Hermitian tensor and quantum mixed state

Abstract: 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.