Uncertainty relations on the joint numerical range of operators

The expectation values of operators drawn from a single quantum state cannot be outside of a particular region, called their allowed region or the joint numerical range of the operators. We present a method to obtain all necessary and sufficient constraints—from Hermiticity, normalization, and positivity of a state and through the Born rule—that analytically defines the allowed region. Then, we present the allowed regions for the Heisenberg–Weyl operators, the angular momentum operators, and for their functions in dimension two to infinity. Especially, we consider three kinds of functions—combinations of powers of the ladder operators, powers of the angular momentum operators, and their anticommutators—and discover the allowed regions of different shapes. Here we also introduce uncertainty measures on the joint numerical range that are different from the standard deviation and the Shannon entropy. With the measures, we achieve a new kind of tight uncertainty relations for the Weyl- and the angular-momentum-operators. Overall, we demonstrate how the joint numerical range and the uncertainty relations change as the dimension grows. We apply the quantum de Finetti theorem to attain the allowed regions and tight uncertainty relations in the limit where the dimension goes to infinity. (paper)