Hermitian Tensor Decompositions
arXiv:1912.07175
Abstract
Hermitian tensors are generalizations of Hermitian matrices, but they have very different properties. Every complex Hermitian tensor is a sum of complex Hermitian rank-1 tensors. However, this is not true for the real case. We study basic properties for Hermitian tensors such as Hermitian decompositions and Hermitian ranks. For canonical basis tensors, we determine their Hermitian ranks and decompositions. For real Hermitian tensors, we give a full characterization for them to have Hermitian decompositions over the real field. In addition to traditional flattening, Hermitian tensors specially have Hermitian and Kronecker flattenings, which may give different lower bounds for Hermitian ranks. We also study other topics such as eigenvalues, positive semidefiniteness, sum of squares representations, and separability.
arXiv comment28 pages
References in corpus (13)
- Eigenvectors of tensors and algorithms for Waring decomposition
- Generic uniqueness conditions for the canonical polyadic decomposition and INDSCAL
- Direct Reconstruction of the Quantum Density Matrix by Strong Measurements
- Effective criteria for specific identifiability of tensors and forms
- General Tensor Decomposition, Moment Matrices and Applications
- The A-truncated K-moment problem
- Measuring the single-particle density matrix for fermions and hard-core bosons in an optical lattice
- How entangled can a multi-party system possibly be?
- Tensor network ranks
- Theoretical and computational aspects of entanglement
- Hermitian tensor and quantum mixed state
- On decompositions and approximations of conjugate partial-symmetric complex tensors
- Quillen property of real algebraic varieties