Property-Preserving Hash (PPH) achieves compression of large-scale data while providing predicate evaluation functionality on hash digests. PPH can be used to construct new cryptographic primitives or directly applied to privacy-sensitive data scenarios. These constructions or applications impose privacy requirements on the information leakage of PPH. The property definition of PPH follows the Direct-Access Robustness. However, the information leakage of PPH hash values concerning the preimage has not been formally analyzed or defined. In this work, we first propose \(\mathcal {L}\) -Simulation, a simulation-based definition to formalize the information disclosure of PPH hash values with respect to the preimage. We introduce a leakage profile in the simulation to quantify the level of preimage leakage. Subsequently, based on this formal definition, we analyze and evaluate the PPH scheme in Order-Revealing Encryption (ORE) and all PPH schemes for the Hamming distance predicate. Our formal proofs demonstrate that the PPH in ORE achieves an ideal leakage bound. However, the PPH schemes by Fleischhacker and Simkin at Eurocrypt 2021 and Fleischhacker et al. at Eurocrypt 2022 suffer from partial preimage leakage. To address this issue, we propose a new PPH construction for the Hamming distance predicate. Our scheme reduces leakage while supporting additive homomorphism and scalar multiplicative homomorphism, enhancing security and broadening its applicability.