**Update alert!:** Several addenda have been inserted inline on 21st and 22nd May 2021, IST.

**Special Note:** This post is just a reply to a particular post made by Dr. Roger Schlafly at his blog.

*Those of you who’ve come here to check out the general happenings from my side, please see my previous post (below this one); I posted it just a couple of days ago.*

**1. Context for this post:**

This is an unplanned post. In fact, it’s a reply to an *update* to a post by Dr. Roger Schlafly. His own post can be found here [^]. Earlier, I had made a couple of comments below that post. Then, later on, Schlafly added an update to the same post, in order to clarify how he was thinking like.

As I began writing a reply to that update, at his blog, my write-up became way too big. Also, I couldn’t completely avoid LaTeX. So, I decided to post my reply here, with a link to be noted at Schlafly’s blog too. …

… I could’ve, perhaps, shortened my reply and posted it right at Schlafly’s blog. However, I also think that the points being discussed here are of a more general interest too.

*Many beginners in QM carry exactly the same or very similar kind of misconceptions concerning superposition and entanglement.* Further, R&D investments in the field of Quantum Computers have grown very big, especially in the recent few years. *Many of the QC enthusiasts come with a CS background and almost nothing on the QM side.* In any case, a lot of them seem to be carrying similar misconceptions. *Even pop-sci write-ups about quantum computing show a similar lack of understanding—all too often*.

Hence this separate, *albeit very context-specific*, post. …~~ This post does not directly focus on the difference between superposition and entanglement (which will take a separate post/document).~~ ~~However, it does touch upon many points concerning the two related, but separate, phenomena.~~ [*Done!*]

**2. What Dr. Schlafly said in his update:**

Since Schlafly’s update is fairly stand-alone, let me copy-paste it here for ease of reference. However, it’s best if you also go through the entirety of his post, and also the earlier replies, for the total context.

Anyway, the update Schlafly noted is this:

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.

Let me cite some excerpts from this passage as we go along…

**3. My reply:**

* 3.1. I will state how the mainstream QM (MSQM) conceptualizes the scenario Schlafly describes*, and leave any comments from the viewpoint of my own new approach, for some other day (after my document is done)…

So, let’s get going with MSQM (I mean the non-relativistic version, unless otherwise noted):

**3.2.**

Excerpt:

“Consider the double-slit experiment with electrons being fired thru a double-slit to a screen, and the screen is divided into ten regions.”

To simplify our discussion, let’s assume that the interference chamber forms an isolated system. Then we can prescribe the system wavefunction to be zero outside the chamber.

(MSQM can handle open systems, but doing so only complicates the *maths* involved; it doesn’t shed any additional light on the issues under the discussion. OTOH, MSQM *agrees* that there is no negative impact if we make this simplification.)

So, let’s say that we have an isolated system.

Electrons are detected at the screen in spatially and temporally discrete events. In MSQM, detectors are characterized classically, and so, these can be regarded as being spatially finite. (The “particle” aspect.)

Denote the time *interval* between two consecutive electron detection events as . In experiment, such time-durations (between two *consecutive* detections) appear to be randomly distributed. So, let be a random variable. The PDF (probability distribution function) which goes with can be reasonably modeled with a distribution having a rapidly decaying and long tail. For bosons (e.g. photons), the detection events are independent and so can be modeled with a Poisson distribution. However, for electrons (fermions), the Poisson distribution won’t apply. Yet, when the electron “gas” is so thin as to have just a few electrons in a volume that is the scale of the wavelength of electrons as in the experiment, the tail of PDF is very long—*indefinitely* long.

That’s why, when you detect some electron at the screen, you can never be sure that the next electron hadn’t already been emitted *and* hadn’t made its way into the interference chamber.

Practically, however, observing that the distribution decays rapidly, people consider the average (i.e. expectation) value for the time-gap , and choose some multiple of it that is reasonably large. In other words, a lot of “screening” is effected (by applying an opposite potential) after the electron gun, before the electrons enter the big interference chamber proper (Five sigma? I don’t know the criterion!)

Thus, assuming a large enough a time-gap between consecutive events, we can make a further simplifying assumption: There is only one electron in the chamber at a time.

**3.3.**

Excerpt:

“Shortly before an electron hits the screen, there is an electron-possibility-thing that is about to hit each of the ten regions.”

In the MSQM, before the lone electron hits the screen, the state of the electron is described by a wavefunction of the form: .

