Entanglement Entropy of two Spin systems

I'm given a quantum system with a density matrix $\rho$ which is broken down into two part systems A and B, which can be entangled. The system has two Spin-$\frac{1}{2}$-degrees of freedom (either up or down) and is in the state:

$|{\psi(\alpha)}\rangle = \cos(\alpha) |A\uparrow\rangle \otimes |B \downarrow\rangle + \sin(\alpha) |A \downarrow\rangle \otimes |B \uparrow \rangle$ ,

and want to calculate the density matrices $\rho(\alpha)$ and $\rho_\text{A}(\alpha)$ using: $\rho_\text{A}(\alpha) = Tr_\text{B}\rho(\alpha)$ and also the entanglement entropy of these two density matrices. I calculated so far the two density matrices using the standard formula ($\rho(\alpha) = |{\psi(\alpha)}\rangle \langle \psi(\alpha)|$) and got two solutions for $\rho(\alpha)$ and $\rho_\text{A}(\alpha)$, which pretty much make sense.

These are my calculated solutions for the density matrices ( I used the arrow notation to simplify, meaning that the first arrow represents the spin of system A and the second one the spin of system B):

$\rho(\alpha) = \cos²(\alpha)|\uparrow \downarrow\rangle \langle \uparrow \downarrow| + \cos(\alpha)\sin(\alpha)|\uparrow\downarrow\rangle\langle\downarrow\uparrow|+\sin(\alpha)\cos(\alpha)|\downarrow\uparrow\rangle\langle\uparrow\downarrow|+\sin²(\alpha)|\downarrow\uparrow\rangle\langle\downarrow\uparrow| \quad$,

$\rho_\text{A}(\alpha)=\cos²(\alpha)|\uparrow\rangle\langle\uparrow|\langle\downarrow|\downarrow\rangle+\cos(\alpha)\sin(\alpha)\left[|\uparrow\rangle\langle\downarrow|\langle\uparrow|\downarrow\rangle+|\downarrow\rangle\langle\uparrow|\langle\downarrow|\uparrow\rangle\right]+\sin²(\alpha)|\downarrow\rangle\langle\downarrow|\langle\uparrow|\uparrow\rangle$

I'm now trying to calculate the entanglement entropy using von-Neumann-Entropy, so my approach was to write $\rho$ in a matrix representation so I can find the eigenvalues $p_i$ and diagonalise it to find the entropy via $S = -k \sum p_i \ln{p_i}$. But my problem is that I'm not certain about what basis to choose. I thought about writing the states as a 4x4-matrix, but I'm not sure if it's allowed to do so because the states depend on the two systems A and B, which are actually independent on each other. I tried to transfer $\rho$ into the $|S,M\rangle$-basis, so that there are only two possible states, $|0,0\rangle$ and $|1,0\rangle$, but my solution was completely different. I'm not quite sure now, so my question is 1) if my solutions for $\rho$ and $\rho_A$ are right in the first place or if there's anything missing, and 2) how I'm supposed to choose my basis to calculate the von_Neumann_entropy? Is it possible to use a shared basis of eigenstates even though they depend on the states of two independent part systems A and B? Thanks in advance.

• But ⟨↓|↑⟩ = ⟨↑|↓⟩ = 0. You need to use that. – Peter Shor Nov 7 '17 at 20:37