On the Origin of Quantum Uncertainty by Chris Adami

"... connect mathematical and quantum uncertainty. It seemed that there were two ways to go about it: show that a quantum measurement really is attempting to solve the halting problem, or else to show that a classically entangled set of Turing machines looks in important ways just like a quantum system."

Have you, or others, attempted to justify Lestone's theory of virtual cross sections by using classically (or semi-classically) entangled sets of Turing machines?

Lestone, J. P. (2018). QED: A different perspective. Los Alamos National Laboratory Report LA-UR-18-29048

Dear Dr. Adami,

it's encouraging to see that the research program of investigating the connections between quantum uncertainty and mathematical undecidability appears to be gaining momentum. To the best of my knowledge, this program started with Wheeler, who proposed as his 'quantum principle' the 'undecidable propositions of mathematical logic', and nearly almost died when he got himself thrown out of Gödel's office for mentioning the matter to him. Ever since, it's been a challenge to lead a serious investigation into the matter, although there have been attempts---I give a brief history of the field in my article in Foundations of Physics, which also seeks to put the matter on more solid footing.

I think there is great insight to be gained from the application of this perspective---in my contribution to this contest, I investigate basically what happens when, as you put it, a system is 'forced to respond to a question without having enough bits to encode the answer' in the context of Bell tests.

Reading your perspective on the matter is thus a very welcome, and well argued, addition to what almost seems to be becoming a little cottage industry of quantum foundations research. In particular, I appreciate you taking the perspective of finite-dimensional systems---the argument I gave, strictly speaking, applies only in the infinitary case; I don't view that as a defect, and I believe that similar arguments ought to be possible involving bounded-strength systems, but having an explicitly finite-dimensional perspective greatly increases the scope of these investigations.

Indeed, it seems that in the two routes towards connecting quantum uncertainty and undecidability you lay out, I have chosen the other---trying to expose the connection between quantum measurement and the halting problem, in a way; although it would be really more apt to say that both quantum measurement, the undecidability of the halting problem, Gödel's incompleteness theorem, and really a host of other such results are manifestations of the same underlying structure, worked out by Lawvere as a particular fixed-point theorem in Cartesian closed categories.

This also brings us back to what you, I think rightly, see as the importance of the impossibility of the copying operation: it's really the presence of a diagonalization N --> N x N, where N denotes the natural numbers, used as indices of e. g. some enumeration of quantum states ("Gödel numbers") that makes this particular connection possible. Its absence in quantum mechanics assures its immunity from paradox---albeit at the price of introducing all the familiar elements of quantum weirdness.

Anyway, I will need to take another pass at your essay to digest your example. Given my own research interests, this is one of the most exciting essays of this year's contest so far. I wish you all the best in the competition!

Cheers

Jochen

"To the best of my knowledge, this program started with Wheeler, who proposed as his 'quantum principle' the 'undecidable propositions of mathematical logic', and nearly almost died when he got himself thrown out of Gödel's office"

It occurs to me that I phrased this poorly. It was the research program of connecting undecidability to quantum mechanics that 'nearly almost died', not Wheeler---to the best of my knowledge, his removal from Gödel's office was not that forceful!

Dear Prof Chris Adami,

Your quantum Uncertainty essay great! I work mainly in Cosmology and Astrophysics, As you are Professor in astrophysics also you will have a good understanding about Cosmology too.

I was raading about Godel's law, I have few questions about it. This law is applicable to Quantum Mechanics, but will this law be applicable to COSMOLOGY.......?????.........

I never encountered any such a problem in Dynamic Universe Model in the Last 40 years, all the the other conditions mentioned in that law are applicable ok

I hope you will have CRITICAL examination of my essay... "A properly deciding, Computing and Predicting new theory's Philosophy".....

Best Regards

=snp

Dear Chris Adami,

You claimed having made Gold while I see myself a bit in the role of a Tschirnhaus. In other words it is not by chance that almost nobody took issue, commented on your essay and rated it. Of course, I have to criticize that cos +i sin but not cos + sin is fundamental to QM (due vote 1). On the other hand (my vote 10) you might be in principle correct with putting attention to a very basic issue at the heart of point set theory. I prefer boldly calling Cantor's paradise rather a dot set theory.

