Efficient factorization with a single pure qubit and $log N$ mixed   qubits
  It is commonly assumed that Shor's quantum algorithm for the efficient factorization of a large number $N$ requires a pure initial state. Here we demonstrate that a single pure qubit together with a collection of $log_2 N$ qubits in an arbitrary mixed state is sufficient to implement Shor's factorization algorithm efficiently. 
S. Parker and M.B. Plenio