  Optical hybrid entanglement can be created between two qubits, one encoded in
a single photon and another one in coherent states with opposite phases. It
opens the path to a variety of quantum technologies, such as heterogeneous
quantum networks, merging continuous and discrete variable encoding, and
enabling the transport and interconversion of information. However, reliable
characterization of the nature of this entanglement is limited so far to full
quantum state tomography. Here, we perform a thorough study of
Clauser-Horne-Shimony-Holt (CHSH) Bell inequality tests, enabling practical
verification of quantum correlations for optical hybrid entanglement. We show
that a practical violation of this inequality is possible with simple photon
number on/off measurements if detection efficiencies stay above 82%. Another
approach, based on photon-number parity measurements, requires 94% efficiency
but works well in the limit of higher photon populations. Both tests use no
postselection of the measurement outcomes and they are free of the
fair-sampling hypothesis. Our proposal paves the way to performing
loophole-free tests using feasible experimental tasks such as coherent state
interference and photon counting, and to verification of hybrid entanglement in
real-world applications.
