Title: State Decomposition for Model-free Partially observable Markov Decision Process. Abstract: As an essential part of partially observable Markov theory, the measurement of the gap between states and observations is an important issue. In this paper, we propose a novel theory of state decomposition and a simple model-free metric algorithm ($\lambda$-algorithm) for estimating the gap between states and observations in the partially observable Markov decision process with a stationary environment with some missing state conditions. To verify our idea, we design a dimension ablation method to simulate different gaps in the cliff-walking experiment with Q-learning and Sarsa. The simulation results show that $\lambda$ increases steadily as more dimensions are ablated. This proves that $\lambda$ can adequately measure the gap.