What is the magic of coherent, squeezed, and cat states in QFT?

Statement

Compute the magic (via the anti-flatness witness $F_2$) of the group’s standard free-field states — coherent, squeezed, and cat states — reduced to a Rindler wedge, using the same modular / Bogoliubov machinery already used for their Araki–Uhlmann relative entropy. Concretely: for each state $\omega$, obtain the wedge reduced density matrix ${\rho }_{\omega }$, evaluate $F_2({\rho }_{\omega })=\mathrm{Tr}({\rho }_{\omega }^3)\mathrm{Tr}({\rho }_{\omega })-\mathrm{Tr}({\rho }_{\omega }^2{)}^2,$ and hence the stabilizer-distance lower bound $F_2/8\le {\min }_{\sigma \in \mathrm{STAB}}\Vert \omega -\sigma \Vert$. Do cat states maximize magic where large-$N$ particle states flatten it?

Why It Matters

Benedetti–Dabholkar–Dalmonte establish that every physical QFT state is magical, and compute anti-flatness explicitly only for free-boson Unruh $N$-particle states — finding, counter-intuitively, that the spectrum becomes flatter as $N\to \infty$ (a singular, non-Fock limit). The group already possesses the exact reduced structure for a different and physically richer family — coherent, squeezed, and cat states — from its 2025 relative-entropy papers and the 2026 cat-states/Bell-CHSH paper (Phys. Rev. D 113, 065008). Assigning those states a magic content is therefore a near-drop-in computation that (i) extends the paper with new, non-trivial examples, (ii) puts magic side-by-side with relative entropy and Bell-CHSH violation for the same states, and (iii) tests a sharp hypothesis: superpositions (cat states) are exactly where non-classicality piles up, so they are the natural candidate for maximal anti-flatness, opposite to the $N\to \infty$ flattening. A clean “cat states are maximally magical” result would be a compact, publishable QI-in-QFT statement.

What We Know

• Flatness ⟺ stabilizer (necessary). A flat entanglement spectrum is necessary for stabilizerness; anti-flatness $F_2$ is a magic witness and lower-bounds the distance to the stabilizer set (Tirrito et al. 2024; Cao et al. 2024). See magic-nonstabilizerness.
• The reduced states are in hand. The Minkowski vacuum is a two-mode squeezed state of left/right Rindler modes; coherent, squeezed, and cat states are built by Weyl / Bogoliubov operations on it, and their wedge reductions are computable from the two-point function — the same input the group uses for $S(\omega \Vert \Omega )$.
• A comparison point exists. For Unruh $N$-particle states $F_n$ peaks at intermediate $q_{\omega }=e^{-\omega /T_U}$ and decreases with $n$ and with $N$; this is the curve to beat with cat states.
• Relative entropy is already computed for these states, so magic vs. relative entropy can be plotted directly, feeding relative-entropy-magic-inequality.

Possible Approaches

1. Direct anti-flatness from the two-point function. Express $\mathrm{Tr}({\rho }_{\omega }^n)$ for coherent/squeezed/cat states via the Gaussian (for coherent/squeezed) and Gaussian-superposition (for cat) structure of the wedge reduced state; evaluate $F_2$ in closed form or numerically per Rindler frequency $\omega$, then sum $S_n(\rho )={\sum }_{\omega }{S}_n({\rho }_{\omega })$ as in the paper.
2. Cat vs. coherent contrast. Since a cat state is a superposition of two coherent states, compute the interference contribution to $F_2$ and check whether it enhances anti-flatness relative to a single coherent state at matched energy.
3. Cross-plot against relative entropy and Bell-CHSH. For the same test-function / squeezing parameters that maximize the Bell-CHSH violation, evaluate magic — a first look at whether the operators that best violate Bell are also the most magical (Gap 1 of type-iii-algebras-across-areas).
4. Sanity check against known limits. Recover the paper’s $N$-particle curve as a special case; verify $F_2\to 0$ on the (unphysical) flat Rindler/Boulware-like construction.

Related Questions

• relative-entropy-magic-inequality — the quantitative magic ↔ relative-entropy ↔ Bell relation this feeds.
• magic-and-the-gribov-horizon — the same computation with a Gribov-restricted vacuum.
• relative-entropy-interacting-theories — extending beyond free fields.
• long-range-bell-decay, bell-chsh-in-holographic-setting — sibling III₁ observables on the same states.