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CRC Press eBooks

- A bit of math: vectors, tensors, Fourier transform, etc.

Grigory E. Vekstein

Published Mar 7, 2013• 0 citations
Publisher

Abstract

A tensor of rank two Tik is a hermitian tensor, if Tki = T ∗ ik, and it is an antihermitian one, if Tki = −T ∗ik. Any tensor Tik can be represented as a superposition of the hermitian tensor T (H) ik and the antihermitian one T Tik = T (H) ik + T (A) ik , T 2 (Tik + T ∗ ki), T 2 (Tik − T ∗ki) Vector ~A is called an eigenvector of a matrix Tik, if TikAk = λAi, where λ is the respective eigenvalue of this matrix ( summation over “dumb” indices, in this case over the index k, is assumed). If matrix Tik is a hermitian one, then all its eigenvalues are real and, vice versa, all eigenvalues of an antihermitian matrix are imaginary.

DOI
10.1201/b16095-3

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A tensor of rank two Tik is a hermitian tensor, if Tki = T ∗ ik, and it is an antihermitian one, if Tki = −T ∗ik. Any tensor Tik can be represented as a superposition of the hermitian tensor T (H) ik and the antihermitian one T Tik = T (H) ik + T (A) ik , T 2 (Tik + T ∗ ki), T 2 (Tik − T ∗ki) Vector ~A is called an eigenvector of a matrix Tik, if TikAk = λAi, where λ is the respective eigenvalue of this matrix ( summation over “dumb” indices, in this case over the index k, is assumed). If matrix Tik is a hermitian one, then all its eigenvalues are real and, vice versa, all eigenvalues of an antihermitian matrix are imaginary.
An order 2m2m2m 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.
An order 2m2m2m 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.
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Apparently new expressions are given for the exponential of a hermitian matrix, A, in the 2 × 2, 3 × 3, and 4 × 4 cases. Replacing A by iA these are explicit formulas for the Fourier transform of eiA. The support of the Fourier transform of eiA, A hermitian, has been long known [1], [2], [3] and [4]. We exhibit explicit expressions for traceless hermitian matrices of sizes 2 × 2, 3 × 3, and 4 × 4. In d-dimensions, where A acts on Cd, we employ matrices, W, hermitian projections of rank one. Thus W may be written as Wij = vi ¯vj, where vi is a unit vector in Cd, and Tr(AW) =< v, Av>. ∫ dΩ denotes a normalized integral over al l such W defined by integrating the associated vi over a unit sphere in C d with the normalized unitary- invariant measure. The expressions we give are not unique, among similar forms. It is clear one can derive such expressions in any number of dimensions. We leave to others the task of creating an appropriate theory. 1 2 − d case e A ∫ dΩe Tr(AW)