The canonical Naimark extension for the Pauli quantum roulette wheel

TL;DR

This paper derives the canonical Naimark extension for qubit quantum roulettes involving Pauli measurements, providing a practical model for implementing such measurements without destroying the quantum state.

Contribution

It explicitly evaluates the Naimark extensions for Pauli quantum roulettes, enabling indirect measurement schemes that preserve the quantum state for qubits.

Findings

Provides a concrete implementation model for quantum roulettes

Enables repeated or transmitted measurements of the quantum state

Applies results to Stern-Gerlach-like experiments

Abstract

We address measurement schemes where certain observables are chosen at random within a set of non-degenerate isospectral observables and then measured on repeated preparations of a physical system. Each observable has a given probability to be measured, and the statistics of this generalized measurement is described by a positive operator-valued measure (POVM). This kind of schemes are referred to as quantum roulettes since each observable 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.