Size-Dependence of Lindblad Eigenoperators

• The topic is defined by the interplay between the Liouvillian superoperator and the Pauli string basis, where operator size reflects spatial support and impacts decay dynamics.
• Local Lindbladians exhibit a bulk spectrum with rapid decay of high-size operators following Ginibre-like random matrix statistics, while low-size modes govern late-time observable behavior.
• The analysis reveals that the mix of single-site and two-site dissipators can lead to anomalous persistence of nonlocal eigenoperators, affecting decoherence and quantum memory stability.

The size-dependence of Lindblad eigenoperators arises from the interplay between the mathematical structure of the Liouvillian superoperator, spatial locality of the system’s Hamiltonian and dissipative processes, and the scaling properties of the operator basis as system size increases. This topic connects quantum open system theory, non-equilibrium statistical mechanics, and random matrix theory, providing key insights into decoherence, relaxation, operator growth, and eigenmode localization in dissipative quantum many-body systems.

1. Operator Basis, Expansion, and Size in Lindbladian Dynamics

For a system of $N$ qubits (or spins), the operator space forms a $D^2$-dimensional Hilbert space ($D=2^N$ for qubits), and operators can be expanded in the orthonormal Pauli string basis,

$$|A\rangle = \sum_{\mathbf{m}} c_{\mathbf{m}} |F_{\mathbf{m}}\rangle$$

where $$F_{\mathbf{m}} = \sigma_1^{\mathbf{m}_1} \otimes \cdots \otimes \sigma_N^{\mathbf{m}_N}$$, with $\sigma_j^0 = \mathbb{I}$ and $\sigma_j^{1,2,3}$ the Pauli matrices. The "size" $\mathtt{S}[\mathbf{m}]$ of a given string is the number of non-identity operators: $$\mathtt{S}[\mathbf{m}] = N - \sum_{j=1}^N \delta_{\mathbf{m}_j, 0}$$ This size is a proxy for the operator’s spatial support or Hamming weight.

The right (and left) eigenoperators $|r_j\rangle$ of the Lindbladian superoperator $\mathcal{L}$, defined by $\mathcal{L}|r_j\rangle = \lambda_j|r_j\rangle$, are characterized by their size distribution,

$$p_s(r_j) = \sum_{\mathbf{m}} |c_{\mathbf{m}}|^2 \delta_{\mathtt{S}[\mathbf{m}], s}$$

System size enters both in the dimension of operator space and the possible sizes of the Pauli strings, ranging from $0$ (totally identity) to $N$ (fully nonlocal).

2. Lindbladian Spectra, Locality, and Random Matrix Theory

The spectral properties of $\mathcal{L}$ and the size distribution of its eigenoperators depend crucially on the locality of the underlying Hamiltonian and jump (dissipative) operators:

• For local Lindbladians (i.e., both the Hamiltonian and dissipators are sums of terms acting on at most $k \ll N$ sites), the spectrum exhibits a "bulk" described by Ginibre-type random matrix theory, with eigenvalues centered at large negative real parts proportional to $N$, and "slow" modes (with small $|\mathrm{Re}(\lambda)|$) lying outside this bulk (Chirame et al., 16 Oct 2025).
• The "bulk" eigenoperators are nearly maximally scrambled within their fixed-size sector; their coefficients $c_{\mathbf{m}}$ behave almost like those of random vectors, and the inverse participation ratio (IPR) approaches that of random states.
• Locality ensures that the action of the Lindbladian can change the size of an operator by at most a constant amount per site per timestep, which fundamentally constrains operator growth and decoherence rates, even for highly nonlocal initial states.

Random matrices with local structure display level statistics and eigenvector delocalization reminiscent of the Ginibre ensemble, but the dynamical relevance of the eigenoperators is tied to their size and locality (Chirame et al., 16 Oct 2025).

3. Correlation Between Operator Size and Decay Rate

A central result is the strong correlation between the size of an eigenoperator (Pauli weight) and its Lindbladian decay rate:

• With single-site jump operators (e.g., local dephasing or amplitude damping), the decoherence rate of an operator is approximately linear in its size:

$$\frac{d}{dt} \mathcal{N}(t) \propto -c\,\mathcal{S}(t)\,\mathcal{N}(t)$$

where $\mathcal{N}(t)$ is the operator norm and $\mathcal{S}(t)$ is the mean size (Chirame et al., 16 Oct 2025). Thus, bulk eigenmodes (rapidly decaying, large $|\mathrm{Re}(\lambda)|$) are comprised of predominantly high-size operators.

