Determining if two qubits in an ensemble are entangled

Question

I know that given two qubits, $A$ and $B$, in a state, $|\psi\rangle_{AB}$, you can find out if $A$ is entangled with $B$ by partial tracing and seeing if $tr_{A}\left(|\psi\rangle\langle\psi|\right)$ is a mixed state.

However, what I want to know is if you're given a state of many qubits, including $A$ and $B$; e.g., $|\phi\rangle_{ABCD\cdots}$, how can you determine if $A$ is entangled specifically with $B$.

It cannot be a simple matter of tracing out $A$ to see if it leaves behind a mixed state as that does not tell you with whom $A$ is entangled. For example:

$$ |\phi\rangle_{ACBD} = |\Phi^{+}\rangle_{AC}\ \otimes\ |\Phi^{+}\rangle_{BD}. $$

If we trace over $A$ then we get a mixed state. If we trace over $B$ then we get a mixed state. However, this does not tell us with whom $A$ and $B$ are entangled. Even if we trace over ever qubit, we just get four mixed states, it still provides no information as to which qubits are entangled with which. The only way I can see to determine this is to trace over every combination of qubits. Then when you trace over $BD$ you'll realise that you get a pure state.

Is there any more straightforward way to find out the entanglement relation between any arbitrary qubits in an ensemble in a general state?

Answer 1

If I understand your question correctly you have a multipartite pure state $|\psi\rangle_{ABCD..}$, but you are only concerned with the properties of the bipartite subsystem $AB$. In that case it is sufficient to consider the reduced density matrix $\rho_{AB}=Tr_{CD..}(\psi_{ABCD..})$. You then want to know if there is a simple way to test if this bipartite state $\rho_{AB}$ is entangled. As ZeroTheHero has suggested, in the case where systems $A$ and $B$ are qubits (or if one system is a qutrit and the other a qubit), then the Peres-Horodecki criterion is both necessary and sufficient for testing if $\rho_{AB}$ is entangled. For higher dimensions the problem becomes harder.

As Norbert Schuch has suggested there are multiple different kinds of entanglement among multipartied systems when you consider more than two parties. However it is not clear to me that you are concerned with any other situation than the two qubit case.

So there is a more straightforward way to find out the entanglement relation between any two arbitrary qubits in an ensemble in a general state. But not between any n arbitrary qubits, because there is no definition of the entanglement between n qubits.

Hope that answers your question! :)