Genuine Multi-Entropy in Quantum Systems

• Genuine multi-entropy is a multipartite entanglement measure that isolates intrinsic three-way or higher-order quantum correlations by subtracting bipartite contributions.
• It employs replica methods, analytic continuation of the Rényi index, and permutation symmetries to derive explicit formulas applicable in Lifshitz theories and topologically ordered states.
• The measure bridges established bipartite metrics like mutual information and logarithmic negativity, validated through its equivalence with dihedral invariants and reflected entropies.

Genuine multi-entropy is a multipartite entanglement measure designed to capture quantum correlations beyond those present in mere bipartite or partially separable states. By focusing on the “genuine” component—subtracting all contributions explainable by bipartitions—this measure probes the intrinsic three-way or higher-order entanglement fundamental to quantum many-body systems. Recent developments have established concrete formulas for genuine multi-entropy via the replica method, analytical continuation in the Rényi index, and explicit connections to invariants built from permutation symmetries, with numerous applications from quantum field theory to topological and stabilizer states (Berthière et al., 30 Aug 2025).

1. Definition and Conceptual Role

Genuine multi-entropy for a tripartite pure state (A, B, C) is defined as the difference between the full multi-entropy, constructed by replica methods, and the average of individual Rényi entropies: $$\mathrm{G}_n^{(3)}(A:B:C) = S_n^{(3)}(A:B:C) - \frac{1}{2}\big[S_n(A) + S_n(B) + S_n(C)\big]$$ where $S_n^{(3)}(A:B:C)$ is computed via a “replica graph” contracting $n^2$ copies of the (reduced) density matrix for %%%%2%%%%, whereas $S_n(A)$ is the standard single-region Rényi entropy. The genuine multi-entropy thus isolates entanglement that cannot be reconstructed from pairwise contributions and vanishes for states factorizing across any bipartition.

This concept addresses the need for multipartite diagnostics in quantum information theory, where simple bipartite entropy or mutual information fails to detect true multi-party quantum correlations.

2. Computational Framework in Lifshitz Theories

In Lifshitz quantum field theories—characterized by nonrelativistic scaling—the ground state wavefunctions (often of Rokhsar-Kivelson type) lend themselves to exact calculation of multipartite entanglement via Gaussian integrals on the replica graph: $$\mathrm{G}_n^{(3)}(A:B:C) = \frac{2-n}{4n} \log\left( \frac{ \sinh(\omega \ell_{AC}) \sinh(\omega \ell_{BC}) }{ \sinh(\omega \ell_{C}) \sinh(\omega \ell) } \right)$$ where $\omega$ is a mass parameter from the Hamiltonian and the $\ell$'s are subsystem lengths or other geometric partition parameters. The Gaussian integral’s determinant encodes the symmetry and permutation structure of the replicas, allowing explicit computation for any integer $n$ and, critically, analytic continuation to noninteger Rényi indices using identities for circulant matrices and Chebyshev polynomials.

The ability to extend these expressions beyond integer $n$ allows for precise paper in, e.g., the von Neumann limit, and for exploration of nonstandard entropy regimes.

3. Analytical Continuation and Invariant Structure

The structure of the replica graph’s determinant facilitates analytic continuation in $n$, enabling evaluation of the multi-entropy at fractional Rényi indices—a procedure central to comparisons with other entanglement measures (such as mutual information at $n=1/2$ or the logarithmic negativity). The paper demonstrates that the mathematical expressions, derived via replica symmetry and permutation operators, admit closed-form analytic expressions in terms of hyperbolic functions for arbitrary $n$.

This analytic flexibility broadens the utility of genuine multi-entropy in theoretical analysis and links it to the broader class of permutation invariants.

4. Relationship to Mutual Information and Logarithmic Negativity

A key result is the explicit reduction of genuine multi-entropy in Lifshitz ground states to established bipartite measures: $$\mathrm{G}_n^{(3)}(A:B:C) = \frac{2-n}{2n}\left[ I_{1/2}(A:B) - 2 \mathcal{E}(A:B) \right]$$ where $I_{1/2}(A:B)$ is the Rényi mutual information at index $\frac{1}{2}$, and $\mathcal{E}(A:B)$ denotes logarithmic negativity (from partial transpose). This relation underscores that genuine multi-entropy, though UV-finite and strictly multipartite, can serve as a bridge connecting multipartite diagnostics directly to the well-understood bipartite framework.

The measure vanishes for biseparable states and is maximal for fully-entangled tripartite (GHZ-type) states.

5. Dihedral Invariants and Reflected Entropies

The “dihedral invariants”—constructed by contracting $2n$ state replicas with permutation operators forming the dihedral group $\mathbb{D}_{2n}$—represent an alternative family of multipartite probes, defined as: $$\mathcal{D}_{2n}(A:B) = \frac{1}{1-n}\log \left( \frac{ \mathcal{Z}_{2n} }{ \mathcal{Z}_2^n } \right)$$ where $\mathcal{Z}_{2n}$ is the partition function with dihedral replica permutations. The authors show that these invariants are mathematically equivalent to the Rényi reflected entropies, $S_{2,n}^{R}(A:C)$, obtained by realigning density matrices. This equivalence links replica symmetry, permutation group theory, and matrix realignment in the multipartite context, demonstrating that dihedral invariants probe the same physical features as reflected entropy.

6. Applications, Implications, and Future Directions

Genuine multi-entropy, especially as realized through explicitly computable formulas in Lifshitz theory, provides a robust, UV-finite tool for quantifying intrinsic multipartite entanglement. Its expressions in terms of linking numbers—as in topologically ordered states or Chern-Simons theory—suggest further extensions into topological phases and quantum gravity contexts.

The relation to mutual information and logarithmic negativity enables cross-verification and suggests methodological compatibility with existing quantum information tools. The direct equivalence between dihedral invariants and Rényi reflected entropies validates the use of permutation-based approaches to multipartite entanglement.

A plausible implication is the ultimately broad applicability of genuine multi-entropy and dihedral invariants for benchmarking quantum correlations in many-body systems, quantum simulators, and topological quantum field theories, as well as for guiding the design of quantum networks relying on multipartite entanglement resources.

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In summary, genuine multi-entropy—defined via replica tricks, permutation symmetries, and complemented by the analytical continuation of the Rényi index—emerges as a mathematically rigorous and operationally meaningful probe of the true multipartite entanglement in quantum many-body systems. Its connections to established bipartite measures, equivalence with group-based invariants, and broad physical applicability position it as a central concept in the paper of quantum correlations beyond the bipartite domain (Berthière et al., 30 Aug 2025).