# Complementarity of Spin?

## Main Question or Discussion Point

I’m a layman who has reviewed some popular literature on QM. Apologies in advance for my naivety.

I’m of the impression that the quantum spin (±½) of an electron can be determined as it travels through a Stern-Gerlach apparatus (SGA) for the single direction in which the SGA is aligned (say, along the X-axis). Spin along other axes is a superposition of those states, with amplitudes corresponding to the angle departing from X. Spin is also said to be complementary. Spin measurement along different axes is mutually exclusive. For example, “A particle cannot have definite values for both X and Z at the same time.[ref]

It seems possible, at least in theory, to obtain values for both X and Z spin at the same time by the following method. Please let me know where it goes wrong.

1). Two electrons are prepared in an entangled, total-spin-zero state (each has spin opposite the other).

2). Two SGAs are oriented perpendicularly, one X-axis aligned and the other Z-axis aligned.

3). Each entangled electron is sent simultaneously through one of the perpendicular SGAs.

4). Spin X is determined for one electron while spin Z is determined for the other, at the same time.

5). Each electron had spin opposite the other, so spin X and spin Z of both electrons were simultaneously determined.

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You are opening up an interesting track here... This is more or less exactly the same reasoning as in the original EPR-setup. Einstein, Podolsky and Rosen then used it to argue that QM had to be incomplete because it seemed possible to simultaneously measure definite values of two non-commuting observables (which should not be possible according to QM as you point out).

Your setup is also very close to the original setup proposed by J.S. Bell when deriving his inequalities. Thanks to Bell and later experiments confirming the strangness of entanglement, we now know what is wrong in the argument.

You (and EPR!) assume that the measurement of one particle does not change the state of the other particle. This does seem very reasonably indeed, but is not true for entangled particles! This is precisely the strange "spooky action at a distance"-behavior of entanglement that is clearly proven in the many experimental verifications of Bells theorem.

I admit that I was rooting for EPR in the chapters leading up to Bell’s theorem (which I am now reviewing again.) Initially, I was curious whether spin complementarity was truly a property of the particle or simply a limitation of the measuring device (SGA). But as the given quote, “A particle cannot have definite values for both X and Z [spin] at the same time.[ref] indicates, QM insists that complementarity is a property of the particle.

Direct simultaneous measurements of a single particle may indeed be impossible. However, two particles sharing a single quantum state would seem to offer the opportunity to examine that state in different (even complementary) ways at once.

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Bill_K
Your reasoning founders on the use of the word "simultaneous".

4a) Suppose spin X is determined for particle A first. This determines the spin X for particle B. Then spin Z for particle B is measured. Particle A is not influenced by this. The two particles wind up with independent values.

4b) Suppose spin Z is determined for particle B first. This determines the spin Z for particle A. Then spin X for particle A is measured. Particle B is not influenced by this. The two particles wind up with independent values.

Same results, you can't tell 4a from 4b. And either way, you have independent values, no entanglement, no contradiction.

Your reasoning founders on the use of the word "simultaneous". ...spin X is determined for particle A first...Then spin Z for particle B is measured.
Are you saying that complementarity prohibits "simultaneous" measurement of two different (though entangled) particles?

I didn’t propose measuring one electron spin first, then the other. They are to each proceed through a separate Stern-Gerlach apparatus for simultaneous X-spin and Z-spin assessments.
Is that not even theoretically possible? What mechanism prevents it?

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Bill_K
No, I am saying that you are confusing yourself by imagining measurements at two different locations to be precisely simultaneous. The term has no operational meaning. The slightest departure from simultaneity in either direction, even one attosecond, spoils the paradox.

...simultaneous. The term has no operational meaning.
Then we might as well drop the term “simultaneous” from our language. But I said, “theoretically possible”, not operationally possible.

Special Relativity makes clear that for any pair of events deemed simultaneous in one inertial reference frame, there are an infinite number of other frames in which they are not simultaneous. Conversely, if we find that two spin measurements are not quite simultaneous in our reference frame, we can be assured that there always exist inertial frames in which they are. Simultaneity is guaranteed, both theoretically and operationally, in this sense.

