The ternary replica collapse for four-party stabilizer states

Abstract

For a normalized pure qubit stabilizer state, ternary replica contractions are positive real numbers. Their four-party genuine multi-entropy is determined by the tripartite information.

Definition 1.1 (Shifting a replica coordinate).

$$\forall (n:\mathbb{N}),\forall (q:\mathbb{N}),\forall (c:\mathrm{Fin}\left(q\right)),\forall (r:(\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(n\right))),\mathrm{shift}\left(n,q,c,r\right)=(i:\mathrm{Fin}\left((q-1)\right))\mapsto \mathrm{ite}\left((i:\mathbb{N})=(c:\mathbb{N}),r\left(i\right)+1,r\left(i\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.shift (✓ std3).

Citation. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v1.

Commentary.

Replica labels are functions Fin(q − 1) → ℤ/nℤ. The shift adds one at the coordinate whose natural index equals the colour c. Colour q − 1 has no such coordinate and acts as the identity. Natural differences such as q − 1 and q − 2 in these formulas use truncated subtraction.

Definition 1.2 (Contracting the replicas).

$$\forall (n:\mathbb{N}),\forall (q:\mathbb{N}),[\mathrm{NeZero}\left(n\right)],\forall (N:\mathbb{N}),\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),Z\left(n,q,col,\psi \right)=\sum _{(x:((\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(n\right))\to (\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))))}\prod _{(r:(\mathrm{Fin}\left((q-1)\right)\to \mathrm{ZMod}\left(n\right)))}\mathrm{star}\left(\psi (x\left(r\right))\right)\cdot \psi (((u:\mathrm{Fin}\left(N\right))\mapsto x\left(\mathrm{shift}\left(n,q,\mathrm{col}\left(u\right),r\right),u\right)))$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.Z (✓ std3).

Citation. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v1.

Commentary.

A configuration x assigns a binary N-qubit string to each replica label r. The conjugated amplitude is evaluated at x(r); the other amplitude takes qubit u from the replica shifted by its colour col(u). Summing the product gives the computational-basis contraction of Appendix A.

Definition 1.3 (The normalized multi-entropy).

$$\forall (n:\mathbb{N}),\forall (q:\mathbb{N}),[\mathrm{NeZero}\left(n\right)],\forall (N:\mathbb{N}),\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),S\left(n,q,col,\psi \right)=\frac{1}{(1-(n:\mathbb{R}))}\cdot \frac{1}{(n:\mathbb{R}{)}^{(q-2)}}\cdot \log \left(\mathrm{Re}\left(\frac{Z\left(n,q,col,\psi \right)}{Z{\left(1,q,col,\psi \right)}^{n^{(q-1)}}}\right)\right)$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.S (✓ std3).

Citation. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v1.

Commentary.

The normalization divides Z(n,q,col,ψ) by the indicated power of Z(1,q,col,ψ). The logarithm is applied to the real part of the quotient, with the prefactors specified in Appendix A. The casts of n to ℝ appear explicitly.

Definition 1.4 (Four cuts minus three cuts).

$$\forall (n:\mathbb{N}),[\mathrm{NeZero}\left(n\right)],\forall (N:\mathbb{N}),\forall (party:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(4\right))),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),I3\left(n,party,\psi \right)=S\left(n,2,(([0,0,0,1]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right)+S\left(n,2,(([0,0,1,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right)+S\left(n,2,(([0,1,0,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right)+S\left(n,2,(([1,0,0,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right)-(S\left(n,2,(([0,0,1,1]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right)+S\left(n,2,(([0,1,0,1]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right)+S\left(n,2,(([0,1,1,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right))$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.I3 (✓ std3).

Citation. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v1.

Commentary.

Label A, B, C, D by 0, 1, 2, 3. Each displayed tuple is a function from Fin 4 to the stated colour set, composed with party. In order, the four positive terms are ABC:D, ABD:C, ACD:B and BCD:A; the three subtracted terms are AB:CD, AC:BD and AD:BC. This is the Rényi tripartite information I₍₃,n₎ in Eq. (1) of the paper.

