Quantum-to-classical correspondence in Krylov complexity

Abstract

We study quantum-to-classical correspondence of the Krylov space for evolutions driven by unitary maps with a classical limit. This entails a proper definition of corresponding quantum and classical operators, inner products and initial states. We prove that with these definitions the purely classical Krylov space is indeed obtained as the asymptotic $\hbar\to 0$ expansion of the quantum Krylov space, and provide several examples of such correspondence. We use these examples to analyze some general aspects about the evolution of the Krylov complexity as they relate to the phase-space representation for the Krylov states. Additionally, we discuss alternative definitions to obtain the correspondence and why they fail. This paper constitutes a first step in understanding complexity and ergodicity of unitary evolution through the Krylov perspective as they relate to classical dynamical notions.

Figures (8)

• Figure 1: Quantum-to-classical correspondence in the Arnoldi sequences of the harmonic oscillator. The black dashed lines are the classical sequences, while the solid curves correspond to the quantum case with values of $\hbar \in \{2^{-4}, 2^{-5}, 2^{-6}, 2^{-7}\}$ (light to dark).
• Figure 2: Quantum-to-classical correspondence in the Krylov complexity of the harmonic oscillator. Top panel: Krylov complexity as a function of time in the classical (black dashed line) and quantum (solid lines). Bottom panel: relative difference (to the classical) between the classical and quantum complexities (solid lines) and their average values (dash-dotted lines). The quantum curves correspond to values of $\hbar \in \{2^{-4}, 2^{-5}, 2^{-6}, 2^{-7}\}$ (light to dark).
• Figure 3: Classical and quantum phase space portraits of the Krylov states generated at various times $t$ for the evolution of the harmonic oscillator. The quantum Krylov states are represented via the Husimi distribution. The colormap in each row is normalized with respect to its first Krylov state.
• Figure 4: Quantum-to-classical correspondence in the Arnoldi sequences of the Harper map. The black dashed lines are the classical sequences, while the solid curves correspond to the quantum case with Hilbert space dimensions $N \in \{2^5, 2^6, 2^7, 2^8\}$ (light to dark).
• Figure 5: Quantum-to-classical correspondence in the Krylov complexity of the Harper map. Top panel: Krylov complexity as a function of time in the classical (black dashed line) and quantum (solid lines). Bottom panel: relative difference (to the classical) between the classical and quantum complexities (solid lines) and their average values (dash-dotted lines). The quantum curves correspond to Hilbert space dimensions $N \in \{2^5, 2^6, 2^7, 2^8\}$ (light to dark).