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.