Entanglement and Quantum Coherence in Krylov Space Dynamics

Entanglement and Quantum Coherence in Krylov Space Dynamics

Overview of Krylov Space Complexity and Quantum Resources

The study presents a rigorous analysis of quantum complexity growth through Krylov space dynamics and its fundamental connections to quantum resources—specifically entanglement and quantum coherence. The authors focus on 'spread complexity' and the inverse participation ratio (IPR) as quantifiers of Krylov-space spreading, which are functions of both the initial quantum state and the governing Hamiltonian. By analytically establishing bounds, this work elucidates the nuanced interplay between Krylov-space delocalization and core quantum-information-theoretic quantities, offering formal constraints and relations between them.

Formalism: Krylov Basis and Complexity Measures

Krylov space techniques, traditionally grounded in the Lanczos algorithm, construct an orthonormal basis from repeated applications of the system’s Hamiltonian to a reference state. The Krylov chain representation enables mapping unitary quantum evolution onto a one-dimensional structure, where the operator or state complexity is recast in terms of average position (spread complexity) and localization (IPR) across the resultant basis states. The spread complexity $K(t)$ is defined as the first moment $\sum_n n |\phi_n(t)|^2$ of the expansion coefficients $\phi_n(t)$, whereas the IPR $\sum_n |\phi_n(t)|^4$ measures localization among the basis states.

Entanglement Bounds from Krylov-Space Delocalization

The paper rigorously derives upper bounds on bipartite entanglement growth in terms of spread complexity and the entanglement content of Krylov basis vectors. For multipartite systems, analogous bounds relate the IPR of an evolved Krylov-space state to geometric entanglement measures. Notably, the authors clarify that nonzero spread complexity does not universally imply entanglement generation—demonstrating circumstances where separable dynamics produce substantial Krylov-space spreading absent any quantum correlations. Upper bounds for von Neumann entropy of reduced subsystems are established as: $$S(\rho_A(t)) \le d_{K}\left[ \sum_{n=0}^{d_K-1} |c_n(t)|^2 S(\rho_A^{(n)}) + f(K) \right]$$ where $f(K)$ is a function of the spread complexity and $d_K$ is the Krylov chain dimension. For multipartite entanglement, derived bounds express IPR in terms of the geometric measure $\mathcal{G}_\psi$ and weighted contributions from the Krylov basis: $$\frac{1 - \sqrt{1 - [\mathcal{G}_\psi - X]^2}}{2} \le \text{IPR} \le \frac{1 + \sqrt{1 - [\mathcal{G}_\psi - X]^2}}{2}$$ where $X = \sum_n |c_n|^4 \mathcal{G}_{|k_n\rangle}$.

Coherence-Spreading Correspondence for Qubits and Qutrits

Analytical relations are established between quantum coherence of the initial state (measured in the energy eigenbasis) and spread complexity during unitary evolution. For qubits, an explicit proportionality is derived: $\sum_n n |\phi_n(t)|^2$0 where $\sum_n n |\phi_n(t)|^2$1 is the $\sum_n n |\phi_n(t)|^2$2 norm of coherence and $\sum_n n |\phi_n(t)|^2$3 is the energy gap. The study generalizes the correspondence for qutrits, expressing $\sum_n n |\phi_n(t)|^2$4 as a sum of terms governed by pairwise and tripartite coherence amplitudes and all energy gaps. These relations formalize the requirement that Krylov-space delocalization is contingent on initial quantum coherence in the Hamiltonian eigenbasis; if the initial state is an eigenstate (zero coherence), Krylov-space complexity remains stationary.

Implications and Future Research Directions

The work clarifies the non-trivial dependence of complexity diagnostics—spread complexity and IPR—on fundamental quantum resources and system dynamics. Strong numerical results include explicit analytic forms for bounds and proportionalities for low-dimensional systems, reinforcing the claim that Krylov complexity is a resource-dependent and context-specific probe. Practical implications are evident in quantum control, measurement, and diagnostics of scrambling and information flow in many-body systems. Theoretically, the established bounds provide constraints for the design and interpretation of complexity growth protocols, with broad applicability spanning quantum chaos, operator growth, and entanglement dynamics.

Future developments could include extending these formal results to open quantum systems, exploring many-body transitions such as localization-chaos crossovers, and integrating Krylov complexity analyses with resource-theoretic entanglement and coherence quantifiers in scalable architectures. The precise interplay uncovered here between complexity, entanglement, and coherence is likely to inform protocols for efficient state engineering and quantum information processing in increasingly high-dimensional quantum systems.

Conclusion

This study provides a comprehensive analytical framework linking Krylov-space complexity growth to the generation and distribution of quantum resources, including entanglement and quantum coherence. Quantitative bounds and explicit functional correspondences are derived, formalizing constraints on complexity growth, delocalization, and correlation development under unitary dynamics. These results strengthen the theoretical foundation for using Krylov-space diagnostics in quantum information, chaos, and many-body physics, revealing avenues for future inquiry into complexity-resource duality and practical quantum technologies.

For further details, see "Entanglement and Quantum Coherence in Krylov Space Dynamics" (2603.26619).