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arXiv:quant-ph/0305140 · Fortschritte der Physik

The quantum way to diagonalize hermitean matrices

Stefan Weigert

Published Feb 26, 2003• 2 citations
arXivHTMLPDF

Abstract

Abstract An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. The method is based on the measurement of quantum mechanical observables which provides the computational resource. In brief, quantum mechanics is able to directly address and output eigenvalues of hermitean matrices. Here, the example of (2×2) matrices is considered in detail. This simple case allows one to illustrate the conceptual features of the general method which is applicable to (N×N) hermitean matrices.

DOI
10.1002/prop.200310035

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Abstract An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. The method is based on the measurement of quantum mechanical observables which provides the computational resource. In brief, quantum mechanics is able to directly address and output eigenvalues of hermitean matrices. Here, the example of (2×2) matrices is considered in detail. This simple case allows one to illustrate the conceptual features of the general method which is applicable to (N×N) hermitean matrices.
To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource: Hermitean (N ×N) matrices can be diagonalized, in principle, by performing appropriate quantum mechanical measurements. To do so, one considers the given matrix as an observable of a single spin with appropriate length s which can be measured using a generalized Stern-Gerlach apparatus. Then, each run provides one eigenvalue of the observable. As the underlying working principle is the `collapse of the wavefunction' associated with a measurement, the procedure is neither a digital nor an analogue calculation - it defines thus a new example of a quantum mechanical method of computation.
To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource : Hermitean ( N × N ) matrices can be diagonalized, in principle, by performing appropriate quantum mechanical measurements. To do so, one considers the given matrix as an observable of a single spin with appropriate length s which can be measured using a generalized Stern-Gerlach apparatus. Then, each run provides one eigenvalue of the observable. As the underlying working principle is the `collapse of the wavefunction' associated with a measurement, the procedure is neither a digital nor an analogue calculation - it defines thus a new example of a quantum mechanical method of computation.
We present a prescription for forming matrices with specified eigenvalues and known eigenvectors. With this method, we can form Hermitian, anti-Hermitian, symmetric and general matrices with arbitrary eigenvalues. In addition we propose an algorithm for diagonalizing such matrices. The functions required for the realization of this are probability amplitudes connecting observables with discrete eigenvalue spectra and can be obtained from spin theory. For the example case of 5×55\times 55×5 matrices, these functions are given.
The unitary matrix that brings a Hermitian H into block-diagonal form can be uniquely determined under very simple and transparent conditions. Int his work the block-diagonalization problem is investigated in the framework of the second quantization formalism. Starting with an operator Ḩ which in any n-particle Fock space has a well-defined matrix representation an attemt was made to answer the question whether the transformation matrices T which can be separately given in the various n-particle spaces can be considered matrix representations of the same operator. T̂. Interestingly, the very important result was reached that the block-diagonalization operator T̂ exists and is unique. As a particular example, attention was concentrated on the case of an operator Ĥ given by a one-particle operator. In this case the block-diagonalization operator can be constructed and given in explicit from. This approach is applied to the theory of Green’s functions where the block-diagonalization of the Hamiltonian has interesting consequences that are illustrated in some details.
The original version of matrix mechanics is little used in applications nowadays. In that early theory position, momentum and other quantities were represented by infinite matrices, whereas many modern applications of quantum mechanics represent such observables as operators. Infinite matrices are unwieldy to use and have some disturbing properties, e.g. they do not obey the associative rule. By far the most commonly appearing matrix in quantum mechanics is the matrix of the energy operator as set up in some finite basis of trial functions. The procedure has been used for perturbed oscillator problems; the recurrence relation approach is particularly simple when the matrix has only one non-zero element beyond the diagonal element in each row. It seems to the author that there are still many hybrid methods which need investigation.