Operators and Matrix Elements

George C. Turrell

Yaakov S. Weinstein, C. Stephen Hellberg

We explore connections between an operator's matrix-element distribution and its entanglement generation. Operators with matrix-element distributions similar to those of random matrices generate states of high multipartite entanglement. This occurs even when other statistical properties of the operators do not coincide with random matrices. Similarly, operators with some statistical properties of random matrices may not exhibit random matrix element distributions and will not produce states with high levels of multipartite entanglement. Finally, we show that operators with similar matrix-element distributions generate similar amounts of entanglement.

J S Briggs

Reduced matrix elements of a tensorial product of n tensor operators in the basis of N-particle angular-momentum eigenstates are expanded into a sum of products of n 1-particle reduced matrix elements and a single (N + n)-particle recoupling coefficient. Application of the formula is illustrated by specific examples. The method leaves the coupling schemes of N-particle states undisturbed, which allows summation over intermediate states in a product of matrix elements to be made for any number of factors. A formula is given for the sum of products of two matrix elements and the extension to a greater number of matrix elements is illustrated by an example which is reduced to a form suitable for numerical evaluation. Such summations over products of matrix elements occur in the perturbation theory of configuration interaction and have hitherto been discussed in terms of effective operators. The connection of the method used here with the effective-operator approach is demonstrated.

Dževad Belkić

The exact calculation of the matrix elements is extraordinarily difficult beyond the second order in a perturbation expansion. Therefore, it is of crucial importance to examine the operator and vector series, with the purpose of gaining some insight into convergence of the pertinent physical matrix elements. A great advantage is that convergence of an operator expansion automatically guarantees the convergence of the corresponding series of vectors and matrix elements. The chapter examines in some detail a connection which exists between operator and vector expansions, thus establishing the quoted relation of the corresponding convergence radii. The importance of the convergence problem of the Neumann-Born expansions based upon the outlined concept of a comparative study of series of operators, vectors and matrix elements is in its application to the cases when the kernel K is not a compact operator.