Spontaneous parametric down-conversion (SPDC) is a widely used method to generate entangled photons, enabling a range of applications from secure communication to tests of quantum physics. Integrating SPDC on a chip provides interferometric stability, allows to reduce a physical footprint, and opens a pathway to true scalability. However, dealing with different photon polarizations and wavelengths on a chip presents a number of challenging problems. In this work, we demonstrate an on-chip polarization beam-splitter based on z-cut titanium-diffused lithium niobate asymmetric adiabatic couplers (AAC) designed for integration with a type-II SPDC source. Our experimental measurements reveal unique polarization beam-splitting regime with the ability to tune the splitting ratios based on wavelength. In particular, we measured a splitting ratio of 17 dB over broadband regions (>60 nm) for both H- and V-polarized lights and a specific 50%/50% splitting ratio for a cross-polarized photon pair from the AAC. The results show that such a system can be used for preparing different quantum polarization-path states that are controllable by changing the phase-matching conditions in the SPDC over a broad band. Furthermore, we propose a fully integrated electro-optically tunable type-II SPDC polarization-path-entangled state preparation circuit on a single lithium niobate photonic chip.