Approximating incompatible von Neumann measurements simultaneously

Abstract

We study the problem of performing orthogonal qubit measurements simultaneously. Since these measurements are incompatible, one has to accept additional imprecision. An optimal joint measurement is the one with the least possible imprecision. All earlier considerations of this problem have concerned only joint measurability of observables, while in this work we also take into account conditional state transformations (i.e., instruments). We characterize the optimal joint instrument for two orthogonal von Neumann instruments as being the Lüders instrument of the optimal joint observable.

Incompatibility of unbiased qubit observables and Pauli channels

“…This means that if we want to implement a joint measurement of X s and Z t with s, t = 0, the joint measurement process cannot contain a Lüders channel L Zt or L Xs . We remark that it has been earlier shown that a joint measurement process cannot contain both L Zt and L Xs [17]. Our new result hence strengthens that observation.…”

An invitation to quantum incompatibility

“…The concept of an instrument gives rise also to another sort of compatibility, as two operations may or may not belong to the range of a single instrument [43]. Further, the limitations on approximate joint measurability of two measurements become different if we our aim is to approximate not only measurement outcome probabilities but also state transformations [36].…”

Uniqueness of the joint measurement and the structure of the set of compatible quantum measurements

“…However, as already pointed in Ref. [12], this tuple admits more than one joint measurement. The 8-outcome POVMs M (+) , M (−) given by 1, 1), (−1, −1, 1), (−1, 1, −1), (1, −1, −1)} and M ± abc = 0 otherwise are also joint measurements for Φ t * (A (xyz) ), obtained by adding the marginal-preserving perturbation D ± given by D ± abc = ±abcM abc .…”

“…Summary of relations between properties of measurements, tuples of measurements, and joint measurements (in the case where the tuples are jointly measurable). In blue, the previously known results in the literature of quantum measurements [7,12].…”

“…with a = ±1, w = x, y, z, associated to the Pauli observables. The tuple Φ t (A (xyz) ) is jointly measurable for any t < t * = 1/ √ 3 [12]. At visibility t * , an 8-outcome joint measurement M is given by the effects…”