Araki–Uhlmann Relative Entropy

• Araki–Uhlmann relative entropy is a measure defined via modular theory that quantifies the distinguishability of quantum states in von Neumann algebras.
• It unifies various quantum divergences, including Umegaki’s relative entropy, sandwiched Rényi, and max-relative entropy, through a common algebraic framework.
• Its monotonicity under completely positive maps and applications in quantum field theory and statistical mechanics underscore its operational and theoretical significance.

The Araki–Uhlmann relative entropy is a central functional in the theory of operator algebras, quantum information, and quantum field theory. It extends the notion of quantum relative entropy to the setting of von Neumann algebras and plays a pivotal role in quantum statistical mechanics, entanglement theory, and the paper of quantum channels. On the algebraic level, it unifies and generalizes distinguishability measures for quantum states, encompassing Umegaki’s quantum relative entropy, Uhlmann’s fidelity, sandwiched Rényi divergences, and max-relative entropy as limiting cases. Its robust operational and mathematical properties are deeply intertwined with Tomita–Takesaki modular theory and complex interpolation of noncommutative $L_p$ spaces, leading to a unification of entropic quantities across finite and infinite-dimensional quantum systems.

1. Mathematical Formulation and Modularity

In the context of von Neumann algebras, the Araki–Uhlmann relative entropy between two normal faithful states $\varphi$ and $\psi$ is defined using modular theory. For representations where these states correspond to cyclic and separating vectors $|\xi_\varphi\rangle$ and $|\xi_\psi\rangle$, one introduces the relative modular operator $\Delta_{\varphi,\psi}$ via the polar decomposition $$s_{\varphi,\psi} = J_{\varphi,\psi} \Delta_{\varphi,\psi}^{1/2}$$, where $s_{\varphi,\psi}$ is the closed anti-linear operator satisfying $$s_{\varphi,\psi} a|\xi_\psi\rangle = a^\ast|\xi_\varphi\rangle$$ for all $a$ in the algebra.

The Araki–Uhlmann relative entropy is then

$$S(\varphi\|\psi) = -\langle \xi_\varphi, \log\Delta_{\varphi,\psi} \, \xi_\varphi \rangle,$$

provided that $\mathrm{supp}(\varphi) \leq \mathrm{supp}(\psi)$, and $S(\varphi\|\psi) = +\infty$ otherwise. In the type I (matrix) case, this reduces to the Umegaki formula $$S(\rho\|\sigma) = \operatorname{Tr}[\rho (\log\rho - \log\sigma)]$$.

A crucial feature is that $S(\varphi\|\psi)$ depends only on the states and the algebra, not on the particular Hilbert space realization. Moreover, via the spectral resolution of the modular operator, one obtains functional calculus representations that are essential for analysis in infinite-dimensional settings (Berta et al., 2016).

2. Relationship to Rényi Divergences and Weighted Noncommutative $L_p$ Spaces

The Araki–Uhlmann relative entropy arises as a limiting case of a one-parameter family of nonlinear quantum divergences, notably the sandwiched Rényi divergences, when these are generalized to a fully algebraic setting. Araki and Masuda introduced weighted noncommutative vector-valued $L_p$-spaces to define the so-called Araki–Masuda divergences, which, for a parameter $\alpha$ (with $p=2\alpha$), are

$$D_\alpha(\rho\|\sigma) = \frac{1}{\alpha-1} \log \|\rho\|_{2\alpha,\sigma}^2,$$

with

$$\|\rho\|_{p,\sigma} = \| D_\sigma^{1/p - 1/2} D_\rho^{1/2} \|_p$$

in finite dimensions. The limit $\alpha\to1$ yields the Araki–Uhlmann relative entropy, while $\alpha\to 1/2$ and $\alpha\to\infty$ yield minus the logarithm of Uhlmann’s fidelity and the max-relative entropy, respectively (Berta et al., 2016, Jencova, 2016).

Through complex interpolation (Riesz–Thorin theorem for the weighted $L_p$-spaces), one can prove monotonicity and data-processing inequalities for these divergences, with Umegaki/Araki–Uhlmann relative entropy being central in connecting operational distinguishability, fidelity, and max-entropy (Berta et al., 2016).

3. Fundamental Properties: Monotonicity, Data-Processing, and Sufficiency

The monotonicity property (also known as Uhlmann’s theorem) asserts that the Araki–Uhlmann relative entropy is non-increasing under completely positive, trace-preserving (CPTP) maps: $$S(\varphi\|\psi) \geq S(\varphi\circ\Phi\|\psi\circ\Phi)$$ for any CPTP map $\Phi$ (Pérez-Pardo, 2022, Reible, 8 Jan 2025). This property, proved via the interpolation theory of quadratic forms and modular operator functionals, underpins the operational significance of relative entropy as a measure of statistical distinguishability that cannot be increased by quantum operations.

Equality in the monotonicity (data-processing) inequality is intimately related to the notion of sufficiency or reversibility. If a channel $\Lambda$ satisfies

$$S(\varphi\|\psi) = S(\Lambda(\varphi)\|\Lambda(\psi)),$$

then there exists a recovery map (e.g., Petz map) such that the original states can be reconstructed, characterizing the reversibility (sufficiency) of the channel (Jencova, 2016).