If, statistically, there are *two* electrons in the chamber at the same time (i.e. a less effective screening), then the assumed system wavefunction would have the form:

,

where and are *not* the positions of the two electrons, but the two vector coordinates of the configuration space (i.e. six degrees of spatial freedom in all).

Should we assume some such a thing?

If you literally apply MSQM to the universe, then in principle, *all* electrons in the universe are *always* interacting with each other, no matter how far apart. Further, in the *non-relativistic* QM, *all* the interactions are *instantaneous*. In the relativistic QM the interactions are not instantaneous, but we need not consider relativity here, simply because the chamber is so small in extent. [I am not at all sure about this part though! I don’t have any good intuition about relativity; in fact I *don’t* *know* it! I should have just said: Let’s ignore the relativistic considerations, as a first cut!]

So, keeping out relativity, the electron-to-electron interactions are modeled via the Coulomb force. This force decays rapidly with distance, and hence, is considered negligibly small if the distance is of the order of the chamber (i.e., practically speaking, the internal cavity of a TEM (transmission electron microscope)).

*Aside:* In the scenarios where the interaction is *not* negligibly small, then the two-particle state cannot be expressed as a tensor product of two one-particle states . In other words, entanglement between the two electrons can no longer be neglected.

Let us now assume that in between emission and absorption there is only one electron in the chamber.

Now, sometimes, it can so happen that, due to some statistical fluke, there may be two (or even three, four…) electrons in the chamber. However, we now have a *stronger* argument for assuming that there is always only one particle in the chamber, when detection occurs. Reason: We are now saying is that the *magnitude* of the interaction between the two electrons (the one which was intended to be in the chamber, and the additional one(s) which came by fluke) is so *small* that these interactions can be assumed to be zero. We can make that assumption simply because the electrons are so far apart in the TEM chamber—as compared to their wavelengths as realized in this experiment.

*So, at this point, we assume that a wavefunction of the form applies.*

Note, the *configuration* space now has a *single* variable vector , and so, there is no problem *interpreting* it as the coordinate of the ordinary *physical* space. So, we can say that *wavefunction* (which describes a *wave—*a distributed entity) is, in this case, defined right over the *physical* space (the same space as is used in NM / EM). *Note:* We still aren’t interpreting this as the particle-position of the electron!

**3.4.**

Excerpt:

“Assuming locality, these electron-possibility-things cannot interact with each other.”

The wavefunction for the lone electron always acts as a single entity over the entire domain at the same time. (The “wave” aspect.)

The wavefunction has support all over the domain, and the evolution of each of the energy eigenstates comprising it occurs, by Fourier theory, at all points of space *simultaneously*.

In short: *The wavefunction evolution is necessarily “global”.* That’s how the theory works—I mean, the *classical* theory of Fourier’s.

[*Addendum made on 2021.05.21:* BTW, there can be no interaction between the energy eigen-states comprising the total wavefunction because all eigenfunctions of a given basis are always orthogonal to each other. Addendum over.]

**3.5.**

“Each one causes an electron-screen detection event to be recorded, or disappears.”

Great observation! I mean this part: “or disappears”. Most (may be or more, including *some* PhD physicists) would miss it!

OK.

*Assume* that the detector efficiency is .

Assuming a less-than-perfect detector-efficiency doesn’t affect the foundational arguments in any way; it only makes the maths a bit more complicated. Not much, but a shade more complicated. Like, by a multiplying factor of the square-root of something… But why have any complications if we can avoid them?

[*Addendum made on 2021.05.21:* Clarification: May be, I mis-interpreted Schlafly’s write up here. He could easily be imagining here that there are ten components in the total wavefunction of a single electron, and that only one component remains and the other disappear. OTOH, I took the “disappearing” part to be the electron itself, and not the components entering into that superposition which is the system wavefunction . … So, please read these passages accordingly. The explanation I wrote anyway has covered decomposing the system wavefunction into two different eigenbases: (i) the total energy (i.e. the Hamiltonian) operator, and (ii) the position operator. Addendum over.]

**3.6.**

Excerpt:

“These electron-possibility-things must be entangled, because each group of ten results in exactly one event, and the other nine disappear.”

Well…

*Bohr* insisted that the detector be described classically (i.e. using the ideas of classical EM), by insisting on his Correspondence principle. (BTW, Correspondence is *not* the same idea as the Complementarity principle. (BTW, IMO, the abstract idea of the Correspondence principle is good, though not how it is concretely applied, as we shall soon touch upon.))

