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.

arXiv comment18 pages; submitted for publication; comments are welcome

References in corpus (4)

• Finding the Spectral Radius of a Nonnegative Tensor
• How entangled can a multi-party system possibly be?
• The geometric measure of entanglement of pure states with nonnegative amplitudes and the spectral theory of nonnegative tensors
• On decompositions and approximations of conjugate partial-symmetric complex tensors

Cited by in corpus (5)

• Tensor Expander Chernoff Bounds
• Separability of Hermitian Tensors and PSD Decompositions
• Hermitian Tensor Decompositions
• The decompositions and positive semidefiniteness of fourth-order conjugate partial-symmetric tensors with applications
• Hanson-Wright Inequality for Random Tensors under Einstein Product