Based on my limited and poor understanding, a quantum circuit in the "one clean qubit" model of quantum computation generally acts on a single pure (clean) qubit tensored with $n-1$ qubits in a maximally mixed state. The clean qubit can serve as a control to apply an arbitrary unitary $U$ to the maximally mixed qubits, and measuring this clean qubit in the $X$ or $Y$ basis enables characterizations of some properties of $U$ (such as its real and imaginary trace, which is likely classically difficult.)
I was inspired to consider the one clean qubit model of computation by a recent (2021) video of Yoganathan on her "Looking Glass Universe" channel, wherein she describes and summarizes the approach she and her colleagues took to answering an open problem on the power of such a model in the absence of entanglement. The headline is that the one clean qubit model without entanglement is classically tractable.
But as I begin to study their paper, it's not quite clear to me what's even meant by the one clean qubit model with and without entanglement.
What, in particular, is and is not entangled in the one clean qubit model?
The pure qubit would be entangled with mixed qubits after application of $U$, right? Or am I wrong, and even after the controlled application of $U$ the clean qubit need not be entangled with/is separable from the mixed qubits? If so, what is entangled in the simple circuit in the Wikipedia article?
EDIT: I am wrong! While nonetheless carrying in its amplitude some properties of $U$ (such as the real/imaginary trace), a (bi)partition between the clean qubit and the other qubits after application of $U$ has no entanglement, no matter what $U$ is. When one clean qubit is able to calculate the trace of $U$ efficiently, the the one clean qubit may have entanglement in other partitions wherein [the clean qubit $\otimes$ some other mixed qubits] is entangled with the remaining mixed qubits, but there is no entanglement in the "obvious" partition! This is what @DaftWullie has emphasized.