Complexity in multi-qubit and many-body systems

Characterizing complexity and criticality in quantum systems requires diagnostics that are both computationally tractable and physically insightful. We apply 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 Renyi entropy. This quantity has already been used earlier termed as structural entropy and it is particularly powerful as the Renyi entropy is directly related to state purity, linear entropy, and the inverse participation ratio, providing a clear physical grounding. While other Renyi 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.

💡 Research Summary

The paper introduces a quantitative measure of quantum state complexity, called “entropic complexity” (SC), defined as the difference between the von Neumann (or Shannon) entropy S and the second‑order Rényi entropy R₂: SC = S − R₂ = −Tr ρ ln ρ + ln Tr ρ². Because R₂ is directly related to the purity Tr ρ² and the inverse participation ratio (IPR), SC simultaneously captures global mixedness (via S) and the degree of concentration or localization of the state (via R₂). The authors argue that a good complexity diagnostic should be non‑negative, easy to evaluate, and vanish at trivial limits (pure or maximally mixed states). SC satisfies all three criteria.

First, the authors validate SC on simple noise models. For a single‑qubit depolarizing channel, the state ρ(p) = (1 − p)|Φ⁺⟩⟨Φ⁺| + p I/2 yields eigenvalues λ₁ = 1 − p/2, λ₂ = p/2. Plugging these into the definitions gives an analytic SC(p) that is zero at p = 0 (pure Bell‑like state) and p = 1 (completely mixed), with a maximum around p ≈ 0.31. The same pattern appears for the two‑qubit Werner state and its n‑qubit generalizations ρ(p) = (1 − p)|Φ⁺⟩⟨Φ⁺| + p I/2ⁿ. As the number of qubits n grows, the optimal mixing parameter p* shifts toward smaller values (approximately 1 − p* ∝ n⁻¹) and the peak value SC(p*) scales roughly linearly with n (exponent δ ≈ 0.14). This demonstrates that SC naturally identifies an “intermediate‑complexity window” that widens with system size.

The core contribution is applying SC to many‑body physics. The authors study deformed random‑matrix ensembles (including the two‑body random interaction ensemble, TBRE) and a disordered Heisenberg spin chain that exhibits a many‑body localization (MBL) transition. By projecting eigenstates onto the computational (Fock) basis, they compute the probability distribution over basis states and evaluate SC. In the localized phase, the IPR is large (states are concentrated) and SC is low; in the fully chaotic (ergodic) phase, both S and R₂ are large and their difference again becomes small. Strikingly, at the MBL transition the SC shows a pronounced peak, indicating that the eigenstates are neither fully localized nor fully thermalized but possess maximal structural richness—non‑ergodic yet extended. This peak precisely marks the critical region where the system transitions from integrable/localized to quantum‑chaotic/ergodic behavior.

Within the chaotic regime, the authors further correlate SC with the survival probability of a local excitation, P(t) = |⟨ψ(0)|ψ(t)⟩|². They find that higher SC values correspond to slower decay of P(t), suggesting that the complexity measure also encodes information about thermalization rates. This provides a novel link between an information‑theoretic diagnostic and dynamical properties of many‑body systems.

From an experimental standpoint, SC is attractive because it requires only two quantities: the von Neumann entropy (obtainable via full or partial state tomography) and the purity Tr ρ² (measurable with SWAP tests or randomized measurements). Thus, the method is feasible for current quantum platforms, ranging from superconducting qubits to trapped‑ion chains.

The paper also discusses extensions. One could define SCₘ = S − Rₘ for higher‑order Rényi entropies, which would shift the location of the peak and probe robustness of the intermediate‑complexity window. However, higher‑order Rényi entropies are experimentally more demanding and lack the clear operational meaning that purity enjoys, justifying the focus on m = 2.

In summary, the work establishes entropic complexity as a simple, versatile, and physically grounded probe for identifying nontrivial quantum regimes. It successfully captures the crossover between localized and ergodic phases in many‑body systems, highlights the critical region of maximal structural richness, and links static informational measures to dynamical thermalization behavior. This provides a valuable new tool for both theoretical analyses of quantum chaos and practical diagnostics in noisy intermediate‑scale quantum (NISQ) devices.