  We define predictive states and predictive complexity for quantum systems
composed of distinct subsystems. This complexity is a generalization of
entanglement entropy. It is inspired by the statistical or forecasting
complexity of predictive state analysis of stochastic and complex systems
theory, but is intrinsically quantum. Predictive states of a subsystem are
formed by equivalence classes of state vectors in the exterior Hilbert space
that effectively predict the same future behavior of that subsystem for some
time. As an illustrative example, we present calculations in the dynamics of an
isotropic Heisenberg model spin chain and show that, in comparison to the
entanglement entropy, the predictive complexity better signifies dynamically
important events, such as magnon collisions. We discuss how this quantity may
usefully characterize a variety of symmetries in quantum systems in an
information-theoretic way, and comment on possible applications and extensions.
