  We present a recursive quantum mechanical model for the fragmentation of a
string stretched between a quark and an antiquark with entangled spin states.
The quarks are assumed to be produced in the $e^+e^-$ annihilation process via
the exchange of a virtual photon and the correlations between their spin states
are described by a joint spin density matrix. The string fragmentation process
is formulated at the amplitude level by using the splitting matrices of the
recent string+${}^3P_0$ model of polarized quark fragmentation with
pseudoscalar and vector meson emissions, and accounts for the systematic
propagation of the spin correlations in the fragmentation chain. The model is
formulated as a recursive recipe suitable for a Monte Carlo implementation. It
reproduces the expected angular correlation, due to the Collins effect, between
back-to-back pseudoscalar and/or vector mesons. For the latter, this
correlation also involves the momenta of the decay products. We use the model
for studying the sign of the Collins asymmetry for back-to-back vector and
pseudoscalar mesons.
