According to the standard definition of "Entropy of Entanglement"


one starts from the density matrix of a pure state $$ \rho=|\psi\rangle\langle\psi|, $$ then divides the system into two parts, $A$ and $B$, traces away the degrees of freedom of one of the two subsystems, say subsystem B, and thus obtains the reduced density matrix of the remaining subsystem $$ \rho_A=\mathrm{Tr}_B(\rho). $$ Eventually, the entanglement between subsystem $A$ and subsystem $B$ is given by the Von Neumann entropy of $\rho_a$: $$ S(\rho_a)=-\mathrm{Tr}[\rho_A\,\log\rho_A]. $$

My question is: does the choice of the bipartition play an essential role? I think that the final result strongly depends on how one chooses subsystem $A$ (and, of course, the complementary subsystem $B$). To my knowledge, in fact, books do not emphasize how important is the choice of the partition. Is there a region?

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    $\begingroup$ Of course the choice of the partition (as you say) is crucial. But this is generally quite acknowledged. For example, there is a quite famous result for a GS of a semi-infinite chain. Partition the system such that A is the first L sites and B the rest. Then the entanglement between A and B goes like $\log(L)$ if the GS is critical, while it saturates to a constant if there is a gap above the GS. $\endgroup$ – lcv Feb 14 '20 at 7:33
  • $\begingroup$ Thank you very much! So it was me that ignored that this is well known. Just another curiosity: is there any restriction connected to the choice of A and B, i.e. constraints concerning locality or some other physical properties? $\endgroup$ – AndreaPaco Feb 14 '20 at 7:37
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    $\begingroup$ No restrictions, just your imagination. BTW the entanglement we are talking about is formulated in a Galieleian framework. But indeed entanglement shows quantum non-locality. $\endgroup$ – lcv Feb 14 '20 at 7:43
  • $\begingroup$ So, in the case of a chain, subsystem A could be site 1 + site 27 and subsystem B could be all the remaining sites? $\endgroup$ – AndreaPaco Feb 14 '20 at 7:57
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    $\begingroup$ Yes absolutely. To be precise you can also ask what is the entanglement between A and B whatever they are (they don't need to fill the whole space). But then A and B are described by a mixed density matrix and we don't know how to compute the entanglement of that (except for a few cases). $\endgroup$ – lcv Feb 14 '20 at 8:03

Indeed, the definition is pointless without a choice of partition of the underlying space. It is true that sometimes this is not made explicit. For example, people often talk about "entangled particles", but they should be really talking about specific properties of the particles being entangled, not the particles themselves. Similarly, you can have entanglement between different degrees of freedom of the same particle (though you might not have "nonlocality" in such cases).


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