We propose a formalization of C.G. Jung’s theory of personality using a four-dimensional Hilbert-space for the representation of two qubits. The first qubit relates to Jung’s four psychological functions: Thinking, Feeling, Sensing and iNtuition, which are represented by two groups of projection operators, {T, F} and {S, N}. The operators in each group are commuting but operators of different groups are not. The second qubit represents Jung’s two perspectives of extraversion and introversion. It is shown that this system gives a natural explanation of the 16 psychological types that are defined in the Jungian tradition. Further, the system accounts for the restriction posed by Jung concerning the possible combination of psychological functions and perspectives. The empirical consequences of the present theory are discussed, and the results of a pilot study are reported with the aim to check some basic predictions of the theory. In addition, it is shown why the present praxis of personality diagnostics based on classical statistics is insufficient.