Fig. 6: The results of applying the classical shadows and machine learning methods to the reconstructed density matrices from NMR data.

It shows no clear advantage for ρ_A = [12] in a due to the small subsystem size. However, a significant advantage emerges for ρ_A = [123] in b. The difference becomes more obvious with larger subsystems (see Fig. 4 in the main text). Here, the average error, defined as the mean distance between the real entropy S^NMR (obtained through quantum state tomography) and the predicted entropy S^ML,CS (obtained through machine learning and classical shadow methods), is calculated as \(\epsilon =\frac{1}{M}\mathop{\sum }\nolimits_{m = 1}^{M}| {S}_{m}^{{\rm{NMR}}}({\rho }_{A})-{S}_{m}^{{\rm{ML,CS}}}({\rho }_{A})|\). M denotes the number of reconstructed density matrices in the experiments.