  We demonstrate that, in the case of Shor's algorithm for factoring, highly
mixed states will allow efficient quantum computation, indeed factorization can
be achieved efficiently with just one initial pure qubit and a supply of
initally maximally mixed qubits (S. Parker and M. B. Plenio, Phys. Rev. Lett.
85, 3049 (2000)) . This leads us to ask how this affects the entanglement in
the algorithm. We thus investigate the behaviour of entanglement in Shor's
algorithm for small numbers of qubits by classical computer simulation of the
quantum computer at different stages of the algorithm. We find that
entanglement is an intrinsic part of the algorithm and that the entanglement
through the algorithm appears to be closely related to the amount of mixing.
Furthermore, if the computer is in a highly mixed state any attempt to remove
entanglement by further mixing of the algorithm results in a significant
decrease in its efficiency.
