Can relative entropy computations be extended beyond free field theories?

Statement

All closed-form results for araki-uhlmann-relative-entropy in the research program have been obtained for free fields (free massive scalar in (1+1)d and higher dimensions, with coherent, squeezed, and cat state excitations). Can these methods be extended to interacting theories — and if so, which classes of interactions are tractable? In particular, can the relative entropy between gauge-invariant states in Yang-Mills theory, or between states in the refined-gribov-zwanziger vacuum, be computed analytically or semi-analytically?

Why It Matters

The araki-uhlmann-relative-entropy $S(\omega ||\phi )$ is the most operationally fundamental measure of distinguishability in quantum information theory: it controls the asymptotic error exponent for hypothesis testing (quantum Stein lemma), and it satisfies the strongest monotonicity properties (data processing inequality). In free-field QFT, it is computable because the relative modular operator ${\Delta }_{\Phi ,\Omega }$ can be expressed in terms of the two-point function (Bogoliubov transformation structure). In interacting theories, the modular operator involves all $n$-point correlators, and no closed form is known in general.

Extending relative entropy computations to interacting theories would:

• Enable the program of using relative entropy as a phase-transition probe (see entanglement-as-confinement-probe).
• Allow computation of the embezzlement-cost-relative-entropy for physically realistic gauge theories.
• Connect to the tomita-takesaki-modular-theory program in constructive QFT, where the modular operator for interacting theories is an active area of mathematical physics research.
• Make contact with the holographic-relative-entropy program (Faulkner, Lewkowycz, Maldacena), where relative entropy in CFTs is computed via the Ryu-Takayanagi formula.

What We Know

Free-field results (established in the program):

• For the free massive scalar, the relative modular operator factorizes: ${\Delta }_{\Phi ,\Omega }={\Delta }_{\Phi }{\Delta }_{\Omega }^{-1}$, and both factors can be expressed in terms of the two-point function via Bogoliubov transformations.
• Relative entropy of coherent states: $S(W(f)\Omega ||\Omega )\propto \text{Im}\Delta (f,f)$ where $\Delta (f,g)$ is the smeared pauli-jordan-distribution (Nucl. Phys. B 1018, 2025).
• Relative entropy of squeezed states: closed form in terms of Bogoliubov coefficients (Eur. Phys. J. C 85, 2025).
• Cat states: the interference terms introduce non-Gaussian corrections; results exist but are structurally more complex.
• Mass and dimension dependence: studied systematically in arXiv:2511.20244.

Structural results from mathematical physics:

• bisognano-wichmann-theorem: for wedge regions in the vacuum of a local QFT satisfying haag-kastler-axioms, the modular operator is ${\Delta }_{\Omega }=e^{-2\pi K}$ where $K$ is the boost generator (Rindler Hamiltonian). This holds for any QFT (free or interacting), but gives the modular operator of the vacuum, not the relative modular operator for an excited state.
• Araki’s perturbation theory: if $\Phi =\exp (h)\cdot \Omega$ for a self-adjoint $h$ affiliated with the algebra, the relative entropy can be computed perturbatively in $h$. This applies to coherent states in free theories but is formal for interacting theories.
• For 2d integrable theories (S-matrix bootstrap), some properties of the modular operator have been studied using the KMS condition and TCP invariance, but no closed-form relative entropy is known.

Perturbative interacting theories:

• At weak coupling $\lambda {\varphi }^4$, corrections to free-field relative entropy can in principle be computed using the Schwinger-Dyson equations and Wick’s theorem. The leading correction is $O(\lambda )$ and involves the connected four-point function. No such computation has been carried out in the program.

Possible Approaches

1. Perturbation theory in coupling constant: For $\lambda {\varphi }^4$ theory or for Yang-Mills at weak coupling, compute relative entropy to first order in the coupling. The perturbative expansion of the relative modular operator generates a series in the connected $n$-point functions. At $O(\lambda )$, only the four-point connected function contributes. This is technically demanding but conceptually straightforward.

2. 2d integrable theories: In integrable field theories (sine-Gordon, Thirring, Ising model off-critical), form factor methods give exact $n$-point functions. Use these to construct the modular operator perturbatively around the free fixed point, and compute relative entropy between particle states and the vacuum. The form factor bootstrap provides a systematic expansion in terms of the particle spectrum.

3. RGZ vacuum as non-perturbative input: In the refined-gribov-zwanziger framework, the gluon propagator is known non-perturbatively (it fits lattice data). Treat the RGZ propagator as the effective two-point function and compute relative entropy in the “quasi-free” approximation where all higher connected correlators are neglected. This is not exact but gives a physically motivated estimate of the corrections from the Gribov structure.

4. Holographic approach: For strongly coupled large-$N$ gauge theories with a holographic dual, the relative entropy between excited states and the ground state is computable via the Ryu-Takayanagi formula extended to excited states (Faulkner-Lewkowycz-Maldacena). This gives non-perturbative, strongly coupled results — though in a regime far from QCD.

5. Numerical approach: Extend the numerical framework already developed for free fields (Nucl. Phys. B 1018, 2025) to interacting theories regulated on a lattice. The modular Hamiltonian can be approximated numerically by the reduced density matrix on a spatial interval, from which relative entropy is computed as $S(\rho ||\sigma )=\text{tr}(\rho \log \rho )-\text{tr}(\rho \log \sigma )$ using the eigenvalue decomposition.

6. Monotonicity constraints: Even without a closed-form result, the data-processing inequality and monotonicity of relative entropy under coarse-graining constrain the interacting-theory relative entropy to lie within bounds set by the free-field result. Determining how tight these bounds are is a tractable preliminary step.

Related Questions

• embezzlement-cost-relative-entropy — Embezzlement cost in physically realistic (interacting) QFTs requires this extension
• entanglement-as-confinement-probe — Phase-transition probe via relative entropy requires extending to the confined phase of Yang-Mills
• bell-inequalities-with-gribov-horizon — Bell violations in the RGZ vacuum is the complementary question using the same interacting framework
• gribov-copies-physical-observables — Identifying physical observables for relative entropy computation in gauge theories