the paper draws the following conclusion:

The Butterfly Effect does not apply to QM as such; it applies only to classical mechanics.

Actually, the paper is a bit more technical than that. In fact, I didn’t go through it fully because even if I were to, I wouldn’t understand all of it. QC is not a topic of my primary research interests, and I have never studied it systematically.

But still, yes, I do think that the above is the sort of logic on which the paper relies, to draw the conclusion which it draws.


2. My take on the paper:

2.1 It’s an over-statement:

Based on what I know, and my above, first, take, I do think that:

The paper makes an over-statement. The press release then highlights this “over” part. Finally, the news item fully blows up the same, “over” part.

Why do I think so? Here is my analysis..

If the Butterfly Effect produced due to nonlinearity is fully confined only to making an irreversible (or at least exponentially divergent) change only to a single pixel in the FFT representation of the original image (or alternatively, even if we didn’t look into alternative analgoy in which what Bob tampers is the original photograph but each subsequent processing involves only an FFT-ed version), then any and all of the further steps of linear and reversible transformations wouldn’t magnify the said tampering.

Why not?

Because all the further steps are prescribed to be linear (and in fact even reversible), that’s why!

In other words, what the paper says boils down to a redundancy (or, a re-statement of the same facts):

A linear and reversible transformation is emphatically not a non-linear and exponential divergent one (as in the butterfly effect).

That’s what the whole point of the paper seems to be!

2.2. The actual processing described in the paper does not at all involve the butterfly effect:

Realize, the only place the butterfly effect can at all occur during the entire processing is as a mechanism by which Bob might tamper with that single pixel.

Now, of course, the paper doesn’t say so. The paper only says that there is a tampering of a qubit via a measurement effected on it (with all other qubits, constituting “the bath” being left alone).

But, yes, I have proposed this idea that the measurement process itself progresses, within the detector, via the butterfly effect. I identified it as such during my Outline document posted at iMechanica, here (PDF) [^].

Of course, I stand ready to be corrected, if I am wrong anywhere in the fundamentals of my analysis.

2.3. I didn’t say anything about the “time-travel” part:

That’s right. The reason is: there is no real time-travel here anyway!

Hmmm… Explaining why would unnecessarily consume my time. … Forget it! Just remember: There is no time-travel here, not even a time-reversal, for that matter. In the first half of the processing by Alice (and may be with tampering by Bob), each step occurs some finite time after the completion of previous step. In the second half of the processing, again, each step of the inverse-processing occurs some finite time after the completion of the previous step. What reverses is the sequence of operators, but not time. Time always flows steadily in the forward direction.

Enough said.

2.4. Does my critique reflect on the paper taken as a whole?

I did manage to avoid Betteridge’s law [^] thus far, but can’t, any more!

The answer seems to be: “no”, or at least: “I didn’t mean that“.

The thing is this: This is a paper from the field of Quantum Computer/Quantum Information Science—which is not at all a field of my knowledge (let alone expertize). The paper reports on a simulation the authors conducted. I am unable to tell how valuable this particular simulation is, in the overall framework of QInfoScience.

However, as a computational modelling and simulation engineer myself, I can tell this much: Some times, even a simple (stupid!)-looking simulation actually is implemented merely in order to probe on some aspect that no one else has thought of. The simulation is not an end in itself, but merely a step in furthering research. The idea is to explore a niche and to find / highlight some gap in knowledge. In topics that are quite complicated, isolation of one aspect at a time, afforded by a simulation, can be of great help.

(I can cite an example of a very simple-looking simulation, actually a stupid-looking simulation, from my own PhD time research: I had a conference paper on simulating a potential field using random walk and comparing its results with a self-implemented FEM solver. The rather coluorful Gravatar icon which you see (the one which appears in the browser bar when you view my posts here) was actually one of the results I had reported in this preliminary exploration of what eventually became my PhD research.)

Coming back to this paper, it’s not just possible but quite likely that the authors are reporting something that has implications for much more “heavy-duty” topics, say topics like quantum error corrections, where and when they are necessary, the minimum efficiency they must possess, in what kind of architecture/processing, and whatnot. I can’t tell, but this is the nature of simulations. Sometimes, they look simple, but their implications can be quite profound. I am in no position to judge the merits of this paper, from this viewpoint.

At the same time, I also think that probing this idea of measuring just one qubit and tracing its effects on the nearby “bath” of qubits can have good merits. (I vaguely recall the discussions, some time ago, of “pointer states” and all that.)

Yet, of course, I do have a critical comment to make regarding this paper. But my comment is entirely limited to what the paper says regarding the foundational aspects of QM and the relevance of chaos / nonlinear science in QM. With the kind of nonlinearity in \Psi which I have proposed [^], I can clearly see that you can’t say that just because the mainstream QM theory is linear, therefore everything about quantum phenomena has to be linear. No, this is an unwarranted assumption. It was from this viewpoint that I thought that the implication concerning the foundational aspects was not acceptable. That’s why (and how) I wrote the tweets-series and this post.

