Quantum Probability Theoretic Asset Return Modeling: A Novel Schr\"odinger-Like Trading Equation and Multimodal Distribution

Quantum theory provides a comprehensive framework for quantifying uncertainty, often applied in quantum finance to explore the stochastic nature of asset returns. This perspective likens returns to microscopic particle motion, governed by quantum probabilities akin to physical laws. However, such approaches presuppose specific microscopic quantum effects in return changes, a premise criticized for lack of guarantee. This paper diverges by asserting that quantum probability is a mathematical extension of classical probability to complex numbers. It isn't exclusively tied to microscopic quantum phenomena, bypassing the need for quantum effects in returns.By directly linking quantum probability's mathematical structure to traders' decisions and market behaviors, it avoids assuming quantum effects for returns and invoking the wave function. The complex phase of quantum probability, capturing transitions between long and short decisions while considering information interaction among traders, offers an inherent advantage over classical probability in characterizing the multimodal distribution of asset returns.Utilizing Fourier decomposition, we derive a Schr\"odinger-like trading equation, where each term explicitly corresponds to implications of market trading. The equation indicates discrete energy levels in financial trading, with returns following a normal distribution at the lowest level. As the market transitions to higher trading levels, a phase shift occurs in the return distribution, leading to multimodality and fat tails. Empirical research on the Chinese stock market supports the existence of energy levels and multimodal distributions derived from this quantum probability asset returns model.