# Do entangled particles exist in superposition?

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Perhaps someone can show me some simple math showing two states in superposition and entanglement so I can see how entanglement relates to superposition. Thanks.

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vanhees71
Gold Member
2019 Award
What is entangled? It doesn't make sense to say something is entangled. It's like saying something is related. If you don't tell related to what, it doesn't make any sense.

A famous example is the Stern-Gerlach (SG) experiment. After running through the magnetic field of the SG apparatus, you have a particle, whose spin-##z## component is entangled with its position, i.e., if the particle state started as a pure state which is a superposition of spin up and spin down, after running through the apparatus, it's in the state
$$|\Psi \rangle=|\phi_1 \rangle \otimes |\sigma_z=+1/2 \rangle+|\phi_2 \rangle \times| \sigma_z=-1/2 \rangle,$$
where ##|\phi_1 \rangle## and ##|\phi_2 \rangle## are the spatial part of the state, describing wave packets being centered around different positions.

bhobba
Mentor
First you need to understand superposition. It is this - if |a> and |b> are quantum states then c1*|a> + c2*|b> where c1 and c2 are complex numbers is also a state.

Entanglement is an extension of superposition to different systems. Suppose two systems can be in state |a> and |b>. If system 1 is in state |a> and system 2 is in state |b> that is written as |a>|b>. If system 1 is in state |b> and system 2 is in state |a> that is written as |b>|a>. But we now apply the principle of superposition so that c1*|a>|b> + c2*|b>|a> is a possible state, The systems are entangled - neither system 1 or system 2 are in a definite state - its in a peculiar non-classical state the combined systems are in.

Thanks
Bill

What is entangled? It doesn't make sense to say something is entangled. It's like saying something is related. If you don't tell related to what, it doesn't make any sense.

A famous example is the Stern-Gerlach (SG) experiment. After running through the magnetic field of the SG apparatus, you have a particle, whose spin-##z## component is entangled with its position, i.e., if the particle state started as a pure state which is a superposition of spin up and spin down, after running through the apparatus, it's in the state
$$|\Psi \rangle=|\phi_1 \rangle \otimes |\sigma_z=+1/2 \rangle+|\phi_2 \rangle \times| \sigma_z=-1/2 \rangle,$$
where ##|\phi_1 \rangle## and ##|\phi_2 \rangle## are the spatial part of the state, describing wave packets being centered around different positions.
Thank you. So am I to understand, then, that entanglement does not necessarily require two (or more) particles, that the properties of a single particle can be entangled (spin z-component with position)? I always thought it required two particles to be entangled, spooky action at a distance (between two particles). But maybe I misunderstood.

bhobba
Mentor
Thank you. So am I to understand, then, that entanglement does not necessarily require two (or more) particles, that the properties of a single particle can be entangled (spin z-component with position)? I always thought it required two particles to be entangled, spooky action at a distance (between two particles). But maybe I misunderstood.
Entanglement is an extension of superposition to different systems.
Spooky action at a distance is an interpretive thing - its not part of the formalism of entanglement.

Thanks
Bill

For example, what mechanism is it that creates a pair of entangled photons? Is there a special way of entangling a pair of photons emanating from a single source? Or will any process that results in a pair of photons from a single interaction guarantee that the photons are entangled?

bhobba
Mentor
For example, what mechanism is it that creates a pair of entangled photons?
The mechanisms, as has been trashed out in a thread a while ago, are very complex and cant be explained at the lay level.

Thanks
Bill

The mechanisms, as has been trashed out in a thread a while ago, are very complex and cant be explained at the lay level.

Thanks
Bill
OK, so not every interaction that results in a pair of photons of the same energy results in their being entangled, right? I suspect that the complication has something to do with what frame of reference you're using. What else could screw it up?

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bhobba
Mentor
OK, so not every interaction that results in a pair of photons of the same energy results in their being entangled, right?
Yes

Thanks
Bill

vanhees71
Gold Member
2019 Award
Thank you. So am I to understand, then, that entanglement does not necessarily require two (or more) particles, that the properties of a single particle can be entangled (spin z-component with position)? I always thought it required two particles to be entangled, spooky action at a distance (between two particles). But maybe I misunderstood.
Yes. You can also have entanglement between two particles or photons, but then in a sense they build a system as a whole. One thing that you can completely exclude from contemporary relativistic quantum field theory are instantaneous interactions. The theory is built by construction such that all interactions are local and do not occur instantaneously over space-like distances in Minkowski space (principle of microcausality).

One thing that you can completely exclude from contemporary relativistic quantum field theory are instantaneous interactions. The theory is built by construction such that all interactions are local and do not occur instantaneously over space-like distances in Minkowski space (principle of microcausality).
So... no "spooky action at a distance"? This would appear to rule out MWI and "collapse" theories in general. Yes?

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So I'm understanding that two or more amplitudes can be in superposition. And those amplitudes (for lack of a better term) don't usually have anything to do with each other. They may simply interfere at some point. But entanglement means that those amplitudes are somehow correlated whether they interfere with each other or not. Is this right? So there necessarily have to be two or more states in superposition before you can even talk about entanglement. Right?

vanhees71
Gold Member
2019 Award
So... no "spooky action at a distance"? This would appear to rule out MWI and "collapse" theories in general. Yes?
In my opinion MWI cannot be ruled out, and "collapse" is an unnecessary assumption in certain flavors of the Copenhagen interpretation. For precisely the reason that the collapse hypothesis contradicts the causality structure of relativistic spacetime, one should avoid it. That's why I'm a follower of the ensemble interpretation, where such quibbles don't occur.

bhobba
Perhaps someone can show me some simple math showing two states in superposition and entanglement so I can see how entanglement relates to superposition. Thanks.
you could argue that entangled states are a particular subgroup of superposition states of many particle systems, namely superpositions that you can not express as a product (generally speaking: tensor product) of single particle wave functions.
It is not easy to "quantify" entanglement when it involves more than 2 particles. For 2 particles you can define the "concurence" which is a measure of the degree of entanglement.
What it means physically, is that measrurement results are correlated. This kind of correlation is different from a "classical correlation" because entangled states violate the locality principle.