  We present a systematic procedure to obtain all necessary and sufficient
(quantum) constraints on the expectation values for any set of qudit's
operators. These constraints---arise form Hermiticity, normalization, and
positivity of a statistical operator and through Born's rule---analytically
define an allowed region. A point outside the admissible region does not
correspond to any quantum state, whereas every point in it come from a quantum
state. For a set of observables, the allowed region is a compact and convex set
in a real space, and all its extreme points come from pure quantum states. By
defining appropriate concave functions on the permitted region and then finding
their absolute minimum at the extreme points, we obtain different tight
uncertainty relations for qubit's and spin observables. In addition, quantum
constraints are explicitly given for the Weyl operators and the spin
observables.