This is the reason why the MSQM *does not* describe the ten detectors at the screen *quantum mechanically*, to begin with.

MSQM also *cannot. *Even if we were to describe the ten detectors quantum mechanically, problems would remain.

According to MSQM, the *quantum-mechanical* system would now consist of {1 electron + 10 detectors (with all their constituent quantum mechanical particles)}.

This entire huge system would be described via a single wavefunction. Just keep adding , as many of them as needed. Since there no longer is a classical-mechanical detector in the description, the system would forever go oscillating, with its evolution exactly as dictated by the Schrodinger evolution. Which implies that there won’t be this one-time big change of a detection event, in such a description. MSQM cannot accomodate an irreversible change in the state of the {the 1e + 10 detectors} system. By postulation, it’s linear. (Show some love to Bohr, Dirac, and von Neumann, will you?)

Following the lead supplied by Bohr (and all Nobel laureates since), the MSQM models our situation as the following:

There is a single *quantum-mechanically *described electron. It is described by a *wavefunction* which evolves according to the Schrodinger equation. Then, there are those 10 *classical* detectors that do not *quantum mechanically* interact with the electron (the system wavefunction) at all, for any and all instants, until the detection event actually happens.

Then, the detection event happens, and it occurs at one and only one detector. Which detector in particular? “At random”. What is the mechanism to describe it? Blank-out!

But let’s continue with the official view (i.e. MSQM)…

The detection event has two parts: (1) The randomly “chosen” detector irreversibly changes its always *classical* state, from “quiscent” to “detected”. At the same time, (2) the *quantum-mechanical* wavefunction *“collapses”* into that particular *eigenfunction* of the position operator which has the associated *eigenvalue* of that Dirac’s delta which is situated at the detector (which happened to undergo the detection event).

What is a collapse? Refer to the above. It refers to a single eigenfunction remaining from among a *superposition* of all eigenfunctions that were there. (The wave was spread out, i.e. having an infinity of Dirac’s delta positions; after the collapse, it became a single Dirac’s delta.)

What happened to the other numerous (here, an infinity of) eigenfunctions that were not selected? Blank out.

What is the *mechanism* for the collapse? Blank out. (No, really!)

How much time does it take for the detection event to occur? Blank out. (No, really!)

To my limited knowledge, MSQM is actually silent about the time lapse. Even Bohr himself, I think, skirted around the issue in his more official pronouncements. However, he also never gave up the idea of those sudden “quantum jumps”—which idea *Schrodinger* hated.

So, MSQM is silent on the time taken for collapse. But people (especially the PhD physicists) easily rush in, and will very confidently tell you: “infinitesimally small”. Strictly speaking, that’s their own interpretation. (Check out the QM Postulates document [^], or the original sources.)

One more point.

Carefully note: *There were no ten events existing prior to a detection anywhere in the above description.* That’s why the question of the nine of them then disappearing simply cannot arise. MSQM doesn’t describe the scenario the way Schlafly has presented (and many people believe it does)—at all.

IMO, MSQM does that with good reason. You can’t equate a *potential* event with an *actual* event.

Perhaps, one possible source of the confusion is this: People often seem to think that probabilities superpose. But it’s actually only the complex amplitudes (the wavefunctions) that superpose.

[*Addendum made on 2021.05.21:* Clarification: Even if we assume that by ten things we mean ten components of the wavefunction and not ten events, the rest of the write-up adequately indicates the decomposition of into eigenbasis of the Hamiltonian (total energy) operator as well the position operator. Addendum over.]

**3.7.**

Excerpt:

“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.”

There are no ten events in the first place; there is only one. So, there is no *correlation* to speak of.

[*Addendum made on 2021.05.21: *Clarification. Just in case the ten things refer to the ten components (not a complete eigenbasis, but components in their own right, nevertheless) of the wavefunction and not ten events, there still wouldn’t be correlations to speak of between them, because all of them would collapse to a single Dirac’s delta at the time of the single detection event. Addendum over.]

That’s why, we can’t even begin talking of any numerical characteristics (or relative “strengths”) of the so-supposed correlations. Not in single-particle experiments.

