Open quantum system simulations are essential for exploring novel quantum phenomena and evaluating noisy quantum circuits. In this Letter, we investigate whether mixed states generated from noisy quantum circuits can be efficiently represented by locally purified density operators (LPDOs). We map an LPDO of $N$ qubits to a pure state of size $2\times N$ defined on a ladder and introduce a unified method for managing virtual and Kraus bonds. We numerically simulate noisy random quantum circuits with depths of up to $d=40$ using fidelity and entanglement entropy as accuracy measures. The LPDO representation is effective in describing mixed states in both the quantum and classical regions; however, it encounters significant challenges at the quantum-classical critical point, restricting its applicability to the quantum region. In contrast, matrix product operators (MPO) successfully characterize the entanglement trend throughout the simulation, while the truncation in MPOs breaks the positivity condition required for a physical density matrix. This work advances our understanding of efficient mixed-state representations in open quantum systems and provides insights into the entanglement structure of noisy quantum circuits.