Incompatibility probability of random quantum measurements

Abstract

Incompatibility of quantum measurements is of fundamental importance in quantum mechanics. It is closely related to many nonclassical phenomena such as Bell nonlocality, quantum uncertainty relations, and quantum steering. We study the necessary and sufficient conditions of quantum compatibility for a given collection of n measurements in d-dimensional space. From the compatibility criterion for two-qubit measurements, we compute the incompatibility probability of a pair of independent random measurements. For a pair of unbiased random qubit measurements, we derive that the incompatibility probability is exactly 3 5 . Detailed results are also presented in figures for pairs of general qubit measurements.

Uncertainty regions of observables and state-independent uncertainty relations

“…where, in the second equality, we used the probability density function of inner product of two random unit vectors, a result has been already obtained in [35], we obtain Let (α , β , γ ) = (α + γκ a , β + γκ b , γ), then the Jacobian of (α,…”

Random positive operator valued measures

“…random vectors were considered in [AL16], in a situation where the number of outcomes is larger than the dimension. Finally, in the work [ZXLJF19] (which appeared after the preprint version of our work was made available online), the authors compute several probabilities for the compatibility of independent dichotomic qubit POVMs, parametrized by points on the Bloch sphere.…”