  The Talbot effect in quantum physics is known to produce intricate patterns
in the probability distribution of a particle, known as "quantum carpets",
corresponding to the revival and replication of the initial wave function.
Recently, it was shown that one can encode a $D$-level qudit, in such a way
that the Talbot effect can be used to process the $D$-dimensional quantum
information [Far\'{\i}as et al, PRA (2015)]. Here we introduce a scheme to
produce free-propagating "entangled quantum carpets" with pairs of photons
produced by spontaneous parametric down-conversion. First we introduce an
optical device that can be used to synthesize arbitrary superposition states of
Talbot qudits. Sending spatially entangled photon pairs through a pair of these
devices produces an entangled pair of qudits. As an application, we show how
the Talbot effect can be used to test a $D$-dimensional Bell inequality.
Numerical simulations show that violation of the Bell inequality depends
strongly on the amount of spatial correlation in the initial two-photon state.
We briefly discuss how our optical scheme might be adapted to matter wave
experiments.
