Measurement of qubits

TLDR

The study focuses on qubits encoded in the polarization of entangled photons, but the discussion is general to any physical realization. The authors present the theory underlying measurement of density matrices for a pair of qubits. They compare tomographic reconstruction and maximum‑likelihood estimation, and provide a detailed error analysis for derived quantities such as entropy and entanglement. Illustrative examples from down‑conversion experiments demonstrate the applicability of the proposed methods.

Abstract

We describe in detail the theory underpinning the measurement of density matrices of a pair of quantum two-level systems (``qubits''). Our particular emphasis is on qubits realized by the two polarization degrees of freedom of a pair of entangled photons generated in a down-conversion experiment; however, the discussion applies in general, regardless of the actual physical realization. Two techniques are discussed, namely, a tomographic reconstruction (in which the density matrix is linearly related to a set of measured quantities) and a maximum likelihood technique which requires numerical optimization (but has the advantage of producing density matrices that are always non-negative definite). In addition, a detailed error analysis is presented, allowing errors in quantities derived from the density matrix, such as the entropy or entanglement of formation, to be estimated. Examples based on down-conversion experiments are used to illustrate our results.