The Quantum Way to Diagonalize Hermitean Matrices

An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. Here, the genuinely quantum mechanical approach is considered in detail for (2× 2) matrices. 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. The simple low-dimensional case allows one to illustrate the conceptual features of the general method which applies to (N ×N) hermitean matrices. (Fortschr. Phys. 51, 248 (2003)) PACS: 03.67.-a, 03.65Sq Introduction A new attitude towards quantum theory has emerged in recent years. The focus is no longer on attempts to come to terms with counter-intuitive quantum features but to capitalize on them. In this way, surprising methods have been uncovered to solve specific problems by means which have no classical equivalent: quantum cryptography, for example, allows one to establish secure keys for secret transmission of information [1]; entanglement [2] is used as a tool to set up powerful quantum algorithms which do factor large integers much more efficiently than any presently known classical algorithm [3]. Throughout, these techniques make use of the measurement of quantum mechanical observables as an unquestioned tool. This is also true for many (but not all [4]) proposals of quantum error correction schemes [3, 5], required to let a potential algorithm run. The purpose of the present contribution is to study the simplest situation in which a quantum mechanical measurement, i.e. the bare ‘projection’ [2], “does” the computation. The computational task is to determine the eigenvalues of hermitean (2 × 2) matrices by quantum means alone. Although the answer to this problem can be given analytically, it is useful to discuss this particular case since there is no conceptual difference between diagonalizing (2× 2) or (N ×N) hermitean matrices along these lines [6]. Before turning to the explicit example, consider briefly the traditional view on quantum mechanical measurements: a measurement is thought to confirm or reveal some information about the state of the system. The measured observable  is assumed to be known entirely,