  We study the problem of state synthesis in the DQC1 (One Clean Qubit) model
of quantum computation, which provides a single pure qubit and $n$ maximally
mixed qubits, and after applying any quantum circuit some subset of the qubits
are measured or discarded. In the case of discarding, we show that it is
impossible to prepare additional pure qubits, and that it is impossible to
prepare very low-temperature Gibbs states on additional qubits. In the case of
measurements, we show that the probability of synthesizing $m$ additional
qubits is bounded by $2^{1-m}$, and that the probability of preparing
low-temperature Gibbs states is bounded by $2^{2-m}$. As a consequence, we give
a lower-bound the runtime of a recently studied class of repeated interaction
quantum algorithms. The techniques used study states and circuits at the level
of entries of their respective density and unitary matrices.
