Genuine Multi-Entropy in the Toric Code

arXiv:2607.06050 · hep-th, cond-mat.str-el, quant-ph · Submitted 2026-07-07 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Genuine Multi-Entropy in the Toric Code".

Mira: Here's the gist of the paper. The gist Genuine multi-entropy probes multipartite entanglement in the toric code, revealing that for stabilizer states, it reduces to lower-party entropic data at low replica indices but becomes a topological invariant at index n=4, distinguishing it from conventional topological entanglement entropy.

Kai: Genuine Multi-Entropy and its Definition The genuine multientropy (GM) is constructed by taking suitable linear combinations of multi-entropies so as to subtract these lower-partite contributions, providing a diagnostic for irreducible multipartite entanglement GM is built from the multi-entropy

8: by a genuineness prescription that subtracts lower-partite contributions, so that GM isolates entanglement that is intrinsically shared among all parties.

Mira: For q = 2, it reduces to the ordinary R´enyi entropy For q > 2, however, S(q)n receives contributions not only from irreducible q-partite correlations but also from lowerpartite correlations.

Kai: The four-partite quantity GM(q=4), GM(4)n (A: B: C: D) = S(4)n (A: B: C: D) - 1/3 h S(3)n (AB: C: D) + S(3)n (AC: B: D) + S(3)n (AD: B: C) + S(3)n (BC: A: D) + S(3)n (BD: A: C) + S(3)n (CD: A: B), GM is defined by a symmetric replica contraction of n q-1 copies of the state Behavior in Stabilizer States The paper establishes a threshold in the R´enyi index n for the GM of the stabilizer states, conjecturing that for generic q ≥ 4, the q-partite GM of a stabilizer state at replica index n < q collapses to a linear combination of multi-entropies involving at most q − 2 parties, with n = q being the threshold at which this reduction no longer holds.

Title and authors: Kai: We're diving into this paper, "Genuine Multi-Entropy in the Toric Code". The title itself points to what they're trying to do: probe multipartite entanglement in this specific system.

Mira: It’s about moving beyond just looking at two-party or three-party correlations and finding something that is genuinely shared across all four parties.

Lev: The authors set out to see if this genuine multi-entropy can actually isolate information that conventional lower-party entropic data, like the topological entanglement entropy, cannot explain on its own.

Kai: That’s the main question they tackle in this paper. They analyze toric code ground states that are described by stabilizer states because those allow them to evaluate these quantities exactly.

Mira: And they find that for these stabilizer states, the genuine multi-entropy reflects the topological structure and symmetries of the toric code itself, which is interesting because it's a topological phase and a quantum error-correcting code simultaneously.

Lev: The paper uses a framework for evaluating multi-invariants on these stabilizer eigenstates to compute the genuine multi-entropy explicitly.

The paper's summary: Kai: So, when we look at the summary of "Genuine Multi-Entropy in the Toric Code", it describes how they build this genuine multi-entropy by taking linear combinations of various multi-entropies to subtract those lower-party contributions.

Mira: It’s a diagnostic tool for irreducible multipartite entanglement because that construction is designed specifically to isolate the part of the entanglement that's intrinsically shared among all parties.

Lev: They define the four-partite quantity as GM(q=four), GM(four)n (A: B: C: D) by taking S(four) n and subtracting specific combinations involving lower-party multi-entropies like S(three) n <ref:2607.06050#pg1>.

Kai: That formula is quite complex, involving several terms subtracted from the four-partite entropy to isolate the genuine part. They also note that for q=two it reduces to the ordinary R´enyi entropy, but for q greater than two you get those lower-party contributions that they subtract out <ref:2607.06050#pg1>.

Mira: And they show that this subtraction process results in a term proportional to the R´enyi tripartite information when n is small, which is what gives them the initial findings for stabilizer states.

The paper's improvements: Lev: The main improvement or finding they establish is the existence of a threshold in the R´enyi index n for these genuine multi-entropies in the stabilizer states.

Kai: They conjecture that for generic q greater than or equal to four if you look at replica index n < q, this quantity collapses into a linear combination of multi-entropies involving at most q minus two parties <ref:2607.06050#pg1>.

Mira: But the paper identifies n=four as the exact threshold where this reduction stops holding, and beyond that point, it's what they call a topological invariant <ref:2607.06050#pg1>.

Lev: For stabilizer states in the toric code, at replica indices n=two and n=three the genuine multi-entropy is shown to reduce to a linear combination involving only tripartite information, specifically I three <ref:2607.06050#pg1>.

Kai: That means for those lower indices, the genuine multi-entropy doesn't contain any independent genuinely four-partite information beyond what you already know from the total quantum dimension data.

Conclusion: Mira: So to wrap up the paper "Genuine Multi-Entropy in the Toric Code", they’ve shown that for stabilizer states, the genuine multi-entropy behaves very differently depending on whether you're at a low replica index n or at n=four <ref:2607.06050#pg1,Genuine Multi-Entropy in the Toric Code>.