All in all, my critique is limited to saying that a nonlinearity in \Psi, and hence the butterfly effect, is not only possible in QM, but it is crucial in correctly addressing the measurement problem right. I don’t have any other critique to offer regarding any of the other aspects of the reported work.

Hope this clarifies.

And, to repeat: Of course, I stand ready to be corrected, if I have gone wrong anywhere in the fundamentals of my analysis regarding the foundational issues too.


3. An update on my own research:

3.1. My recent tweets:

Recently (on 23 July 2020), I also tweeted a series [^] regarding the on-going progress in my new approach. Let me copy-paste the tweets (not because the wording is great, but because I have to finish writing this post, somehow!). I have deleted the tweet-continuation numbers, but otherwise kept the content as is:

Regarding my new approach to QM. I think I still have a lot of work to do. Roughly, these steps:

1. Satisfy myself that in simplest 1D toy systems (PIB, QHO), x-axis motion of the particle (charge-singularity) occurs as it should, i.e., such that operators for momentum, 1/

position, & energy have *some* direct physical meaning.

2. Using these ideas, model the H atom in a box with PBCs (i.e., an infinite lattice of repeating finite volumes/cells), and show that the energy of electron obtained in new approach is identical to that in the std. QM (which uses reduced mass, nucleus-relative x, no explicit particle positions, only electron’s energy).

3. Possibly, some other work/model.

4. Repeat 2. for modelling two *interacting* electrons (or the He atom) in a box with PBCs.

Turned out that I got stuck for the past one month+ right at step no. 1!

However, looks like I might have finally succeeded in putting things together right—with one e- in a box, at least.

In the process, found some errors in my post from the ontologies series, part 10.

Will post the corrections at my blog a bit later.

Tentatively, have decided to try and wrap up everything within 4–6 weeks.

So, either I report success with my new approach by, say 1st week of September or so, or I give up (i.e. stop work on QM for at least few months).

But yes, there does seem to be something to it—to the modelling ideas I tried recently. Worth pursuing for a few weeks at least.

… At least I get energy and probability right, which in a way means also position. But I am not fully happy with momentum, even though I get the right numerical values for it, and so, the thinking required is rather on the conceptual–physical side … There *are* “small” issues like these.

But yes,

(1) I’m happy to have spotted definite errors in my own previous documentation—Ontologies series, part 10, as also in the Outline doc. (PDF here [^]) ,
and,
(2) I’m happy to have made a definite progress in the modelling with the new approach.

Bottomline: I don’t have to give up my new approach. Not right away. Have to work on it for another month at least.

3.2. Some comments on the tweets:

I need to highlight the fact that I have spotted some definite errors, both in the Ontologies series (part 10), and in the Outline document.

In particular:

3.2.1. In the Ontologies series, part 10, I had put forth an argument that it’s a complex-valued energy that gets conserved. I am not so sure of that any more, and am going over the whole presentation once again. (The part up covering the PIB modelling from that post is more or less OK.)

3.3.2. In the Outline document, I had said:

“The measurement process is nondestructive of the state of the System. It produces catastrophic changes only in the Instrument”

I now think that this description is partly wrong. Yes, a measurement has to produce catastrophic changes in the Instrument. But now, the view I am developing amounts to saying that the state of the System also undergoes a permanent change during measurement, though such a change is only partial.

3.3. Status of my research:

I am working through the necessary revision of all such points. I am also working through simulations and all. I hope to have another document and a small set of simulations (as spelt in the immediately preceding Twitter thread) soon. The document would still be preliminary, but it is going to be more detailed.

In particular, I would be covering the topic of the differences between the tensor product states (e.g. two non-interacting electrons) in a box vs. the entangled states of two electrons. Alternatively, may be, treating the proton as a quantum object (having its own wavefunction), and thus, simulating only the Hydrogen atom, but with my new approach. Realize, when you treat the proton quantum mechanically, allowing its singularity to move, it becomes a two-particle system.

So, a two-particle system is the minimum required for validation of my new approach.

For convenience of simulation, i.e. especially to counter the difficulties (introduced by boundaries due to discretization of space via FDM mesh), I am going to put the interacting pair of particles inside a box but with periodic boundary conditions (PBCs for short). Ummm… This word, “PBCs”, has been used in two different senses: 1. To denote a single, representative, finite-sized unit cell from an infinitely repeated lattice of such cells, and 2. Ditto, but with further physical imagination that this constitutes a “particle on a ring” so that computations of the orbital angular momentum too must enter the simulation. Here, I am going to stick to PBCs in the sense 1., and not in the sense 2.

I hope to have something within a few weeks. May be 3–4 weeks, perhaps as early as 2–3 weeks. The trouble is, as I implement some simulation, some new conceptual aspects crop up, and by the time I finish ironing out the wrinkles in the conceptual framework, the current implementation turns out to be not very suitable to accommodate further changes, and so, I have to implement much of the whole thing afresh again. Re-implementation is not a problem, at least not a very taxing one (though it can get tiring). The real problems are the conceptual ones.

