able to reproduce the overall thermodynamic conditions properly. Regarding the density in Fig.2(c), the deviations with respect to the reference value (above all in the transition region) are the same obtained in the classical case [21]. As for the classical AdResS case, one must be sure that indeed there is exchange of molecules between the two regimes. In fact it may occur situations where the two regimes are in equilibrium because some barriers, artifact of the method, would hinder the free exchange so that the molecules are reflected back at the transition region. Fig.3 shows that this is not the case and the molecules diffuse between the two regions in a proper way. Fig.4 shows the results for T$_{2}$ , and also in this case the agreement with the reference full path integral simulation is satisfactory (the diffusion profile, not reported here, shows also the same behaviour as for the case T$_{1}$ ). The overall agreement make us confident that the principles of the adaptive algorithm can be extended to the case of path integral representation. Conclusions: In general for physical systems one should not expect the agreement found here although it can be expected a reasonable accuracy in the same way (and with similar limitations) of the classical AdResS. Having shown that the idea works in a rather reasonable way, it remains to address the question of why such an approach may be useful for appli- cations and not a mere conceptual exercise. In general, there are several problems in soft condensed matter where relevant properties are the results of the interconnection between the local bonding and the larger scale molecular packing. In such cases classical models may give a satisfactory description of the large scale packing but are not sufficient for the high resolution required locally and the path integral description becomes crucial [30]; this is the situation where our approach would represent an optimal tool. Another important field of application is that involving liquid water and above all water as a solvent of large molecules [31]. In particular in cases where the solute-water hydrogen bond plays a crucial role [32–34] and classical model can only partially describe the water mediated process. Moreover, com- bined with specific force fields [35], would be ideal to study local protonation/deprotonation of molecules in solution. In these cases the adaptive method would require that only the region around the solute should be treated with the path integral resolution while the rest can be treated in a coarse-grained fashion as shown here or with a hierarchy of models as already done in part within AdResS [24, 25, 36]. Needless to say that this is only a first step towards the development of a generic scheme for quantum-classical coupling and serve as a basis for further work. 5