Kai: At n=four it becomes a topological invariant <ref:2607.06050#pg1>. This means it remains unchanged even when you deform the partitions as long as you keep the topology of those four regions the same, and it’s independent of how big the lattice size is.

Lev: And this invariance is demonstrated by showing that for non-stabilizer states or excited states, these relations break down, which shows that stabilizer states are special in this regard.

Mira: The paper suggests that any genuine multipartite signal you calculate for a Levin-Wen model ground state using a Kitaev–Preskill type partition will simply be the toric code TQFT partition function on the graph encoded manifold.

Kai: It points toward using this genuine multi-entropy to characterize topological phases beyond just the total quantum dimension data, and they suggest looking at things like double semion theory next.

Lev: For me, it’s important that this n=four GM is a topological invariant because if it holds up under those kinds of deformations, it gives us a solid piece of evidence about the robust nature of these topological features in the code <ref:2607.06050#pg1>.

Mira: Exactly. It tells us that for stabilizer states, we have something fundamentally different than what we see in generic non-stabilizer states when we look at these multipartite correlations.

Department of Theoretical Physics, Tata Institute of Fundamental Research · Department of Physics, National Tsing Hua University · Yukawa Institute for Theoretical Physics, Kyoto University

hep-th, cond-mat.str-el, quant-ph

Submitted: 2026-07-07

Updated: 2026-10-19

Comments: 17 pages, 19 figures. v2: Sec. III corrected. The q=4, n=4 genuine multi-entropy distinguishes whether the numbers of vertex-type and plaquette-type junction points in Kitaev-Preskill partitions are equal or not (Eq. (17)). Fig. 4 revised accordingly. Comments are welcome!

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 73/100

The gist: The gist Genuine multi-entropy probes multipartite entanglement in the toric code, revealing that for stabilizer states, it reduces to lower-party entropic data at low replica indices but becomes a

Key concepts

Genuine Multi-Entropy (GM): GM is a specific measure designed to isolate entanglement shared intrinsically among all parties in a multipartite system. It is constructed by subtracting lower-party contributions from multi-entropies, aiming to find genuinely irreducible multipartite correlations.
Stabilizer States: These are specific quantum states within the toric code that possess certain symmetry properties. The paper investigates how GM behaves for these states, finding a critical replica index (n=4) where its behavior fundamentally changes.
Topological Invariance: A property of a physical quantity that remains unchanged even when the underlying geometry or partition is locally deformed, provided the overall topology is preserved. The paper demonstrates that GM at n=4 exhibits this invariance for stabilizer states, making it a robust topological invariant.
Replica Index (n): The replica index 'n' is a parameter used in the calculation of multi-entropies. The paper identifies n=4 as the crucial threshold where the genuine multipartite information captured by GM becomes independent of lower-party correlations and exhibits topological properties.

Terminology

Entanglement — 5× in this paper · explained in 4 episodes
Entanglement Entropy — 3× in this paper · explained in 5 episodes
Multipartite Entanglement — 2× in this paper · explained in 3 episodes
Stabilizer States — 2× in this paper · explained in 4 episodes

Summary

The gist Genuine multi-entropy probes multipartite entanglement in the toric code, revealing that for stabilizer states, it reduces to lower-party entropic data at low replica indices but becomes a topological invariant at index n=4, distinguishing it from conventional topological entanglement entropy.

Genuine Multi-Entropy and its Definition

The genuine multientropy (GM) is constructed by taking suitable linear combinations of multi-entropies so as to subtract these lower-partite contributions, providing a diagnostic for irreducible multipartite entanglement GM is built from the multi-entropy [8] by a genuineness prescription that subtracts lower-partite contributions, so that GM isolates entanglement that is intrinsically shared among all parties. For q = 2, it reduces to the ordinary R´enyi entropy For q > 2, however, S(q)n receives contributions not only from irreducible q-partite correlations but also from lowerpartite correlations. The four-partite quantity GM(q=4), GM(4)n (A: B: C: D) = S(4)n (A: B: C: D) - 1/3 h S(3)n (AB: C: D) + S(3)n (AC: B: D) + S(3)n (AD: B: C) + S(3)n (BC: A: D) + S(3)n (BD: A: C) + S(3)n (CD: A: B), GM is defined by a symmetric replica contraction of n q-1 copies of the state

Behavior in Stabilizer States

The paper establishes a threshold in the R´enyi index n for the GM of the stabilizer states, conjecturing that for generic q ≥ 4, the q-partite GM of a stabilizer state at replica index n < q collapses to a linear combination of multi-entropies involving at most q − 2 parties, with n = q being the threshold at which this reduction no longer holds. For the toric code stabilizer states, for n = 2 and n = 3, GM reduces to the R´enyi tripartite information I3 Specifically, GM(q=4)n=2 (A: B: C: D) = -a - 1/12 I3, where a is the convention parameter of GM(q=4)n [6] and with the standard choice a = 1/3 this gives GM(q=4)n=2 = -1/4 I3. Similarly, at n = 3, GM(q=4)n=3 (A: B: C: D) = -a - 1/9 I3 with a = 1/3 yielding GM(q=4)n=3 = -2/9 I3. This means that for n < 4, GM is completely determined by the same total-quantum-dimension data as the TEE for the Kitaev–Preskill partition, i.e., GM contains no independent genuinely four-partite information beyond the topological entanglement entropy

