If one accepts as true, as some have asserted, that quantum entanglement is not a well understood consequence of both quantum mechanics and relativity, then it seems that there are at least two possible reasons. One is that the consistency of entanglement with all other known physical law is testable yet at present insufficiently tested to achieve that understanding. Another is that no such test can be envisioned and implemented to produce a conclusive result that does not contradict some tenet of accepted physical theory; i.e., that the mathematical modeling component of entanglement is an instance of Gödelian incompleteness. Having searched and not found a clear answer, I invite discussion aimed at a way either to determine the decidability whether quantum entanglement is consistent with all generally accepted physical theory, or else to find reason to question the premise that this phenomenon is not well understood in the aforesaid way.
The quantum entanglements and the apparent quantum nonlocality were explained.
Entanglements are nothing else than multi-particle wave-functions in which part of the products of states were destroyed. The nonlocality is apparent - probabilities are non-local, joint amplitudes are products of local, single-particle amplitudes.
For details see
Article A problem with Bell-type Inequalities, the origin of the qua...
Quantum Entanglement has absolutely nothing to do with Gödelian Incompleteness. The former is a consequence of the mathematical fact that non-commuting operators cannot be simultaneously diagonalised [1]. Even if one considers quantum mechanics an incomplete theory insofar as the physical reality is concerned (as Einstein did), quantum entanglement is perfectly consistent and fully understood within the framework of quantum mechanics. In contrast, Gödelian Incompleteness concerns statements within a mathematical/logical framework whose truth or falsehood cannot be established on the basis of the axioms of that framework.
Incidentally, quantum entanglement has a classical analogue [2]. Suppose one has a blue and a red card and puts these in two non-transparent envelopes. He/she subsequently blindly takes one of these envelopes and gives it to a friend who subsequently flies to a far-off place. Now, the moment this person opens the envelope that has been left with him/her and sees the colour of the card, he/she knows the colour of the card given to the friend. One can extend this idea with three (and more) cards of different colours, two of which (put inside non-transparent envelopes) have been selected randomly and given to two friends who have subsequently flown to two far-off places. In this case, each of the persons opening his/her envelope cannot know with any certainty the colour of the card with each of the two friends; to remove this uncertainty, at least two of the friends must communicate with each other. Now, the difference between these classical cases and their quantum counterparts is that the colour of the card one has in one's envelope is unchangeable, whereas in the quantum case the relevant state before observation is in a superposition (think of the celebrated Schrödinger cat).
Lastly, there are many good books on quantum entanglement, two of which are [3] and [4].
[1] To be explicit, considering an eigenstate of a system consisting of two quantum spins, since the x-, y- and z-components of the spin operator corresponding to each spin do not commute with each other, the said eigenstate cannot be a simultaneous eigenstate of the said three components of the said spin operator.
[2] I owe this example to the late Professor NG van Kampen from whom I heard it around 2002.
[3] B Thaller, Advanced Visual Quantum Mechanics (Springer, New York, 2005). See in particular Chapter 6: Quantum Information Theory.
[4] S Weinberg, Lectures on Quantum Mechanics, 2nd edition (Cambridge University Press, 2005). See in particular Chapter 12: Entanglement.
Dear Behnam Farid ,
" Even if one considers quantum mechanics an incomplete theory insofar as the physical reality is concerned (as Einstein did) "
Why should one consider QM incomplete? Because of Einstein? But we are in the 21st century, not in the 20th century. Since Einstein, a lot of research was done - we didn't remain at the stage of the beginning of QM.
What one has to do is to read the recent corresponding literature.
Sofia D. Wechsler : I did not say that one should blindly follow Einstein. Please reread my text.
Dear Sofia D. Wechsler,
In the abstract of your interesting article " A problem with Bell-type Inequalities ..." comes this summarizing observation that seems relevant to the question at hand: "The real purpose here is to point to a severe impasse: assuming the existence of a preferred frame contradicts the theory of relativity, while refuting the full/empty waves idea one runs into other insurmountable difficulties." Would it be correct to interpret this observation as support for the original premise of this question "that quantum entanglement is not a well understood consequence of both quantum mechanics and relativity"?
