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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.

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This blog post may be of use for you: http://backreaction.blogspot.co.nz/2016/03/dear-dr-b-what-is-difference-between.html -- written by physicist Sabine Hossenfelder.

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bhobba

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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.

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Bill

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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.

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.

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bhobba

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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.

Spooky action at a distance is an interpretive thing - its not part of the formalism of entanglement.Entanglement is an extension of superposition todifferent systems.

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Bill

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bhobba

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

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Bill

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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?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

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bhobba

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

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Bill

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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).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.

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So... no "spooky action at a distance"? This would appear to rule out MWI and "collapse" theories in general. Yes?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).

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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.So... no "spooky action at a distance"? This would appear to rule out MWI and "collapse" theories in general. Yes?

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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.

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