This paper investigates the interplay between Krylov complexity ($C_K$) and Spread complexity ($C_S$) for density matrix operators in quantum systems. The authors analyze various analytical and numerical examples, including two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories. Key findings include: 1. **Correspondence between Moment-Generating Functions and Survival Amplitudes**: For generic pure states, there is a correspondence between the moment-generating functions of Lanczos coefficients and survival amplitudes. 2. **Equivalence of $C_K$ and $2C_S$ at Early Times**: For maximally entangled pure states, $C_K$ and $2C_S$ are equivalent at early times. 3. **Deviations in Intermediate Times**: Deviations between $C_K$ and $C_S$ are observed in intermediate times, highlighting subtleties in the averaging approach. 4. **Spectral Form Factor**: For maximally entangled pure states, the moment-generating function of $C_K$ becomes the Spectral Form Factor. 5. **Repackaging of Lanczos Coefficients**: The Lanczos coefficients are repackage for density matrix evolution, and their relationship with the Krylov basis is discussed. 6. **Random Matrix Theories**: The paper explores the relationship between $C_K$ and $C_S$ in random matrix theories, providing insights into universal features of chaotic systems. The study provides a comprehensive analysis of the interplay between $C_K$ and $C_S$, offering new perspectives on the complexity of quantum states and operators.This paper investigates the interplay between Krylov complexity ($C_K$) and Spread complexity ($C_S$) for density matrix operators in quantum systems. The authors analyze various analytical and numerical examples, including two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories. Key findings include: 1. **Correspondence between Moment-Generating Functions and Survival Amplitudes**: For generic pure states, there is a correspondence between the moment-generating functions of Lanczos coefficients and survival amplitudes. 2. **Equivalence of $C_K$ and $2C_S$ at Early Times**: For maximally entangled pure states, $C_K$ and $2C_S$ are equivalent at early times. 3. **Deviations in Intermediate Times**: Deviations between $C_K$ and $C_S$ are observed in intermediate times, highlighting subtleties in the averaging approach. 4. **Spectral Form Factor**: For maximally entangled pure states, the moment-generating function of $C_K$ becomes the Spectral Form Factor. 5. **Repackaging of Lanczos Coefficients**: The Lanczos coefficients are repackage for density matrix evolution, and their relationship with the Krylov basis is discussed. 6. **Random Matrix Theories**: The paper explores the relationship between $C_K$ and $C_S$ in random matrix theories, providing insights into universal features of chaotic systems. The study provides a comprehensive analysis of the interplay between $C_K$ and $C_S$, offering new perspectives on the complexity of quantum states and operators.