  The existence of primordial black holes (PBHs), which may form from the
collapse of matter overdensities shortly after the Big Bang, is still under
debate. Among the potential signatures of PBHs are gravitational waves (GWs)
emitted from binary black hole (BBH) mergers at redshifts $z\gtrsim 30$, where
the formation of astrophysical black holes is unlikely. Future ground-based GW
detectors, Cosmic Explorer and Einstein Telescope, will be able to observe
equal-mass BBH mergers with total mass of $\mathcal{O}(10-100)~M_{\odot}$ at
such distances. In this work, we investigate whether the redshift measurement
of a single BBH source can be precise enough to establish its primordial
origin. We simulate BBHs of different masses, mass ratios and orbital
orientations. We show that for BBHs with total masses between $20~M_{\odot}$
and $40~M_{\odot}$ merging at $z \geq 40$ one can infer $z>30$ at up to 97\%
credibility, with a network of one Einstein Telescope, one 40-km Cosmic
Explorer in the US and one 20-km Cosmic Explorer in Australia. A smaller
network made of one Einstein Telescope and one 40-km Cosmic Explorer in the US
measures $z>30$ at larger than 90\% credibility for roughly half of the sources
than the larger network. We then assess the dependence of this result on the
Bayesian redshift priors used for the analysis, specifically on the relative
abundance of the BBH mergers originated from the first stars, and the
primordial BBH mergers.
