Operator delocalization in disordered spin chains via exact MPO marginals

1. Problem Statement

The paper addresses the challenge of quantifying operator delocalization and scrambling in quantum many-body systems, specifically focusing on the distinction between Anderson localization (non-interacting disordered systems) and Many-Body Localization (MBL, interacting disordered systems).

• Context: In thermalizing systems, local operators evolve into extended "operator strings" that spread across the system, dispersing information (scrambling). In localized systems, this spreading is hindered.
• Limitations of Existing Tools:
  • OTOC (Out-of-Time-Order Correlator): While useful for detecting scrambling, OTOC provides space-time information without reference to a specific operator basis and often requires stochastic sampling or full tomography for detailed distribution analysis.
  • Stabilizer Rényi Entropy (SRE): Measures "nonstabilizerness" (magic) but typically requires stochastic sampling of the Pauli spectrum, which scales exponentially with system size.
• Goal: To introduce complementary, computable measures of operator complexity that can be calculated exactly and deterministically without stochastic sampling, allowing for the study of full probability distributions in large systems.

2. Methodology

A. New Quantities: Operator Mass and Operator Length

The authors define two metrics based on the expansion of a time-evolved operator $\hat{O}(t)$ in the Pauli-string basis $\{ \hat{Q} \}$:
$$\hat{O}(t) = \frac{1}{2^L} \sum_{\hat{Q}} A_{\hat{Q}}(t) \hat{Q}$$
where $A_{\hat{Q}} = \text{Tr}[\hat{Q}\hat{O}]$. The squared amplitudes $|A_{\hat{Q}}|^2$ form a probability distribution.

1. Operator Mass ($m$): The number of non-identity Pauli matrices in a string.
   $$m_{\hat{Q}} = \sum_{j=1}^L (1 - \delta_{\alpha_j, 0})$$
   The average mass $m(t)$ quantifies the number of sites the operator acts on non-trivially.
2. Operator Length ($h$): The position of the rightmost non-identity site in a string.
   $$h_{\hat{Q}} = \max \{ j : \alpha_j \neq 0 \}$$
   The average length $h(t)$ quantifies the spatial extent (support) of the operator along the chain.

B. Computational Framework: Matrix Product Operators (MPO)

The core technical innovation is the calculation of these quantities using Matrix Product States (MPS) in the Choi-Jamiołkowski representation (mapping operators to states in a doubled Hilbert space).

• Exact Marginals: Instead of sampling the Pauli spectrum, the authors derive exact formulas to compute the full probability distributions $P(m)$ and $P(h)$ directly from the MPO tensors.
  • Length: Computed via unnormalized Rényi-2 entropies of the operator state across bipartitions, utilizing a recurrence relation.
  • Mass: Computed by treating the mass operator as a diagonal superoperator and using a discrete Fourier transform (generating function approach) to extract the distribution.
• Efficiency: This method avoids stochastic sampling. The computational cost scales polynomially with system size ($L$) and bond dimension ($\chi$). Crucially, in MBL regimes, the required bond dimension grows only linearly with time, making long-time simulations feasible.

C. Model System

The authors apply this framework to the disordered XXZ spin-1/2 chain:
$$\hat{H} = \sum_{j} \left[ \frac{J}{2}(\hat{\sigma}^+_j \hat{\sigma}^-_{j+1} + \text{H.c.}) + \frac{\Delta}{4}\hat{\sigma}^z_j \hat{\sigma}^z_{j+1} \right] + \sum_j \frac{h_j}{2}\hat{\sigma}^z_j$$

• Initial State: A local operator $\hat{O} = \hat{\sigma}^z_1$ (acting only on the first site).
• Regimes:
  • Non-interacting ($\Delta = 0$): Maps to Anderson localization (free fermions).
  • Interacting ($\Delta \neq 0$): MBL regime for strong disorder $W$.

3. Key Results

A. Distinct Behaviors in Localization Regimes

1. Anderson Localization ($\Delta = 0$):
   • Operator mass, length, and entanglement entropy rapidly saturate to constant values.
   • The operator remains strictly localized; no scrambling occurs.
2. Many-Body Localization ($\Delta \neq 0$, Strong Disorder):
   • All quantities (mass, length, entanglement entropy) exhibit a robust logarithmic growth in time: $h(t) \sim \ln t$.
   • This confirms the existence of a logarithmic light cone for operator spreading, consistent with the phenomenology of MBL.
   • The growth persists for arbitrarily weak interactions ($\Delta > 0$), indicating that Anderson localization is singular at $\Delta=0$.

B. System Size and Disorder Dependence

• Time-Averaged Scaling: When averaging over time windows proportional to system size $L$, the quantities collapse when rescaled by $\ln L$.
• Non-monotonicity: The time-integrated mass shows a non-monotonic dependence on disorder strength $W$, peaking at intermediate disorder, likely due to the transition from the Bethe-Ansatz integrable point ($W=0$) to the localized regime.

C. Theoretical Validation

• $\ell$-bit Model: The authors derive an analytical $\ell$-bit model (localized integrals of motion with exponentially decaying interactions).
• Agreement: The analytical solution of the $\ell$-bit model perfectly reproduces the logarithmic growth of the operator length observed in the numerical XXZ chain simulations, confirming that the slow dynamics are driven by weak dephasing interactions between emergent localized degrees of freedom.

4. Significance and Contributions

1. Methodological Advancement:

   • Provides the first exact, deterministic method to compute full probability distributions of operator mass and length using MPOs.
   • Eliminates the need for stochastic sampling, which is a bottleneck for studying operator complexity in large systems.
   • Demonstrates that MPO simulations in MBL regimes are highly efficient due to the slow (logarithmic) growth of entanglement.

2. Physical Insights:

   • Establishes Operator Length as a robust, complementary probe to Operator Mass and Entanglement Entropy for characterizing delocalization.
   • Quantitatively confirms the logarithmic light cone in MBL systems, distinguishing it sharply from the saturation in Anderson localization.
   • Highlights the singular nature of interactions: Even infinitesimal interactions destroy the strict saturation of Anderson localization, inducing slow, unbounded operator spreading.

3. Experimental Feasibility:

   • Proposes a concrete experimental protocol using classical shadows and Bell-pair measurements on system-ancilla pairs (Choi states).
   • The required operations (local Clifford gates, controlled time evolution, and Bell measurements) are native to current quantum simulation platforms (superconducting qubits, trapped ions, Rydberg atoms).
   • This allows for the direct experimental reconstruction of operator marginals without full tomography.

Conclusion

The paper successfully bridges the gap between theoretical operator complexity measures and practical computation/experiment. By introducing "operator length" and leveraging exact MPO marginals, the authors provide a powerful tool to distinguish between different localization phases and characterize the slow, logarithmic dynamics of information scrambling in many-body localized systems.