Hermitian tensor and quantum mixed state

Abstract

An order 2m complex tensor H is said to be Hermitian ifIt 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.

Hankel Tensor Decompositions and Ranks

“…Hermitian tensors are natural generalizations of Hermitian matrices, while they have very different properties. This concept was introduced by Ni [30]. For an array u, we use u to denote the complex conjugate of u.…”

“…The same result holds for Hermitian tensors over the complex field. For every H ∈ C [n1,...,nm] , Ni [30] showed that there exist vectors u j i ∈ C nj and real scalars λ i ∈ R, i = 1, . .…”

“…For two tensors A, B ∈ C [n1,...,nm] , their inner product is defined as The Hilbert-Schmidt norm of A is accordingly defined as ||A|| := A, A . If A, B are Hermitian, then A, B is real [30]. For convenience of operations, we define multilinear matrix multiplications for tensors (see [28]).…”

“…If each Q k is real and orthogonal, the tensor B is said to be orthogonally congruent to A. Unitary and orthogonal congruent transformations preserve norms of Hermitian tensors [30].…”

“…Hermitian tensors have important applications in quantum physics [30]. An mpartite pure state |ψ of a quantum system can be represented by a rank-1 tensor in C n1ו••×nm .…”