The minimum of the long-time algebra OTOC

F. Andreadakis, E. Dallas, P. Zanardi, Long-time Quantum Scrambling and Generalized Tensor Product Structures, Phys. Rev. A 109, 052424 (2024), arXiv:2312.13386v2, the Conjecture of Section IV:

Let $H$ be an NRC Hamiltonian and $\mathcal{A}f$ be the equivalence class of isomorphic algebras of the form $\mathcal{A}\cong \oplus{J=1}^{d_\mathcal{Z}} \mathds{1}{n_J} \otimes \mathcal{L}(\mathbb{C}^{d_J} )$. Then, the algebra corresponding to a gTPS (\cref{structure_H}) for which the Hamiltonian eigenstates have the form ${\ket{\psi_J}\otimes\ket{\phi_J} , \vert , J=1,\dots d\mathcal{Z}; , \psi_J = 1,\dots, n_J; , \phi_J = 1,\dots , d_J }$ minimizes the $\mathcal{A}$-OTOC LTA over the family of isomorphic algebras. The minimum value is $\overline{G_{\mathcal{A}}}^{NRC}{min}=1-\frac{1}{d} \left( \sum_J d_J + \sum_J n_J - d\mathcal{Z} \right).$

The long-time average under the non-resonance condition (NRC) is Eq. (6) (Proposition 1 (ii)): for the eigenbasis ${\phi_k}$ of $H$ and $\Pi_k=|\phi_k\rangle\langle\phi_k|$, $$\overline{G}^{NRC}=1-\frac1d\sum_{\mathcal X\in{\mathcal A,\mathcal A'}} \Big(\operatorname{Tr}(R^{(0),\mathcal X}R^{(1),\mathcal X'})-\tfrac12 \operatorname{Tr}(R^{(0),\mathcal X}DR^{(1),\mathcal X'}D)\Big),$$ with $R^{(0),\mathcal X}{lk}=|\mathbb P{\mathcal X}(|\phi_k\rangle \langle\phi_l|)|2^2$, $R^{(1),\mathcal X}{kl}=\langle\mathbb P_{\mathcal X}(\Pi_k),\mathbb P_{\mathcal X}(\Pi_l)\rangle$, $\mathbb P_{\mathcal A}=\oplus_J\frac{\mathbb 1_{n_J}}{n_J}\otimes \operatorname{Tr}{n_J}$ and $\mathbb P{\mathcal A'}=\oplus_J \operatorname{Tr}{d_J}\otimes\frac{\mathbb 1{d_J}}{d_J}$. The paper proves the minimum for Hamiltonians that are block diagonal in the sectors, and for $d_\mathcal Z=1$ over all NRC Hamiltonians (Proposition 2); its Section IV A reports numerical evidence up to $d=40$.

The $\mathcal A$-OTOC measures how fast a unitary evolution scrambles information between a subalgebra of observables and its commutant, for generalized tensor product structures that include superselection sectors and quantum reference frames. The conjecture identifies the least long-time scrambling over all non-resonant Hamiltonians of a given algebra class, and says it is attained by the distinguished product basis, the vectors $|a\rangle\otimes|b\rangle$ of each sector. The frozen declaration D5/S3/Quantum/Information/AlgebraOTOCLongTimeMinimum.result proves it.

Issue #12247 classifies the conjecture as Tier 1 and records the checks made before any Lean:

• arXiv:2312.13386 has versions v1 and v2, and v2 and the published version state the conjecture;
• the citing works found (arXiv:2509.13519, arXiv:2506.22218, a book chapter) and later work of the same group (arXiv:2510.06389, the bipartite case; arXiv:2606.29137, the $\mathbb Z_2$-symmetric case with $n_J=1$) do not prove the general mixed-block statement;
• an adversarial check of later papers of the group and of equivalent formulations (conditional expectations, Wedderburn blocks, noiseless subsystems, operator-space entanglement, coherence-generating power, frame potentials) found no proof or refutation.

These are orchestrator-reported readings and seat-reported readings, not-found-in-searched-scope; they do not establish exhaustive worldwide novelty. Step 2 of the Route extends the bipartite reduced-state Gram method of Styliaris–Anand–Zanardi, PRL 126, 030601 (2021), Prop. 4, and Proposition 2 of the paper; the new parts are the bound of the cross kernels by the identity for an arbitrary eigenbasis, which mixes the sectors, and the per-sector inequalities of steps 3–4.

