Araki Relative Entropy

Araki relative entropy compares two normal states directly on a von Neumann algebra, without first factorizing the Hilbert space or assigning either state a local density matrix. It recovers $\operatorname{tr}\rho(\log\rho-\log\sigma)$ in a finite factor, is nonnegative, and decreases when observables are discarded. In continuum QFT it often yields a finite, regulator-independent comparison even when individual entanglement entropies diverge—but it is fundamentally extended-valued, and support mismatch or excessive ultraviolet difference can give $+\infty$.

Required background. Quantum Information supplies matrix relative entropy, support projections, and data processing.

Helpful background. Flat-Spacetime Quantum Field Theory provides local operator algebras; Partial Differential Equations and Green Functions defines the distributional state regularity behind ultraviolet comparisons; Causality and Global Structure establishes nested causal regions; Quantum Fields in Curved Spacetime provides normal Hadamard states and local von Neumann algebras.

For matrices, the support $s(\rho)$ is the span of eigenvectors with nonzero eigenvalue. The logarithm is taken only on that support. The condition $s(\rho)\le s(\sigma)$ prevents $\rho$ from assigning probability to an event that $\sigma$ declares impossible; its algebraic version will control finiteness below.

Relative modular data

Let $\mathcal M$ be a von Neumann algebra in standard form and let $\rho$ and $\sigma$ be normal states, represented by vectors $\xi_\rho$ and $\xi_\sigma$ in the natural positive cone. First suppose both are faithful. On the dense set $\mathcal M\xi_\rho$, define the relative antilinear operator

$$S_{\sigma\mid\rho}(A\xi_\rho)=A^*\xi_\sigma.$$

Its closure determines the relative modular operator

$$\Delta_{\sigma\mid\rho} =S_{\sigma\mid\rho}^*S_{\sigma\mid\rho}.$$

The order of the subscripts is consequential. With this convention, the reference state $\sigma$ appears first in $\Delta_{\sigma\mid\rho}$, while the expectation defining $S(\rho\Vert\sigma)$ is taken in $\xi_\rho$.

Araki relative entropy is

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

The logarithm is defined by spectral calculus. In the nonfaithful case, support projections and the general relative modular construction replace inverses; if $s(\rho)\not\le s(\sigma)$, the definition gives $+\infty$. Even when support is included, the spectral integral of $-\log\Delta_{\sigma\mid\rho}$ may diverge positively. Araki’s original construction and its extension beyond faithful functionals are developed across Araki 1976, pp. 809–833 and Araki 1977, pp. 173–192.

Recovery of the density-matrix formula

Take $\mathcal M=\mathcal B(\mathbb C^n)$ in its Hilbert–Schmidt standard representation, with $\xi_\rho=\rho^{1/2}$ and $\xi_\sigma=\sigma^{1/2}$. For faithful matrices,

$$\Delta_{\sigma\mid\rho}=L_\sigma R_{\rho^{-1}},$$

where $L$ and $R$ denote left and right multiplication. Since these commuting operators have logarithm

$$\log\Delta_{\sigma\mid\rho}=L_{\log\sigma}-R_{\log\rho},$$

the algebraic definition becomes

$$\begin{aligned} S_{\mathcal M}(\rho\Vert\sigma) &=-\operatorname{tr}\!\left[\rho^{1/2} \big((\log\sigma)\rho^{1/2}-\rho^{1/2}\log\rho\big)\right]\\ &=\operatorname{tr}\!\left[\rho(\log\rho-\log\sigma)\right]. \end{aligned}$$

This calculation checks both the sign and the order of the relative modular operator. If $\rho$ has support outside that of $\sigma$, regularize $\sigma$ by assigning a small eigenvalue $\epsilon$ there. The corresponding term behaves as $-\rho_i\log\epsilon\to+\infty$, exactly reproducing the support rule rather than hiding it.

Positivity and monotonicity

For normalized states,

$$S_{\mathcal M}(\rho\Vert\sigma)\ge0,$$

with equality precisely when the two states agree. In a finite factor this is Klein’s inequality; Araki’s theorem establishes the operator-algebraic result without requiring a trace on $\mathcal M$ Araki 1977, Lemma 3.2 and Theorem 3.6(1), pp. 177–179, PDF. Extended values are allowed, so positivity includes $+\infty$.

For faithful states the sign also follows directly from the relative modular operator. Its spectral measure in the normalized vector $\xi_\rho$ is a probability measure, and

$$\langle\xi_\rho,\Delta_{\sigma\mid\rho}\xi_\rho\rangle =\lVert S_{\sigma\mid\rho}\xi_\rho\rVert^2 =\lVert\xi_\sigma\rVert^2=1,$$

where the first expression is understood as a quadratic form. Scalar Jensen applied to the convex function $-\log x$ gives $S(\rho\Vert\sigma)\ge-\log1=0$. This short proof uses spectral calculus and normalization, so it does not smuggle a trace into a Type III algebra. Monotonicity under a change of algebra is a further theorem, not a consequence of this one-line inequality.

If $\mathcal N\subset\mathcal M$ is a von Neumann subalgebra sharing the same identity, restrictions of normal states remain normalized and obey

$$S_{\mathcal N}(\rho|_{\mathcal N}\Vert\sigma|_{\mathcal N}) \le S_{\mathcal M}(\rho\Vert\sigma).$$

This is monotonicity under inclusion, the algebraic form of data processing. The inclusion is a unital $*$-homomorphism, so Uhlmann’s theorem applies without a conditional expectation or faithful-state assumption Uhlmann 1977, §5, Propositions 17–18, pp. 30–31, PDF.

