Canonical Naimark extension for generalized measurements involving sets of Pauli quantum observables chosen at random
Carlo Sparaciari, Matteo G. A. Paris · Physical Review A · 2013
We address measurement schemes where certain observables ${X}_{k}$ are chosen at random within a set of nondegenerate isospectral observables and then measured on repeated preparations of a physical system. Each observable has a probability ${z}_{k}$ to be measured, with ${\ensuremath{\sum}}_{k}{z}_{k}=1$, and the statistics of this generalized measurement is described by a positive operator-valued measure. This kind of scheme is referred to as quantum roulettes, since each observable ${X}_{k}$ is chosen at random, e.g., according to the fluctuating value of an external parameter. Here we focus on quantum roulettes for qubits involving the measurements of Pauli matrices, and we explicitly evaluate their canonical Naimark extensions, i.e., their implementation as indirect measurements involving an interaction scheme with a probe system. We thus provide a concrete model to realize the roulette without destroying the signal state, which can be measured again after the measurement or can be transmitted. Finally, we apply our results to the description of Stern-Gerlach-like experiments on a two-level system.