Araki–Uhlmann Relative Entropy in Quantum Theory

• Araki–Uhlmann relative entropy is an operator-algebraic extension of the Kullback–Leibler divergence, defined via Tomita–Takesaki modular theory on von Neumann algebras.
• It exhibits monotonicity under quantum channels, ensuring the data processing inequality and robust operational properties in quantum information settings.
• This framework enables precise quantification of quantum distinguishability and the derivation of thermodynamic bounds in both finite and infinite-dimensional systems.

The Araki–Uhlmann relative entropy is a fundamental concept in mathematical physics, quantum information theory, and operator algebras, providing an operator-algebraic generalization of the classical Kullback–Leibler divergence and Umegaki’s relative entropy to the setting of possibly infinite-dimensional von Neumann algebras without any restriction to trace-class operators or density matrices. This notion, originally introduced by Araki and Uhlmann, is defined intrinsically through Tomita–Takesaki modular theory and provides a basis for quantum distinguishability measures, operational monotonicity properties (such as the data processing inequality), and thermodynamic functionals in both quantum statistical mechanics and algebraic quantum field theory.

1. Formal Definition and Mathematical Framework

Given a von Neumann algebra $\mathcal{M}$ in standard form, with two faithful, normal positive normalized states (or more generally, normal positive functionals) $\omega$ and $\phi$, the Araki–Uhlmann relative entropy is defined via the relative modular operator $\Delta_{\omega,\phi}$ constructed from the polar decomposition of the closable anti-linear Tomita operator $S_{\omega,\phi}$, which acts as $S_{\omega,\phi}(a\Omega_\phi) = a^* \Omega_\omega$ for $a \in \mathcal{M}$. The relative entropy is then

$$S(\omega \| \phi) = -\langle \Omega_\omega, \log \Delta_{\omega,\phi} \,\Omega_\omega \rangle$$

where $\Omega_\omega$ and $\Omega_\phi$ are cyclic and separating vectors for $\mathcal{M}$ corresponding to $\omega$ and $\phi$.

In type I (finite-dimensional) settings, this reduces to

$$S(\rho \| \sigma) = \operatorname{Tr}(\rho \log \rho - \rho \log \sigma)$$

where $\rho$, $\sigma$ are density operators, but the Araki–Uhlmann construction is independent of such a representation and is well-defined on all von Neumann algebras, even when type III structure precludes the existence of a density operator.

The construction is robust under various mathematical equivalences: it can be formulated in the standard form of $\mathcal{M}$ or via the spatial derivative and spectral calculus (with equivalence proved between these forms), and scales correctly under affine rescaling of the functionals: $$S(\lambda \omega, \mu \phi) = \lambda S(\omega, \phi) - \lambda \omega(\mathbf{1}) \log (\mu/\lambda)$$ for $\lambda,\mu > 0$ (Reible, 8 Jan 2025).

2. Fundamental Properties and Monotonicity

A central property of the Araki–Uhlmann relative entropy is its monotonicity under positive unital (more generally, Schwarz or completely positive) maps, commonly referred to as the data processing inequality. For a normal, completely positive, unital map $\Phi: \mathcal{M}_1 \rightarrow \mathcal{M}_2$ and states $\omega_2, \phi_2$ on $\mathcal{M}_2$, with $\omega_1 = \omega_2 \circ \Phi$ and $\phi_1 = \phi_2 \circ \Phi$, the monotonicity theorem asserts: $$S_{\mathcal{M}_1}(\omega_1, \phi_1) \leq S_{\mathcal{M}_2}(\omega_2, \phi_2)$$ Equality is achieved if and only if $\Phi$ is sufficient (i.e., there exists a recovery map reversing the action of $\Phi$ for $\omega_2$ and $\phi_2$) (Jencova, 2016).

The data processing inequality underlies operational interpretations in quantum information theory and ensures that no quantum channel (CPTP map) can increase the distinguishability of two quantum states (Pérez-Pardo, 2022, Reible, 8 Jan 2025). Further, vector-induced monotonicity holds: for a contraction $V$ on $\mathscr{H}$, $$S_{\mathcal{M}}(V\psi,V\psi) \leq S_{\mathcal{M}}(\psi,\psi)$$ (Reible, 8 Jan 2025).

3. Connections to Quantum Information Quantities and Limiting Cases

The Araki–Uhlmann relative entropy arises as a limiting case of various interpolating quantum divergences. In particular, sandwiched Rényi divergences $D_\alpha(\rho\|\sigma)$, defined via noncommutative $L_p$-spaces or weighted $L_p$-norms, converge to the Araki–Uhlmann relative entropy as $\alpha \to 1$: $$\lim_{\alpha\to1} D_\alpha(\rho\|\sigma) = \langle \rho, \log \Delta_{\rho,\sigma}\,\rho \rangle = S(\rho\|\sigma)$$ (Berta et al., 2016, Jencova, 2016). Other limits recover well-known quantum information measures:

• $\alpha \to \tfrac{1}{2}$: $- \log$ Uhlmann fidelity (quantifies pure-state overlap as $$F(\rho,\sigma) = \sup\{ |\langle \rho|U\sigma \rangle|^2 : U\ \text{contraction intertwiners} \}$$),
• $\alpha \to \infty$: max-relative entropy (worst-case distinguishability).

In chiral CFT or free fermion models, the Araki–Uhlmann relative entropy reproduces finite mutual information, overcoming the divergence of von Neumann entropy in type III algebras (Longo et al., 2017, Xu, 2018).

