Abstract
The quantum measurement incompatibility is a distinctive feature of quantum mechanics. We investigate the incompatibility of a set of general measurements and classify the incompatibility by the hierarchy of compatibilities of its subsets. By using the approach of adding noises to measurement operators, we present a complete classification of the incompatibility of a given measurement assemblage with n members. Detailed examples are given for the incompatibility of unbiased qubit measurements based on a semidefinite program.
Highlights
The incompatible measurements are one of the striking features in quantum physics, and can be traced back to Heisenberg’s uncertainty principle [1] and wave-particle duality [2,3]
The incompatibility of quantum measurements is known to be a powerful tool in many branches of quantum information theory [13,14,15,16,17,18,19]
[20], quantum incompatibility has been the object of intense research [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]
Summary
The incompatible measurements are one of the striking features in quantum physics, and can be traced back to Heisenberg’s uncertainty principle [1] and wave-particle duality [2,3]. Qin, et al [41] formulated state-independent tight uncertainty relations, satisfied by three measurements, in terms of their triple joint measurability. Another way to quantify the joint measurability of a set of measurements is to add noise to the measurement operators, and numerically calculate the noise threshold for the measurements to be jointly measurable [21,25,26]. Similar to the quantum multipartite entanglement or non-locality, we classify the measurement incompatibility for a given set of measurements, and present a hierarchy of quantum measurement incompatibilities.
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