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Complexity in multi-qubit and many-body systems

Imre Varga

TL;DR

The paper defines entropic complexity $S_C = S - R_2$ for $n$-qubit states and shows how it captures the structure of quantum states across decoherence and many-body dynamics, bridging purity and localization through a density-matrix framework.Analytic and numerical results for depolarization, phase damping, and Werner-state generalizations illustrate how $S_C$ peaks at intermediate noise levels, marking quantum-classical crossovers with clear scaling in system size.In many-body settings, $S_C$ reveals the fractal regime and crossover between ergodic and many-body localized phases, with scaling laws in random-matrix (GOE/TBRE) models and finite-size Heisenberg chains, and it connects to the survival probability dynamics in chaotic systems.Overall, entropic complexity provides a simple, versatile, and physically grounded diagnostic that complements spectral and dynamical indicators, with potential for experimental accessibility and insight into quantum information processing under realistic noise and interactions.

Abstract

Characterizing complexity and criticality in quantum systems requires diagnostics that are both computationally tractable and physically insightful. We introduce a measure of quantum state complexity for n-qubit systems, defined as the divergence between the Shannon or von Neumann entropy of the computational basis distribution and the second-order Rényi entropy. This quantity is particularly powerful as the latter Rényi entropy is directly related to state purity, linear entropy, and the inverse participation ratio, providing a clear physical grounding. While other Rényi orders could be used, the second order offers a deep and established connection to these key physical quantities. We first validate the measure in canonical noise channels, showing it peaks at the boundary between quantum and classical regimes. We then demonstrate its power in many-body physics. For systems exhibiting a many-body localization transition - including deformed random matrix ensembles and a disordered Heisenberg spin chain - the complexity measure reliably signals the crossover from integrable/localized to quantum-chaotic/ergodic phases. Crucially, the maximum complexity occurs in the non-ergodic yet extended states at the transition, precisely capturing the critical region where the system is neither fully localized nor thermalized. Furthermore, within the chaotic phase, the measure correlates with the survival probability of local excitations, revealing a spectrum of thermalization properties. Our results establish that the entropic complexity is a simple, versatile, and effective probe for identifying nontrivial quantum regimes and transitions giving a new and alternative insight into such systems.

Complexity in multi-qubit and many-body systems

TL;DR

The paper defines entropic complexity $S_C = S - R_2$ for $n$-qubit states and shows how it captures the structure of quantum states across decoherence and many-body dynamics, bridging purity and localization through a density-matrix framework.Analytic and numerical results for depolarization, phase damping, and Werner-state generalizations illustrate how $S_C$ peaks at intermediate noise levels, marking quantum-classical crossovers with clear scaling in system size.In many-body settings, $S_C$ reveals the fractal regime and crossover between ergodic and many-body localized phases, with scaling laws in random-matrix (GOE/TBRE) models and finite-size Heisenberg chains, and it connects to the survival probability dynamics in chaotic systems.Overall, entropic complexity provides a simple, versatile, and physically grounded diagnostic that complements spectral and dynamical indicators, with potential for experimental accessibility and insight into quantum information processing under realistic noise and interactions.

Abstract

Characterizing complexity and criticality in quantum systems requires diagnostics that are both computationally tractable and physically insightful. We introduce a measure of quantum state complexity for n-qubit systems, defined as the divergence between the Shannon or von Neumann entropy of the computational basis distribution and the second-order Rényi entropy. This quantity is particularly powerful as the latter Rényi entropy is directly related to state purity, linear entropy, and the inverse participation ratio, providing a clear physical grounding. While other Rényi orders could be used, the second order offers a deep and established connection to these key physical quantities. We first validate the measure in canonical noise channels, showing it peaks at the boundary between quantum and classical regimes. We then demonstrate its power in many-body physics. For systems exhibiting a many-body localization transition - including deformed random matrix ensembles and a disordered Heisenberg spin chain - the complexity measure reliably signals the crossover from integrable/localized to quantum-chaotic/ergodic phases. Crucially, the maximum complexity occurs in the non-ergodic yet extended states at the transition, precisely capturing the critical region where the system is neither fully localized nor thermalized. Furthermore, within the chaotic phase, the measure correlates with the survival probability of local excitations, revealing a spectrum of thermalization properties. Our results establish that the entropic complexity is a simple, versatile, and effective probe for identifying nontrivial quantum regimes and transitions giving a new and alternative insight into such systems.

Paper Structure

This paper contains 13 sections, 41 equations, 10 figures.

Table of Contents

  1. Introduction
  2. The derivation of the complexity based on the density matrix
  3. Multi-qubit state in noise induced decoherence
  4. The depolarization channel of a single qubit
  5. The 2-qubit Werner state
  6. The n-qubit generalization
  7. Multi-qubit state in noise induced phase damping
  8. Complexity in Many-body Systems
  9. Random Matrix Theory and Ergodicity Breaking
  10. Deformed Two-Body Random Interaction Ensemble
  11. Many-body localization in 1D Heisenberg model
  12. Complexity of survival probability: Time Evolution in Chaotic Many-Body Systems
  13. Conclusions

Figures (10)

  • Figure 1: Comparison of the $2$-qubit Werner state combined with a totally mixed state as given in Eq. (\ref{['werner_nsc']}) as a function of parameter $p$ using several quantities, the entropic complexity, $S_C(p)$, the negativity, $N(p)$, and the concurrence, $C(p)$. The important limits of CHSH and entanglement edges are marked by vertical dotted and dashed lines.
  • Figure 2: Complexity of the $n$-qubit state combined with a totally mixed state as given in Eq. (\ref{['werner_nsc']}) as a function of parameter $p$ for small values of $n$.
  • Figure 3: The value of parameter $p^*$ with maximum complexity as a function of $n$, the number of qubits. The fit is approximately $1-p^*\sim n^{\gamma}$ with $\gamma\approx 1.05$ for $n\gg 1$. In the inset we plot the value of the maximum complexity at $p=p^*$ vs number of qubits, $n$. The fit is approximately $S_C(p^*)\sim n^{\delta}$ with $\delta\approx 0.14$.
  • Figure 4: The entropic complexity of the $n$-qubit system under the effect of dephasing with probability $p$. The inset shows the value of the maximum complexity as a function of the number of qubits, $n$ together with the best fit, $p^\ast \sim 1/n$.
  • Figure 5: Deformed GOE: (a) spectral statistics, (b) $R_2$, and (c) entropic complexity, $S_C$ for different $N$s
  • ...and 5 more figures