  We characterize entanglement subject to its definition over real and complex,
composite quantum systems. In particular, a method is established to assess
quantum correlations with respect to a selected number system, illuminating the
deeply rooted, yet rarely discussed question of why quantum states are
described via complex numbers. With our experiment, we then realize two-photon
polarization states that are entangled with respect to the notion of two
rebits, comprising two two-level systems over real numbers. At the same time,
the generated states are separable with respect to two complex qubits. Among
other results, we reconstruct the best approximation of the generated states in
terms of a real-valued, local expansion and show that this yields an incomplete
description of our data. Conversely, the generated states are shown to be fully
decomposable in terms of tensor-product states with complex wave functions.
Thereby, we probe paradigms of quantum physics with modern theoretical tools
and experimental platforms that are relevant for applications in quantum
information science and technology and connected to the fundamentals of the
quantum description of nature.
