What is the Araki-Uhlmann relative entropy of opening a traversable wormhole?

Statement

The Gao–Jafferis–Wall double-trace deformation $V=g{\mathcal{O}}_L{\mathcal{O}}_R$ that makes the eternal-black-hole wormhole traversable is a Connes-cocycle perturbation of the thermofield double modular flow (made explicit by Ahmad–Jefferson). The group already computes Araki–Uhlmann relative entropy $S(\omega \Vert \phi )=-⟨\Omega ,\log {\Delta }_{\Phi ,\Omega }\Omega ⟩$ in closed form for free-field states. Question: in a free-field two-wedge analogue of the TFD, perturbed by $V=gW(f_L)W(f_R)$, what is the relative entropy $S({\omega }_V\Vert \omega )$ between the deformed and undeformed states to $O(g^2)$, and can it be read as the “cost of opening the throat”?

Why It Matters

• It turns the wormhole-traversability story into a calculation the group can actually do, using exactly the Weyl operator + cocycle machinery of the relative-entropy program and AQFT course Sem I Wk 7 / Sem II Wk 12.
• It would give an information-theoretic meaning to traversability: relative entropy measures distinguishability between the open-throat and closed-throat states, and its $g$-dependence quantifies how far the deformation pushes the state.
• It is the controlled, flat-space mirror of Ahmad–Jefferson’s holographic generalized-entropy corrections — a way to separate the algebraic content (which the group owns) from the holographic content (taken on trust). See wormholes-and-quantum-information-qft (project P1).

What We Know

• For a single coherent state $W(f)|0⟩$, $S({\omega }_f\Vert {\omega }_0)\propto \Delta (f,f)$, the smeared Pauli–Jordan distribution (Nucl. Phys. B 1018, 2025). The two-coherent-state and squeezed-state cases are also closed-form.
• The Connes cocycle $(D{\omega }_V/D\omega {)}_t$ has a known perturbative expansion (Sem I Wk 7; Ahmad–Jefferson §2–3). The first non-trivial relative-entropy correction is $O(g^2)$ because the $O(g)$ term is a one-point function that vanishes by symmetry for a suitable $V$.
• The Bisognano–Wichmann identification (vacuum on two wedges = TFD at the Unruh temperature) guarantees the free-field analogue is a faithful model of the boundary modular data.

Possible Approaches

1. Direct cocycle expansion. Expand $\log {\Delta }_{{\Phi }_V,\Omega }$ in $g$ using the cocycle series; the bilinear Weyl perturbation $V=gW(f_L)W(f_R)$ keeps every term a Gaussian integral against the two-point/Pauli–Jordan data, so the $O(g^2)$ coefficient should be a double smearing of $\Delta$ and the Wightman function.
2. Reuse the squeezed-state technology. $W(f_L)W(f_R)$ acting on the TFD is closely related to a two-mode squeezing of the wedge pair; the existing squeezed-state relative-entropy result (EPJC 85, 2025) may port over with the modular data of the wedge replacing the single-mode Bogoliubov coefficients.
3. Cross-check numerically. The group’s (1+1)d numerical framework (Nucl. Phys. B 1018) can validate the analytic $O(g^2)$ coefficient and explore the mass / wedge-separation dependence — the same knobs already mapped for coherent states.

Related Questions

• bell-chsh-across-traversable-wormhole — the Bell-CHSH counterpart of the same deformation.
• relative-entropy-interacting-theories — the GJW ${\mathcal{O}}_L{\mathcal{O}}_R$ coupling is the kind of interaction that motivates going beyond free fields.
• embezzlement-cost-relative-entropy — relative entropy as the cost measure for type III${}_1$ resource manipulation.

Tracked under the wormholes-and-quantum-information-qft connection.