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Bill_K
No, special relativity guarantees lack of simultaneity. Any argument that depends on the attainment of precise simultaneity fails, because the events are nonsimultaneous in all other rest frames.

However I assume we are talking about Schrodinger quantum mechanics, which is nonrelativistic. If you perform the stated measurements on particles A and B, and find that particle A has a definite value of Z spin, while particle B has a definite value of X spin, how can you claim that a basic principle has been violated?

Answer: you would have to show that the measurements took place at precisely the same time. PRECISELY the same time. There is no way to do this. Measurement of a time interval can only be done to some finite precision, and regardless of whether it is an attosecond or a millionth of an attosecond (whatever that is called!) it cannot be zero. And if it is not zero, quantum mechanics is saved!

Thanks Bill, but I don’t think QM has much to fear from me!
...Schrodinger quantum mechanics, which is nonrelativistic. ...measurements... at precisely the same time. There is no way to do this.
I would suggest that for any two separate events in space-time (not having an event horizon between them), there exists a set of locations having equal intervals from both events. Observers occupying these will interpret the two events as simultaneous. However, I concede that QM is often claimed to be non-relativistic. (Notwithstanding that the Dirac equation improved upon Schrodinger’s by applying relativistic considerations to the electron.)

It seems inconsistent to, on the one hand deny simultaneity (of measurements) and on the other hand, require it. What is entanglement if not particles sharing a state simultaneously?

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Entanglement is still fairly mysterious. Perhaps it was not the best initial approach to the complementarity of quantum spin (a.k.a. intrinsic spin). Switching to a single particle would avoid the relativity of simultaneity, for now. My question might be restated this way (pertaining to an electron):

Quantum spin is not “classical” spin. The same spin magnitude is always observed on every axis measured. If no device can measure quantum spin on more than a single axis at a time, how can a claim that full quantum spin occurs on every observable axis at once, be refuted?

DrChinese
Gold Member
Are you saying that complementarity prohibits "simultaneous" measurement of two different (though entangled) particles?

I didn’t propose measuring one electron spin first, then the other. They are to each proceed through a separate Stern-Gerlach apparatus for simultaneous X-spin and Z-spin assessments.
Is that not even theoretically possible? What mechanism prevents it?
You can perform this experiment, and simultaneity is irrelevant to the results. You now know Alice's X and Bob's Z. Cool.

Now the \$64,000 question is whether you also know Alice's Z and Bob's X. The answer to that question is NO. And it would need to be YES for your idea to be useful. Learning Alice's X resets her Y and Z to indeterminate values. So you have accomplished nothing. It is simply a philosophical exercise to assert you know both Alice's X and Z components when no experiment can confirm that. (And further that experiments will show the Z to match the "expected" value only randomly.)

Bill_K
Thank you, DrChinese. This, believe it or not, is what I was struggling to say. The result is the same regardless of the time sequence.

So, also just elaborating on DrChinese's answer above, the argument in the original post goes wrong at:
5). Each electron had spin opposite the other, so spin X and spin Z of both electrons were simultaneously determined.
This is not true, as DrChinese says. But I think the confusion here is about the non-local aspects of entanglement. In the original EPR paper they argued (just as you do!) that since a measurement at Alice's place could not possibly affect what they called "the physical reality" at Bob's place, Alice's X-measurement should also let us know Bob's X. But it is exactly this assumption - that a Alice measurement does not affect outcomes at Bill's - that leads to the various Bells inequalities. And they are clearly violated by QM predictions and in real experiments. So the (strange!) conclusion is that we can't assume that, and hence Alice's X-measurement does not determine Bob's X.

