# What is entanglement, both classical and quantum and what is the difference between these ?

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So the title gives the topic for discussion.

Here is my introduction.

An entanglement occurs when at least two bodies posses properties where observation (interaction) of that property value on one body (automatically) identifies the property value on the other body.

A classical example would be a bag containing one red and one blue ball.
Withdrawing one ball would automatically identify the colour of the ball left in the bag.

But knowing the temperature one the one ball would not tell you anything about the temperature of the other ball.

A quantum example would be two electrons in a covalent bond.
Knowing all the quantum numbers of one electron would automatically define all the quantum numbers of the other electron. (Pauli)

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There is no such thing as classical entanglement.

Entanglement is a purely quantum phenomenon. There are several ways to talk about it. The one I prefer is the most general one. There is entanglement whenever you have two particles in a pure quantum state (maximally determined) and you cannot factor out the common state as a product of one state (for one particle) times another (for the other). The probabilities do not check with those of independent statistical collectivities. Some people call entanglement what really is maximal entanglement (maximum maximal confusion, or equal probabilities between the 1-2 and the 2-1 --exchanged-- states), which is peculiar in and of itself. I see no end to the confusion of terminologies.

But as Swansont and Markus have stated before, it's not the hallmark of a distant interaction, but of a past one.

Another way people like to characterise it is by saying that the state of both particles is more determined (or exhaustive) than the state of just one of them. It's It checks with what I know.

This you cannot do with classical fields, because, eg., the electromagnetic field at one point is just one entity that's built up from the contributions of all the sources in the universe, making one big vector thing at that point. In QM, on the contrary, you have a phase space of "thingies" (1)x(2)x(3)... etc., so it's nothing like the classical case. You can have things like (1)x(2)+(1)'x(2)'. "Identity", so to speak, can be "scrambled". This is very peculiar.

Some people talk about "classical entanglement" simply because they confuse the principle of superposition for classical fields with the principle of superposition for quantum states (that only when combined with composite states being so-called "tensor products" produces this situation.

Entanglement is a consequence of the fact that the simplest physical systems are particles (the 1-thing, the 2-thing, the 3-thing,...) and fields (their values everywhere) at the same time.

I'm not being very clear, and I know it, but I'm ready to be corrected/clarified/completed by other users, including you, of course.

I apologise, @studiot, for my handwavy and cursory way to put it, but I find your topic fascinating and I hope to be able to contribute more significantly later --hopefully.

One last thing before I say something stupid on account of being too tired today: There is no such thing as the quantum numbers. Quantum systems have incompatible sets of those. None is better than the other. That's why Pauli's definition:

45 minutes ago, studiot said:

Knowing all the quantum numbers of one electron would automatically define all the quantum numbers of the other electron. (Pauli)

doesn't really cut it. Particularly severely for spin.

Really looking forward to continuing this discussion.

Edited by joigus
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Entanglement means the particles can’t be described by separate wave functions. There is a wave function that describes the composite system.

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23 minutes ago, joigus said:
54 minutes ago, studiot said:

Knowing all the quantum numbers of one electron would automatically define all the quantum numbers of the other electron. (Pauli)

doesn't really cut it. Particularly severely for spin.

Really looking forward to continuing this discussion.

5 minutes ago, swansont said:

Entanglement means the particles can’t be described by separate wave functions. There is a wave function that describes the composite system.

Fair statement.

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I would have to agree with Joigus and Swansont.
'Classical entanglement' is better ( and more accurately ) known as a correlation

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40 minutes ago, joigus said:

making one big vector thing at that point.

making one big anti-symmetric tensor thing at that point.

Sorry, @Markus Hanke.

Edited by joigus
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To me entanglement is a correlation between measurement outcomes, due to non-separability of the system. Note that non-separability does not equal non-locality. Also note that it is meaningless to talk of ‘correlation’ in a quantum system unless there’s an observer there who measures both parts, and compares results (remember counterfactual definiteness) - thus entanglement is a relationship between parts as much as a relationship of a system with its environment.
Lastly, an entangled system seems to me a good example of where reductionism arguably becomes problematic, because knowledge of the system does not imply knowledge of the parts. You need to know the correlation plus at least one measurement outcome on one part.

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So is anyone else brave enough to suggest that two electrons in a bonding orbital are not entangled ?

Note they conform to swansont's definition here, but not to his earlier statement about being exclusively a historic event since the bond has continued existence.

Edited by studiot
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42 minutes ago, swansont said:

Entanglement means the particles can’t be described by separate wave functions. There is a wave function that describes the composite system.

That's a neat example of emergence.

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9 hours ago, Markus Hanke said:

To me entanglement is a correlation between measurement outcomes, due to non-separability of the system. Note that non-separability does not equal non-locality. Also note that it is meaningless to talk of ‘correlation’ in a quantum system unless there’s an observer there who measures both parts, and compares results (remember counterfactual definiteness) - thus entanglement is a relationship between parts as much as a relationship of a system with its environment.
Lastly, an entangled system seems to me a good example of where reductionism arguably becomes problematic, because knowledge of the system does not imply knowledge of the parts. You need to know the correlation plus at least one measurement outcome on one part.

Quantum mechanics is puzzling in every which way you look at it when you approach it with a classical mind. Counterfactual definiteness is certainly one of the most bizarre ones. In order to clarify to anyone not totally familiar with the term, as well as check that we're talking about the same thing: Quantum systems have the ability to reveal their information and react accordingly (mostly by breaking their interference patterns) even when a detector that hasn't clicked is placed somewhere in the setup where the wave function takes significant values. The detector that hasn't clicked but was here, thus reveals that the other detector would have clicked, had we bothered to put it there

This happens even when no entanglement is involved (for just one particle: Elitzur-Vaidman bomb tester, etc.). It also happens, of course, for numbers of entangled particles. So yes, it happens when there's an observer (detector) even if it doesn't observe anything!

