Entanglement between what?

Question

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?

Answer 1

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).