In many cases, monotonicity extends from completely positive maps to Schwarz maps or unital 2-positive maps, emphasizing the robustness of this property in the operator algebraic framework (Pérez-Pardo, 2022, Reible, 8 Jan 2025).

4. Applications in Quantum Field Theory, Statistical Mechanics, and Subfactor Theory

In algebraic quantum field theory (AQFT), the Araki–Uhlmann relative entropy provides a rigorous framework where even for type III von Neumann algebras (arising in QFT), relative entropy remains finite and meaningful, while the usual von Neumann entropy diverges (Longo et al., 2017, Xu, 2018). Relative entropy quantifies mutual information, provides a finite measure of entanglement between regions, and is related to the index of subfactor inclusions in conformal field theories.

Explicit computations for quasi-free fermion and bosonic states, as well as for coherent and squeezed state excitations in free quantum field theories, yield analytic or numerically robust expressions for $S(\varphi\|\psi)$ in terms of the smeared Pauli–Jordan distribution or energy-momentum tensor, with positivity, monotonicity in region size, and decay with increasing mass (Casini et al., 2019, Garbarz et al., 2022, Guimaraes et al., 13 Feb 2025, Guimaraes et al., 17 Apr 2025, Caribé et al., 23 Aug 2025).

In further developments, the framework extends to quantized energy inequalities, entropy production, and the paper of equilibrium and nonequilibrium (KMS and NESS) states in statistical mechanics. Perturbative constructions and the analysis of the adiabatic limit yield rigorous statements about relative entropy densities and entropy production (Drago et al., 2017).

The relation of relative entropy to the index of representations and subfactor theory is particularly notable in conformal field theory: the logarithm of the index appears as a limiting value of regularized relative entropy between vacuum states and those compressed by conditional expectations (Xu, 2018).

5. Generalizations: Petz–Rényi Relative Entropy and Noncommutative Extensions

Recent advances generalize the Araki–Uhlmann construction to the Petz–Rényi relative entropies in the algebraic framework. For cyclic and separating vectors $|\Psi\rangle$, $|\Phi\rangle$, the Petz–Rényi divergence is

$$S_\alpha(\Psi\|\Phi) = \frac{1}{\alpha-1} \log\langle \Psi|\Delta_{\Psi|\Phi}^{1-\alpha}|\Psi\rangle,$$

recovering the Araki–Uhlmann entropy as $\alpha\to 1$ (Fröb et al., 14 Nov 2024). This generalization is essential for QFT and infinite systems, as the Petz–Rényi divergence is sensitive to both the antisymmetric (symplectic) and symmetric structure of the two-point function, thus capturing genuine quantum (as opposed to classical) corrections.

Within completely positive bimodule maps and quantum channels on finite von Neumann algebras, the framework interpolates smoothly between the logarithm of the Pimsner–Popa index and the Connes–Størmer entropy using suitable sandwiched Rényi divergences, establishing bridges between index theory and entropy in quantum information contexts (Zhao, 2023).

6. Operational and Physical Interpretation

Operationally, the Araki–Uhlmann relative entropy quantifies the optimal error exponent in quantum hypothesis testing and is directly linked to the distinguishability and reversibility of quantum processes (Kudler-Flam, 2021). Its monotonicity properties guarantee the non-increase of distinguishability under noisy evolution, underpinning the design of quantum communication, error correction, and data processing protocols. In quantum field theory, relative entropy encodes entanglement, energy conditions, and Noether charges in geometrically nontrivial spacetimes (e.g., de Sitter, Rindler wedges) (Fröb et al., 2023).

Further, results such as regularized Uhlmann-type theorems establish how single-system divergences can be realized as asymptotic limits over extensions or purifications, connecting single-shot and many-copy scenarios in quantum information (Mazzola et al., 3 Feb 2025).

7. Numerical and Analytic Aspects, and Open Directions

Substantial numerical studies based on the explicit integral representations from modular theory confirm that the relative entropy between coherent or squeezed states and the vacuum is positive, monotonic with respect to region size, and decreases with increasing mass (Guimaraes et al., 13 Feb 2025, Guimaraes et al., 17 Apr 2025, Caribé et al., 23 Aug 2025). For more complex situations (e.g., multi-region, multi-particle, or across phase transitions), these modular integral techniques provide a concrete path for both analytical and numerical evaluation.

Current research continues to explore generalizations to the Petz–Rényi and sandwiched Rényi divergences in non-type I and infinite-dimensional algebras, behavior under general quantum channels, connections to bulk/boundary correspondences in holography, and applications to entropy production and energy inequalities in nonequilibrium steady states.

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In summary, the Araki–Uhlmann relative entropy, constructed via modular theory, is a cornerstone of algebraic quantum information, offering a unified, operationally robust, and mathematically rigorous framework for quantifying quantum distinguishability and entanglement in operator algebras and quantum field theory. As both its finite and infinite-dimensional manifestations are now fully understood in terms of noncommutative $L_p$-space geometry, monotonicity, and recovery, it serves as the essential reference point for more general quantum divergences, quantum information inequalities, and algebraic entanglement measures.