For instance, it’s only recently that I’ve realized that there is actually a parallel in my approach to Feynman’s idea of an electron “smelling” its neighbourhood around. In Feynman’s version, the electron not only “smells” but also “runs everywhere” at the same time, with the associated “amplitudes” cancelling out / reinforcing at various places. So, he had a picture of the electron that is not a localized particle and yet smells only a local neighbourhood at each point in the domain. He could not remove this contradiction.

I thought that I had fully removed such contradictions, but only to realize, at this (relatively “late”) stage that while “my electron” is a point-particle (in the sense that the singularity in the potential energy field is localized at a point), it still retains the sense of the “smell”. The difference being, now it can smell the entire universe (action at a distance, i.e. IAD). I knew that so long as I use the Fourier theory the IAD would be there. But it was part-surprise and part-delight for me to notice that even “my” electron must have such a “nose”.

Another thing I learnt was that even if I am addressing only the spinless electron, looks like, my framework seems to very easily and naturally incorporates the spin too, at least so long as I remain in 1D. I had just realized it, and soon (within days) came Dr. Woit’s post “What is “spin”?” [^]. I don’t understand it fully, but now that I see this way of putting things, that’s another detour for me.

All in all, working out conceptual aspects is taking time. Further, simply due to the rich inter-connections of concepts, I am afraid that even if I publish a document, it’s not going to be “complete”, in the sense, I wouldn’t be able to insert in it everything that I have understood by now. So, I am aiming to simply put out something new, rather than something comprehensive. (I am not even thinking of having anything “well polished” for months, even an year or so!)

Alright, so there. May be I won’t be blogging for a couple of weeks. But hopefully, I will have something to put out within a month’s time or so…

In the meanwhile, take care, and bye for now…


A song I like:

(Hindi) जादूगर तेरे नैना, दिल जायेगा बच के कहाँ (“jaadugar tere nainaa, dil jaayegaa…”)
Singers: Kishore Kumar, Lata Mangeshkar
Music: Laxmikant-Pyarelal
Lyrics: Rajinder Krishen

[Another song from my high-school days that somehow got thrown up during the recent lockdowns. … When there’s a lockdown in Pune, the streets (and the traffic) look (and “hear”) more like the small towns of my childhood. May be that’s why!]


[Some very minor editing may be effected, but I really don’t have much time—rather, any enthusiasm—for it! So, drop a line if you find something confusing… Take care and bye for now…]

 

Do you really need a QC in order to have a really unpredictable stream of bits?

0. Preliminaries:

This post has reference to Roger Schlafly’s recent post [^] in which he refers to Prof. Scott Aaronson’s post touching on the issue of the randomness generated by a QC vis-a-vis that obtained using the usual classical hardware [^], in particular, to Aaronson’s remark:

“the whole point of my scheme is to prove to a faraway skeptic—one who doesn’t trust your hardware—that the bits you generated are really random.”

I do think (based on my new approach to QM [(PDF) ^]) that building a scalable QC is an impossible task.

I wonder if they (the QC enthusiasts) haven’t already begun realizing the hopelessness of their endeavours, and thus haven’t slowly begun preparing for a graceful exit, say via the QC-as-a-RNG route.

While Aaronson’s remarks also saliently involve the element of the “faraway” skeptic, I will mostly ignore that consideration here in this post. I mean to say, initially, I will ignore the scenario in which you have to transmit random bits over a network, and still have to assure the skeptic that what he was getting at the receiving end was something coming “straight from the oven”—something which was not tampered with, in any way, during the transit. The skeptic would have to be specially assured in this scenario, because a network is inherently susceptible to a third-party attack wherein the attacker seeks to exploit the infrastructure of the random keys distribution to his advantage, via injection of systematic bits (i.e. bits of his choice) that only appear random to the intended receiver. A system that quantum-mechanically entangles the two devices at the two ends of the distribution channel, does logically seem to have a very definite advantage over a combination of ordinary RNGs and classical hardware for the network. However, I will not address this part here—not for the most part, and not initially, anyway.

Instead, for most of this post, I will focus on just one basic question:

Can any one be justified in thinking that an RNG that operates at the QM-level might have even a slightest possible advantage, at least logically speaking, over another RNG that operates at the CM-level? Note, the QM-level RNG need not always be a general purpose and scalable QC; it can be any simple or special-purpose device that exploits, and at its core operates at, the specifically QM-level.

Even if I am a 100% skeptic of the scalable QC, I also think that the answer on this latter count is: yes, perhaps you could argue that way. But then, I think, your argument would still be pointless.

Let me explain, following my approach, why I say so.


2. RNGs as based on nonlinearities. Nonlinearities in QM vs. those in CM:

2.1. Context: QM involves IAD:

QM does involve either IAD (instantaneous action a distance), or very, very large (decidedly super-relativistic) speeds for propagation of local changes over all distant regions of space.

