Complete and incomplete state estimation via the simultaneous unsharp measurement of two incompatible qubit operators

Abstract

We consider the simultaneous coupling of a qubit system to two qubit probes, designed to measure noncommuting qubit observables when working in isolation. While the single meter model usually corresponds to an unsharp measurement of a qubit operator, the double meter model usually describes an informationally complete measurement. The double meter positive operator measurement can be characterized by three vectors associated with the space of Hermitian tracesless observables of the system. The dimension of the linear space generated by these vectors (Bloch rank) is equal to the number of independent operators (or the number of independent Bloch-vector components) which can be estimated. $\mathbb{S}$, the subspace of parameters associated with informationally incomplete measurements (IICs), is characterized either analytically or numerically, and their Bloch rank is used to classify the IICs. Bloch rank of the ``simultaneous measurement'' of two Pauli matrices is shown to be 2 if the isolated measurements are projective and 3 if they are weak; however, two-meter weak measurements are expected to converge more slowly than their single-meter counterparts.