What is entanglement if not particles sharing a state simultaneously?
Entanglement is difficult to grasp. It is not really two particles "sharing a state", but rather a two-particle system that can only be viewed as one single QM system. In the case of two spin-half particles, we have to view them as one single 4-state system. (Which actually has one more degree of freedom than two separate 2-state systems!) And all predictions about measurements on the two individual particles and their correlations are independent of the order or simultaneousness in which measurements are done.

If no device can measure quantum spin on more than a single axis at a time, how can a claim that full quantum spin occurs on every observable axis at once, be refuted?
You have to be carefull with the word "occurs" here. If there is no way to measure it, how could you conclude that "it" occurs? I would say that it is the very statement that there is no conceivable measurement device that could (even in principle) measure more then one spin component at a time that is the reason for refuting the idea of simultaneous spin components of a single particle.

DrChinese
Gold Member
Entanglement is difficult to grasp. It is not really two particles "sharing a state", but rather a two-particle system that can only be viewed as one single QM system. In the case of two spin-half particles, we have to view them as one single 4-state system. (Which actually has one more degree of freedom than two separate 2-state systems!) And all predictions about measurements on the two individual particles and their correlations are independent of the order or simultaneousness in which measurements are done.
Yes, this part is hard. Made extra hard because we don't at all understand the process by which the 2 individual particles emerge from a 2 particle system. When and how do the 2 individual particles appear? There is no single answer that is really completely satisfying.

DrChinese
Gold Member
Thank you, DrChinese. This, believe it or not, is what I was struggling to say. The result is the same regardless of the time sequence.
Your words were good, thought I might put a different way to help the OP follow the line of reasoning provided. Sometimes a reader picks up something said one way, while another picks up on a different analogy.

You can perform this experiment [simultaneous measurement of entangled X- and Z-spin.] You now know Alice's X and Bob's Z.
Thanks for the clarification.

[Do] you also know Alice's Z and Bob's X? …NO.”
It seems the very essence of entanglement, that learning Alice’s Z instantly gives us Bob’s entangled Z. (If not, what is it that we are claiming is entangled in a "total-spin-zero" state?) Conversely, learning Bob’s X instantly gives us Alice’s entangled X. And you have graciously conceded the simultaneity of these! With all due respect (having read several of your other, quite helpful posts), all caps doesn’t make “NO” correct.

Learning Alice's X resets her Y and Z to indeterminate values.
Now that’s “irrelevant”. In the moment before resetting*, we knew both X and Z, for Alice and Bob, at once.

*Do we actually believe “resetting” is a real physical process (as opposed to a model)?

…no experiment can confirm that…<->… you know both Alice's X and Z components [at once].
Exactly! It seems that the Principle of Complementarity boils down to denial of falsifiablity. That’s a serious shortcoming for any model.

QM asserts that “A particle cannot have definite values for both X and Z at the same time.”[ref]. The Principle of Complementarity conveniently prevents us from disproving this.

Under the same guise, it may be irrefutably claimed that quantum spin occurs in all observable directions at once. No matter which direction is measured, full spin is found.

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Entanglement is difficult to grasp.
Thanks for your trouble Dali. That’s why, in post #10 and the end of #16, I'm focusing instead on a single particle. After I reread Bell’s inequality about six more times, I’m sure I’ll have something to say about it in a separate thread. As you have guessed, despite overwhelming verification of Bell, I’m still rooting for EPR! And yet I’m not denying QM, I just feel it's still growing.

If there is no way to measure it [simultaneous spin in all observable directions], how could you conclude that "it" occurs?
The evidence is as I say in post #16, “No matter which direction is measured, full spin is found.” It has already been established that quantum spin is not “classical” spin, so we should expect something to be different.

DrChinese
Gold Member
Now that’s “irrelevant”. In the moment before resetting*, we knew both X and Z, for Alice and Bob, at once.

*Do we actually believe “resetting” is a real physical process (as opposed to a model)?

Exactly! It seems that the Principle of Complementarity boils down to denial of falsifiablity. That’s a serious shortcoming for any model.