From the experimental evidence we have, it seems that there have to be very, very high speeds of propagation, for even smallest changes that can take place in the \Psi and V fields. The Schrodinger equation assumes infinitely large speeds for them. Such obviously cannot be the case—it is best to take the infinite speeds as just an abstraction (as a mathematical approximation) to the reality of very, very high actual speeds. However, the experimental evidence also indicates that even if there has to be some or the other upper bound to the speeds v, with v \gg c, the speeds still have to be so high as to seemingly approach infinity, if the Schrodinger formalism is to be employed. And, of course, as you know it, Schrodinger’s formalism is pretty well understood, validated, and appreciated [^]. (For more on the speed limits and IAD in general, see the addendum at the end of this post.)

I don’t know the relativity theory or the relativistic QM. But I guess that since the electric fields of massive QM particles are non-uniform (they are in fact singular), their interactions with \Psi must be such that the system has to suddenly snap out of some one configuration and in the same process snap into one of the many alternative possible configurations. Since there are huge (astronomically large) number of particles in the universe, the alternative configurations would be {astronomically large}^{very large}—after all, the particles positions and motions are continuous. Thus, we couldn’t hope to calculate the propagation speeds for the changes in the local features of a configuration in terms of all those irreversible snap-out and snap-in events taken individually. We must take them in an ensemble sense. Further, the electric charges are massive, identical, and produce singular and continuous fields. Overall, it is the ensemble-level effects of these individual quantum mechanical snap-out and snap-in events whose end-result would be: the speed-of-light limitation of the special relativity (SR). After all, SR holds on the gross scale; it is a theory from classical electrodynamics. The electric and magnetic fields of classical EM can be seen as being produced by the quantum \Psi field (including the spinor function) of large ensembles of particles in the limit that the number of their configurations approaches infinity, and the classical EM waves i.e. light are nothing but the second-order effects in the classical EM fields.

I don’t know. I was just loud-thinking. But it’s certainly possible to have IAD for the changes in \Psi and V, and thus to have instantaneous energy transfers via photons across two distant atoms in a QM-level description, and still end up with a finite limit for the speed of light (c) for large collections of atoms.

OK. Enough of setting up the context.

2.2: The domain of dependence for the nonlinearity in QM vs. that in CM:

If QM is not linear, i.e., if there is a nonlinearity in the \Psi field (as I have proposed), then to evaluate the merits of the QM-level and CM-level RNGs, we have to compare the two nonlinearities: those in the QM vs. those in the CM.

The classical RNGs are always based on the nonlinearities in CM. For example:

  • the nonlinearities in the atmospheric electricity (the “static”) [^], or
  • the fluid-dynamical nonlinearities (as shown in the lottery-draw machines [^], or the lava lamps [^]), or
  • some or the other nonlinear electronic circuits (available for less than $10 in hardware stores)
  • etc.

All of them are based on two factors: (i) a large number of components (in the core system generating the random signal, not necessarily in the part that probes its state), and (ii) nonlinear interactions among all such components.

The number of variables in the QM description is anyway always larger: a single classical atom is seen as composed from tens, even hundreds of quantum mechanical charges. Further, due to the IAD present in the QM theory, the domain of dependence (DoD) [^] in QM remains, at all times, literally the entire universe—all charges are included in it, and the entire \Psi field too.

On the other hand, the DoD in the CM description remains limited to only that finite region which is contained in the relevant past light-cone. Even when a classical system is nonlinear, and thus gets crazy very rapidly with even small increases in the number of degrees of freedom (DOFs), its DoD still remains finite and rather very small at all times. In contrast, the DoD of QM is the whole universe—all physical objects in it.

2.3 Implication for the RNGs:

Based on the above-mentioned argument, which in my limited reading and knowledge Aaronson has never presented (and neither has any one else either, basically because they all continue to believe in von Neumann’s characterization of QM as a linear theory), an RNG operating at the QM level does seem to have, “logically” speaking, an upper hand over an RNG operating at the CM level.

Then why do I still say that arguing for the superiority of a QM-level RNG is still pointless?


3. The MVLSN principle, and its epistemological basis:

If you apply a proper epistemology (and I have in my mind here the one by Ayn Rand), then the supposed “logical” difference between the two descriptions becomes completely superfluous. That’s because the quantities whose differences are being examined, themselves begin to lose any epistemological standing.

The reason for that, in turn, is what I call the MVLSN principle: the law of the Meaninglessness of the Very Large or very Small Numbers (or scales).

What the MVLSN principle says is that if your argument crucially depends on the use of very large (or very small) quantities and relationships between them, i.e., if the fulcrum of your argument rests on some great extrapolations alone, then it begins to lose all cognitive merit. “Very large” and “very small” are contextual terms here, to be used judiciously.

Roughly speaking, if this principle is applied to our current situation, what it says is that when in your thought you cross a certain limit of DOFs and hence a certain limit of complexity (which anyway is sufficiently large as to be much, much beyond the limit of any and every available and even conceivable means of predictability), then any differences in the relative complexities (here, of the QM-level RNGs vs. the CM-level RNGs) ought to be regarded as having no bearing at all on knowledge, and therefore, as having no relevance in any practical issue.