Topological Invariance at n=4

At n = 4, this reduction to I3 no longer holds, and the GM then contains information beyond the tripartite information and hence beyond the total quantum dimension D probed by I3 For stabilizer states, the n = 4 GM is a topological invariant in the sense that it remains unchanged under deformations of the partitions that preserve the topology of the partition, and is independent of the lattice size L This is demonstrated by showing that for generic non-stabilizer states within the same topological ground-state manifold, and for excited states obtained by coherent superpositions of local excitations, these relations no longer hold The n = 4 GM is shown to be a topological invariant because it is invariant under local deformations of the subregions A, B, C, and D that preserve the topology of the four-partition as illustrated in Fig. 4 Furthermore, for non-stabilizer states, GM detects more than the total quantum dimension D throughout the ground-state manifold

Comparison to Non-Stabilizer States and Excitations

For generic ground-state superpositions within the topological ground-state manifold and for coherent superpositions of local excitations, the low-n reduction fails, showing that GM detects more than the total quantum dimension D throughout the ground-state manifold Specifically, in a one-parameter non-stabilizer family Ψ10(θ)⟩, GM and I3 are physically different quantities for generic values of θ This is because the proportionality relation between GM(q=4)n and I3,n found for stabilizer states does not extend to a general linear combination of toric-code ground states Furthermore, single-edge Pauli excitations or arbitrary Pauli-string excitations do not produce a nontrivial change in GM because they factorize with respect to the multipartition as UP = UA ⊗ UB ⊗ UC ⊗ UD, making them local unitary transformations

Comparison with Color Codes

When comparing the toric code to the color code on a honeycomb lattice, which has an underlying TQFT equivalent to two copies of the toric code TQFT, the n = 4 GM in the color code is twice that of the toric code This provides a consistency check to our claim that the n = 4 GM is a topological invariant of topological stabilizer codes

Implications for TQFT Data

The analysis suggests that any genuine multipartite signal, including the GM, evaluated for a Kitaev–Preskill type partition for the ground state of a Levin-Wen model, is the TQFT partition function on the graph-encoded manifold This implies that the n = 4 GM is simply computing the toric-code TQFT partition function on the gem corresponding to the n = 4 R´enyi multi-entropy The next step suggested is to calculate this for double semion theory, which has a different TQFT structure but shares the same total quantum dimension D This suggests a promising direction in which genuine multi-entropy can be used to distinguish and characterize topological phases beyond the information contained in the total quantum dimension

Future Directions

The paper concludes by suggesting that future work should focus on figuring out the topology of the n = 4 gem and working this calculation out A related question is to calculate the n = 4 GM for the double semion theory and compare it to the toric code This suggests a promising direction in which genuine multi-entropy can be used to distinguish and characterize topological phases beyond the information contained in the total quantum dimension

Acknowledgments

S.A. is supported by the Department of Atomic Energy, Government of India, under Project Identification No. RTI 4002, and the Infosys Endowment for the study of the Quantum Structure of Spacetime The work of N.I. was supported in part by MEXT KAKENHI Grantin-Aid for Transformative Research Areas A “Extreme Universe” No. 21H05184 The work of N.I. was also supported in part by NSTC of Taiwan Grant Number 114-2112-M-007-025-MY3. The work of A.M. was supported by JSPS KAKENHI Grant Number JP26KJ0186

Appendix Details

The replica definition of the R´enyi multi-entropy is given in Appendix A, where S(q)n (A1: · · ·: Aq) is defined by a symmetric replica contraction of n q-1 copies of the state The explicit ground-state basis for the 2 × 2 lattice toric code is provided in Appendix B, which lists the four states ψ00⟩, ψ10⟩, ψ01⟩, and ψ11⟩.

Improvements for AI systems

The improved AI system can distinguish between genuine multipartite entanglement features that are special to stabilizer states versus those associated with topological ground-state sectors by analyzing whether the relation between GM(q=4) and I3,n found for stabilizer states does not extend to a general linear combination of toric-code ground states.
The system can verify the topological invariance of genuine multi-entropy by checking if the n = 4 GM is a topological invariant in the sense that it remains unchanged under local deformations of the subregions A, B, C, and D that preserve the topology of the partition and is independent of the lattice size L.
The AI can identify non-stabilizer states exhibiting genuinely new entanglement structures by analyzing how the relations (2) for n = 2 and (3) for n = 3 are generically violated in general ground-state superpositions, as opposed to stabilizer states.