An extremely intriguing feature of quantum entanglement is a property that has been called the "monogamy of entanglement": a particle can only be maximally entangled with precisely one other particle. \({ }^{23}\) In this problem, we will see this monogamy explicitly in systems of two and three spin-1/2 particles.

(a) Consider two spin-1/2 particles, call them 1 and 2. Determine the pure states that exhibit maximum entanglement; that is, the states that have entropy of entanglement equal to \(\log 2\), as we found in Eq. (12.74).

(b) Is it possible for a partial trace of the density matrix of two spin-1/2 particles to be a pure state? Show that the only way that a pure state density matrix of two spin-1/2 particles reduces to a pure state of a single particle by tracing out the other particle is if the state was initially separable.

(c) Now, consider a system of three spin-1/2 particles, call them 1,2, and 3. Consider the most general density matrix of the three particles, \(ho_{123}\). Show that it is impossible for the reduced-density matrices \(ho_{12}\) and \(ho_{23}\) to simultaneously describe a pure state with maximum entropy of entanglement. Assume that \(ho_{12}\) describes a maximally entangled pure state. What was the initial state of the three particles?