Both QM-level and CM-level RNGs would be far too complex for you to devise any algorithm or a machine that might be able to predict the sequence of the bits coming out of either. Really. The complexity levels already grow so huge, even with just the classical systems, that it’s pointless trying to predict the the bits. Or, to try and compare the complexity of the classical RNGs with the quantum RNGs.

A clarification: I am not saying that there won’t be any systematic errors or patterns in the otherwise random bits that a CM-based RNG produces. Sure enough, due statistical testing and filtering is absolutely necessary. For instance, what the radio-stations or cell-phone towers transmit are, from the viewpoint of a RNG based on radio noise, systematic disturbances that do affect its randomness. See random.org [^] for further details. I am certainly not denying this part.

All that I am saying is that the sheer number of DOF’s involved itself is so huge that the very randomness of the bits produced even by a classical RNG is beyond every reasonable doubt.

BTW, in this context, do see my previous couple of posts dealing with probability, indeterminism, randomness, and the all-important system vs. the law distinction here [^], and here [^].


4. To conclude my main argument here…:

In short, even “purely” classical RNGs can be way, way too complex for any one to be concerned in any way about their predictability. They are unpredictable. You don’t have to go chase the QM level just in order to ensure unpredictability.

Just take one of those WinTV lottery draw machines [^], start the air flow, get your prediction algorithm running on your computer (whether classical or quantum), and try to predict the next ball that would come out once the switch is pressed. Let me be generous. Assume that the switch gets pressed at exactly predictable intervals.

Go ahead, try it.


5. The Height of the Tallest Possible Man (HTPM):

If you still insist on the supposedly “logical” superiority of the QM-level RNGs, make sure to understand the MVLSN principle well.

The issue here is somewhat like asking this question:

What could possibly be the upper limit to the height of man, taken as a species? Not any other species (like the legendary “yeti”), but human beings, specifically. How tall can any man at all get? Where do you draw the line?

People could perhaps go on arguing, with at least some fig-leaf of epistemological legitimacy, over numbers like 12 feet vs. 14 feet as the true limit. (The world record mentioned in the Guinness Book is slightly under 9 feet [^]. The ceiling in a typical room is about 10 feet high.) Why, they could even perhaps go like: “Ummmm… may be 12 feet is more likely a limit than 24 feet? whaddaya say?”

Being very generous of spirit, I might still describe this as a borderline case of madness. The reason is, in the act of undertaking even just a probabilistic comparison like that, the speaker has already agreed to assign non-zero probabilities to all the numbers belonging to that range. Realize, no one would invoke the ideas of likelihood or probability theory if he thought that the probability for an event, however calculated, was always going to be zero. He would exclude certain kinds of ranges from his analysis to begin with—even for a stochastic analysis. … So, madness it is, even if, in my most generous mood, I might regard it as a borderline madness.

But if you assume that a living being has all the other characteristic of only a human being (including being naturally born to human parents), and if you still say that in between the two statements: (A) a man could perhaps grow to be 100 feet tall, and (B) a man could perhaps grow to be 200 feet tall, it is the statement (A) which is relatively and logically more reasonable, then what the principle (MVLSN) says is this: “you basically have lost all your epistemological bearing.”

That’s nothing but complex (actually, philosophic) for saying that you have gone mad, full-stop.

The law of the meaningless of the very large or very small numbers does have a certain basis in epistemology. It goes something like this:

Abstractions are abstractions from the actually perceived concretes. Hence, even while making just conceptual projections, the range over which a given abstraction (or concept) can remain relevant is determined by the actual ranges in the direct experience from which they were derived (and the nature, scope and purpose of that particular abstraction, the method of reaching it, and its use in applications including projections). Abstractions cannot be used in disregard of the ranges of the measurements over which they were formed.

I think that after having seen the sort of crazy things that even simplest nonlinear systems with fewest variables and parameters can do (for instance, which weather agency in the world can make predictions (to the accuracy demanded by newspapers) beyond 5 days? who can predict which way is the first vortex going to be shed even in a single cylinder experiment?), it’s very easy to conclude that the CM-level vs. QM-level RNG distinction is comparable to the argument about the greater reasonableness of a 100 feet tall man vs. that of a 200 feet tall man. It’s meaningless. And, madness.


6. Aaronson’s further points:

To be fair, much of the above write-up was not meant for Aaronson; he does readily grant the CM-level RNGs validity. What he says, immediately after the quote mentioned at the beginning of this post, is that if you don’t have the requirement of distributing bits over a network,

…then generating random bits is obviously trivial with existing technology.

However, since Aaronson believes that QM is a linear theory, he does not even consider making a comparison of the nonlinearities involved in QM and CM.

I thought that it was important to point out that even the standard (i.e., Schrodinger’s equation-based) QM is nonlinear, and further, that even if this fact leads to some glaring differences between the two technologies (based on the IAD considerations), such differences still do not lead to any advantages whatsoever for the QM-level RNG, as far as the task of generating random bits is concerned.

