Quantum electromagnetic X-waves

We show that two distinct quantum states of the electromagnetic field can be associated to a classical vector X wave or a propagation-invariant solution of Maxwell equations. The difference between the two states is of pure quantum mechanical origin since they are internally entangled and disentangled, respectively and can be generated by different linear or nonlinear processes. Detection and generation of Schr\"odinger-cat states comprising two entangled X-waves and their possible applications are discussed.