Mitigating sloppiness in joint estimation of successive squeezing parameters

When two successive squeezing operations with the same phase are applied to a field mode, reliably estimating the amplitude of each is impossible because the output state depends solely on their sum. In this case, the quantum statistical model becomes sloppy, and the quantum Fisher information matrix turns singular. However, estimation of both parameters becomes feasible if the quantum state is subjected to an appropriate scrambling operation between the two squeezing operations. In this work, we analyze in detail the effects of a phase-shift scrambling transformation, optimized to reduce sloppiness and maximize the overall estimation precision using only classical probes. We also compare the optimized precision bounds of joint estimation with those of stepwise estimation methods, finding that joint estimation retains an advantage despite the quantum noise induced by the residual parameter incompatibility. Finally, we analyze the precision achievable by general-dyne detection and find that it approaches the optimal precision when one of the two squeezing amplitudes is large enough.