As to the task of transmitting them over a network is concerned, Aaronson then notes:

If you do have the requirement, on the other hand, then you’ll have to do something interesting—and as far as I know, as long as it’s rooted in physics, it will either involve Bell inequality violation or quantum computation.

Sure, it will have to involve QM. But then, why does it have to be only a QC? Why not have just special-purpose devices that are quantum mechanically entangled over wires / EM-waves?

And finally, let me come to yet another issue: But why would you at all have to have that requirement?—of having to transmit the keys over a network, and not using any other means?

Why does something as messy as a network have to get involved for a task that is as critical and delicate as distribution of some super-specially important keys? If 99.9999% of your keys-distribution requirements can be met using “trivial” (read: classical) technologies, and if you can also generate random keys using equipment that costs less than $100 at most, then why do you have to spend billions of dollars in just distributing them to distant locations of your own offices / installations—especially if the need for changing the keys is going to be only on an infrequent basis? … And if bribing or murdering a guy who physically carries a sealed box containing a thumb-drive having secret keys is possible, then what makes the guys manning the entangled stations suddenly go all morally upright and also immortal?

From what I have read, Aaronson does consider such questions even if he seems to do so rather infrequently. The QC enthusiasts, OTOH, never do.

As I said, this QC as an RNG thing does show some marks of trying to figure out a respectable exit-way out of the scalable QC euphoria—now that they have already managed to wrest millions and billions in their research funding.

My two cents.


Addendum on speed limits and IAD:

Speed limits are needed out of the principle that infinity is a mathematical concept and cannot metaphysically exist. However, the nature of the ontology involved in QM compels us to rethink many issues right from the beginning. In particular, we need to carefully distinguish between all the following situations:

  1. The transportation of a massive classical object (a distinguishable, i.e. finite-sized, bounded piece of physical matter) from one place to another, in literally no time.
  2. The transmission of the momentum or changes in it (like forces or changes in them) being carried by one object, to a distant object not in direct physical contact, in literally no time.
  3. Two mutually compensating changes in the local values of some physical property (like momentum or energy) suffered at two distant points by the same object, a circumstance which may be viewed from some higher-level or abstract perspective as transmission of the property in question over space but in no time. In reality, it’s just one process of change affecting only one object, but it occurs in a special way: in mutually compensating manner at two different places at the same time.

Only the first really qualifies to be called spooky. The second is curious but not necessarily spooky—not if you begin to regard two planets as just two regions of the same background object, or alternatively, as two clearly different objects which are being pulled in various ways at the same time and in mutually compensating ways via some invisible strings or fields that shorten or extend appropriately. The third one is not spooky at all—the object that effects the necessary compensations is not even a third object (like a field). Both the interacting “objects” and the “intervening medium” are nothing but different parts of one and the same object.

What happens in QM is the third possibility. I have been describing such changes as occurring with an IAD (instantaneous action at a distance), but now I am not too sure if such a usage is really correct or not. I now think that it is not. The term IAD should be reserved only for the second category—it’s an action that gets transported there. As to the first category, a new term should be coined: ITD (instantaneous transportation to distance). As to the third category, the new term could be IMCAD (instantaneous and mutually compensating actions at a distance). However, this all is an afterthought. So, in this post, I only have ended up using the term IAD even for the third category.

Some day I will think more deeply about it and straighten out the terminology, may be invent some or new terms to describe all the three situations with adequate directness, and then choose the best… Until then, please excuse me and interpret what I am saying in reference to context. Also, feel free to suggest good alternative terms. Also, let me know if there are any further distinctions to be made, i.e., if the above classification into three categories is not adequate or refined enough. Thanks in advance.


A song I like:

[A wonderful “koLi-geet,” i.e., a fisherman’s song. Written by a poet who hailed not from the coastal “konkaN” region but from the interior “desh.” But it sounds so authentically coastal… Listening to it today instantly transported me back to my high-school days.]

(Marathi) “suTalaa vaadaLi vaaraa…”
Singing, Music and Lyrics: Shaahir Amar Sheikh

 


History: Originally published on 2019.07.04 22:53 IST. Extended and streamlined considerably on 2019.07.05 11:04 IST. The songs section added: 2019.07.05 17:13 IST. Further streamlined, and also further added a new section (no. 6.) on 2019.07.5 22:37 IST. … Am giving up on this post now. It grew from about 650 words (in a draft for a comment at Schlafly’s blog) to 3080 words as of now. Time to move on.

Still made further additions and streamlining for a total of ~3500 words, on 2019.07.06 16:24 IST.

Determinism, Indeterminism, Probability, and the nature of the laws of physics—a second take…

After I wrote the last post [^], several points struck me. Some of the points that were mostly implicit needed to be addressed systematically. So, I began writing a small document containing these after-thoughts, focusing more on the structural side of the argument.

