  The role of entanglement and quantum correlations in complex physical systems
and quantum information processing devices has become a topic of intense study
in the past two decades. In this work we present new tools for learning about
entanglement and quantum correlations in dynamical systems where the quantum
states are mixed and the eigenvalue spectrum is highly degenerate. We apply
these results to the Deterministic quantum computing with one qubit (DQC1)
computation model and show that the states generated in a DQC1 circuit have an
eigenvalue structure that makes them difficult to entangle, even when they are
relatively far from the completely mixed state. Our results strengthen the
conjecture that it may be possible to find quantum algorithms that do not
generate entanglement and yet still have an exponential advantage over their
classical counterparts.
