I am wondering why superluminal communication would be possible if a quantum cloner would exist?
The common argument (FLASH) goes as follows: Alice and Bob share the Bell state $$ |\psi^-\rangle = \frac{1}{\sqrt 2}(|01\rangle - |10\rangle) $$ Now if Bob could make $2N$ copies of his qubit, he could find out in which basis Alice has measured before a signal with this information could reach him.
Let's consider Bob makes $2N$ copies of his qubit: $$ \frac{1}{\sqrt 2}(|01\rangle - |10\rangle) \quad \longrightarrow \quad \frac{1}{\sqrt 2}(|0\underbrace{1...1}_{2N}\rangle - |1\underbrace{0...0}_{2N}\rangle) $$ Now if Alice measures in the z-basis $\{|0\rangle,|1\rangle\}$ Bob will be left with either $|1...1\rangle$ or $|0...0\rangle$. So every qubit is in a pure state: either $|1\rangle$ or $|0\rangle$. So he now measures $N$ qubits in the z-basis and $N$ in the x-basis $\{|+\rangle,|-\rangle\}$. Since he will get $N$ times the same result for the measurements in the z-basis he concludes that Alice has measured in the z-basis.
However, if Alice measures in the x-basis Bob will be left with either $$ \frac{1}{\sqrt 2}(|1...1\rangle - |0...0\rangle) $$ or $$ \frac{1}{\sqrt 2}(|1...1\rangle + |0...0\rangle) $$ Thus all of his qubits are in a completely mixed state $\frac 1 2 (|0\rangle\langle 0| + |1\rangle\langle 1|) = \frac 1 2 (|+\rangle\langle +| + |-\rangle\langle -|)$. So if Bob now measures $N$ qubits in the z-basis and $N$ in the x-basis he will never get $N$ times the same result for one particular basis. So he can conclude nothing.
So what am I doing wrong?