Monday, May 17, 2021

Rethinking entanglement of a single particle

Dr. Bee has caused me to rethink entanglement, and reader Ajit sends a paper on Entanglement isn't just for spin
Quantum entanglement occurs not just in discrete systems such as spins, but also in the spatial wave functions of systems with more than one degree of freedom.
It is sometimes said that Einstein discovered entanglement in 1935, and it was immediately recognized as the central defining feature of quantum mechanics. But as the above paper notes, the word was not in common use until about 1987, and did not find its way into textbooks until after that.

As the article explains, entanglement is not some peculiarity of tricky spin experiments. It is a property of all quantum systems.

Entanglement is explained as the thing that makes quantum mechanics nonlocal, and hence the essence of why the theory is non-classical and mysterious.

Paul Dirac one said:

Quantum-mechanically, an interference pattern occurs due to quantum interference of the wavefunction of a photon. The wavefunction of a single photon only interferes with itself. Different photons (for example from different atoms) do not interfere.
This is not an exact quote, but he said something similar.

This is a confusing statement, and I would not take it too literally. But in a similar spirit, I would say that a quantum particle can be entangled with itself.

Entanglement is often introduced by describing creation of a pair of particles with equal and opposite spins. But it is much more common. In any atom with several orbital electrons, those electrons are entangled. Nearby particles usually are. The case of the equal and opposite pair is interesting because that gives distant entanglement, but nearby entanglement occurs all the time.

Consider a stream of particles being fired into a double slit. Each particle is interfering with itself, and is entangled with itself. The interference results in the interference pattern on the screen.

The entanglement results in each particle hitting the screen exactly once. If you purely followed the probabilities, there are many places on the screen where the particle might hit. Those possibilities are entangled. If the particle is detected in one spot, it will not be detected in any other.

You cannot understand the experiment as localized probabilities in each spot of the screen.

Viewed this way, I am not sure the 2-particle entanglement story is any more mysterious than the 1-particle story. Maybe explanations of entanglement should just stick to the 1-particle story, as the essence of the matter.

Update: Reader Ajit suggests that I am confusing entanglement with superposition. Let me explain further. Consider the double-slit experiment with electrons being fired thru a double-slit to a screen, and the screen is divided into ten regions. Shortly before an electron hits the screen, there is an electron-possibility-thing that is about to hit each of the ten regions. Assuming locality, these electron-possibility-things cannot interact with each other. Each one causes an electron-screen detection event to be recorded, or disappears. These electron-possibility-things must be entangled, because each group of ten results in exactly one event, and the other nine disappear. There is a correlation that is hard to explain locally, as seeing what happens to one electron-possibility-thing tells you something about what will happen to the others. You might object that the double-slit phenomenon is observed classically with waves, and we don't call it entanglement. I say that when a single electron is fired, that electron is entangled with itself. The observed interference pattern is the result.


  1. The computed correlations between the timings and positions of recorded *events* can be said to be caused by "the field" (if you like, I'm not saying you must, but you will eventually have to take on board that it matters a lot that the field is *noisy*), instead of saying that each event is caused by a particle and that those particles are entangled. If we change an experimental apparatus slightly, then "the field" changes slightly because its boundary conditions have changed, and the computed correlations between the timings and positions of the recorded *events* will change slightly.
    It's always seemed to me an enormous leap to say that every recorded event is a measurement of a particle, particularly when at the same time we say that our best theories are field theories: even without that appeal to authority, however, a particle has a trajectory but in modern experiments we only see patterns of events.
    To me, and perhaps to a few other people, it seems helpful to think of the mathematics of Hilbert spaces and operators that act on them as giving us a very good way to describe multiple experiments and computations applied to post-selected collections of events. In general algorithms are precisely transformations of the raw data of times and positions of recorded events, and not all such transformations commute.
    Koopman's Hilbert space formalism for classical mechanics is one moderately good hook for such a deflationary approach, but the mathematics of quantum probability or of generalized probability theory works pretty well. The key, I think, is that state spaces and transformations are classically natural and indeed were always part of classical physics and classical probability. We've been distracted by some quite perverse thinking, however, part of which is the idea that events are always caused particles.
    Another comment into the void, I expect, though a steady de-weirdification of the quantum theory literature has been coming along nicely for the last 20 years at least, but thanks, as always, for your interesting posts.

  2. Dear Roger,

    0. Thanks for your interest!

    1. The simplest system that can in principle show the quantum entanglement has at least *two* particles in it (and they may be spinless, in which case the system has to be prepared carefully).

    A system of just one particle *cannot* show *entanglement* although it *can* show *superposition*.

    2. Dr. Sabine Hossenfelder has covered the difference between superposition and entanglement in a blog post in 2016 ( ) as well as more recently ( ).

    However, in avoiding equations, these write-ups, IMO, do not hit the essence of the difference so satisfactorily. They do get the reader going, but to reach the destination, you just can't avoid at least some equations. (The second link is much clearer about the mathematical issues than the first one, however.)

