Quantum Diagonalization of 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. It allows one, in principle, to diagonalize hermitean (N × N) matrices by quantum mechanical measurements only. 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 it is based on the ‘collapse of the wave function’ associated with a measurement, the procedure is neither a digital nor an analog calculation—it defines thus a new quantum mechanical method of computation. Non-classical features of quantum mechanics such as Heisenberg’s uncertainty relation and entanglement have intrigued physicists for several decades. From a classical point of view, quantum mechanics imposes constraints on the ways to talk about nature. An electron does not “have” position and momentum as does a billiard ball. Similarly, if a photon is entangled with a second one—possibly very far away—one cannot ascribe properties to it as is done for an individual classical particle. The lesson to be learned is that classical intuition about the macroscopic world simply does not extrapolate into the microscopic world. In recent years, an entirely different attitude towards quantum theory has been put forward. The focus is no longer on attempts to come to terms with its strange features but to capitalize on its both counter-intuitive and well-established properties. 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