Interpolation of Density Matrices and Matrix-Valued Measures: The   Unbalanced Case
  In this note, we propose an unbalanced version of the quantum mechanical version of optimal mass transport that was based on the Lindblad equation. We formulate a natural interpolation framework between density matrices and matrix-valued measures via a quantum mechanical formulation of Fisher-Rao information and the matricial Wasserstein distance. 
Yongxin Chen, Tryphon T. Georgiou, Allen Tannenbaum
math.FA