Krylov Entropy: Quantum Chaos Diagnostic

• Krylov entropy is an information-theoretic measure that quantifies the delocalization of quantum states in the Krylov (Lanczos) basis during time evolution.
• It serves as a diagnostic tool for quantum chaos, tracking operator growth, ergodicity breaking, and phase transitions in both Hermitian and non-Hermitian systems.
• The measure bridges quantum complexity, entanglement, and statistical mechanics, providing insights into weak ergodicity breaking and reciprocity metrics.

Krylov entropy is an information-theoretic measure quantifying the spread of a quantum state or operator in the Krylov (Lanczos) basis during time-evolution. Its development provides a powerful diagnostic for quantum chaos, operator growth, and ergodicity breaking, especially in non-Hermitian quantum many-body systems. Closely associated with quantum complexity and entanglement, Krylov entropy bridges operator dynamics, statistical mechanics, and quantum information theory.

1. Mathematical Formulation of Krylov Entropy

Krylov entropy is defined for a state or operator $|\varphi(t)\rangle$ expanded in the orthonormal Krylov basis $\{\,|K_n\rangle\,\}$ as

$$|\varphi(t)\rangle = \sum_n \varphi_n(t)\,|K_n\rangle,$$

where $|\varphi_n(t)|^2$ is interpreted as the probability of the system in the $n$-th Krylov basis vector. The Krylov entropy (also called spread entropy or K-entropy) is then

$$S_K(t) = -\sum_n |\varphi_n(t)|^2 \log |\varphi_n(t)|^2.$$

This quantity captures the effective (Shannon) entropy of the probability distribution of the Krylov components, measuring the delocalization in Krylov space.

For non-Hermitian systems, the Krylov basis is generated using the bi-Lanczos algorithm: left and right Krylov bases are constructed to account for the non-Hermiticity of the Hamiltonian $H$ (or Liouvillian), with tridiagonalizations leading to complex, non-conjugate hopping elements between basis vectors, $(b_n, c_n)$. The time-evolving state is governed by a non-Hermitian tridiagonal effective Hamiltonian in Krylov space.

2. Krylov Entropy and Quantum Chaos Diagnostics

Krylov entropy directly reflects operator or state spreading:

• Chaotic regimes: $S_K(t)$ grows rapidly and saturates at large values, indicating broad support over Krylov space, in parallel with growth in bipartite entanglement entropy.
• Non-chaotic or localized regimes: $S_K(t)$ saturates quickly at lower values, reflecting localization or restricted spreading in Krylov space.

In non-Hermitian quantum spin chains, $S_K(t)$ is suppressed as the strength of non-Hermitian disorder (e.g., gain/loss or dissipative terms) increases, tracking the suppression in Krylov complexity $C_K(t) = \sum_n n |\varphi_n(t)|^2$. Both Krylov entropy and entanglement entropy exhibit critical transitions as a function of disorder, with critical points from scaling analyses found in close agreement.

3. Reciprocity, Non-Hermiticity, and Phase Boundaries

Non-Hermitian dynamics introduce unique features, most notably reciprocity breaking in Krylov space. In Hermitian systems, the off-diagonal tridiagonal matrix elements satisfy $b_n = c_n^*$, leading to reciprocal hopping; in non-Hermitian cases, these are no longer conjugate. This manifests in a Krylov reciprocity metric

$$R_K^{(d)} = \frac{1}{d}\sum_{n=1}^{d}\cos\theta_n,\quad \theta_n = \arg(b_n c_n),$$

which tracks the degree of non-reciprocity. A sign flip in $R_K^{(d)}$ marks the chaos-to-non-chaos transition, coinciding with sharp drops in both $S_K$ and $C_K(t)$ and with the phase transition identified by entanglement entropy variance and complex level spacing ratios (CSR). In the fully localized phase, $S_K$ is minimized and operator evolution is strongly suppressed.

4. Connection to Standard Entropy and Spectral Measures

Krylov entropy aligns with standard quantum chaos diagnostics:

• Bipartite entanglement entropy: Both $S_K$ and entanglement entropy grow rapidly in chaotic regimes, saturate at lower values in localized regimes, and provide consistent critical disorder strengths.
• Complex level spacing statistics (CSR): Transition from Wigner-Dyson (RMT-type) statistics in the chaotic phase to Poisson statistics in the localized phase matches the transition in $S_K$ and $C_K$.
• Variance of entanglement entropy: The peak (divergence) of standard deviation in entanglement entropy marks the chaos-to-localization transition, coinciding with the sharp changes in Krylov entropy and reciprocity metrics.

These findings establish $S_K(t)$ as qualitatively and quantitatively consistent with established entropy and spectral measures, but also sensitive to mechanistic features unique to non-Hermitian dynamics.

5. Phase Structure and Weak Ergodicity Breaking

Varying the non-Hermitian disorder yields two principle transitions:

1. Krylov localization/weak ergodicity breaking ($W_\gamma = W_L$): $C_K$ and $S_K$ saturate at lower values than in the fully chaotic regime, corresponding to prethermalization or partial localization while still not fully integrable.
2. Chaos-to-non-chaos transition ($W_\gamma = W_C$): Strong disorder or dissipation drives both $S_K$ and $C_K$ to drop, indicating strong localization and complete suppression of chaos.

The intermediate regime, characterized by partial Krylov localization, signals weak ergodicity breaking.

Aspect	Krylov Metric	Entropy/Spectral Measure	Physical Interpretation
Chaotic	$S_K$ large, delocalized	High entropy, RMT statistics	Ergodic operator spreading
Weakly ergodic	$S_K$ suppressed, localized	Reduced entropy, plateau CSR	Prethermal/weakly localized
Non-chaotic	$S_K$ strongly suppressed	Low entropy, Poisson stats	Full localization, integrability

6. Broader Implications and Applications

Krylov entropy and related Krylov-space diagnostics provide a physically transparent set of tools for studying phase transitions, chaos suppression, and ergodicity breaking, with notable advantages in non-Hermitian and dissipative quantum systems. The reciprocity metric is especially precise in identifying phase boundaries. This framework is adaptable to a range of ergodicity-breaking phenomena, including many-body localization, Hilbert space fragmentation, and quantum scars, and is directly accessible in experimental systems such as Rydberg atom arrays.

Comparison with alternative diagnostics demonstrates that Krylov entropy not only corroborates but also enhances standard approaches by illuminating the mechanisms—particularly reciprocity breaking and operator delocalization—underlying transitions in complex, open quantum systems.

7. Summary and Key Formulas

Key mathematical relations:

• Krylov entropy: $$\displaystyle S_K = -\sum_n |\varphi_n(t)|^2 \log |\varphi_n(t)|^2$$
• Krylov complexity: $\displaystyle C_K = \sum_n n |\varphi_n(t)|^2$
• Krylov variance: $$\displaystyle \sigma_K^2 = \text{Var}\left\{\ln\frac{j_{2n-1}}{j_{2n}}\right\}$$ with $j_n = |b_n c_n|^{1/2}$
• Reciprocity metric: $$\displaystyle R_K^{(d)} = \frac{1}{d}\sum_{n=1}^{d}\cos\theta_n$$

Krylov entropy provides a robust, mechanism-sensitive diagnostic for quantum chaos and localization phenomena in both Hermitian and non-Hermitian quantum systems, enabling finer characterization of operator growth, information delocalization, and the breakdown of ergodicity in many-body dynamics (Zhou et al., 27 Jan 2025).