However, I don’t find time to convert these points + statements into a proper write-up. At the same time, I want to get done with this topic, at least for now, so that I can better focus on some other tasks related to data science. So, let me share the write-up in whatever form it is in, currently. Sorry for its uneven tone and all (compared to even my other writing, that is!)


Causality as a concept is very poorly understood by present-day physicists. They typically understand only one sense of the term: evolution in time. But causality is a far broader concept. Here I agree with Ayn Rand / Leonard Peikoff (OPAR). See the Ayn Rand Lexicon entry, here [^]. (However, I wrote the points below without re-reading it, and instead, relying on whatever understanding I have already come to develop starting from my studies of the same material.)

Physical universe consists of objects. Objects have identity. Identity is the sum total of all characteristics, attributes, properties, etc., of an object. Objects act in accordance with their identity; they cannot act otherwise. Interactions are not primary; they do not come into being without there being objects that undergo the interactions. Objects do not change their respective identities when they take actions—not even during interactions with other objects. The law of causality is a higher-level view taken of this fact.

In the cause-effect relationship, the cause refers to the nature (identity) of an object, and the effect refers to an action that the object takes (or undergoes). Both refer to one and the same object. TBD: Trace the example of one moving billiard ball undergoing a perfectly elastic collision with another billiard ball. Bring out how the interaction—here, the pair of the contact forces—is a name for each ball undergoing an action in accordance with its nature. An interaction is a pair of actions.


A physical law as a mapping (e.g., a function, or even a functional) from inputs to outputs.

The quantitative laws of physics often use the real number system, i.e., quantification with infinite precision. An infinite precision is a mathematical concept, not physical. (Expect physicists to eternally keep on confusing between the two kinds of concepts.)

Application of a physical law traces the same conceptual linkages as are involved in the formulation of law, but in the reverse direction.

In both formulation of a physical law and in its application, there is always some regime of applicability which is at least implicitly understood for both inputs and outputs. A pertinent idea here is: range of variations. A further idea is the response of the output to small variations in the input.


Example: Prediction by software whether a cricket ball would have hit the stumps or not, in an LBW situation.

The input position being used by the software in a certain LBW decision could be off from reality by millimeters, or at least, by a fraction of a millimeter. Still, the law (the mapping) is such that it produces predictions that are within small limits, so that it can be relied on.

Two input values, each theoretically infinitely precise, but differing by a small magnitude from each other, may be taken to define an interval or zone of input variations. As to the zone of the corresponding output, it may be thought of as an oval produced in the plane of the stumps, using the deterministic method used in making predictions.

The nature of the law governing the motion of the ball (even after factoring in aspects like effects of interaction with air and turbulence, etc.) itself is such that the size of the O/P zone remains small enough. (It does not grow exponentially.) Hence, we can use the software confidently.

That is to say, the software can be confidently used for predicting—-i.e., determining—the zone of possible landing of the ball in the plane of the stumps.


Overall, here are three elements that must be noted: (i) Each of the input positions lying at the extreme ends of the input zone of variations itself does have an infinite precision. (ii) Further, the mapping (the law) has theoretically infinite precision. (iii) Each of the outputs lying at extreme ends of the output zone also itself has theoretically infinite precision.

Existence of such infinite precision is a given. But it is not at all the relevant issue.

What matters in applications is something more than these three. It is the fact that applications always involve zones of variations in the inputs and outputs.

Such zones are then used in error estimates. (Also for engineering control purposes, say as in automation or robotic applications.) But the fact that quantities being fed to the program as inputs themselves may be in error is not the crux of the issue. If you focus too much on errors, you will simply get into an infinite regress of error bounds for error bounds for error bounds…

Focus, instead, on the infinity of precision of the three kinds mentioned above, and focus on the fact that in addition to those infinitely precise quantities, application procedure does involve having zones of possible variations in the input, and it also involves the problem estimating how large the corresponding zone of variations in the output is—whether it is sufficiently small for the law and a particular application procedure or situation.

In physics, such details of application procedures are kept merely understood. They are hardly, if ever, mentioned and discussed explicitly. Physicists again show their poor epistemology. They discuss such things in terms not of the zones but of “error” bounds. This already inserts the wedge of dichotomy: infinitely precise laws vs. errors in applications. This dichotomy is entirely uncalled for. But, physicists simply aren’t that smart, that’s all.


“Indeterministic mapping,” for the above example (LBW decisions) would the one in which the ball can be mapped as going anywhere over, and perhaps even beyond, the stadium.

Such a law and the application method (including the software) would be useless as an aid in the LBW decisions.

However, phenomenologically, the very dynamics of the cricket ball’s motion itself is simple enough that it leads to a causal law whose nature is such that for a small variation in the input conditions (a small input variations zone), the predicted zone of the O/P also is small enough. It is for this reason that we say that predictions are possible in this situation. That is to say, this is not an indeterministic situation or law.


Not all physical situations are exactly like the example of the predicting the motion of the cricket ball. There are physical situations which show a certain common—and confusing—characteristic.