Common sense has it: Who is dying is still alive. The cat cannot be dead and alive at a time. Precisely measured elapsed time is strictly speaking either positive or negative with no state in between. Einstein's blur "past, present, and future" mingles different categories.

Mathematicians will further down-vote my essay which humbly suggests being careful: Do calculate just as if Cantor-Hilbert space and denial of causality were adequate mirrors of reality. Actually, even the best map is not the territory, and mathematics must not be reduced merely to countable numbers. Already the Greeks were aware of this. Kronecker and Brouwer failed. Notice: Intuitionism is named after the Urintuition of counting.

You got it by pointing out that densities may vanish. Why not reminding that division by zero yields nonsense too?

Thank you,

Eckard Blumschein

There is missing one highly relevant reference.

At present even the assumption of "time evolution of a quantum system is deterministic" is not obvious and is subject to investigation as the temporal order might in principle be entangled.

When invoking Turing machines in the QM context one has to resist temptation that QM is like a computer and 'computing'. In QM dealing with qubits there are indeed computations going one and quantum computers are possible. But this is trivial QM as the real QM is inherently infinitely dimensional and the really real QM is quantum field theory.

Best regards,

Irek

Chris,

Quantum uncertainty being a manifestation of the indeterminism inherent in mathematical logic is, of course, a theoretical embodiment. My concept of the quantum world is quite vague and based on a non-math perspective, but I tend to reject the Copenhagen interpretation. Nevertheless, your view is well crafted logically, and as near as I can tell mathematically. I find your suggestion of a quantum classical presence by insinuation as interesting. I find studies of bridging the classical and quantum worlds quite interesting and cite a study by US and Austrian physicists (https://phys.org/news/2020-01-strange-metals.html) in my essay, as I find studies in quantum biology of interest too. It mentions the quantum entanglement nature of quantum criticality. I hope you will visit my essay as well: https://fqxi.org/community/forum/topic/3396.

Regards,

Jim

Hi Chris,

Very inspiring essay. On your notation, the origin of quantum uncertainty implicitly assumes more than two-bits. Is this true? If so, what do you consider the uncertainty of the output from quantum devices as pointed in my essay?

Best wishes,

Yutaka

Wow! Most of the essays I've seen here (including mine) draw some broad links between uncertainty in its various forms and undecidability/uncomputability, but yours is the first I've seen that draws a convincing direct link. Very interesting central idea about the relationship between quantum measurement and the halting problem. I like also your point about how the distinction between 'system' and 'measuring device' is artificial, and really just for our own convenience.

Is there some history of related ideas? In particular, do there exist models where people try to describe (ideally in some level of microscopic detail) the interaction between a 'system' and 'measuring device' from both directions: by treating the 'measuring device' as doing the measurement, and by treating the 'system' as doing the measurement? This is the first time I've heard about this, and it seems like an important thing to think about in trying to understand the measurement problem.

John

Hello Chris!

I really enjoyed your essay!

When I read the title, I had my doubts. But your presentation is very clear and your arguments are simple and convincing. Clearly, you have been thinking about this for twenty years, as you say. And while I call your arguments "simple and convincing", it is going to take me a while to really internalize this work!

I must say that I was delighted to read your treatment of the measurement process with the target and detector as a joint system. So often, it seems, that people do not understand that the measurement occurs via symmetric interaction. And people often forget that the phase is randomized after a measurement. My friend John Skilling and I are especially sensitive to this since we used this fact to help derive the Born rule in .

Although we have a cleaner derivation in the paper we are working on now.

It is fascinating how you demonstrate that this randomization of the phase is due to the joint target-detector system being asked to measure itself.

Do I understand that correctly?

In a paper we are currently writing, we write:

>

We use this as justification for the Pair Postulate which states that two numbers are needed to describe an object:

>

We then go on to use symmetries, such as associativity and distributivity (as I describe in my essay), to derive the Feynman rules.

