3 qubit maximally entangled state $\frac{1}{2}(|000\rangle + |110\rangle+ |011\rangle + |101\rangle)$

Question

Is there any general criteria for maximal entanglement in 3 qubit system. I have encountered this problem-"Suppose you have a state $\frac{1}{2}(|000\rangle + |110\rangle+ |011\rangle + |101\rangle)$. If the three qubits are labeled a, b, c, obtain the reduced density matrices $\rho_{ab}$ and $\rho_a$. Argue why this is a maximally entangled state."

I have calculated density matrices $\rho_{ab}$, $\rho_a$, $\rho_b$, $\rho_c$. $$\rho_{ab}=\frac{1}{2}(|00\rangle+|11\rangle)(\langle 00|+\langle 11|)+\frac{1}{2}(|01\rangle+|10\rangle)(\langle 01|+\langle 10|)$$$\rho_a=\rho_b=\rho_c=\frac{I}{2}$. But I don't know how to argue from this matrices that it is maximally entangled. I thought that I should calculate Von-Neumann entropy. If it is 1 then it is maximally entangled. This is true for bipartite case. I don't know if it is valid for 3 qubit case. What is the criteria of maximal entanglement for 3 qubit system? Can someone help?

Answer 1

The question might just be asking you to show that the state is 'locally maximally entangled', which simply means what you already figured out, that the reduced density matrices on the three subsystems are all maximally mixed, i.e. all $\rho_S = I/2$.

You are correct that for bipartite pure states, the von Neumann entropy uniquely characterizes entanglement. But for mixed states or more subsystems this is no longer true. There are criteria for bipartite entanglement of mixed states, e.g. PPT criterion. For multipartite systems, characterizing entanglement is an open and very difficult problem (for instance, see this paper).

But for three qubits, the situation is a little simpler. There is a sense in which there are two classes of maximally entangled pure states of three qubits: GHZ-type and W-type (I believe these can be thought of as two extremes in the space of states), as well as four classes of separable states. See this paper as well as Sec 8 and 11 of this review for nice explanations. Your state above will either be locally equivalent to one of the two classes of entangled states (W- or GHZ-type) or will be separable.