4. Applications in Quantum Field Theory and Statistical Mechanics

In algebraic quantum field theory (AQFT), the Araki–Uhlmann relative entropy enables the rigorous definition and computation of mutual information and geometric entanglement measures, even in the absence of type I structure. For two disjoint spacetime regions, the mutual information between their vacuum-restricted algebras is defined as $S(\omega, \omega_1 \otimes \omega_2)$, which is finite and encodes quantum correlations (Xu, 2018).

In perturbative AQFT, the Araki–Uhlmann approach extends to non-Gaussian, interacting KMS states, with positivity and additivity properties preserved. Relative entropy densities can be defined via suitable adiabatic limits to handle infinite-volume cases (Drago et al., 2017). In operator-algebraic statistical mechanics, the relative entropy provides bounds for thermodynamic functionals such as the free energy and underlies the Bogoliubov inequality (Reible, 8 Jan 2025).

Additionally, the connection to subfactor theory is deep: the mutual information regularized via Araki–Uhlmann entropy detects the Jones index of representation inclusions (Longo et al., 2017, Xu, 2018).

5. Explicit Calculations: Free Theories, Coherent and Squeezed States

For quasi-free states on CCR algebras (bosons) and self-dual CAR algebras (fermions), the Araki–Uhlmann relative entropy between a reference state and its coherent or multi-excitation is computable in terms of underlying single-particle data.

• Bosonic case: For a coherent excitation $|\Psi_f\rangle = W(f)|\Omega\rangle$, the entropy is

$S(\omega_f\|\omega) = -\langle f, K f\rangle$

where $K$ is the single-particle modular Hamiltonian (Bostelmann et al., 2020, Garbarz et al., 2022). For a wedge region in Minkowski, $K$ is a boost generator and $S(\omega_f\|\omega)$ corresponds to a Noether (boost) charge (Casini et al., 2019, Fröb et al., 2023). Numerical implementations confirm that for massive free fields: - The entropy decreases monotonically with increasing mass (suppression of long-range correlations) (Guimaraes et al., 13 Feb 2025, Guimaraes et al., 17 Apr 2025). - The entropy increases monotonically with the spatial region size or geometric separation between the regions of excitation (Caribé et al., 23 Aug 2025). - For squeezed states, the relative entropy is governed by the proportionality to the smeared Pauli–Jordan distribution (Guimaraes et al., 17 Apr 2025).

• Fermionic case: For a CAR algebra with KMS (quasi-free) state $\omega$, the entropy for a multi-excitation state is expressible as a sum or Pfaffian formula over single-particle data (Galanda et al., 2023, Galanda, 2022).

6. Extensions: Quantum Channels, Subalgebras, and Generalized Divergences

The Araki–Uhlmann relative entropy extends to the paper of quantum channels and bimodule maps in finite von Neumann algebra inclusions, where analogous entropy functionals interpolate between the Pimsner–Popa index and the Connes–Størmer entropy (Zhao, 2023). For such channels $\Phi$, the relative entropy, and more generally the family of sandwiched Rényi entropies $S_p(\Phi,\Psi)$, feature critical monotonicity, convexity, and order relations, providing both operational and structural invariants for subfactor theory and noncommutative probability.

Furthermore, the Petz–Rényi relative entropy generalizes the Araki–Uhlmann formula by replacing the logarithm with an $\alpha$-parametrized exponential average of the relative modular operator: $$S_\alpha(\Psi \| \Phi) = \frac{1}{\alpha - 1} \log \langle \Psi | \Delta_{\Psi|\Phi}^{1-\alpha} | \Psi \rangle$$ This reduces to the Araki–Uhlmann entropy at $\alpha \to 1$ and, for general $\alpha$, incorporates both the symplectic (antisymmetric) and symmetric parts of the two-point functions, reflecting genuine quantum effects beyond the classical regime (Fröb et al., 14 Nov 2024).

7. Operational Significance, Convexity, and Physical Bounds

The Araki–Uhlmann relative entropy satisfies key operational inequalities and functional properties:

• Monotonicity under quantum channels and restrictions to subalgebras (data processing inequality).
• Positivity: $S(\omega \| \phi) \geq 0$, with equality iff $\omega = \phi$.
• Convexity: Several convexity results, e.g., convexity of the entropy along modular tunnels or parameterized families of subspaces, are proved (these underpin quantum energy inequalities such as QNEC) (Ciolli et al., 2021, Garbarz et al., 2022).
• Physical bounds: In QFT, entropic analogs of Bekenstein bounds and other geometric-energy inequalities can be derived from relative entropy (e.g., $S_I(f) \leq (\pi L/2) E(f)$ for interval $I$) (Garbarz et al., 2022).

These properties underpin the interpretational role of the relative entropy as a physically meaningful, representation-independent measure of quantum distinguishability and resource in both quantum field theory and operator algebraic statistical mechanics.

Table: Key Mathematical Objects in the Araki–Uhlmann Relative Entropy

Mathematical entity	Role/Definition	Context
$\Delta_{\omega,\phi}$	Relative modular operator via Tomita–Takesaki theory	Operator algebras
$S(\omega \| \phi)$	$$- \langle \Omega_\omega, \log \Delta_{\omega,\phi}\, \Omega_\omega \rangle$$	Relative entropy
$S_{\alpha}(\Psi \|\Phi)$	$$\frac{1}{\alpha-1} \log \langle \Psi| \Delta_{\Psi|\Phi}^{1-\alpha}|\Psi\rangle$$	Petz–Rényi entropy
$\mathcal{N}(O)$	Local von Neumann algebra (from Weyl operators on $O$)	AQFT

The Araki–Uhlmann relative entropy thus provides a unifying mathematical infrastructure for understanding quantum distinguishability, thermodynamic inequalities, quantum channel capacities, and entanglement measures across both finite- and infinite-dimensional quantum systems in the language of operator algebras.