Rényi Entanglement Entropies

• Rényi entanglement entropies are a one-parameter family of measures generalizing the von Neumann entropy to capture quantum correlations and spectral properties in diverse systems.
• They are computed via spectral analysis and advanced Monte Carlo techniques, yielding both leading extensive contributions and universal subleading corrections.
• These entropies play a critical role in characterizing quantum phase transitions, non-equilibrium dynamics, and resource quantification in quantum many-body theory.

Rényi entanglement entropies constitute a one-parameter family of quantum information measures that generalize the von Neumann entanglement entropy, providing refined diagnostics of quantum correlations, spectral properties of reduced density matrices, and signatures of quantum phase transitions and non-equilibrium dynamics. Central to quantum many-body theory, condensed matter physics, and quantum information, these entropies capture not just leading extensive features but also universal subleading corrections, support advanced numerical and experimental probes of complex quantum states, and enable resource-theoretic quantification of entanglement for both pure and mixed states.

1. Mathematical Definitions and Characterizations

The α-th Rényi entanglement entropy for a subsystem $A$ with reduced density matrix $\rho_A$ is defined as

$$S_A^{(\alpha)} = \frac{1}{1-\alpha}\ln \operatorname{Tr}[\rho_A^\alpha].$$

This generalizes the von Neumann entropy, recovered as $\alpha\rightarrow 1$: $S_A = -\operatorname{Tr}[\rho_A \ln \rho_A].$ For $\alpha\to\infty$, $S_A^{(\infty)} = -\ln \lambda_{\max}$ yields the single-copy entanglement. In free-fermionic systems, Rényi entropies can be computed directly from the spectrum $\{\lambda_m\}$ of the correlation matrix $𝔠_A$ restricted to $A$ via

$$S_A^{(\alpha)} = \sum_m e_\alpha(\lambda_m),\quad e_\alpha(\lambda) = \frac{1}{1-\alpha}\ln(\lambda^\alpha + (1-\lambda)^\alpha).$$

This spectral formula underpins both analytical calculations and computational algorithms for free and mapped-to-free systems.

2. Physical Contexts: Non-Equilibrium and Integrable Systems

A paradigmatic application involves quantum quenches in integrable models, such as the Lieb–Liniger Bose gas. After a quench from free to hard-core bosons (the so-called Tonks-Girardeau limit), the post-quench steady-state is governed by a Generalized Gibbs Ensemble (GGE). The stationary two-point fermionic correlation function for particle density $n$ is

$$C(x-y) = \langle \Psi^\dagger(x)\Psi(y)\rangle = n\,\exp(-2n|x-y|).$$

Entanglement entropies are computed by:

• Direct evaluation via trace expansions and cumulant techniques for $𝔠_A$.
• Spectral analysis, reducing the eigenvalue problem for $𝔠_A$ (with kernel $n\,e^{-2n|x-y|}$) on $A=[0,\ell]$ to second-order differential equations:

$$v''(x) = -\omega^2 v(x); \qquad \tan(2n\ell\Omega) = \frac{2\Omega}{\Omega^2-1},\quad \Omega = \frac{\omega}{2n},$$

with eigenvalues $\lambda_m = 1/(1+\Omega_m^2)$.

The leading and subleading terms in the large-$\ell$ expansion of the Rényi entropies are obtained,

$$S_A^{(\alpha)} = s_\alpha^1 n\ell + s_\alpha^0 + \mathcal{O}(e^{-4n\ell}),$$

with

$$s_\alpha^1 = \frac{4}{\alpha-1} \sum_{p=1}^{\lfloor \alpha/2\rfloor}\left[1+\sin\left(\frac{\pi(1-2p)}{2\alpha}\right)\right] = \frac{2[\csc(\pi/(2\alpha))-\alpha]}{1-\alpha}.$$

In the von Neumann case,

$$S_A = 2n\ell + 2\ln 2 - 1 + \mathcal{O}(e^{-4n \ell}).$$

Crucially, the thermodynamic GGE entropy density is found to coincide with the extensive part of the stationary entanglement entropy. The diagonal entropy, associated with the non-local structure of the post-quench state, is exactly half of the GGE entropy, emphasizing the role of pairwise excitation structure.

3. Quantum Monte Carlo and Algorithmic Implementations

Efficient Monte Carlo techniques for Rényi entropies have been developed for both lattice and continuum systems.

