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One can see that the wavefunction for a system of two electrons (not very far apart) is one that cannot be written as a tensor product of individual states.

The same is true for a bosonic state.

For instance, production of an electron-positron pair has a zero total spin, the state of which looks identical to an entangled system of two electrons.

Is it possible to have entanglement between indistinguishable particles?

Does one have to first make them distinguishable like taking them far apart and then try to entangle them?

The question arises from the fact that the fermionic state looks like an entangled state. And I don't understand whether this is something that is just a coincidence or is there something else to derive from this.

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    $\begingroup$ Does this answer your question? Are identical particles always entangled even when not interacting? $\endgroup$
    – Rococo
    Commented Jan 7, 2022 at 15:11
  • $\begingroup$ If a state cannot be written as a tensor product of individual states on subsystems $A$ and $B$, then there is entanglement between $A$ and $B$. So if you can't write your fermionic/bosonic state as a tensor product of individual fermions/bosons, then the particles are entangled - simple as that. You can see this from the Schmidt decomposition ( en.wikipedia.org/wiki/Schmidt_decomposition ) and its relation to Von Neumann entropy. $\endgroup$
    – Jordan Taylor
    Commented Jan 7, 2022 at 23:40

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Is it possible to have entanglement between indistinguishable particles?

Yes.

Does one have to first make them distinguishable like taking them far apart and then try to entangle them?

No.

An instructive (counter) example is parametric down-conversion, which is commonly used to create entangled pairs of photons:
A single photon with spin up or down is send through a crystal (glass), where it is converted into a pair of two photons with the same spin but slightly different directions. Those can be separated and then used as an entangled pair. Since the spin of the initial photon is unknown, the wavefunction after down-conversion is: $$|\psi\rangle = \frac{1}{\sqrt{2}}( |\uparrow\rangle \otimes |\uparrow\rangle + |\downarrow\rangle \otimes |\downarrow\rangle). $$ The photons are still "distinguishable" by their position, although nothing would be changed if you swapped them and the wave-function is symmetric. And they are entangled even before you can distinguish them, so they do not have to be separated to be entangled. In fact, you can only create entangled pairs of particles from a common origin (they have to be close in the beginning).

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  • $\begingroup$ "And they are entangled even before you can distinguish them, so they do not have to be separated to be entangled" Right but they do get separated or distinguishable by their position right? (even if later on) Is it possible that you have entanglement between particles that remain indistinguishable (would it make sense ?). If let's say I consider two electrons constrained in a box. A wavefunction for the system would fermionic. Now can the electrons be entangled also ? $\endgroup$
    – Lost
    Commented Jan 8, 2022 at 14:49
  • $\begingroup$ Also, physics.stackexchange.com/questions/405155/… ....the top voted answer here says "Entanglement is only a meaningful concept when there is a well-defined notion of subsystems, which generally means spatially separated subsystems" so it seems entanglement only becomes meaningful if you make them distinguishable somehow (spatially seperate etc). $\endgroup$
    – Lost
    Commented Jan 8, 2022 at 15:00
  • $\begingroup$ I have to admit that the answer you refer to and the cited papers are a bit over my head. But here are my 2 cents: In principle, there are no two particles. The universe can be described by a single pure state evolving in time. But this state is a non-local object. If we consider a single subsystem, we can trace out the rest of the universe and descibe the subsystem. When we consider two subsystems, we again trace out the rest of the universe, but they might not be independent of each other. Entanglement is how we describe this non-local dependence of the subsystems on each other. $\endgroup$
    – Cream
    Commented Jan 9, 2022 at 12:23
  • $\begingroup$ In that sense, it is not an objective truth but more a concept useful to us. For what I mean with "traced out", see the sections Ensembles and Reduced density matriced on WP. $\endgroup$
    – Cream
    Commented Jan 9, 2022 at 12:31

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