Entanglement and Quantum Coherence in Krylov Space Dynamics

The spreading of quantum states in Krylov space under unitary dynamics provides a natural framework for characterizing quantum complexity. Quantifiers of this spreading, such as the spread complexity and the inverse participation ratio, depend explicitly on both the Hamiltonian and the initial state, rendering their connection to fundamental quantum resources such as entanglement and quantum coherence subtle. We establish quantitative bounds relating Krylov-space spreading to the entanglement of the evolved state and to the quantum coherence of the initial state. For bipartite systems, we have shown that the entanglement of the evolved state is upper bounded in terms of the entanglement of the Krylov basis vectors and the spread complexity. In the case of multipartite systems, analogous bounds are obtained for the inverse participation ratio, a quantifier of the delocalization of a quantum state in the Krylov basis, in terms of the geometric measures. Furthermore, for qubit and qutrit systems, we derive relations between the quantum coherence of the initial state in the energy eigenbasis and the spread complexity, valid for arbitrary Hamiltonians. Our results provide quantitative constraints linking Krylov-space complexity growth to fundamental quantum resources.

💡 Research Summary

This paper establishes a rigorous, quantitative framework connecting the dynamical spreading of quantum states in Krylov space to fundamental quantum resources: entanglement and quantum coherence.

The study focuses on two primary Krylov-space measures: “Spread Complexity” (K), which quantifies the average position of a state on the Krylov chain generated via the Lanczos algorithm, and the “Inverse Participation Ratio” (IPR), which measures the localization of the state within the Krylov basis. While these metrics are useful for diagnosing quantum chaos and information scrambling, their direct relationship to entanglement generation and coherence consumption remained unclear.

The core analytical results are presented in two main propositions. First, for bipartite systems, the entanglement entropy S(ρ_A(t)) of the evolved state is shown to be upper-bounded by a sum of two terms: (i) the weighted average of the entanglement entropies of the individual Krylov basis states, and (ii) a function f(K) = (1+K)log(1+K) - K log K of the spread complexity. This bound explicitly limits entanglement growth based on both the inherent entanglement of the Krylov basis and the extent of dynamical spreading.

Second, for multipartite systems, analogous bounds are derived for the IPR in terms of geometric measures of entanglement (such as the Generalized Geometric Measure). This links the degree of delocalization in Krylov space to the multipartite entanglement structure of the state.

Furthermore, the paper derives an explicit, direct relationship between quantum coherence and spread complexity for low-dimensional systems (qubits and qutrits). It is proven that, for an arbitrary Hamiltonian, the spread complexity K(t) is proportional to the ℓ1-norm of coherence of the initial state with respect to the energy eigenbasis. This provides a clear, quantitative statement that quantum coherence in the initial state acts as a fuel for complexity growth in Krylov space.

Overall, this work moves beyond qualitative observations by providing strict mathematical inequalities that constrain how quantum resources (entanglement, coherence) and dynamical complexity measures (spread complexity, IPR) can co-evolve under unitary dynamics. The results offer new tools for analyzing quantum many-body systems, quantum information protocols, and the fundamental limits of complexity growth in closed quantum systems.