They involve interactions that are deterministic when occurring between two (or few) bodies. Thus, the laws governing a simple interaction between one or two bodies are deterministic—in the above sense of the term (i.e., in terms of infinite precision for mapping, and an existence of the zones of variations in the inputs and outputs).

But these physical situations also involve: (i) a nonlinear mapping, (ii) a sufficiently large number of interacting bodies, and further, (iii) coupling of all the interactions.

It is these physical situations which produce such an overall system behaviour that it can produce an exponentially diverging output zone even for a small zone of input variations.

So, a small change in I/P is sufficient to produce a huge change in O/P.

However, note the confusing part. Even if the system behaviour for a large number of bodies does show an exponential increase in the output zone, the mapping itself is such that when it is applied to only one pair of bodies in isolation of all the others, then the output zone does remain non-exponential.

It is this characteristic which tricks people into forming two camps that go on arguing eternally. One side says that it is deterministic (making reference to a single-pair interaction), the other side says it is indeterministic (making reference to a large number of interactions, based on the same law).

The fallacy arises out of confusing a characteristic of the application method or model (variations in input and output zones) with the precision of the law or the mapping.


Example: N-body problem.

Example: NS equations as capturing a continuum description (a nonlinear one) of a very large number of bodies.

Example: Several other physical laws entering the coupled description, apart from the NS equations, in the bubbles collapse problem.

Example: Quantum mechanics


The Law vs. the System distinction: What is indeterministic is not a law governing a simple interaction taken abstractly (in which context the law was formed), but the behaviour of the system. A law (a governing equation) can be deterministic, but still, the system behavior can become indeterministic.


Even indeterministic models or system designs, when they are described using a different kind of maths (the one which is formulated at a higher level of abstractions, and, relying on the limiting values of relative frequencies i.e. probabilities), still do show causality.

Yes, probability is a notion which itself is based on causality—after all, it uses limiting values for the relative frequencies. The ability to use the limiting processes squarely rests on there being some definite features which, by being definite, do help reveal the existence of the identity. If such features (enduring, causal) were not to be part of the identity of the objects that are abstractly seen to act probabilistically, then no application of a limiting process would be possible, and so not even a definition probability or randomness would be possible.

The notion of probability is more fundamental than that of randomness. Randomness is an abstract notion that idealizes the notion of absence of every form of order. … You can use the axioms of probability even when sequences are known to be not random, can’t you? Also, hierarchically, order comes before does randomness. Randomness is defined as the absence of (all applicable forms of) orderliness; orderliness is not defined as absence of randomness—it is defined via the some but any principle, in reference to various more concrete instances that show some or the other definable form of order.

But expect not just physicists but also mathematicians, computer scientists, and philosophers, to eternally keep on confusing the issues involved here, too. They all are dumb.


Summary:

Let me now mention a few important take-aways (though some new points not discussed above also crept in, sorry!):

  • Physical laws are always causal.
  • Physical laws often use the infinite precision of the real number system, and hence, they do show the mathematical character of infinite precision.
  • The solution paradigm used in physics requires specifying some input numbers and calculating the corresponding output numbers. If the physical law is based on real number system, than all the numbers used too are supposed to have infinite precision.
  • Applications always involve a consideration of the zone of variations in the input conditions and the corresponding zone of variations in the output predictions. The relation between the sizes of the two zones is determined by the nature of the physical law itself. If for a small variation in the input zone the law predicts a sufficiently small output zone, people call the law itself deterministic.
  • Complex systems are not always composed from parts that are in themselves complex. Complex systems can be built by arranging essentially very simpler parts that are put together in complex configurations.
  • Each of the simpler part may be governed by a deterministic law. However, when the input-output zones are considered for the complex system taken as a whole, the system behaviour may show exponential increase in the size of the output zone. In such a case, the system must be described as indeterministic.
  • Indeterministic systems still are based on causal laws. Hence, with appropriate methods and abstractions (including mathematical ones), they can be made to reveal the underlying causality. One useful theory is that of probability. The theory turns the supposed disadvantage (a large number of interacting bodies) on its head, and uses limiting values of relative frequencies, i.e., probability. The probability theory itself is based on causality, and so are indeterministic systems.
  • Systems may be deterministic or indeterministic, and in the latter case, they may be described using the maths of probability theory. Physical laws are always causal. However, if they have to be described using the terms of determinism or indeterminism, then we will have to say that they are always deterministic. After all, if the physical laws showed exponentially large output zone even when simpler systems were considered, they could not be formulated or regarded as laws.

In conclusion: Physical laws are always causal. They may also always be regarded as being deterministic. However, if systems are complex, then even if the laws governing their simpler parts were all deterministic, the system behavior itself may turn out to be indeterministic. Some indeterministic systems can be well described using the theory of probability. The theory of probability itself is based on the idea of causality albeit measures defined over large number of instances are taken, thereby exploiting the fact that there are far too many objects interacting in a complex manner.


A song I like:

(Hindi) “ho re ghungaroo kaa bole…”
Singer: Lata Mangeshkar
Music: R. D. Burman
Lyrics: Anand Bakshi