Probability representation and quantumness tests for qudits and two-mode light states

Using the tomographic-probability representation of spin states, the quantum behavior of qudits is examined. For a general j-qudit state, we propose an explicit formula for quantumness witness whose negative average value is incompatible with classical statistical model. The probability representations of quantum and classical (2j + 1)-level systems are compared within the framework of quantumness tests. In view of the Jordan–Schwinger map, the method is extended for checking the quantumness of two-mode light states.