# Entanglement between what?

According to the standard definition of "Entropy of Entanglement"

https://en.wikipedia.org/wiki/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?

• 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.
– lcv
Feb 14, 2020 at 7:33
• 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? Feb 14, 2020 at 7:37
• 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.
– lcv
Feb 14, 2020 at 7:43
• So, in the case of a chain, subsystem A could be site 1 + site 27 and subsystem B could be all the remaining sites? Feb 14, 2020 at 7:57
• 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).
– lcv
Feb 14, 2020 at 8:03