  Originally conceived as a theory of consciousness, integrated information
theory (IIT) provides a theoretical framework intended to characterize the
compositional causal information that a system, in its current state, specifies
about itself. However, it remains to be determined whether IIT as a theory of
consciousness is compatible with quantum mechanics as a theory of microphysics.
Here, we present an extension of IIT's latest formalism to evaluate the
mechanism integrated information ($\varphi$) of a system subset to
finite-dimensional quantum systems (e.g., quantum logic gates). To that end, we
translate a recently developed, unique measure of intrinsic information into a
density matrix formulation, and extend the notion of conditional independence
to accommodate quantum entanglement. The compositional nature of the IIT
analysis might shed some light on the internal structure of composite quantum
states and operators that cannot be obtained using standard
information-theoretical analysis. Finally, our results should inform
theoretical arguments about the link between consciousness, causation, and
physics from the classical to the quantum.
