Araki Relative Entropy, Monotonicity, and Data Processing

Araki relative entropy is the trace-free measure of distinguishability for normal positive functionals on an arbitrary von Neumann algebra. It is defined from a relative modular operator, takes the value $+\infty$ when the required support relation fails, and decreases under restriction or a normal unital completely positive channel. The theorem does not license density matrices for a sharp type-III region.

Required background. Normal von Neumann-algebra channels provide the maps to which data processing applies; Tomita–Takesaki theory supplies relative modular operators. Helpful background. Standard forms and modular automorphisms fix canonical implementing vectors, while type-III entropy limits explain the trace-free setting. The information-theoretic relative-entropy overview, regulated Araki entropy, data processing, first-law expansion, state susceptibility, Bekenstein bounds, horizon laws, and generalized second law develop applications.

Relative modular data and support

Put $\mathcal M$ in standard form $(\mathcal M,\mathcal H,J,\mathcal P)$ and let normal positive functionals $\varphi,\psi$ have natural-cone representatives $\xi_\varphi,\xi_\psi$. On its natural core, the relative Tomita operator is

$$S_{\psi\mid\varphi,0}(A\xi_\varphi+\eta) =s(\varphi)A^*\xi_\psi, \qquad \eta\in(1-s(\varphi))\mathcal H,$$

and its closure has polar square $\Delta_{\psi\mid\varphi}=S_{\psi\mid\varphi}^*S_{\psi\mid\varphi}$. With the convention that the first argument is the state being tested, define

$$S_{\mathcal M}(\varphi\Vert\psi) =-\langle\xi_\varphi,\log\Delta_{\psi\mid\varphi}\,\xi_\varphi\rangle.$$

The logarithm is understood spectrally and the expression is extended-valued. If $s(\varphi)\nleq s(\psi)$, it is $+\infty$. For faithful density matrices in a type-I algebra this reduces to $\operatorname{Tr}\rho(\log\rho-\log\sigma)$, but that formula is a check, not the definition. Araki established positivity and its equality condition in the faithful case Araki 1976, §§3–4, pp. 818–824 and treated nonfaithful functionals by support reduction in the sequel Araki 1977, §§2–3, pp. 176–184.

Data processing

Let $\Phi:\mathcal M\to\mathcal N$ be normal, unital, and completely positive, and let $\varphi,\psi\in\mathcal N_*^+$. Then

$$S_{\mathcal M}(\varphi\circ\Phi\Vert\psi\circ\Phi) \leq S_{\mathcal N}(\varphi\Vert\psi).$$

Restriction to a von Neumann subalgebra is the special case in which $\Phi$ is the inclusion. The statement remains meaningful when one or both sides are infinite. Its proof is modular rather than trace-combinatorial: a contraction induced between the relative GNS cores intertwines left actions; operator monotonicity of the resolvent, followed by an integral representation of $-\log$, compares the quadratic forms of the relative modular operators. This is the mechanism behind Araki’s monotonicity theorem Araki 1976, §5, pp. 824–827.

Complete positivity is a robust physical hypothesis and is needed for stability under ancillas. Some strengthened mathematical versions work under weaker Schwarz-type assumptions, but bare positivity alone is not the theorem stated here. Normality ensures that both pulled-back functionals are normal and that the standard-form construction stays in the declared category.

Equality is not automatically recovery. To infer a recovery map one must state the reference functional, support conditions, and the relevant family of states; those hypotheses are developed on the sufficiency and Petz recovery page.

The orientation can be checked in a two-point commutative algebra. Take $\rho=(3/4,1/4)$ and $\sigma=(1/2,1/2)$, then apply the channel that forgets which point occurred. The input relative entropy is

$$\frac34\log\frac32+\frac14\log\frac12>0,$$

whereas both output states are the unique state on $\mathbb C$ and have relative entropy zero. A coarse graining therefore lowers distinguishability in the same direction as the modular theorem. This elementary case also shows that strict decrease is generic and that equality contains additional structure.

Coherent excitation in a Rindler wedge

Let $W_R=\{x^1>|x^0|\}$ and let $\omega_0$ be the Minkowski vacuum of the free scalar field. A real compactly supported classical solution $f$ produces a Weyl-coherent state $\omega_f=\omega_0\circ\operatorname{Ad}W(f)^*$. For data supported in the wedge, the vacuum modular flow is the boost flow. With canonical stress tensor and initial surface $x^0=0$, the coherent-state relative entropy is

$$S_{\mathcal A(W_R)}(\omega_f\Vert\omega_0) =2\pi\int_{x^1>0}x^1T_{00}^{\mathrm{cl}}[f](0,\mathbf x) \,\mathrm d^{d-1}\mathbf x.$$

The calculation uses cancellation of the vacuum entanglement term and identifies the remaining modular-energy difference with the classical boost energy Casini, Grillo, and Pontello 2019, §§II–IV, article 125020. Positivity is independently checked because $T_{00}^{\mathrm{cl}}\geq0$ for the minimally coupled massive scalar on the initial surface.

Translate the wedge inward to $W_a=\{x^1-a>|x^0|\}\subset W_R$. Restriction of both states gives

$$S_{\mathcal A(W_a)}(\omega_f\Vert\omega_0) \leq S_{\mathcal A(W_R)}(\omega_f\Vert\omega_0).$$

When the classical data are supported in $x^1>a$, the explicit formula changes the weight from $x^1$ to $x^1-a$, so the difference is $2\pi a$ times the nonnegative classical energy. This model calculation independently reproduces the abstract ordering required by data processing.

Two adversarial changes expose the boundary. First choose $s(\omega_f)\nleq s(\omega_0)$ in a nonfaithful representation: the entropy is infinite and a finite modular-energy formula is unjustified. Second replace restriction by a nonnormal coarse graining: the output functional may be singular, so the asserted normal-state theorem has no object to compare.

Exercises

1. Classical two-level check. For commuting densities $\rho=\operatorname{diag}(p,1-p)$ and $\sigma=\operatorname{diag}(q,1-q)$, verify the modular definition reduces to binary relative entropy and becomes infinite when $q=0<p$.

Solution

In the diagonal standard representation, $\Delta_{\sigma\mid\rho}$ acts by the ratios $q/p$ and $(1-q)/(1-p)$ on the two components of $\xi_\rho$. Therefore $-\langle\xi_\rho,\log\Delta_{\sigma\mid\rho}\xi_\rho\rangle=p\log(p/q)+(1-p)\log((1-p)/(1-q))$. If $q=0<p$, the first ratio vanishes and the corresponding contribution is $+\infty$, exactly matching the support rule.

2. Wedge restriction. Assume the coherent data lie in $x^1>a>0$. Derive the difference between the two wedge entropies.

Solution

The modular weight for $W_a$ is $x^1-a$. Subtraction gives $S_{W_R}-S_{W_a}=2\pi a\int T_{00}^{\mathrm{cl}}\,\mathrm d^{d-1}\mathbf x\geq0$. Thus the explicit solution obeys data processing and also shows strict inequality whenever the excitation has positive energy.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11 (1976): 809–833. DOI.
• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras II.” Publications of the Research Institute for Mathematical Sciences 13 (1977): 173–192. DOI.
• Casini, Horacio, Eduardo Testé Grillo, and Diego Pontello. “Relative Entropy for Coherent States from Araki Formula.” Physical Review D 99 (2019): 125020. DOI.