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arxiv: 2509.00593 · v3 · submitted 2025-08-30 · ✦ hep-th · cond-mat.str-el· quant-ph

Genuine multientropy, dihedral invariants and Lifshitz theory

Cl\'ement Berthi\`ere , Paul Gaudin This is my paper

Pith reviewed 2026-05-18 19:16 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords multientropydihedral invariantsLifshitz theoryRényi entropyreflected entropyentanglement monotonesreplica methodstabilizer states
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The pith

Genuine multientropy for Lifshitz ground states equals mutual information plus logarithmic negativity and continues analytically to non-integer Rényi indices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors compute the genuine multientropy, a local-unitary invariant of state replicas, for Lifshitz ground states in tripartite pure states. They obtain its analytic continuation to non-integer values of the Rényi index and express it in terms of mutual information and logarithmic negativity. This same relation holds for stabilizer states. For general tripartite pure states, they relate dihedral invariants to Rényi reflected entropies by showing that dihedral permutations of replicas match the reflected construction or realignment of density matrices.

Core claim

For Lifshitz ground states the genuine multientropy is computed explicitly via the replica method and continued analytically to non-integer Rényi indices. It reduces to a combination of mutual information and logarithmic negativity, a reduction that also holds for stabilizer states. Dihedral invariants on replica states for any tripartite pure state are equivalent to Rényi reflected entropies because dihedral permutations realize the reflected replica construction or, equivalently, the realignment operation on density matrices.

What carries the argument

Genuine multientropy as a multi-invariant of state replicas, computed through the replica construction in Lifshitz ground states.

If this is right

  • The reduction allows multientropy to be evaluated from standard bipartite entanglement measures in Lifshitz systems and stabilizer states.
  • Analytic continuation supplies a definition of multientropy for non-integer Rényi indices without further regularization.
  • Dihedral invariants supply an alternative replica route to Rényi reflected entropies for any tripartite pure state.
  • Replica permutations under the dihedral group are equivalent to the reflected replica construction or to realignment of density matrices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the reduction extends beyond Lifshitz and stabilizer states it would simplify multipartite entanglement calculations across a broader class of many-body systems.
  • Linking multientropy to reflected entropies could unify different replica-based probes of higher-order correlations in quantum field theory.
  • The same techniques might be tested on other scale-invariant critical points to extract universal multipartite features.

Load-bearing premise

The replica construction for Lifshitz ground states admits a well-defined analytic continuation in the Rényi index that preserves the quantity as a valid entanglement monotone.

What would settle it

An explicit calculation of genuine multientropy for a concrete Lifshitz ground state at a non-integer Rényi index that fails to match the combination of mutual information and logarithmic negativity would falsify the reduction.

Figures

Figures reproduced from arXiv: 2509.00593 by Cl\'ement Berthi\`ere, Paul Gaudin.

Figure 1
Figure 1. Figure 1: FIG. 1. Examples of tripartitions, with subsystem sizes [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Graphical representation of the calculations leading to the multi-entropy for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
read the original abstract

Multi-invariants are local-unitary invariants of state replicas introduced as potential probes of multipartite entanglement and correlations in quantum many-body systems. In this paper, we investigate two multi-invariants for tripartite pure states, namely multientropy and dihedral invariant. We compute the (genuine) multientropy for Lifshitz groundstates, and obtain its analytical continuation to noninteger values of R\'enyi index. We show that the genuine multientropy can be expressed in terms of mutual information and logarithmic negativity, a relation that also holds for stabilizer states. For general tripartite pure states, we demonstrate that dihedral invariants are related to R\'enyi reflected entropies. In particular, we show that the dihedral permutations of replicas are equivalent to the reflected construction, or alternatively to the realignment of density matrices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript investigates multi-invariants for tripartite pure states, focusing on multientropy and the dihedral invariant. It computes the genuine multientropy for Lifshitz ground states and obtains its analytical continuation to non-integer values of the Rényi index. The authors show that the genuine multientropy can be expressed in terms of mutual information and logarithmic negativity, a relation that also holds for stabilizer states. For general tripartite pure states, they relate dihedral invariants to Rényi reflected entropies, demonstrating that dihedral permutations of replicas are equivalent to the reflected construction or realignment of density matrices.

