  We consider an arbitrary qubit channel depending on a single parameter, which
is to be estimated by a physical process. Using the quantum Fisher information
per channel invocation to quantify the estimation accuracy, we consider various
estimation protocols when the available initial states are mixed with very low
purity. We compare a protocol using a single channel invocation on one out of
$n$ qubits prepared in a particular correlated input state to the optimal
protocol using uncorrelated input states, with the same initial state purity.
We show that, to lowest order in initial-state purity, for a unital channel
this correlated state protocol enhances the estimation accuracy by a factor
between $n-1$ and $n.$ We also show that to lowest order in initial-state
purity, a broad class of non-unital channels yields no gain regardless of the
input state.
