Does opening a traversable wormhole degrade the Bell-CHSH violation between its two sides?

Statement

A maximal Bell-CHSH violation (Tsirelson value $2\sqrt{2}$) between two regions in a QFT vacuum — the Summers–Werner result the group works with — relies on the two algebras being mutually commuting (spacelike, no signaling). A traversable wormhole is precisely a configuration in which a signal can pass between the two sides: the Gao–Jafferis–Wall coupling $V=g{\mathcal{O}}_L{\mathcal{O}}_R$ partially breaks the strict commutativity of ${\mathcal{A}}_L$ and ${\mathcal{A}}_R$. Question: as the coupling $g$ is turned on (opening the throat), does the achievable Bell-CHSH violation between the two boundary/wedge algebras decrease, and is there a quantitative trade-off “more traversability ⇒ less violation”?

Why It Matters

• It is a sharp, distinctive question: it crosses the group’s signature Bell-CHSH program with the wormhole literature, in a way no existing wiki page covers (the holographic-bell-program and bell-chsh-in-holographic-setting consider Bell tests on a fixed TFD; this asks what deforming it does).
• It probes a genuine conceptual tension at the heart of ER=EPR: if entanglement “is” the wormhole, what happens to the entanglement diagnostic when we use that entanglement to signal? A monotone trade-off would be a clean, quantitative face of the monogamy-like intuition that “using” the channel costs correlation.
• The flat-space version is computable now (vacuum on two wedges = TFD, plus a Weyl-bilinear perturbation), so the question is testable, not just rhetorical.

What We Know

• Undeformed baseline: complementary wedges in the vacuum/TFD give maximal violation via modular conjugation $J$ (Summers–Werner; group’s Phys. Rev. D 108, 085026). This is the $g=0$ endpoint.
• Signaling breaks the premise: once $[{\mathcal{A}}_L,{\mathcal{A}}_R]\ne 0$ effectively (the coupling correlates the two sides through dynamics, not just the state), the CHSH operator’s algebraic derivation of the bound no longer applies unchanged. Whether the value drops, and how fast, is open.
• Monogamy heuristic: in finite systems, entanglement that is “spent” on a channel is unavailable for maximal Bell violation. Whether a type III${}_1$ analogue holds — and whether embezzlement evades it — is unknown.

Possible Approaches

1. Perturbed correlator. Compute $⟨\mathrm{TFD}|e^{iV}\mathcal{C}{e}^{-iV}|\mathrm{TFD}⟩$ for the Weyl-built CHSH operator $\mathcal{C}$ with $V=gW(f_L)W(f_R)$, in the free-field two-wedge model, to $O(g^2)$. Look for a definite-sign correction to the Tsirelson value.
2. Modular re-derivation. Track how the coupling deforms the modular conjugation/relative modular operator (same cocycle as relative-entropy-wormhole-opening) and feed the deformed $J$ into the Bob-operator construction; see whether the optimized violation is bounded by a $g$-dependent ceiling.
3. Toy/finite check. A two-mode (coupled-oscillator) caricature of the TFD with a tunable $L$–$R$ coupling, where both the “signaling capacity” and the CHSH value are elementary, to establish the qualitative trade-off before the field-theory computation.

Progress (2026-06-04): leading-order result and a corrected framing

A leading-order pass corrects the hypothesis above (“the violation drops”). Model the opening as $U=e^{-iV}$, $V=g\varphi (f_L)\varphi (f_R)$. In the Heisenberg picture Alice’s operator $A(g)=U^{†}AU$ ($A\in {\mathcal{A}}_R$) develops a nonzero commutator with Bob’s $B\in {\mathcal{A}}_L$:

$[A(g),B]=ig[{\mathcal{O}}_L,B][{\mathcal{O}}_R,A]+O(g^2)=-ig\Delta (f_L,f_B)\Delta (f_R,f_A)BA+O(g^2),$

using $[\varphi (f),W(h)]=i\Delta (f,h)W(h)$ in the free field. So the inter-wedge signaling capacity is a product of two Pauli–Jordan smearings,

$\kappa (g)\equiv \Vert [A(g),B]\Vert \simeq g|\Delta (f_L,f_B)||\Delta (f_R,f_A)|,$

nonzero iff the coupling’s right support is causally linked to Alice’s region and its left support to Bob’s. This $\kappa$ is the regenesis signal.

Corrected conclusion. The Tsirelson proof (${\mathcal{C}}^2=41-[A_1,A_2][B_1,B_2]$) needs $[A_i,B_j]=0$. Once $[A_i,B_j]=O(g)$, the ceiling rises into the PR-box regime $(2\sqrt{2},4]$ — opening the wormhole does not lower the CHSH number (type III${}_1$ keeps $\ge 2\sqrt{2}$ available, and signaling can push higher). What degrades is the certification: a CHSH value only witnesses entanglement under no-signaling, which now fails at order $\kappa (g)$. The sharp statement is therefore: traversability converts Bell-certified wormhole entanglement into communicable correlation at rate $\kappa (g)$ — the entanglement that is the ER bridge is exactly what stops being certifiable once the bridge is used to signal. The order parameter is the gap between $⟨\mathcal{C}⟩$ and the signaling-fakeable part $f(\kappa )=O(\kappa )$.

Next: evaluate $\kappa (g)$ and the re-optimized $⟨\mathcal{C}{⟩}_g$ in an explicit (1+1)d geometry (fixed $m$, chosen coupling supports/boost-time separation) with the group’s Pauli–Jordan numerics; plot certified-violation vs. $\kappa$. Shares all inputs with relative-entropy-wormhole-opening.

Related Questions

• relative-entropy-wormhole-opening — same deformation, relative-entropy observable; shares the cocycle computation.
• bell-chsh-in-holographic-setting — Bell-CHSH on the undeformed holographic TFD.
• higher-spin-bell-inequalities — extending the Weyl/CHSH construction to the operators ${\mathcal{O}}_{L,R}$ that appear in the coupling.

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