Simultaneous measurement of two quantum observables: Compatibility, broadcasting, and in-between

One of the central features of quantum theory is that there are pairs of quantum observables that cannot be measured simultaneously. This incompatibility of quantum observables is a necessary ingredient in several quantum phenomena, such as measurement uncertainty relations, violation of Bell inequalities, and steering. Two quantum observables that admit a simultaneous measurement are, in this respect, classical. A finer classification of classicality can be made by formulating four symmetric relations on the set of observables that are stronger than compatibility; they are broadcastability, one-side broadcastability, mutual nondisturbance, and nondisturbance. It is proven that the five relations form a hierarchy and their differences in terms of the required devices needed in a simultaneous measurement are explained. All four relations that are stronger than compatibility are completely characterized in the case of qubit observables.

Figure 1

The whole area depicts the set of all pairs of quantum observables. The strictest condition for a pair of observables is broadcastability and the loosest is compatibility. The other three properties are between these two.

Figure 2

(a) In the one-side broadcasting scenario two approximate copies of the input state are produced and the target observables are measured on these copies. (b) In a nondisturbing sequential measurement one of the measured observables is allowed to be different from the corresponding target observable. The auxiliary observable can operate on a Hilbert space different from the target observable. (c) In the most general setup a global measurement is allowed. Two observables can be obtained in this way exactly when they are compatible.