• The slowest-decaying eigenoperators (those with eigenvalues near zero) are heavily composed of low-size (local) Pauli strings. These modes dominate the late-time dynamics of local observables (expectation values of one- or few-site operators).
• When two-site jump operators (e.g., Lindblad dissipators acting on neighboring sites) dominate, this monotonic correspondence can be violated: some slowly decaying eigenoperators are found to have large Pauli weight (and correspondingly broad spatial support). This is a signature of "anomalous" operator persistence, in contrast to the expectation from single-site dissipation.

Thus, size-dependence is not merely a combinatorial feature of operator space, but is dynamically enforced by the Lindbladian structure, and can be strongly modulated by the mix of one- and two-site (or more extended) dissipators.

4. Dynamical Consequences for Observables and Purity

The sensitivity of observable evolution to the size-dependence of Lindblad eigenoperators manifests in several ways:

• Nonlinear observables, such as the purity $P(t) = \text{Tr}(\rho(t)^2)$ or higher-order correlations, are dominated in early time by the "bulk" eigenoperators. For generic, highly entangled initial states, their overlaps with the bulk produce quasi-universal early-time decay, nearly independent of the specific initial state. Such universality is a hallmark of Ginibre-type RMT statistics in the Lindbladian spectrum (Chirame et al., 16 Oct 2025).
• Conversely, linear observables, e.g., the expectation value of a local operator $A^{(\text{loc})}$, have significant overlap only with low-size eigenoperators outside the bulk. These modes determine the late-time approach to steady state and are highly sensitive to locality and boundary effects. Their relaxation rates are often independent of or only weakly dependent on system size in local models, in stark contrast to the exponentially fast decay of bulk, nonlocal (large-size) eigenoperators.
• In settings where two-site dissipation is dominant, the breakdown of the monotonic size-decay ordering may imply that some highly nonlocal observables retain memory over long times, potentially challenging assumptions about dissipative erasure of nonlocal correlations.

5. Mathematical Framework and Diagnostic Quantities

Quantitative analysis of size-dependence relies on several diagnostic measures:

(a) Size-resolved Participation Ratio

$$p_s(r_j) = \sum_{\mathbf{m}} |c_{\mathbf{m}}|^2 \delta_{\mathtt{S}[\mathbf{m}], s}$$

Characterizes how each eigenoperator distributes its norm over Pauli string of various sizes.

(b) Decoherence Rate Model

For single-site dissipators, the effective decay rate for a given eigenoperator (in operator norm) is often approximately additive over its size: $$\frac{d}{dt} \mathcal{N}(t) = 2 \langle A(t)| \mathcal{L}^\dagger |A(t)\rangle \sim -c\, s\, \mathcal{N}(t)$$ where $s$ is the Pauli weight.

(c) Early-Time Universal Decay of Purity

$$P(t) = \langle \rho_0| e^{\mathcal{L}^\dagger t} e^{\mathcal{L} t} | \rho_0 \rangle$$

Exponential decay for early times, set by averages over bulk eigenvalues associated with large-size eigenoperators.

(d) Mean-Field or Large-N Approximations

For models where analytic treatment is possible, the scaling of decay rates with size and system size can be derived explicitly, further confirming numerics and random-matrix predictions.

6. Impact of System Size and Implications for Dissipative Phases

System size $N$ enters both in the dimensionality of operator space ($4^N$ for qubits) and in the scaling of the Lindbladian spectrum:

• In the bulk, decay rates scale extensively with $N$: $\mathrm{Re}(\lambda) \sim -c N$.
• The number of slow modes (long-lived, low-size eigenoperators) does not grow extensively, implying that late-time local observable dynamics often appears insensitive to system size.
• The presence of slow, large-size eigenmodes when two-site (or higher-order) dissipation dominates suggests the possible emergence of novel dissipative phases or operator localization phenomena, especially in the thermodynamic limit (Chirame et al., 16 Oct 2025).

A key implication is that, despite the apparent randomness of the Lindbladian spectrum for generic local systems, the structure of the eigenoperators—in particular, their size distribution—remains strongly constrained. This has consequences for operator growth, stabilization of quantum memory, and the universality of dissipative relaxation.

7. Summary Table: Size–Dependence Features in Local Lindbladian Systems

Dissipator type	Size–dependence of eigenoperators	Dynamics of observables
Single-site	Decay rate $\propto$ Pauli weight $s$	Local observables relax via low-$s$ modes; early-time purity set by RMT bulk ($s \sim N$)
Two-site (dominant)	Can support slow large-size eigenoperators	Anomalous long-lived nonlocal observables possible
Random Lindbladian	Bulk spectrum follows Ginibre RMT; eigenoperator participation scrambled within size sector	Universal early-time decay for generic initial states; late-time behavior depends on slow modes

Systematic paper of the size-dependence of Lindblad eigenoperators thus provides a powerful framework for understanding open quantum dynamics in many-body systems, revealing the tensions and connections between locality, randomness, and operator complexity in the presence of dissipation.