In this note, we survey some elementary theorems and proofs concerning dynamical matrices theory. Some mathematical concepts and results involved in quantum information theory are reviewed. A little new result on the matrix representation of quantum operation are obtained. And best separable approximation for quantum operations is presented.

arXiv:1009.2210Sep 12, 20101 citation

Entanglement with centers

Chen-Te Ma

arXiv:1511.02671Journal of High Energy PhysicsJan 1, 201626 citations

AIP conference proceedings

Fluctuation studies on the finite interval matrix representations of operator products and their decompositions

Ercan Gürvіt, N. Abdülbaki Baykara, Metin Demіralp

Published Jan 1, 2012• 0 citations

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Abstract

DOI: 10.1063/1.4771818

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Fluctuation studies in the infinite interval matrix representations of operator products and their decompositions

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The approximate representation of operators by finite matrices is analysed in terms of accuracy and convergence. The identity operator, for example, can be reconstructed using a basis of harmonic oscillator states leading to a narrow peak approximation of the $δ$ function, but this peak may be perturbed by small, residual, oscillations. The peak does not shrink nor grows quickly, and the oscillations only diminish slowly as the size of the matrix increases. For the kinetic energy operator, a triple peak (one positive, two negative) representation of $-δ''$ is obtained, but that is affected also by residual oscillations. Again, convergence is slow as the matrix dimension increases. We find compact formulas to explain such oscillations. Similar observations are found for representations of local interactions, while separable potentials are better represented. As a comparison, in the context of a toy model, the effects of choosing an alternative single particle basis are studied. A formal approach for the approximation of operators is considered for comparison. We conclude with a word of caution for (finite) matrix approximations of operators.The approximate representation of operators by finite matrices is analysed in terms of accuracy and convergence. The identity operator, for example, can be reconstructed using a basis of harmonic oscillator states leading to a narrow peak approximation of the $δ$ function, but this peak may be perturbed by small, residual, oscillations. The peak does not shrink nor grows quickly, and the oscillations only diminish slowly as the size of the matrix increases. For the kinetic energy operator, a triple peak (one positive, two negative) representation of $-δ''$ is obtained, but that is affected also by residual oscillations. Again, convergence is slow as the matrix dimension increases. We find compact formulas to explain such oscillations. Similar observations are found for representations of local interactions, while separable potentials are better represented. As a comparison, in the context of a toy model, the effects of choosing an alternative single particle basis are studied. A formal approach for the approximation of operators is considered for comparison. We conclude with a word of caution for (finite) matrix approximations of operators.

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δ

function, but this peak may be perturbed by small, residual, oscillations. The peak does not shrink nor grows quickly, and the oscillations only diminish slowly as the size of the matrix increases. For the kinetic energy operator, a triple peak (one positive, two negative) representation of

-δ''

is obtained, but that is affected also by residual oscillations. Again, convergence is slow as the matrix dimension increases. We find compact formulas to explain such oscillations. Similar observations are found for representations of local interactions, while separable potentials are better represented. As a comparison, in the context of a toy model, the effects of choosing an alternative single particle basis are studied. A formal approach for the approximation of operators is considered for comparison. We conclude with a word of caution for (finite) matrix approximations of operators.