Fix the algebra in its distinguished basis and let the columns $\phi_k$ of a unitary $U$ be the eigenbasis (every orthonormal basis is the eigenbasis of an NRC Hamiltonian, and by the duality of Section IV this is the family of the conjecture). Write $C_{Jk}$ for the $n_J\times d_J$ block of $\phi_k$, $q_{Jk}=|C_{Jk}|F^2$, $p{Jk}=\operatorname{tr} [(C_{Jk}C_{Jk}^\dagger)^2]$, and $F=D(1-\overline G^{NRC})$ with $D=\sum_Jn_Jd_J$.

1. $R^{(1),\mathcal A'}{kl}=\sum_J\operatorname{tr}(\operatorname{Tr}{d_J} \Pi_k\operatorname{Tr}{d_J}\Pi_l)/d_J$. For coefficients $c_k$ the quadratic form $\sum{kl}\bar c_kc_lR^{(1),\mathcal A'}{kl}$ equals $\sum_J|\operatorname{Tr}{d_J}Y_J|_F^2/d_J$ with $Y=U\operatorname{diag}(c)U^\dagger$; by Cauchy–Schwarz in each block it is at most $|Y|_F^2=\sum_k|c_k|^2$. So $R^{(1),\mathcal A'}\le I$, and likewise $R^{(1),\mathcal A}\le I$.
2. $R^{(0),\mathcal A}$ and $R^{(0),\mathcal A'}$ are sums of Gram matrices, so $\operatorname{Tr}(R^{(0),\mathcal X}R^{(1),\mathcal X'}) \le\operatorname{Tr}R^{(0),\mathcal X}$; their diagonal entries are $\sum_Jp_{Jk}/n_J$ and $\sum_Jp_{Jk}/d_J$, at most $r_k=\sum_Jq_{Jk}^2/n_J\le1$ and $w_k=\sum_Jq_{Jk}^2/d_J\le1$.
3. $x+y-xy$ is increasing on $[0,1]^2$ and $r_kw_k\ge\sum_J q_{Jk}^4/(n_Jd_J)$, so with $\sum_Jq_{Jk}=1$ and $\sum_kq_{Jk}=n_Jd_J$, $F\le\sum_J\big((n_J+d_J)\sum_kq_{Jk}^2-\sum_kq_{Jk}^4\big)/(n_Jd_J)$.
4. For $M=n_J+d_J\ge3$, $(M-1)x-Mx^2+x^4=x(1-x)(M-1-x-x^2)\ge0$ on $[0,1]$, so the $J$ term is at most $n_J+d_J-1$; for $n_J=d_J=1$, $f(x)=2x^2-x^4$ satisfies $f(a+b)-f(a)-f(b)=ab(4-4(a+b)^2+2ab)\ge0$ for $a,b\ge0$, $a+b\le1$, so $\sum_kf(q_{Jk})\le f(1)=1$.
5. Hence $F\le\sum_J(n_J+d_J-1)$, the conjectured bound. At $U=1$ the kernels are indicators of equal first (respectively second) coordinates within a sector, scaled by $1/n_J$ or $1/d_J$; the cross traces are $\sum_Jd_J$ and $\sum_Jn_J$ and both diagonal traces are $r$, so $U=1$ attains the bound.

The kernel-checked result states, for every number of sectors $r$ and all $n_J,d_J\ge1$, that $1-(\sum_Jd_J+\sum_Jn_J-r)/D$ is the least element of the values of Eq. (6) over the unitary group of $\mathbb C^D$. It reads Eq. (6) with a single minus sign before the diagonal term (the source display prints one at the end of its first line and one at the start of its second); with a plus sign the value at the product basis would lie below the paper's own minimum of Proposition 2 by $2d_\mathcal Z/d$. It fixes the sector type $(n_J,d_J)_J$, not only the abstract algebra: $(I_2\otimes M_1)\oplus(I_2\otimes M_2)$ and $(I_4\otimes M_1)\oplus (I_1\otimes M_2)$ are both $\mathbb C\oplus M_2$ but have minima $1/6$ and $0$.