Definition 1.5 (The genuine four-party combination).

$$\forall (n:\mathbb{N}),[\mathrm{NeZero}\left(n\right)],\forall (a:\mathbb{R}),\forall (N:\mathbb{N}),\forall (party:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(4\right))),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),GM4\left(n,a,party,\psi \right)=S\left(n,4,party,\psi \right)-\frac{1}{3}\cdot (S\left(n,3,(([0,0,1,2]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(3\right)))\circ party),\psi \right)+S\left(n,3,(([0,1,0,2]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(3\right)))\circ party),\psi \right)+S\left(n,3,(([0,1,2,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(3\right)))\circ party),\psi \right)+S\left(n,3,(([1,0,0,2]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(3\right)))\circ party),\psi \right)+S\left(n,3,(([1,0,2,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(3\right)))\circ party),\psi \right)+S\left(n,3,(([1,2,0,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(3\right)))\circ party),\psi \right))+\frac{1}{3}\cdot (S\left(n,2,(([0,0,0,1]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right)+S\left(n,2,(([0,0,1,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right)+S\left(n,2,(([0,1,0,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right)+S\left(n,2,(([1,0,0,0]:(\mathrm{Fin}\left(4\right)\to \mathrm{Fin}\left(2\right)))\circ party),\psi \right))-a\cdot I3\left(n,party,\psi \right)$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.GM4 (✓ std3).

Citation. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v1.

Commentary.

The six three-colour tuples merge AB, AC, AD, BC, BD and CD, respectively. Their entropies have coefficient −1/3, while the four singleton cuts have coefficient 1/3. The remaining term is −a times the tripartite information, for an arbitrary real parameter a, as in §2 of the paper.

Definition 1.6 (Positivity and the n = 3 collapse).

$$claim\iff (\forall (N:\mathbb{N}),\forall (party:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(4\right))),\forall (\psi :((\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right))\to \mathbb{C})),\mathrm{StabilizedBy}\left(pauliSet,\psi \right)\Rightarrow (\sum _{(x:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(2\right)))}{\Vert \psi (x)\Vert }^2=1\Rightarrow ((\forall (q:\mathbb{N}),\forall (col:(\mathrm{Fin}\left(N\right)\to \mathrm{Fin}\left(q\right))),0<\mathrm{Re}\left(Z\left(3,q,col,\psi \right)\right)\wedge (\mathrm{Im}\left(Z\left(3,q,col,\psi \right)\right)=0))\wedge (\forall (a:\mathbb{R}),GM4\left(3,a,party,\psi \right)=-(a-\frac{1}{9})\cdot I3\left(3,party,\psi \right)))))$$

Formalization. D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.claim (✓ std3).

Citation. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v1.

Commentary.

The proposition quantifies over every qubit count, every assignment to four labelled parties and every amplitude. Its two assumptions are Pauli stabilization and unit squared norm. The first conclusion includes every colour count and colouring and asserts both strict positivity of the real part and a zero imaginary part. The second conclusion is Eq. (3) for every real a, with I₃ interpreted as I₍₃,₃₎. Empty parties are allowed.

Theorem 1.7 (The stabilizer collapse).

$$claim$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.result (✓ std3). ∎

Resolves. Problems/akella-2026-stabilizer-multi-entropy-n3-collapse (proved) by D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.result.

Source. Repository-derived.

Acknowledgement. S. Akella, N. Iizuka, A. Miyata (2026). Genuine Multi-Entropy in the Toric Code. DOI: 10.48550/arXiv.2607.06050. URL: https://arxiv.org/abs/2607.06050v1.

Commentary.

A stabilizer state is a scalar multiple of a local-unitary image of a graph amplitude. Replica contractions are unchanged by the identical local unitaries on all copies. For graph amplitudes, the ternary translation form decomposes into bilinear blocks indexed by opposite nonzero Fourier modes. A block contributes the inverse power of two determined by its rank, giving positive real contractions. Each rank depends on the partition of the four party labels induced by its mode. The weighted counts of these partitions give the coefficient 1/9, establishing the displayed identity and the counting argument asked for in §6.

References

• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.GM4
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.I3
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.S
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.Z
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.claim
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.result
• Truth anchor: D5/S3/Quantum/Entanglement/StabilizerMultiEntropyCollapse.shift
• Dependency: D5/S3/QuadraticForms/TernaryTranslationQuadraticNormalForm
• Dependency: D5/S3/Quantum/Information/BinaryStabilizerGraphNormalForm