# Entanglement correlations, singlet spin state

Pio2001:

eaglelake: The question I was asking is what forces the result to be one of the 2 you mentioned. In my previous posts I explained how I came to the conclusion that other states could result, while _seemingly_ following the basic rules of QM. Evidentially I did break a fundamental rule somewhere in that logic, I just don't see where. I know people have pointed out that the example state I mentioned breaks the symmetrization rules and is contrary to the literature which says the state must collapse to one of the two you mentioned, ... but I THINK those things are supposed to follow from the fundamental rules of QM, right? I don't think those things themselves are supposed to be the fundamental QM rules. In the case of the symmetrization requirement Griffths seems to imply that once a system is in an antisymmetric state (like the initial one) the state will automatically remain antisymmetric when evolving under the QM collapse/evolution rules (i.e. symmetrization shouldn't have to be added as a new rule, it should be automatic, a consequence of the more fundamental rules).

One of the postulates of quantum mechanics states that the only possible results of a measurement are the eigenvalues of the observable being measured. The observable is an Hermitian operator. For spin ½, the Pauli spin matrices are the Hermitian operators for each component of the spin. If you solve the eigenvalue equation for any one of the Pauli spin matrices you will get eigenvalues +1 and -1, i.e. spin up and spin down. These are the only values ever obtained in a spin ½ measurement.

Quantum experiments confirm the correctness of this view. As far as I know it has never before been challenged. There is nothing that "forces" this to happen; it simply is a fact. I don't know how else to explain it!

zonde
Gold Member
When we measure the z-component of the spin there are only two possible results for the superposition state given:

Result 1 Alice obtains spin up, which tells us that if Bob measures the same component, he gets spin down.
Result 2 Alice obtains spin down, which tells us that if Bob measures the same component, he gets spin up.
The question I was asking is what forces the result to be one of the 2 you mentioned.
Isn't it Pauli exclusion principle that states that two fermions can't occupy the same quantum state? With it somehow extended to entangled particles?

and then the STATE COLLAPSES TO THE CORRESPONDING EIGENSTATE. This is p106 of Intro to QM 2nd Ed by Griffths. So I think this explanation in Griffths is bad -
Unless it gives a definition of observable that is not matched by the operator $$\hat{S}_z \otimes \hat{I}$$, or it gives a special rule for degenerated eigenvalues (eigenvalues that have several different eigenstates).

eaglelake: I know what you said in your most recent post, and the state i had mentioned was an eigenstate of the observable, that wasn't the problem. I think spectracat and truecrimson pointed out the problem, which is that the state is not just ANY (normalized) eigenstate of the observed eigenvalue (in the case of degeneracy there are many) but is in fact the projection of the initial state onto the eigenspace of the observed eigenvalue. Of course the latter must be in the eigenspace and therefore must be an eigenvector, but that's a much more specific rule than the one given in my QM book and prevents eigenstates like the one i had mentioned.

zonde: Pauli exclusion refers to the overall state (including the spin and wave function) i think. In this case the two electrons are considered to be in 2 different places so, even if they have the same spin, they are not in the same overall state. also, I think Pauli exclusion is another property that automatically follows from the more basic rules anyway (i.e. we shouldn't have to apply it, it should just happen).

zonde
Gold Member
Pauli exclusion refers to the overall state (including the spin and wave function) i think. In this case the two electrons are considered to be in 2 different places so, even if they have the same spin, they are not in the same overall state.
For two electrons to be entangled they have to originate from single place and then move to two different places by unitary evolution. So it's preservation of wave function under unitary evolution that is required as well.

also, I think Pauli exclusion is another property that automatically follows from the more basic rules anyway (i.e. we shouldn't have to apply it, it should just happen).
I think you have it backwards. Pauli exclusion principle comes from observations and successful theory should incorporate accepted observational facts. So if you have more basic rule that unambiguously explains this property - fine, but it's unreasonable to say that it should just happen - you have to make it happen.

SpectraCat
For two electrons to be entangled they have to originate from single place and then move to two different places by unitary evolution. So it's preservation of wave function under unitary evolution that is required as well.
I'm not sure what makes you say that ... electrons in a singlet state are always entangled, right? In fact, in atomic and molecular systems, aren't *all* of the electrons entangled with each other under normal circumstances? This entanglement appears as the exchange integral in electronic structure calculations, for example, which needs to be handled properly to get results that agree with experiment.

I guess what you were saying above applies to macroscopic entanglement experiments
like Aspect etc., but I don't think it is generally correct.

zonde
Gold Member
I'm not sure what makes you say that ... electrons in a singlet state are always entangled, right? In fact, in atomic and molecular systems, aren't *all* of the electrons entangled with each other under normal circumstances? This entanglement appears as the exchange integral in electronic structure calculations, for example, which needs to be handled properly to get results that agree with experiment.

I guess what you were saying above applies to macroscopic entanglement experiments
like Aspect etc., but I don't think it is generally correct.
I do not quite understand what are your objections.
Entangled particles do not appear from nowhere at two remote places. There has to be some preparation procedure (source) of entangled pair in experiment, right?

zonde: regarding the pauli exclusion principle thing, what i am saying is that i don't think the pauli explusion principle is something you should have to explicitly apply when doing analysis, it should be automatic, a consequence of the more basic rules for constructing wave functions. see, for example p204 of Griffths "Intro to QM, 2nd Ed"

SpectraCat