    There also is the MIT Prof. (and Nobel laureate) Dr. Frank Wilczek writing for the Quanta Mag, here: .

    This write-up does touch upon some (actually, many) of the crucial details of the maths in an admirably simple manner. But still, IMO, the write-up is such that, it *can* leave people confused. The mathematical details provided *are* simple, but they don't directly bring out some essential aspects. At least IMO.

    3. So, the question we have is:

    Why is it that a one-particle system can show superposition but not entanglement---even if the lone particle is made to pass through two alternative paths in space, as in the single-particle double-slit interference experiment---whereas a two-particle system can show not just superposition but also entanglement?

    The above refs + Schroeder's paper *could* perhaps be enough.

    4. Just in case you can't get to a *fully* satisfactory answer with these and similar references---and *I* couldn't, during my *self*-studies in isolation, at least not in enough generality---then, for a really good jotting down of the necessary theory and history, see Dr. Travis Norsen's book on the Foundations of QM.


  3. [...Contd from my previous comment]

    5. Since working through the question is such a fun, people should give it a try, first.

    However, if you / someone else wants me to write a simplest explanation that is also general enough (*IMO*), then do let me know. (I began writing this reply in that spirit, but it became too big, and so I dropped it.)

    6. Aside: One crucial element in understanding entanglement is the fact that the configuration space for an N-particle system in 3D physical space is 3N-dimensional.

    Chronologically speaking, Lorentz was the first to notice that wave mechanics has this nature; Einstein was (to my knowledge) second. Lorentz communicated this fact and its implications to Schrodinger right in first half of 1926, before Einstein did (which was in August 1926).

    (BTW, Lorentz was in his 70s then; Schrodinger in his late 30s. To my limited knowledge at least, the younger QM physicists like Heisenberg, Pauli or Dirac weren't the first ones to notice it / highlight it.)

    Anyway, so... Once the EPR paper came in 1935, Schrodinger was already ready with all the insights about the issue (with some ideas tracing back to his correspondence with Lorentz in 1926). It was Schrodinger (and not Einstein, Bell, or Feynman) who coined both the original term in German as also its translation in English (viz. "entanglement"). This was in mid-1930s.

    Bell revived the issue in 1960s, but it didn't hit exponential popularity until some time after Feynman proposed the QC in early 1980s.

    7. Yes, interactions imply entanglement.

    States without entanglement are an abstraction from reality, considered to simplify models. However, reality itself isn't actually like that.

    Someone (I forgot who) has said something relevant here; it was to this effect: "The entire universe is a single, giant, molecule".

    9. An aside: IMO, it's best to ignore photons as particles at least initially. Even someone knowledgeable like Dr. Motl talks of the "photon wavefunction" in his reply at the StackExchange thread you mentioned in the post. However, strictly speaking, the Schrodinger wavefunction formalism cannot describe photons.

    You are quite on safe grounds with *electrons*, however.

    8. To conclude: Feel absolutely free to ask for the document I said I could write; it won't take more than a few hours to write it (and won't adversely affect my studies / research schedule).


  4. Over time it has become more and more apparent why you oppose the general concepts of entanglement and nonlocality, as it seems you have not yet studied them fully, beyond Bee's blog.

    You write, "It is sometimes said that Einstein discovered entanglement in 1935, and it was immediately recognized as the central defining feature of quantum mechanics. But as the above paper notes, the word was not in common use until about 1987, and did not find its way into textbooks until after that."


    Please read:'entanglement,555)%3A&text=I%20would%20not%20call%20that,from%20classical%20lines%20of%20thought.

    Schrödinger coined the term ‘entanglement’ to describe this peculiar connection between quantum systems (Schrödinger, 1935; p. 555):

    When two systems, of which we know the states by their respective representatives, enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before, viz. by endowing each of them with a representative of its own. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two representatives [the quantum states] have become entangled.
    He added (Schrödinger, 1935; p. 555):

    Another way of expressing the peculiar situation is: the best possible knowledge of a whole does not necessarily include the best possible knowledge of all its parts, even though they may be entirely separate and therefore virtually capable of being ‘best possibly known,’ i.e., of possessing, each of them, a representative of its own. The lack of knowledge is by no means due to the interaction being insufficiently known — at least not in the way that it could possibly be known more completely — it is due to the interaction itself.
    Attention has recently been called to the obvious but very disconcerting fact that even though we restrict the disentangling measurements to one system, the representative obtained for the other system is by no means independent of the particular choice of observations which we select for that purpose and which by the way are entirely arbitrary. It is rather discomforting that the theory should allow a system to be steered or piloted into one or the other type of state at the experimenter’s mercy in spite of his having no access to it.

    Hope this helps!!

  5. That article confirms what I said. What are you claiming is wrong?

    Note that almost all of the references in that article are to 1935, or after 1980. The only exceptions are Bell and Everett.

  6. Dear Roger,

    Re.: Your update.

    I began writing a reply here, but it became too big. So I made a separate post at my blog, mostly (but not exclusively) in reference to what you have noted in the update.

    My post can be found here: .

    Hope it helps.