Minimal disturbing implementation of symmetric generalized measurements for quantum registers and Schroedinger waves
Thomas Decker, Dominik Janzing · arXiv (Cornell University) · 2005
In a previous paper we have presented a general scheme for the implementation of symmetric generalized measurements (POVMs) on a quantum computer. This scheme is based on representation theory of groups and methods to decompose matrices that intertwine two representations. We extend this scheme in such a way that the measurement is minimal disturbing. A minimal disturbing measurement for a POVM changes the state vector |\\Psi> of a system to \\sqrt{\\Pi}|\\Psi> where \\Pi is the positive operator corresponding to the measured result. As an example we construct quantum circuits for measurements with Heisenberg-Weyl symmetry and generalize these circuits for infinite dimensional Hilbert spaces. The momentum and position of a Schroedinger particle in one dimension can be measured simultaneously in a minimal disturbing way by position measurements on two ancilla particles in one dimension. The initial state of the ancillas determine the uncertainties of the momentum and position measurements. We describe how the required transformations can be generated using harmonic potentials and the free evolution of the particles.