# Can one quantify entanglement between different parts of a system?

Consider some state $$|\psi\rangle$$ of $$n$$ qubits. One can take any subsystem $$A$$ and compute its density matrix $$\rho_A =Tr_{B} |\psi\rangle \langle\psi|$$. The entanglement between subsystem $$A$$ and the rest of the system $$B$$ is quantified for example by von Nemumann entropy associated with the density matrix. For example, the subsystem $$A$$ can be first $$k$$ qubits and subsytem $$B$$ the remaining $$n-k$$.

Now, is it possible to associate in a similar way some measure of entaglment between two particular subsystems that do not sum up to the total system? Say, is it in any way reasonable to ask what is entanglement between qubit 1 and qubit 2? Intuitively, if the state of the system is something like $$\sqrt{2}|\psi\rangle=|000000\dots\rangle+|110000\dots\rangle$$ then I should be able to say that the first two qubits are maxiamally entangled with each other but not at all with the remaining qubits. But can one do this in general?

• Take, for example $a|000\rangle+b|111\rangle$. You can freely choose how much entanglement there is between the $1|23$ partition by selecting $a$. But the reduced density matrices are $\rho_{12}=\rho_{13}=|a|^2|00\rangle\langle 00|+|b|^2|11\rangle\langle 11|$ and are therefore not at all entangled. Thus, the same entanglement properties of the pairs of qubits can lead to arbitrary entanglement of the composite system. Jun 25, 2021 at 8:57