Krylov complexity of density matrix operators

TL;DR

This work probes the Krylov complexity of density-matrix operators and its relation to Spread complexity, revealing that for generic pure states the moment-generating function of Lanczos coefficients equals the survival amplitude and that CK initially doubles CS (C_K ≈ 2 C_S). In finite-dimensional systems (notably N=2), CK coincides with twice CS for all times, while at late times CK scales as N times CS in random-matrix ensembles, linking spectral properties to complexity growth. The study also emphasizes the role of degeneracies in shaping intermediate-time behavior and cautions against naive averaging over return amplitudes, noting connections to spectral form factors and potential holographic interpretations. Across solvable models, QFT-inspired setups, and RMT, the results illuminate how CK and CS encode chaos, entanglement, and spectrum-driven dynamics in quantum systems.

Abstract

Quantifying complexity in quantum systems has witnessed a surge of interest in recent years, with Krylov-based measures such as Krylov complexity ($C_K$) and Spread complexity ($C_S$) gaining prominence. In this study, we investigate their interplay by considering the complexity of states represented by density matrix operators. After setting up the problem, we analyze a handful of analytical and numerical examples spanning generic two-dimensional Hilbert spaces, qubit states, quantum harmonic oscillators, and random matrix theories, uncovering insightful relationships. For generic pure states, our analysis reveals two key findings: (I) a correspondence between moment-generating functions (of Lanczos coefficients) and survival amplitudes, and (II) an early-time equivalence between $C_K$ and $2C_S$. Furthermore, for maximally entangled pure states, we find that the moment-generating function of $C_K$ becomes the Spectral Form Factor and, at late-times, $C_K$ is simply related to $NC_S$ for $N\geq2$ within the $N$-dimensional Hilbert space. Notably, we confirm that $C_K = 2C_S$ holds across all times when $N=2$. Through the lens of random matrix theories, we also discuss deviations between complexities at intermediate times and highlight subtleties in the averaging approach at the level of the survival amplitude.

Figures (10)

• Figure 1: $C_K^{\text{Max}}$ in \ref{['CKMAX']} where $A_{22} = 1-A_{11}$. The yellow surface represents the case for the mixed state, while the blue line corresponds to $C_K^{\text{Max}}$ at $A_{12}^2+B_{12}^2 = A_{11} A_{22}$, indicating a pure state.
• Figure 2: The numerical results for quantum harmonic oscillators with $(\beta,\,\omega)=(\pi/2,\,3)$. In the right panel, $C_K$ is depicted as the black line, while $C_S$ is with the blue line.
• Figure 3: Lanczos coefficients for the inverted harmonic oscillator with $\Omega=3$. The black dots represent the numerical data and the blue solid line is the analytic result \ref{['IHOANARE']}.
• Figure 4: The average Krylov complexity $\bar{C}_K$ is illustrated in the left panel, showcasing its values for various classes: GUE (red), GOE (green), GSE (blue), and IID (black), specifically when $\beta=0$, as per \ref{['N2RESULT4']}. On the right panel, the $\beta$ dependence of $\bar{C}_K$ is displayed for the case of GUE, with $\beta=0,1,2$ (red, orange, yellow). This $\beta$-dependent data is computed numerically with $\mathcal{N}_{\text{it}}=10^4$. Notably, the consistency between the red (analytical) data on the left and the red (numerical) data on the right panel is evident.
• Figure 5: Average Krylov and Spread complexities at $\beta=0$. In the left column, $\bar{C}_K$ is presented for different dimensions: $N=3$ (a), $N=4$ (c), $N=8$ (e), and $N=10$ (g). The right column depict $N \bar{C}_S$ when: $N=3$ (b), $N=4$ (d), $N=8$ (f), and $N=10$ (h). In all figures, various colors denote the different classes: GUE (red), GOE (green), GSE (blue), and IID (black).