Relative Entropy and RG Irreversibility

Relative entropy turns distinguishability into an irreversibility diagnostic because it decreases under restriction or a quantum channel. In QFT, however, the two states must be normal states on one algebra—or be supplied with a controlled common regulator and continuum limit. There is no canonical relative entropy between “the UV theory” and “the IR theory” as abstract theories.

Required background. Relative entropy in QFT supplies the algebraic definition and finiteness conditions; separating scales fixes the regional and regulator limits.

Helpful background. Information measures along RG flows distinguishes region-size comparisons from RG trajectories.

One algebra, two states

For faithful regulated density matrices,

$$D(\rho\Vert\sigma) =\operatorname{Tr}\rho(\log\rho-\log\sigma) =\Delta\langle K_\sigma\rangle-\Delta S,$$

where $K_\sigma=-\log\sigma$ and $\Delta$ means “value in $\rho$ minus value in $\sigma$.” In the continuum, the same relation is expressed through relative modular theory when both states belong to the chosen local algebra.

The von Neumann-algebraic definition, support condition, and monotonicity theorem originate in Araki 1976, §§ 3–4.

The reference $\sigma$ matters. For a deformed vacuum $\rho_g$ and a UV-CFT vacuum $\sigma_0$ restricted to the same ball, $D(\rho_{g,R}\Vert\sigma_{0,R})$ measures distinguishability at resolution $R$. It is not a distance between Lagrangians and it is not symmetric under exchanging the states.

Relative entropy can enter either a finite crossover on a common regional algebra or a channel-loss statement. The common algebra and reference state precede both interpretations. Schematic and not to scale.

Region inclusion and monotonicity direction

If $A\subset B$, restriction gives a channel from $\mathcal A(B)$ to $\mathcal A(A)$. Data processing implies

$$D(\rho_A\Vert\sigma_A) \le D(\rho_B\Vert\sigma_B).$$

Thus distinguishability normally increases with the size of the accessible region. This is the opposite of the slogan that “information decreases toward the IR” if one simply identifies larger $R$ with lower energy. An RG irreversibility construction must define a different function, subtraction, or map whose scale direction is unambiguous.

For a ball and a CFT vacuum reference, the modular Hamiltonian is local:

$$K_{\sigma,R} =2\pi\int_{lvert\mathbf x vert<R} d^{d-1}x\, \frac{R^2-lvert\mathbf x vert^2}{2R},T_{00}(\mathbf x) +\text{constant}.$$

This makes $D=\Delta\langle K\rangle-\Delta S$ calculable. Positivity then bounds the entropy change by the weighted energy change. The formula relies on the CFT vacuum and ball geometry; a generic reference has a nonlocal modular Hamiltonian.

The relative-entropy form of the Bekenstein bound is derived in Casini 2008, §§ 2–3, and the corresponding CFT state calculations are illustrated in Lashkari 2014, Eqs. (7)–(13).

Comparing deformed theories

A regulator can provide a common algebra. For example, put free scalar fields of masses $m$ and $m_0$ on the same lattice, use the same canonical variables and region, and compare the two Gaussian reduced states. The regulated relative entropy is well defined. One then refines the lattice at fixed $mR$ and $m_0R$ and tests whether the limit is finite or requires an allowed subtraction.

This construction is extra data. Different UV completions or field identifications can yield inequivalent embeddings. In the continuum, vacua of different theories may lie in inequivalent representations. Writing $D(\rho_{\rm UV}\Vert\rho_{\rm IR})$ without a common algebra suppresses exactly the information needed to define the expression.

Irreversibility and sufficiency

For an explicit channel $\mathcal N$,

$$\Delta D =D(\rho\Vert\sigma) -D(\mathcal N\rho\Vert\mathcal N\sigma) \ge0.$$

Equality means the pair is sufficient for the channel and can be recovered by a Petz-type map under the usual support conditions. A small positive loss can bound approximate recovery. This is a precise operational statement about a pair of states and a map; it is stronger than observing that one scalar “degree count” decreases.

Validity map

The first gate is decisive for relative entropy: different theories require an explicit embedding or shared regulator. The channel branch applies only after an actual CPTP map and retained algebra are identified. Schematic and not to scale.

Common pitfalls

Comparing abstract theories as if they were density matrices in one Hilbert space. Supply the shared algebra, field identification, and limiting procedure.

Reversing the region-inclusion inequality. More accessible observables can only increase distinguishability. Restriction to a smaller algebra decreases it.

Equating positivity with an a-theorem proof. Relative-entropy positivity is a key ingredient in several arguments, but the geometric and anomaly hypotheses that isolate $a$ remain essential.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.
• Casini, Horacio. “Relative Entropy and the Bekenstein Bound.” Classical and Quantum Gravity 25 (2008): 205021. DOI.
• Lashkari, Nima. “Relative Entropies in Conformal Field Theory.” Physical Review Letters 113 (2014): 051602. DOI.