Dear Behnam Farid,
I applaud your richly detailed commentary, and especially references that (notably) date back to 2005 and earlier. Perhaps we can agree that "one should blindly follow Einstein" would be bad advice, or at least, contrary to the goal of achieving new knowledge. Then let us also for the sake of discussion accept the propositions (1) that quantum entanglement is a phenomenon dependent on the physics of quantum waves, and that (2) the system of mathematical modeling that describes the theory of quantum entanglement (let’s call it QEMM) either: (2a) is complete or (2b) is not complete. In case (2b) QEMM is incomplete. Then might it be so simply because there are as yet undiscovered or unproved consequences of axioms? We can call this situation “unconstrained incompleteness” (as distinct from “necessary incompleteness”). Taking case (2a), might it be complete because all that can be proved has been proved in QEMM, implying incompleteness that is necessarily so given Gödel’s principle?
While it is my understanding that according to that same principle a system of mathematical logic is necessarily incomplete if it contains no falsehoods, let me also grant for the sake of this discussion your premise "Gödelian Incompleteness concerns statements within a mathematical/logical framework whose truth or falsehood cannot be established on the basis of the axioms of that framework". Then under Gödelian Incompleteness as you characterized it, the axioms of QEMM cannot be used to establish certain statements within QEMM, from which it follows that those statements are necessarily unprovable within the system. In this sense, QEMM would have necessary or Gödelian incompleteness. Although I would next move to proposition (1), resuming the discussion there and looking at consequences of the Einstein's claim often summarized in his phrase “God does not play dice”, I must first ask you: Are we in agreement so far? (I suspect not but, if not, would be grateful if you would be so kind as to explain why.)
Albert G. Di Canzio : First, for the second time I have to point out that the statement "one should blindly follow Einstein" is not mine - please kindly reread my first comment. The point I was trying to make was that even if one would look at quantum theory as a purely mathematical theory, detached from physics and the physical reality (whatever this may be), entanglement can be proved explicitly on account of the operator nature of the observables in quantum mechanics - operators that generally do not commute amongst each other. As I pointed out, entanglement is not specific to quantum mechanics; we have also entanglements in the classical world, albeit that they are distinct from quantum entanglements.
Second, we know already that quantum mechanics is not a complete theory of nature, since it leaves out quantum gravity. Aside from this, the quantum mechanics as described by the Schrödinger equation is non-relativistic. Extension of this to the Dirac equation (concentrating for argument's sake on electrons, and more generally on fermions), while provides us with a quantum theory that is consistent with the principles (or symmetries) of the special theory of relativity (flat space-time), it is fraught with problems regarding stability of matter (Dirac sea is unstable) [1]. To circumvent these, one has to go over to the full theory of quantum electrodynamics, QED (I do not go into the fact that this theory was also for a long time fraught with difficulties associated with mathematical infinities - divergent perturbative contributions). Even this is not a complete theory, since it leaves out the strong interactions inside nucleons, for whose description one needs to go over to quantum chromodynamics, QCD. Both QED and QCD are coupled field theories. To make a long story short, no one is under any illusion that the quantum theory as described by the non-relativistic Schrödinger equation were complete. Despite this fact, we know regions where this quantum theory is valid asymptotically. Many predictions of this theory, some of which counter-intuitive, have been experimentally verified. You may wish to consult for instance [2] (you may then wish to make a literature search on the works co-authored by in particular Professor Anton Zeilinger - his 2008 Newton Lecture [3] is very relevant and I warmly recommend it to all).
To summarise, I maintain that my earlier statement, namely that 'Quantum Entanglement has absolutely nothing to do with Gödelian Incompleteness.', is correct, as quantum entanglement is not an undecided or undecidable fact within the mathematical framework of quantum mechanics (one can explicitly prove it without reliance on axioms that are not part of the axioms of quantum mechanics). Further discussions on the subject take us beyond the scope of your original question on this page.
[1] JJ Sakurai, Advanced Quantum Mechanics (Addison-Wesley, London, 1967).
[2] J Handsteiner et al., Cosmic Bell Test: Measurement Settings from Milky Way Stars, Phys. Rev. Lett. 118, 060401 (2017).