In one-particle situations, we can’t even address issues like: Whether the correlations are of the same strengths as what QM predicts (as for the entangled particles); or they are weaker than what QM predicts (which is what happens with the predictions made using some NM- / EM-inspired “classical” models of the kind Bell indicated, i.e., with the NM / EM ontologies), or they are *stronger* than what QM predicts. (Experiments say that the correlations are *not* stronger either!)

*Correlations* become possible once you have at least *two* electrons at the same time in a system.

Even if, in MSQM, the two electros have a *single* wavefunction governing their evolution, the configuration space then has *two* 3D vectors as independent variables. That’s how the theory changes (in going from one particle to two particles).

As to experiments: There is *always* only *one* detection event *per* particle. Also, *all* detection events must occur—i.e. *all* particles must get *detected*—before the presence or absence of entanglement can be demonstrated.

One final point. Since all particles in the universe are always interconnected, they are always interacting. So, the “absence of entanglement” is only a theoretical abstraction. The world is not like that. When we say that entanglement is absent, all that we say is that the strength of the correlation is so weak that it can be neglected.

[*Addendum made on 2021.05.21: *

BTW, even in the classical theories like the Newtonian gravity, and even the Maxwell-Lorentz EM, all particles in the universe are always interconnected. In Newtonian gravity, the interactions are instantaneous. In EM (and even in GR for that matter), the interactions are time-delayed, but the amount of delay for any two particles a finite distance apart is always finite, not infinite.

So, the idea of the universe as being fully interconnected is not special to QM.

One classical analog for the un-entangled particles is this: Kepler’s law says that each planet moves around the Sun in a *strictly* elliptical orbit. If we model this empirical law with the Newtonian mechanics, we have to assume that the interactions in between the planets are to be neglected (because they are relatively so small). We also neglect the interactions of the planets with everything else in the universe like the distant stars and galaxies. In short, each planet *independently *interacts with the Sun and *only* with the Sun.

So, even in classical mechanics, for the first cut in our models, for simplification, we do neglect some interactions even if they are present in reality. Such models are abstractions, not reality. Ditto, for the un-entangled states. They are abstractions, not reality.

Addendum over.]

**4. But what precisely is the difference?**

This section (# 4.) is actually a hurriedly written addendum. It was not there in my comment/reply. I added it only while writing this *post*.

I want to make only this point:

**All non-trivial entangled states are superposition states. But superposition does not necessarily mean entanglement. Entanglement is a special kind of a superposition.**

Here is a brief *indication* of how it goes, in reference to a concrete *example*.

Consider the archetypical example of an entangled state involving the spins of two electrons (e.g., as noted in this paper [^], which paper was noted in Prof. (and Nobel laureate) Franck Wiczek’s Quanta Mag article [^]). Suppose the spin-related system state is given as:

[Eq. 1].

The state of the system, noted on the left hand-side of the above equation, is an * entangled* state. It consists of a a linear

*of the following two states,*

**superposition***of which,*

**each***, is*

**taken by itself***:*

**un-entangled**, [Eq. 2.1]

and

[Eq. 2.2].

The preceding two states are * un-entangled* because as the right hand-sides of the above two equations directly show, each can be expressed—in fact, each is

*defined*—as a

**tensor product of***states, which are: , and . Thus, the states which enter into the superposition themselves are factorizable into one-particle states; so, they themselves are un-entangled. But once we superpose them, the resulting state (given on the left hand-side) turns out to be an entangled state.*

**two one-particle**So, the entangled state in this example is a superposition state.

Let’s now consider a * superposition* state that is

*also an entangled state. Simple!*

**not**[Eq. 3].

This state is in a superposition of two states; it is a spin-related analog of the single-particle double-slit interference experiment.

So, what is the essential * difference between entangled states from the “just” superposition states*?

If the “total” state of a * two- (or more-) particle* system can be expressed as a

*single*tensor product of two (or more) one-particle states (as in Eqs. 2.1 and 2.2], i.e., if the total state is

*into one-particle states, then it is an*

**“separable”/”factorizable”***state.*

**independent i.e. un-entangled**** All** other

*states (like that in Eq. 1) are entangled states.*

**two-particle**Finally, all * one-particle* states (including the superpositions states as in Eq. 3) are

*states.*

**un-entangled**One last thing:

The * difference* between the respective

*involved in the*

**superpositions***is this:*

**two-particle states vs. one-particle states**The orthonormal * eigenbasis vectors* for

*states*

**two-particle***are*

**themselves***one-particle states.*

**not**The * eigenvectors* for

*two-particle states (including those for the theoretically non-interacting particles),*

**any***are,*

**themselves**

**always,***states.*

**two-particle***[Addendum made on 2021.05.22:*

But why bother with this difference? I mean, the one between superpositions of two-particle states vs. superpositions of one-particle states?

