How does opening a traversable wormhole change the magic of the two-sided state?

Statement

The two-sided thermofield double (TFD) — dual to the eternal black hole, with the Hartle–Hawking state as its near-horizon vacuum — is a cyclic-separating state on a type III₁ algebra and therefore, by Benedetti–Dabholkar–Dalmonte, carries non-zero magic. Opening a traversable wormhole via a Gao–Jafferis–Wall (GJW) double-trace deformation acts on the two sides as a Connes cocycle that reshapes the modular flow. How does the magic (anti-flatness $F_2$) of the two-sided state change when the wormhole is opened — does traversability increase or decrease it?

Why It Matters

The group’s wormholes ↔ QI thread already asks how the GJW opening affects two III₁-rooted resources — the Bell-CHSH violation and the Araki–Uhlmann relative entropy (as a Connes cocycle). Magic is the newly-identified third resource read off the same modular data, so its behavior under the same operation is the natural next entry — and it comes with a concrete geometric hook: the magic paper’s §5 contrasts flat-spectrum fixed-area (stabilizer-like) states with the non-flat physical states, and the Hartle–Hawking (not Boulware) vacuum is singled out as the magical, physical one. A result of the form “opening the wormhole moves the state toward/away from the stabilizer (fixed-area) sector” would give a quantum-information reading of traversability and slot directly into the existing wormhole questions.

What We Know

• GJW = Connes cocycle. The double-trace deformation implementing the wormhole opening is naturally a cocycle relating the deformed and undeformed modular structures — the framing already used in relative-entropy-wormhole-opening.
• Magic is modular data. Anti-flatness is a functional of the (relative) modular operator, so the cocycle that changes the modular flow is exactly what changes the magic; the computation reuses the relative-entropy-wormhole-opening machinery.
• Fixed-area vs. physical states. The paper shows fixed-area states are flat/stabilizer-like and physical (cyclic-separating) states get non-flat subleading Rényi corrections via the modified cosmic-brane prescription — the language in which a “magic of the wormhole” statement would be phrased.
• TFD baseline. The TFD is a two-mode-squeezed state, so its wedge reduced structure — and hence a baseline anti-flatness — is computable by the same free-field methods as magic-of-coherent-squeezed-cat-states.

Possible Approaches

1. Cocycle computation of $\Delta F_2$. Compute the anti-flatness of the two-sided state before and after the GJW deformation using the Connes-cocycle relation between the two modular operators; extract the sign and magnitude of the change.
2. Free-field TFD model. In a free-field / generalized-free-field model of the two sides, build the deformed reduced density matrix explicitly (as for coherent/squeezed states) and evaluate $F_2$ directly, paralleling the group’s relative-entropy-of-opening computation.
3. Fixed-area basis reading. Express the deformed state in the fixed-area (flat) basis and quantify its departure — the “magic distance” from the nearest fixed-area state — as the traversability is dialed.
4. Triangulate with Bell and relative entropy. Plot magic, Bell-CHSH, and relative entropy of opening together as functions of the GJW coupling to see whether they move coherently (feeding relative-entropy-magic-inequality).

Related Questions

• relative-entropy-wormhole-opening, bell-chsh-across-traversable-wormhole — the sibling resources under the same GJW operation.
• magic-of-coherent-squeezed-cat-states — the free-field anti-flatness machinery reused here.
• holographic-fixed-area-magic-embezzlement — the fixed-area/embezzlement angle on the same holographic magic.
• relative-entropy-magic-inequality — whether the three resources move together.