Source
Yang and Abanin (2025), LIT-1958, arXiv:2506.02970v3, §§II–III and Appendices A–G; read in NOTE-1534.
What was actually shown
Exactly. Under depolarizing noise a Pauli string P obeys D[P] = −γS(P)P, so d‖O‖/dt = −γS(O)‖O‖ with S(O) the weighted number of sites O acts on (eqs. 5, 6). For a Hermitian eigenoperator with real eigenvalue, ‖[H, V]‖² = γ²(⟨S²⟩_V − ⟨S⟩²_V) (eq. A7), so a slow eigenoperator commutes with H up to O(γ) when its size distribution is not unusually spread.
Numerically, at L ≤ 9, periodic boundaries, zero momentum, γ = 10⁻³:
- mixed-field Ising (no local IOM but H): first three eigenoperators in span{I, H, H²} (Fig. 2a);
- transverse-field Ising: first five in the span of its Majorana-bilinear IOMs (Fig. 2b,c);
- XXX: first 25 accounted for by the known IOMs and their products (Fig. 3a, Fig. 10b,c);
- XXZ and MFIM at hx = hz = 0.4: slow modes outside the listed IOMs, matched to approximate ones (spin current, S²ₓ + S²ᵧ, a domain-wall swap operator, a dressed particle number; Figs. 3b,c, 4, 10a).
These could have come out otherwise: the slowest eigenoperators could have been dominated by non-conserved operators that happen to be small, as the polarized-state projector is under dephasing–decay noise in Heisenberg chains (Appendix B), or the listed IOMs could have failed to account for them. In the depolarizing case they did account for them, up to the approximate IOMs the authors then identified.
The ranking moves with γ. At γ = 0.1 in MFIM (hx = 1.5, hz = 0.1), the approximate IOM A₁ overtakes the exact H² (Fig. 6), as expected from a decay rate of roughly Aε² + Bγ for an approximate IOM against one linear in γ for an exact one.
What this does not say
- That it holds at large L. The diagonalization stops at nine sites; the large-system argument is THEORY-1343’s, and Proposed.
- That every integral of motion appears. IOMs larger than a threshold proportional to L, about 3–4 here, sink into the bulk (Appendix D); the quasi-local IOM of the XXX chain was not seen.
- That the slow modes name themselves. Each was matched by overlap to a candidate operator; the method gives a subspace.
- That slowness alone picks out conserved quantities. It does so given a noise model that punishes size, a split into sites that defines size, and dynamics that grow non-conserved operators; and the order among them depends on γ. Under a noise channel that spares a particular large operator, that operator is slow without being small.
- Anything about noise channels in general. Depolarizing noise is the main case; one other channel was checked (Fig. 7).