Recall the postulates. The state of the system prior to measurement can * always* be expressed as a superposition of the eigenstates of any suitable operator. Then, in

*act of*

**any***of an observable, the*

**measurement***states that can at all be observed are the*

**only***of the operator associated with that particular observable. Further, in any*

**eigenstates***measurement,*

**single***of these eigenstates can ever be observed. That’s what the postulates say (and every one else tells you anyway).*

**one and only one**Since every eigenfunction for a two-particle system is a two-particle state, what a * theoretically single* measurement picks out is not a one-particle state like or , but a

*state like or . Only*

**two-particle***of them, but it’s a two-particle state.*

**one**So, the relevant point (which no one ever tells you) is this:

*A theoretically (or postulates-wise) single measurement*, on a two-particle system, itself refers to

**two**distinct observations made in the actual experiment—one each for the two particles. For an -particle systems, number of one-particle detections are involved—for what the theory calls a

*single*measurement!

In entanglement studies, detectors are deliberately kept as far apart as they can manage. Often, the detectors are on the two opposite sides of the initial (source) point. But this need always be the case. *The theory does not demand it.* The two detectors could be spatially anywhere (wherever the spatial part of the total wavefunction is defined). The detectors could be right next to each other. The theory is completely silent about how far the detectors should be.

In short:

All that the theory says is:

*Even for an -particle system, the state which is picked out in a single measurement itself is one of the eigenstates (of the operator in question). *

But you are supposed to also know that:

*Every eigenstate for such a system necessarily is an -particle state. *

Hence the implication is:

For a single observation during an actual experiment, you still must make number of *separate* observation events, anyway!

So…

There are number of particles and number of events. But the theory is still going to conceptualize it as a * single measurement of a single eigenfunction*.

Every one knows it, but no one tells you—certainly not in textbooks / lecture notes / tutorials / YouTube videos / blogs / Twitter / FaceBook / Instagram / whatever. [Yes, please feel challenged. Please do bring to my notice any source which tells it like it is—about this issue, I mean.]

Addendum over.]

For a more general discussion of the mathematical criterion for un-entangled (or factorizable) vs. entangled states (which discussion also is simple enough, i.e. not involving the most general case that can arise in QM), then check out the section “Pure states” in the Wiki on “Quantum entanglement”, here [^].

And, another, last-last, thing!:

Yes, the states comprising the eigenbasis of any two * non-interacting* particles always consist of tensor-product states (i.e. they are separable, i.e. non-entangled).

However, when it comes to **interacting** particles: Especially for systems of large number of particles that interact, and talking of their “total” wavefunctions (including both: the spatial Schrodinger wavefunctions defined over an infinite spatial domain, *and* their spinor functions), I am not sure if *all* their eigenvectors for *all* observables are always represent-able as tensor-product states or not. … I mean to say, I am not clear whether the Schmidt decomposition always applies or not. My studies fall short. The status of my knowledge is such that I am unable to take a definitive position here (for uncountably infinite-dimensional Hilbert spaces of very large number of particles). May be there is some result that does prove something one way or the other, but I am not sure.

That’s why, let me now stop acting smart, and instead turn back to my *studies!*

Best,

–Ajit

**5. To conclude this post…**

….Phew!… So, that was (supposed to be) my “comment” i.e. “reply”. …Actually, the first draft of my “reply” was “only” about 1,500 words long. By the time of publication, this post has now become more than 3,300 word long…

If there is any further correspondence, I plan to insert it too, right here, by updating this post.

… I will also update this post if (and when!) I spot any typo’s or even conceptual / mathematical errors in my reply. [Always possible!] Also, if you spot any error(s), thanks in advance for letting me know.

OK, take care and bye for now…

[No songs section this time around. Will return with the next post.]

**History:**

— 2021.05.20 21:00 IST: Originally published

— 2021.05.21 17:17 IST: Added some clarifications inline. Streamlined a bit. Corrected some typo’s.

— 2021.05.22 13:15 and then also 22:00 IST: Added one more inline explanation, in section 4. Also added a confession of ignorance about relativity, and a missing normalization constant. …Now I am going to leave this post in whatever shape it is in; I am done with it…