QM asserts that “A particle cannot have definite values for both X and Z at the same time.”[ref]. The Principle of Complementarity conveniently prevents us from disproving this.
Ah, sorry, you are making basic mistake after basic mistake. You should follow the entire line of reasoning before making statements like the above.

QM does not assert something which is not falsifiable here, you are. QM states that you cannot predict both Alice's X and Z components simultaneously. Therefore, QM does not assert they are both simultaneously real. It is the realist (that would be you) who asserts this, and then points out that QM does not support it - which is somewhat humorous you must admit. It is you who is asserting something which cannot be demonstrated, that you DO know both Alice's X and Z. And yet, a subsequent experiment will show one of those to be the same and the other to only randomly correlate. Not really the same thing now, is it?

At any rate, the concept that you know Alice's X and Z was precisely the ingredient which makes Bell's Theorem. Extending your idea that you know X and Z, you get to the idea that Y exists too. Which ultimately leads to the Bell contradiction. So the point is, your starting position is that of EPR (where there was a disagreement about the simultaneity issue, as Bill_K referenced earlier). But we now know EPR is demonstrably incorrect.

A little late for you to be rooting for EPR! They didn't have the benefit, as you do, of knowing about Bell. Rosen (the R in EPR) lived long enough to learn of Bell, so his position naturally changed. Sadly, E and P did not.

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Dr C said:
…you are making basic mistake after basic mistake.
That’s what beginners do! I find it preferable to not beginning.
Dr C said:
A little late for you to be rooting for EPR!...<-> we now know EPR is demonstrably incorrect.
By “rooting for EPR”, I mean that at my early stage of learning, and presuming Bell (thus, Bohr) to be correct, I adopt the EPR position as strongly as I can, in order to force the most out of Bell’s argument. Clearly, I’m not there yet.

That said, I would point out, “[Bell’s] inequality is experimentally testable, and there have been numerous relevant experiments,… They have all shown agreement with quantum mechanics rather than the principle of local realism. However, the issue is not finally settled, for each of these experimental tests has left open at least one loophole by which it is possible to question the validity of the results.[ref]
DrC said:
QM states that you cannot predict both Alice's X and Z components simultaneously. Therefore, QM does not assert they are both simultaneously real.
I hope you'll try to understand why a beginner might feel that QM implies simultaneous spin about all observable axes.

I found, “One consequence of the generalized uncertainty principle is that the spin projection operators (which measure the spin along a given direction like x, y, or z), cannot be measured simultaneously. Physically, this means that it is ill defined what axis a particle is spinning about. A measurement of the z-component of spin destroys any information about the x and y components that might previously have been obtained.[ref]

It speaks of an “ill defined” axis which seems akin to the ill-defined position of an electron occupying an orbital. We're taught to consider the electron as spread out over the probability cloud, effectively occupying a set of locations simultaneously.

Further, it speaks of “spin components” about other axes. Though they “cannot be measured simultaneously”, multiple components are clearly implied to exist (about every observable axis).

I also found, “…Schrödinger equation, angular momentum is quantized… so that total spin angular momentum [is] √(3/2)ħ. However, … when the electron is observed along one axis, such as the Z-axis, is quantized in terms of a magnetic quantum number, which can be viewed as a quantization of a vector component of this total angular momentum, which can have only the values of ±½ħ.[ref]

The spin½ we refer to is treated as "a component" of some larger “total angular momentum”. I’m given the impression of a more classical total spin about some axis which distributes over all observable axes. Is that so unreasonable?

DrC said:
QM states that you cannot predict both Alice's X and Z components simultaneously.
What I found was, “the spin projection operators (which measure the spin along a given direction like x, y, or z), cannot be measured simultaneously[ref]
Not really the same thing now, is it?
DrC said:
QM does not assert something which is not falsifiable here, you are.
When I suggest the Principle of Complementarity seems to deny falsifiability, that applies equally to models where spin occurs about all observable axes or where spin occurs about any one measured axis. If we can’t measure two axes simultaneously, we can’t falsify either model.

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