The proof interpolates the positive forms $\rho(AA^*)$ and $\sigma(A^*A)$. Evaluated at the identity, the interpolation gives $Q_t^{\mathcal M}$; for matrices this is $\operatorname{tr}(\rho^{1-t}\sigma^t)$. Restriction gives $Q_t^{\mathcal M}\le Q_t^{\mathcal N}$ for $0\lt t\lt1$. Subtracting from the common state normalization and taking the defining endpoint limit yields

$$\begin{aligned} S_{\mathcal M}(\rho\Vert\sigma) &=\lim_{t\downarrow0}\frac{1-Q_t^{\mathcal M}}{t}\\ &\ge\lim_{t\downarrow0}\frac{1-Q_t^{\mathcal N}}{t} =S_{\mathcal N}(\rho|_{\mathcal N}\Vert\sigma|_{\mathcal N}). \end{aligned}$$

For nonfaithful states the interior limit of $Q_t$ need not be one; a support mismatch can make the entropy limit infinite. The interpolation argument and the identification with relative modular entropy are developed in Uhlmann 1977, §§4–5, pp. 29–31, PDF and Reible 2025, Theorem IV.3.7 and Remark IV.3.8(1), pp. 62–65, PDF. The direction has a simple meaning: a larger algebra offers more tests to distinguish states. Extended values are allowed, but subtraction of unspecified infinities is not.

A controlled finite check makes the loss visible. Let $\rho=|+\rangle\langle+|$ and $\sigma=\mathbf1/2$ on a qubit. On the full matrix algebra, $S(\rho\Vert\sigma)=\log2$. Restrict both states to the diagonal algebra in the computational basis; both then give $(1/2,1/2)$, so the restricted relative entropy is zero. Thus $0\le\log2$ verifies the direction and shows that discarded phase-sensitive observables carried all the distinguishability.

Why relative quantities survive the continuum limit

For a sharp region in continuum QFT, Type III structure removes a canonical trace and reduced density matrix. A cutoff calculation may nevertheless write

$$S(\rho_{O,a}\Vert\sigma_{O,a}) =-S(\rho_{O,a}) -\operatorname{tr}(\rho_{O,a}\log\sigma_{O,a}).$$

Each term on the right can contain boundary-divergent contributions as the cutoff $a\to0$. Relative modular theory defines the left-hand comparison directly on the limiting local algebra. For physically nearby states with the same admissible short-distance structure, common ultraviolet behavior can cancel in the combined quantity, yielding a finite limit with operational meaning. No separate entanglement entropy has to exist.

The cancellation is conditional. A state not normal in the chosen representation, a support mismatch, or a perturbation with too much short-distance energy can produce infinite relative entropy. Hence the correct answer is not “relative entropy always remains finite,” but “it remains well-defined as an extended quantity and is often finite for controlled state pairs even where the individual local entropies are unavailable.”

If a faithful finite reference state is written $\sigma=e^{-K_\sigma}/Z_\sigma$, then

$$S(\rho\Vert\sigma) =\Delta\langle K_\sigma\rangle-\Delta S,$$

where both differences compare $\rho$ with $\sigma$. Positivity turns this identity into an energy–information inequality. In a Type III algebra, this shorthand is retained only when the modular generator and the differences are defined; the Araki expression is primary.

Support examples

Let $\rho=|0\rangle\langle0|$ and $\sigma=q|0\rangle\langle0|+(1-q)|1\rangle\langle1|$ with $0<q<1$. Then $s(\rho)\le s(\sigma)$ and

$$S(\rho\Vert\sigma)=-\log q<\infty.$$

If instead $\sigma=|1\rangle\langle1|$, the supports are orthogonal and $S(\rho\Vert\sigma)=+\infty$. Both states are legitimate and nonfaithful. The contrast shows why one cannot define the general theory merely by inserting ordinary inverses: support data are part of the statement, not a technical afterthought Araki 1977, Definition 3.1, p. 177, PDF.

Solved diagnostic

For $\rho=\operatorname{diag}(3/4,1/4)$ and $\sigma=\mathbf1/2$, calculate $S(\rho\Vert\sigma)$. What happens after restriction to the scalar algebra $\mathbb C\mathbf1$, and what changes if $\sigma$ is replaced by $|1\rangle\langle1|$?

Solution

Direct substitution gives $S(\rho\Vert\sigma)=\tfrac34\log(3/2)+\tfrac14\log(1/2)\simeq0.130812$. On $\mathbb C\mathbf1$, both normalized states agree on the only observable, so the relative entropy is zero; this checks monotonicity because $0\le0.130812$. Replacing the reference by $|1\rangle\langle1|$ makes the support condition fail: $\rho$ assigns probability $3/4$ to $|0\rangle$, where the reference assigns zero. The relative entropy is therefore $+\infty$, not the value of a formula containing an illicit $\log0$ cancellation.

References

• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras II.” Publications of the Research Institute for Mathematical Sciences 13, no. 1 (1977): 173–192. DOI.
• Araki, Huzihiro. “Relative Entropy of States of von Neumann Algebras.” Publications of the Research Institute for Mathematical Sciences 11, no. 3 (1976): 809–833. DOI.
• Reible, Benedikt M. “Monotonicity of the Relative Entropy and the Two-sided Bogoliubov Inequality in von Neumann Algebras.” arXiv:2501.04564v1 (2025), preprint. arXiv. Open PDF.
• Uhlmann, Armin. “Relative Entropy and the Wigner–Yanase–Dyson–Lieb Concavity in an Interpolation Theory.” Communications in Mathematical Physics 54 (1977): 21–32. DOI. Open PDF.