Is there an inequality relating magic, relative entropy, and Bell-CHSH on type III₁ algebras?

Statement

The type III₁ structure of local algebras forces three quantum-information resources to be simultaneously present in any physical state: Bell-CHSH violation, finite Araki–Uhlmann relative entropy $S(\omega \Vert \Omega )$, and now magic (anti-flatness $F_2$). Are they quantitatively related? Is there an inequality of the form $F_2(\omega )\le g(S(\omega \Vert \Omega )),\text{or}\mathcal{B}(\omega )\le f(S(\omega \Vert \Omega ),F_2(\omega )),$ tying the magic, the relative entropy, and the maximal Bell-CHSH value $\mathcal{B}(\omega )$ of a state $\omega$ on a wedge algebra?

Why It Matters

The “type III₁ as the common thread” connection is currently structural — the three resources share a root — but not quantitative: no inequality ties their magnitudes. Benedetti et al. adds magic as a third III₁-forced resource, which makes the case for a unifying inequality stronger and gives a third measurable to correlate. Even a one-directional bound (e.g. “small relative entropy to the vacuum ⇒ bounded magic”) would convert the common-thread narrative into a theorem and would be a genuinely new result at the QI/QFT interface. It directly answers Gap 3 / Project 1 of type-iii-algebras-across-areas — extended from two resources to three.

What We Know

• All three vanish/are-forced together at the structural level. A flat spectrum (no magic) requires trivial modular flow, which contradicts the continuous Bisognano–Wichmann spectrum that also drives Bell violation and gives the relative entropy its modular definition.
• Relative entropy already correlates with Bell in examples. For coherent/squeezed/cat states the group has both $S(\omega \Vert \Omega )$ and the Bell-CHSH value; adding $F_2$ (via magic-of-coherent-squeezed-cat-states) gives the first empirical scatter of all three.
• Finite-dimensional analogues exist. Magic-vs-entanglement separations (Gu–Oliviero–Leone 2025) and stabilizer-distance bounds (Cao et al. 2024) suggest the form such inequalities can take; the challenge is the type III₁ (no density matrix, no trace) setting where only relative entropy and modular quantities survive.
• Modular flow is the shared variable. All three quantities are functionals of the (relative) modular operator, so a modular-theoretic inequality is the natural target.
• One side of the inequality is already modular — exactly, and for free. The relative entropy of magic is a relative-modular optimization: $M_{\mathrm{rel}}(\rho )={\inf }_{\sigma \in \mathrm{STAB}}[-⟨{\rho }^{1/2},\log {\Delta }_{\sigma |\rho }{\rho }^{1/2}⟩]$, with ${\Delta }_{\sigma |\rho }=L_{\sigma }{R}_{{\rho }^{-1}}$ in Type-I standard form, and the expression is unchanged on a type III algebra. That is a rewrite, not a result (quantum-magic-modular-qft-study, Kill test 3) — but it means the target inequality can be posed entirely in $\Delta$-language from the start.
• The obstruction is asymmetry of the two sides. $S(\omega \Vert \Omega )$ is intrinsic to $(\mathcal{A},\omega ,\Omega )$; magic is not — it is frame-dependent, and no automorphism-invariant functional of $(\mathcal{A},\omega )$ can be a faithful magic measure. An inequality relating them is therefore conditional on a declared free set $\mathfrak{F}$, and its content will sit largely in that choice. Posing it without naming $\mathfrak{F}$ compares a frame-dependent quantity to a frame-independent one. See canonical-free-structure-for-magic-in-qft.
• Exact finite-dimensional anchors exist. Verified benchmarks for the noisy $T$-injection line — closed-form $M_{\mathrm{rel}}$, its optimizer, and the full relative-modular spectrum $\{q/p,q/(1-p),(1-q)/p,(1-q)/(1-p)\}$ — are recorded in quantum-magic-modular-qft-study and are the natural sanity check for any conjectured bound before it is pushed to the continuum.

Possible Approaches

1. Empirical first. Use the magic-of-coherent-squeezed-cat-states data to plot $F_2$ vs. $S(\omega \Vert \Omega )$ vs. $\mathcal{B}(\omega )$ across a parametrized family; look for an envelope. A conjectured inequality with a clean saturating family is a strong paper on its own.
2. Modular-operator bounds. Express $F_2$ and $S(\omega \Vert \Omega )$ through the spectrum of the relative modular operator ${\Delta }_{\Omega |\omega }$ and seek an operator inequality (e.g. via monotonicity of relative entropy under the conditional expectation onto a stabilizer-like subalgebra).
3. Clifford-orbit averaging. Since $F_2$ becomes the linearized stabilizer entropy after Clifford averaging, ask whether the QFT analogue of that average is controlled by a relative entropy to the “nearest flat” (Rindler/Boulware-like) reference state.
4. Contrast with non-Gaussianity. Check whether the inequality distinguishes magic from non-Gaussianity — i.e. whether interacting-theory corrections enter through $S(\omega \Vert \Omega )$ differently than through $F_2$.

Related Questions

• canonical-free-structure-for-magic-in-qft — upstream: names the free set this inequality is conditional on.
• magic-of-coherent-squeezed-cat-states — supplies the data.
• embezzlement-cost-relative-entropy — the fourth III₁ resource (embezzlement cost) that a full unification would include.
• long-range-bell-decay — whether magic tracks the $\beta (L)$ decay of the Bell violation with separation.
• relative-entropy-interacting-theories — the interacting-theory regime where the inequality would be tested.