Aren't the states of two identical particles always entangled even if they are not interacting? The states of two identical particles are either symmetric or antisymmetric i.e., cannot be written as product states.
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$\begingroup$ Please read this other post and my answer to it. I think it will help you understand the question you've posted here. $\endgroup$– DanielSankCommented May 10, 2018 at 20:28
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2 Answers
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Entanglement is only a meaningful concept when there is a well-defined notion of subsystems, which generally means spatially separated subsystems. Indeed, the notion of "product state" (or its converse, "entangled state") is only meaningful relative to a given tensor product decomposition of the Hilbert space, which implicitly defines a splitting into subsystems. Hence, it is not obvious that indistinguishable particles with an (anti-)symmetrised wavefunction can be truly regarded as entangled subsystems.
Nevertheless, Killoran, Cramer and Plenio (arXiv link) showed that this is indeed a genuine form of entanglement. That is, the entanglement formally associated with the symmetrised wave function of a system of indistinguishable bosons can be extracted into the equivalent amount of entanglement (asymptotically) between distinguishable modes. I believe a similar result was found also for fermions by Cavalcanti et al. (arXiv link).
answered May 10, 2018 at 20:55
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$\begingroup$ You say that it is not obvious whether antisymmetrized wavefunctions can be truly regarded as entangled. Maybe. But may I ask if you know any example of a valid antisymmetrized two-electron wavefunction that is writable as a product of a state of the first electron and a state of the second electron? I asked a similar question but it got closed. $\endgroup$ Commented Jun 24, 2021 at 16:10
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Yes and no. Yes if you look at the formal definition, then yes, but this is not really what is meant. Only the full state in symmetrized, so when you have one particle going to Alice and one particle going to Bob and they perform e.g. a spin measurement, then already by the 'location' of the particle detection the symmetrization is 'voided', and now the question if the spin degrees are entagnled makes sense.
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1$\begingroup$ An entangled state in one which cannot be written as a direct product state. So entanglement does not require physical interaction. @lalala $\endgroup$– SRSCommented May 10, 2018 at 20:10
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$\begingroup$ @SRS This is one definition of entanglement. Another definition is "a state is entangled, if you can do [favorite task which requires entanglement, e.g. teleportation to an arbitrary (two-level) quantum system] with it". $\endgroup$ Commented May 13, 2018 at 15:18
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$\begingroup$ @SRS I will try to edit to be clearer. Might take a few days though. $\endgroup$– lalalaCommented May 13, 2018 at 16:55