One could imagine performing a measurement on the probe/detector to try to determine its state (in an attempt to remove the uncertainty). But then the second detector one uses to measure the state of the first detector would have an associated uncertainty. And we get ourselves into a state of infinite regress.

I am wondering how you might interpret the perspective I present above in terms of uncomputability?

Thank you again for sharing your wonderful insights and essay!!!

Take care,

Kevin

Somehow, my post got messed up. The gt lt symbols I used for quotes messed things up.

So here goes again!

Hello Chris!

I really enjoyed your essay!

When I read the title, I had my doubts. But your presentation is very clear and your arguments are simple and convincing. Clearly, you have been thinking about this for twenty years, as you say. And while I call your arguments "simple and convincing", it is going to take me a while to really internalize this work!

I must say that I was delighted to read your treatment of the measurement process with the target and detector as a joint system. So often, it seems, that people do not understand that the measurement occurs via symmetric interaction. And people often forget that the phase is randomized after a measurement. My friend John Skilling and I are especially sensitive to this since we used this fact to help derive the Born rule in https://arxiv.org/pdf/1712.09725.pdf.

Although we have a cleaner derivation in the paper we are working on now.

It is fascinating how you demonstrate that this randomization of the phase is due to the joint target-detector system being asked to measure itself.

Do I understand that correctly?

In a paper we are currently writing, we write:

[[

Our knowledge of the material world comes from control and observation of interactions of one object with another, traditionally target" and probe" when we are trying to learn about a target by probing it. However, a probe was itself manufactured through interactions and so was itself somewhat uncertain. Hence objects cannot be adequately described by the traditional single classical parameter. No matter how large or small the object, there must always remain some uncertainty. That means that at least one parameter must be added to the traditional one-number-per-property description of an object.

]]

We use this as justification for the Pair Postulate which states that two numbers are needed to describe an object:

[[

for any object, we need at least two to represent existence and uncertainty.

Existence involves numerical quantification and uncertainty involves probability.

]]

We then go on to use symmetries, such as associativity and distributivity (as I describe in my essay), to derive the Feynman rules.

One could imagine performing a measurement on the probe/detector to try to determine its state (in an attempt to remove the uncertainty). But then the second detector one uses to measure the state of the first detector would have an associated uncertainty. And we get ourselves into a state of infinite regress.

I am wondering how you might interpret the perspective I present above in terms of uncomputability?

Thank you again for sharing your wonderful insights and essay!!!

Take care,

Kevin

Dear Dr Adami,

Excellent analysis of the QM problem, one I've also grappled with for decades. I agree. Well done. But how are you with analysing solutions? I think you may, rarely, be able. Let's see;

Our pairs retain the pre-splitter axis angle, so one 'leads' with , one - . A,B Polariser electrons at random angles absorb and re-emit, interacting at some surface tangent point angle 'of Latitude' (0-90o) from it's own nearest pole.

Now study the momentum vectors on a Poincare sphere; Only 'Curl' (as Maxwells) at the poles, but only linear (up/down) at the equator. Now THAT is a bit 'new' and missed by Bohr! What's more they BOTH change (from 0-1) inversely, and by the COSINE OF THE ANGLE OF LATITUDE!

Now just do some vector additions for momentum exchange in 3D, What you get is actually a new polarisation ellipticity, (in 3D of course) so major and minor axes amplitudes ('intensity') at 90o. Now BOTH hit the photomulipler channels, but only the MAJOR axis can trip one. (The 2nd interaction squaring the intensity Cos value). Where it's near circular we get 50:50 uncertainty.

An independent computer plot has shown that 'discrete field' model ('DFM') sequence violates Bells inequalities perfectly nicely, and does just what Bell predicted. No 'non-locality' nonsense required as A & B change their results independently, but still in the same proportions.

I touch on that this year, and reference the fuller description and experimental proof in last years essay, and the 'Trail' code & plot.

To stop Quantum Physicists running away in panic it seems it needs some nice complicated mathematical formulation which, if you can understand it, I'd like to recruit your collaboration on. First you'd better see if you can get your head round it!

Best of luck.

Peter