A NumPy recomputation of Eq. (6) from the definitions (Haar-random unitaries) reproduces the conjectured value at $U=1$ for 9 sector types with $D\le6$, and 300 random $U$ per type never go below it (worst margin $-2.7\times10^{-15}$); for 15 random $U$ over 5 types the identity of step 1 holds to $10^{-8}$ and the spectrum of $R^{(1),\mathcal A'}$ lies in $[0,1]$ (issue #12247). An orthonormal basis of product vectors need not attain the minimum: for $n=d=2$, $|00\rangle,|01\rangle,|1{+}\rangle, |1{-}\rangle$ gives $3/8>1/4$.

The canonical source is D5/S3/Quantum/Information/AlgebraOTOCLongTimeMinimum.lean. Its public declarations are Idx, projA, projAc, ketBra, R0, R1, term, lta, claim and result; the partial traces are the frozen D5/S3/Quantum/Information/PartialTraceMutualInformation.partialTraceLeft and partialTraceRight. The frozen module state has statement identity sha256:22574a5bd686430954b57d7948d7b3ac30ae6255c4193aeadfb5af4dc137a667. The result declaration has statement identity sha256:bfe216996a3a9c45ed2de65c8aa42644a70dae6df1ea0347403bccf4a4b6d32d. The Freeze event is sha256:365bcb825aa4ba33287220c61a49d7880c1c730157666e41f0d7d1462a636e59; its project-level frozen prerequisite is the Freeze event of D5/S3/Quantum/Information/PartialTraceMutualInformation. The proof uses only the standard axioms propext, Classical.choice and Quot.sound; no sorry, native_decide, or new axiom.

Tier 1 published conjecture; resolution Proved by D5/S3/Quantum/Information/AlgebraOTOCLongTimeMinimum.result. proof_shape: content; admission_basis: open-problem-resolution (issue #12247). Utility kind none (a theorem about every sector type and every eigenbasis).

What the settlement shows

• Proved by result: for every sector type and every eigenbasis, $\overline G^{NRC}\ge1-(\sum_Jd_J+\sum_Jn_J-d_\mathcal Z)/d$, with equality for the distinguished product basis.
• Proved inside the proof of result: both cross kernels $R^{(1),\mathcal A}$ and $R^{(1),\mathcal A'}$ are bounded by the identity as quadratic forms, for every orthonormal eigenbasis; each cross trace $\operatorname{Tr}(R^{(0),\mathcal X}R^{(1),\mathcal X'})$ is at most the trace of $R^{(0),\mathcal X}$; and $D(1-\overline G^{NRC})\le\sum_J\big((n_J+d_J)\sum_kq_{Jk}^2- \sum_kq_{Jk}^4\big)/(n_Jd_J)$ in terms of the sector weights alone.
• Mechanism (read off the proof; not stated in Lean): the bound is controlled by the sector weights $q_{Jk}$. The inequality of step 4 is strict for $0<q_{Jk}<1$ when $n_J+d_J\ge3$, and the superadditivity gap $ab(4-4(a+b)^2+2ab)$ is positive for $a,b>0$ with $a+b=1$, so weight spread over two sectors loses; within a sector the purity bound $p_{Jk}\le q_{Jk}^2$ is an equality only for product vectors. This is the coherence-and-entanglement picture the paper gives for the conjecture.
• Follows from it (not stated in Lean): when every $n_J=1$ (or every $d_J=1$) the minimum is $0$, recovering the abelian case of the paper; for $r=1$ it is Proposition 2 (ii).
• Open here: the characterization of all minimizers (the argument above forces $q_{Jk}\in{0,1}$ for sectors with $n_J+d_J\ge3$, but equality cases with $n_J=d_J=1$ sectors and in steps 1–2 were not analyzed), the analogous minimum under the weaker NRC$^+$ condition of Eq. (5), and finite-time versions.

The literature checks do not establish worldwide priority or absence of an independent answer; the 2025 dissertation of F. Andreadakis was not read (the USC repository returned HTTP 403 and no public full text was found). The Lean kernel verifies the encoded statement and its axiom closure; its correspondence to the paper, including the reading of the family of isomorphic algebras as the unitary group acting on the eigenbasis and the single minus sign of Eq. (6), is checked by reading the source and the definitions.