Article Cosmic Bell Test: Measurement Settings from Milky Way Stars
[3] Newton Lecture 2008: Quantum Information and the Foundations of Quantum Mechanics, by Anton Zeilinger - video http://www.iop.org/resources/videos/lectures/page_42696.html
Dear Behnam Farid,
Thanks again. It is not the case, as you wrote, that "for the second time I have to point out that the statement 'one should blindly follow Einstein' is not mine". If you will kindly reread my comments, you will find that nowhere have I attributed this comment to you in any way. And why would I do that given that you had already denied saying that one should blindly follow Einstein? Your denial, however, did not constitute denial of the proposition "one should blindly follow Einstein", even though, as we all know, blindly following authority is not science.
From the age of about 9, I took up as fun and enjoyment a study of mathematical astronomy and, though having of course no concept of quantum entanglement at the time, I did think of every pair of celestial objects rotating in gravitational interlock around a barycenter as exhibiting a kind of entanglement, an idea reinforced later with the development of physics. Perhaps that case is covered by your statement "we have also entanglements in the classical world "?
Dear Albert G. Di Canzio : since you had enclosed the statement by quotation marks while in conversation with me, the implication was that it was my statement.
Incidentally, should you be interested, I warmly recommend you [1], for two reasons. First, David Mermin is an excellent educator, and, second, he has written the book after having taught himself the subject, so that the book has a personal touch to it. The book deals amongst others with quantum entangled states.
[1] ND Mermin, Quantum Computer Science: An Introduction (Cambridge University Press, 2007).
Dear Behnam Farid,
Yes, I know, from my experience of producing what is arguably the only complete and easily readable scientific biography of Galileo Galilei, that among the best expository works are those that have a "personal touch" and whose content the author has struggled to learn by energetically cutting through all sorts of difficulties with source literature. Duly impressed by your comments about Mermin's book, I thank you most heartily and sincerely for that reference.
Dr. Farid, I believe you are a gentleman as well as a scholar. Thank you for the explanation about quotation marks, which shows me that we are dealing with some subtlety of language leading to a miscommunication that was benign more in intent than in effect. Sometimes in discussing research it is useful and important to take a brief digression to deal with the use of language. As this is one of those times, let's repair the miscommunication. I refer to your response: "since you had enclosed the statement by quotation marks while in conversation with me, the implication was that it was my statement".
No, there was no such implication. (To a mathematician at least, the English phrase "A implies B" means that B logically follows from A, but in that sense you will find no implication in my message that these words constituted a statement of yours.) Rather, the quote marks were necessitated by the simple fact that the quoted words were your words; these words alone do not rise to the level of a statement in the absence of context explicitly imputing their idea to you. In this case, the context in which I used your words did not attribute to you any support for the idea which those words would convey if they were uttered with no context.
To tell you whole truth, in my draft of that message, of which I have an offline copy, I had written "Perhaps we can agree that (though you only denied saying it) 'one should blindly follow Einstein' would be ... contrary to the goal of achieving new knowledge." Then before sending the message I deleted the parenthetical remark to shorten the sentence and, now in hindsight, wish that I had not done that! In any case I accept your explanation and ask you to reciprocally grant me your acceptance of mine so that we can get on with the questions at hand.
Toward that goal, I took your answer to "Are we in agreement so far?" to be not equivalent to a "yes", but even though your response was replete with new information, it did not suggest to me what you may have found wrong or inadequate in my text preceding "Are we in agreement so far?" and ending with "In this sense, QEMM would have necessary or Gödelian incompleteness."
Dear Albert G. Di Canzio, you are very kind. Thank you.
As regards that statement enclosed by quotation marks, I would not have made such a point about it, were it not that these exchanges are public and it cannot be expected that all visitors of this page (as well as other pages on RG) go through all comments, and then sequentially - from top to the bottom of the page. While things are evident to us, for readers who happen to read one or two comments on this page, this cannot be taken for granted.
It appears to me that the discussion is prematurely stalled. Though respondents have been helpful with details on the quantum-physical side, except for Farid's claim "... absolutely nothing to do with Gödelian Incompleteness", and my subsequent attempt to develop a basis to explore that claim, the "incompleteness" side of the issue is undeveloped. Accordingly, I encourage followers to comment on whether the existing system of quantum-physics mathematical modeling meets (or not) the condition "sufficiently explanatory", a constraint of Gödelian incompleteness (first theorem). I see it as bilaterally arguable.
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