An alternative quantum fidelity for mixed states of qudits
  We give an alternative definition of quantum fidelity for two density operators on qudits in terms of the Hilbert-Schmidt inner product between them and their purity. It can be regarded as the well-defined operator fidelity for the two operators and satisfies all Jozsa's four axioms up to a normalization factor. One desire property is that it is not computationally demanding. 
Xiaoguang Wang, Chang-Shui Yu, and X. X. Yi