Entanglement and Quantum Coherence in Krylov Space Dynamics

TL;DR

This paper establishes quantitative bounds connecting Krylov space spreading measures of quantum states to fundamental quantum resources like entanglement and coherence, enhancing understanding of quantum complexity growth.

Contribution

It introduces bounds relating Krylov-space spreading to entanglement and coherence, providing new insights into quantum resource constraints during quantum dynamics.

Findings

Entanglement of evolved states is bounded by Krylov basis entanglement and spread complexity.

Inverse participation ratio bounds relate delocalization to geometric measures in multipartite systems.

Relations between initial coherence and spread complexity are derived for qubit and qutrit systems.

Abstract

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.