The interacting SYK successor: construction versus recovery

This section states what can be constructed for a fixed finite SYK source and what does not follow from that construction.

Let the source contain $2n$ Majorana operators satisfying

Freeze real couplings $\mathbf{J}$ and define

This is a bounded self-adjoint operator on the finite source Hilbert space. Its normalization and coupling realization must be fixed before testing recovery.

One possible lattice bath specification uses complex fermions $b_{i,x}$, $x\ge 1$, and bounded nearest-neighbor quadratic interactions. For example, take

With the fermionic anticommutation conventions, each term is self-adjoint and even.

Group all modes at a common reservoir coordinate into one finite-dimensional supersite, and group the finite source into the site $0$. The interaction is then finite range on a half-line, with bounded local terms for every fixed $n,\mathbf{J},\mathbf{g}$.

The standard fermionic Lieb–Robinson framework constructs the quasi-local Heisenberg dynamics as the thermodynamic limit of finite-volume dynamics NachtergaeleSimsYoung. This is a legitimate construction of the infinite-bath algebraic dynamics. It is not a recovery theorem.

For a finite detector algebra $X$, a fixed logical encoding, and a bath preparation independent of the code, the Heisenberg evolution defines a complete channel to that detector algebra. A mixed bath preparation requires its purification to be retained in any complete complementary-output interpretation.