  An order $2m$ complex tensor $\cH$ is said to be Hermitian if
\[\mathcal{H}_\ijm=\mathcal{H}_\jim ^*\mathrm{\ for\ all\ }\ijm .\] It can be
regarded as an extension of Hermitian matrix to higher order. A Hermitian
tensor is also seen as a representation of a quantum mixed state. Motivated by
the separability discrimination of quantum states, we investigate properties of
Hermitian tensors including: unitary similarity relation, partial traces,
nonnegative Hermitian tensors, Hermitian eigenvalues, rank-one Hermitian
decomposition and positive Hermitian decomposition, and their applications to
quantum states.
