  A novel information entropy of turbulence systems with multiple field
quantities is formulated. Inspired by quantum mechanics, the von Neumann
entropy (vNE) and the entanglement entropy(EE) are derived from a density
matrix for the turbulence state in terms of the multi-field singular value
decomposition (MFSVD). Applying the information-theoretic entropy analyses to
spatio-temporal dynamics in turbulent plasmas with phase-transition-like
behavior, we discover a new nontrivial transition threshold regarding the vNE,
which significantly deviates from the transition threshold of the field energy
considered in the conventional approaches. These findings provide us with new
classifications of the turbulence state in terms of combined energy and
information. It is also shown that the EE for nonlinear interactions in
turbulence simultaneously describes not only the information for the strength
of nonlinear mode couplings but also the direction of net energy transfer.