And I agree that reductionism has a problem with entanglement.

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12 hours ago, studiot said:

A quantum example would be two electrons in a covalent bond.
Knowing all the quantum numbers of one electron would automatically define all the quantum numbers of the other electron. (Pauli)

For spin projection; one is up the other down. By identifying the orbital that's involved in the bonding you've already determined the principal and azimuthal quantum numbers (n and l). The magnetic levels (m) would not be determined if l ≠ 0

11 hours ago, studiot said:

So is anyone else brave enough to suggest that two electrons in a bonding orbital are not entangled ?

Note they conform to swansont's definition here, but not to his earlier statement about being exclusively a historic event since the bond has continued existence.

It's a trivial case, and they became entangled when the atom formed. There is no continued interaction that causes this; the entanglement of the spins arises from the Pauli exclusion principle, not the electromagnetic interaction of the bond.

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11 hours ago, Markus Hanke said:

To me entanglement is a correlation between measurement outcomes, due to non-separability of the system. Note that non-separability does not equal non-locality. Also note that it is meaningless to talk of ‘correlation’ in a quantum system unless there’s an observer there who measures both parts, and compares results (remember counterfactual definiteness) - thus entanglement is a relationship between parts as much as a relationship of a system with its environment.
Lastly, an entangled system seems to me a good example of where reductionism arguably becomes problematic, because knowledge of the system does not imply knowledge of the parts. You need to know the correlation plus at least one measurement outcome on one part.

Definitely some interesting thoughts here and I do appreciate that you you are trying to connect via an abacus or something but I fear that brevity may have klead to some abiguity.

If at all possible I would appreciate amplification of the following points.

11 hours ago, Markus Hanke said:

You need to know the correlation plus at least one measurement outcome on one part.

For what purpose do you 'need to know' these facts ?

How does that affect the ongoing entanglement?

11 hours ago, Markus Hanke said:

To me entanglement is a correlation between measurement outcomes, due to non-separability of the system. Note that non-separability does not equal non-locality.

Yes, depending upon what you mean by 'non separability'.

If you mean what swansont said ie that you cannot analyse the system as two independent parts agreed.

But if you mean you can't physically separate the parts, thereby breaking the entanglement, the no of course you can break the entanglement.

12 hours ago, swansont said:

Entanglement means the particles can’t be described by separate wave functions. There is a wave function that describes the composite system.

That would correspond to the MO method, but surely the LCAO method is a linear proportion sum of the respective parts ?

9 minutes ago, swansont said:

For spin projection; one is up the other down. By identifying the orbital that's involved in the bonding you've already determined the principal and azimuthal quantum numbers (n and l). The magnetic levels (m) would not be determined if l ≠ 0

Strictly you do not need to know all the quantum numbers, just the spin number of one electron.

But then you would not have identified the orbital, just the entanglement.

I seem to have commited the same brevity sin I accused Markus of.

Edited by studiot
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5 minutes ago, studiot said:

Strictly you do not need to know all the quantum numbers, just the spin number of one electron.

Right. That's where the entanglement is. Not the other quantum numbers.

11 minutes ago, studiot said:

That would correspond to the MO method, but surely the LCAO method is a linear proportion sum of the respective parts ?

If you can do a linear combination, you aren't describing an entangled system. But IIRC this method is describing the non-entangled states, and generally doesn't include spin.

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43 minutes ago, swansont said:

If you can do a linear combination, you aren't describing an entangled system.

But that is one method of deriving the wavefunction of a bonding (bonded) orbital.

$\psi = {\psi _A} + \lambda {\psi _B}$

Where lambda minimises

$\frac{{\int {\psi H\psi d\tau } }}{{\int {{\psi ^2}} d\tau }}$

Edited by studiot
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1 hour ago, studiot said:

But that is one method of deriving the wavefunction of a bonding (bonded) orbital.

ψ=ψA+λψB

Where lambda minimises

ψHψdτψ2dτ

Where is the spin state in that equation?

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9 hours ago, swansont said:

Where is the spin state in that equation?

Well you introduced the wavefunction which is the dependent variable in the (Schrodinger et al) equation but it does not depend upon the spin quantum number so I would not expect that to be represented.

I mus say again, thanks to all, this discussion is helping me with my thinking about entanglement.

Edited by studiot
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12 hours ago, studiot said:

For what purpose do you 'need to know' these facts ?

For the purpose of attempting reduction. I may not have formulated this very clearly, I just meant that knowledge of the parts here does not equal knowledge of the whole system; there’s extra information there.

12 hours ago, studiot said:

How does that affect the ongoing entanglement?

Measurement of any part of the system breaks entanglement.

12 hours ago, studiot said:

If you mean what swansont said ie that you cannot analyse the system as two independent parts agreed.

Yes, it’s what I meant.

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11 hours ago, studiot said:

Well you introduced the wavefunction which is the dependent variable in the (Schrodinger et al) equation but it does not depend upon the spin quantum number so I would not expect that to be represented.

I mus say again, thanks to all, this discussion is helping me with my thinking about entanglement.

Spin is kind of an add-on; the spatial wave function is what you get by solving the Schrödinger equation. Those quantum numbers don’t represent the entangled state, so you’re free to do linear combinations. But not so with the spin states.

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