• Path integral ground state (PIGS) algorithms (Herdman et al., 2014) employ the replica trick, with the estimator for the swap/permutation operator $\Pi_2^A$ furnishing direct access to $S_2$ via $S_2 = -\ln\langle\Pi_2^A\rangle$.
• Advanced estimators exploit symmetry (spatial and imaginary time) to dramatically reduce estimator variance and extend feasible system sizes (Luitz et al., 2014).
• For full-system calculations, these methods also deliver thermodynamic Rényi entropies,

$$S_q^{\mathrm{th}}(\beta) = \frac{1}{1-q} \ln\left[\frac{\operatorname{Tr}(e^{-q\beta H})}{[\operatorname{Tr} e^{-\beta H}]^q}\right],$$

and realize high-precision estimates even at minuscule probabilities.

Recent developments relate Rényi entanglement entropies to nonequilibrium work distributions, utilizing Jarzynski's equality to evaluate entropy as a free energy difference between partition functions of differing trace topology, enabling system sizes far beyond previous numerical limits (D'Emidio, 2019).

4. Universal Subleading Corrections and Scaling Laws

Rényi entropies in critical and symmetry-broken systems reveal universal subleading terms:

• For 1D critical systems, the $n$-th Rényi entropy for a subsystem of length $\ell$ embedded in a 2D $L\times L$ system,

$S_q = a_q \ell + \frac{n_G}{2} \ln \ell + \cdots,$

where $n_G$ is the number of Goldstone modes emerging from broken continuous symmetry (Luitz et al., 2015). The logarithmic prefactor $l_q = n_G/2$ is independent of the Rényi index $q$ and microscopic details.

High-precision QMC and finite-size spin-wave analyses confirm that both SU(2) and U(1) symmetry-breaking give rise to such $l_q$ in accordance with the Metlitski–Grover prediction (Metlitski et al., 2011). Non-universal subleading corrections involve logarithms of logarithms and inverse system size terms.

5. Generalizations, Resource Theory, and Quantum Designs

Rényi entanglement entropies have been extended in various directions:

• The entanglement Rényi $\alpha$-entropy (ER$\alpha$E) forms a continuous spectrum of monotonic entanglement measures, reducing to the entanglement of formation (EoF) at $\alpha\to 1$ (Wang et al., 2015). ER$\alpha$E can render entanglement orders incomparable, in contrast with EoF, thus allowing fine-grained LOCC convertibility analysis. Fully analytical expressions are derived for arbitrary two-qubit, Werner, and isotropic states, and convexity properties determine the optimal pure-state decompositions.
• In resource-theoretic contexts, Rényi relative entropies provide a unified description of both entanglement and coherence monotones, exhibiting operational links (e.g., among maximally correlated states, Rényi-α entanglement and coherence measures coincide and specify conditional α-entropies, while their additivity is proven (Zhu et al., 2017).
• Random quantum states drawn from quantum α-designs achieve almost maximal Rényi entanglement, strengthening Page's theorem and supporting the generalized fast scrambling conjecture: ensembles of order $O(\log d)$ in system dimension suffice to reach maximal entanglement across all Rényi indices (Liu et al., 2017).

6. Analytical Behavior in Various Quantum Systems

Rényi entanglement entropies display distinct analytic features depending on system parameters:

• In the Calogero model, Rényi entropies sensitively detect finite support of the one-particle reduced density matrix—exhibiting non-monotonicity, divergent slopes, and kinks in their dependence on interaction strength, in contrast to the smooth behavior of the von Neumann entropy (Garagiola et al., 2016).
• In two-particle Wigner molecules with harmonic anisotropic traps, for long-range interactions, both the von Neumann and Rényi entropies remain finite for anisotropic traps but diverge logarithmically in the isotropic case. Short-range interactions result in divergence for all anisotropy due to vanishing localization widths; the appearance of non-analyticities in $\alpha < 1$ Rényi entropies signals sudden support truncation in the reduced density matrix (Cuestas et al., 2017).

7. Synthesis and Theoretical Significance

Rényi entanglement entropies serve as fundamental tools in characterizing the entanglement structure of quantum many-body systems. Their extensivity and leading coefficients elucidate the relation between quantum entanglement and thermodynamic entropy (e.g., equivalence of GGE entropy and entanglement entropy post-quench in integrable models). Universal subleading logarithmic corrections provide stringent tests of effective field theory predictions and fundamental symmetries. The ability to interpolate between entanglement measures by continuously varying $\alpha$ allows access to operational resource-theoretic aspects, spectral features, and even genuine quantum computational complexity (as in the context of scrambling and designs). Advanced numerical algorithms leveraging Rényi entropies now drive the frontier in probing entanglement in large, interacting, and continuum quantum systems, and allow direct benchmarking with analytic predictions and experimental results.

Collectively, these features position Rényi entanglement entropies at the intersection of quantum information theory, statistical mechanics, and condensed matter physics as a versatile and rich class of entanglement quantifiers with extensive theoretical, numerical, and experimental relevance.