Significance. If the central claims are substantiated with explicit derivations, the work would provide analytical tools for multipartite entanglement in Lifshitz theories with anisotropic scaling, which are relevant to certain condensed-matter systems. The explicit relations to mutual information, logarithmic negativity, and reflected entropies could enable cross-checks with established measures and extend replica techniques beyond relativistic CFTs. The analytic continuation to non-integer Rényi indices, if rigorously justified, would be a useful computational advance.

major comments (1)
  1. [Lifshitz ground states computation and analytic continuation] The central claim of an explicit computation of genuine multientropy for Lifshitz ground states together with its analytic continuation to non-integer Rényi index lacks displayed derivation steps, the explicit continued expression, pole locations, or regularization procedure. Given the anisotropic scaling z ≠ 1, which alters the replica manifold and twist-operator structure relative to relativistic cases, this omission prevents verification that the n → 1 limit remains a valid entanglement monotone. This issue is load-bearing for the main result stated in the abstract.
minor comments (1)
  1. The relation between genuine multientropy and mutual information plus logarithmic negativity is stated to hold for stabilizer states, but a brief explicit check or reference to the stabilizer case would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive feedback. We are pleased that the referee recognizes the potential utility of our results for studying multipartite entanglement in Lifshitz theories. We address the major comment below.

read point-by-point responses

  1. Referee: The central claim of an explicit computation of genuine multientropy for Lifshitz ground states together with its analytic continuation to non-integer Rényi index lacks displayed derivation steps, the explicit continued expression, pole locations, or regularization procedure. Given the anisotropic scaling z ≠ 1, which alters the replica manifold and twist-operator structure relative to relativistic cases, this omission prevents verification that the n → 1 limit remains a valid entanglement monotone. This issue is load-bearing for the main result stated in the abstract.

    Authors: We thank the referee for highlighting this important point. We agree that the derivation steps for the computation of genuine multientropy in Lifshitz ground states, together with the details of the analytic continuation, would benefit from a more explicit presentation to facilitate verification, particularly in light of the anisotropic scaling z ≠ 1 and its effects on the replica manifold and twist operators. In the revised manuscript, we will expand the relevant sections to include step-by-step derivations of the genuine multientropy for Lifshitz ground states. We will provide the explicit form of the analytically continued expression, discuss the locations of poles, and outline the regularization procedure. We will also elaborate on the validity of the n → 1 limit as an entanglement monotone, carefully addressing the modifications arising from the Lifshitz scaling. These revisions will strengthen the substantiation of our central claim. revision: yes

Circularity Check

0 steps flagged

Derivation of genuine multientropy and dihedral invariants is self-contained

full rationale

The paper derives the genuine multientropy for Lifshitz ground states via replica methods and obtains its analytic continuation in the Rényi index, then expresses it in terms of mutual information and logarithmic negativity. It further relates dihedral invariants to Rényi reflected entropies for general tripartite states. These steps are presented as explicit computations and equivalences rather than reductions to fitted parameters or prior self-definitions. No load-bearing self-citations or ansatzes imported from the authors' own work are required for the central results, and the relations hold independently for stabilizer states as an external check. The replica construction is invoked with the standard assumption that the continuation is well-defined, but this does not make the output equivalent to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard replica-trick assumptions for Rényi entropies and on the existence of well-behaved Lifshitz ground states; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Replica trick for Rényi entropies admits analytic continuation to non-integer index while preserving monotonicity properties.
    Invoked when authors obtain the non-integer continuation for genuine multientropy.
  • domain assumption Lifshitz ground states are pure tripartite states whose multi-copy invariants can be computed via standard quantum-field-theory techniques.
    Required for the explicit computation stated in the abstract.

pith-pipeline@v0.9.0 · 5677 in / 1474 out tokens · 31578 ms · 2026-05-18T19:16:55.506459+00:00 · methodology

discussion (0)

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Forward citations

Cited by 2 Pith papers

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    Δ^(3)_p is a non-negative signal detecting genuine tripartite entanglement, extended via the E_w = E_p conjecture to holographic systems in AdS3/CFT2.

Reference graph

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    DisjointA, Bin the bulk C1 A C2 B C3 ℓC1 ℓA ℓC2 ℓB ℓC3 M (3) 3 = φ 6 φ 12 φ 5 φ 11 φ 4 φ 10 φ 3 φ 9 φ 2 φ 8 φ 1 φ 7 FIG

    On a finite interval a. DisjointA, Bin the bulk C1 A C2 B C3 ℓC1 ℓA ℓC2 ℓB ℓC3 M (3) 3 = φ 6 φ 12 φ 5 φ 11 φ 4 φ 10 φ 3 φ 9 φ 2 φ 8 φ 1 φ 7 FIG. A1. Disjoint regionsAandBin the bulk. Right: Replica graph resulting from the multi-entropy symmetry forn= 3. Multi-entropy— We begin by computing the multi-entropy for Dirichlet boundary conditions, and then bri...

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