Unified Formula for Relative Entropy

• Unified Formula for Relative Entropy is a comprehensive framework unifying Umegaki, Rényi, and Tsallis measures across quantum and classical systems.
• It employs variational, integral, and operator-functional methods that facilitate smooth interpolation between additive and non-additive entropy forms.
• The approach underpins diverse applications in quantum mechanics, statistical physics, and resource theories while ensuring consistency with monotonicity and convexity principles.

The unified formula for relative entropy provides a mathematical framework that simultaneously generalizes standard divergence measures, interpolates between multiple entropy-like quantities, and allows for a systematic analysis of quantum, classical, and resource-theoretic distinguishability. The construction encapsulates diverse contexts—quantum information, statistical mechanics, kinetic theory, field theory, and operational resource theories—using core variational, integral, or operator-functional expressions that reduce to Umegaki (von Neumann), Rényi, and Tsallis relative entropies and their physical or operational generalizations.

1. Canonical Definition and Variational Frameworks

The most fundamental quantum relative entropy between density operators $\rho, \sigma$ on a Hilbert space $\mathcal{H}$ is the Umegaki entropy: $$D(\rho\Vert\sigma) := \operatorname{Tr}\left[\rho(\log\rho - \log\sigma)\right],$$ defined when $$\operatorname{supp}\rho\subseteq\operatorname{supp}\sigma$$, and $+\infty$ otherwise. This form unifies the classical Kullback–Leibler divergence and subsumes prominent entropic quantities used across information theory and statistical physics (Frenkel, 2022, Lewin et al., 2013).

Unified functional characterizations extend this definition in two principal directions:

• Parametric (α,β) Families: A two-parameter unified quantum relative entropy for $\alpha\in[0,1]$, $\beta\in\mathbb{R}$,

$$D_{\alpha}^{\beta}(\rho\Vert\sigma) = \frac{1}{(\alpha-1)\beta} \left[\, \left(\operatorname{Tr}\rho^{\alpha}\sigma^{1-\alpha}\right)^{\beta} - 1\,\right], \quad \beta\neq0,\,\alpha<1,$$

with appropriate limiting cases for $\beta\to0$ (Rényi), $\beta=1$ (Tsallis), and $\alpha\to1$ (Umegaki) (Cheng et al., 11 Jun 2025, Wu et al., 11 Jun 2025). These families interpolate smoothly between additive/logarithmic and non-additive/power-law divergences, ensuring broad applicability.

• Convex-Functional and Variational Principles: For convex $\varphi$, $$D_\varphi(\rho\Vert\sigma) := \operatorname{Tr}[\varphi(\rho) - \varphi(\sigma) - \varphi'(\sigma)(\rho-\sigma)]$$, with operator-monotonicity of $\varphi'$ ensuring monotonicity under completely positive trace-preserving maps (Lewin et al., 2013).

Additionally, variational frameworks define a divergence $\Psi_s(\omega\Vert\sigma)$ for normal states $\omega,\sigma$ on a von Neumann algebra, such that

$$\Psi_{s}(\omega\Vert\sigma) = \inf_{x} \left\{\, C_{p} \int_{0}^\infty \left[\omega(x(t)^*x(t)) + t^{-1}F((1-x(t))\sigma(1-x(t))^*)\right] t^{-2/p}\,dt \right\} - 1,$$

recovering, in the limits $s\to1$, $s\to\frac12$, and $s\in(\tfrac12,1)$, the Umegaki entropy, fidelity, and sandwiched Rényi entropy, respectively (Hollands, 2020).

2. Unified Integral and Distribution-Based Representations

Integral and spectral representations provide a unifying route by expressing relative entropies in terms of observable distributions. For quantum states $\rho$, $\sigma$: $$D(\rho\Vert\sigma) = \int_{-\infty}^{+\infty} \frac{|t|}{(t-1)^2} \operatorname{Tr}_-[(1-t)\rho + t\sigma]\,dt,$$ where $\operatorname{Tr}_-$ is the sum over negative eigenvalues—yielding the classical Kullback–Leibler divergence when $\rho$, $\sigma$ commute (Frenkel, 2022).

For Rényi entropy, a distribution-of-observables construct applies: $$D_\alpha(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\left\langle \sigma^{1-\alpha}\right\rangle_{\rho^\alpha},$$ with expectations defined via the pushforward spectral measure. The Umegaki entropy fits as: $$D(\rho\Vert\sigma) = \left\langle -\log\sigma \right\rangle_\rho - \left\langle -\log\rho \right\rangle_\rho,$$ applicable in both finite and infinite dimensions, including Fock spaces, and for continuous variables (Androulakis et al., 2022).

3. General Properties and Special Cases

The unified formulas yield the following universal properties for $D(\cdot\Vert\cdot)$ and its generalizations (Lewin et al., 2013, Wu et al., 11 Jun 2025):

• Nonnegativity and Faithfulness: $D(\rho\Vert\sigma)\geq0$, with equality iff $\rho=\sigma$.
• Data Processing Inequality (DPI): $$D(\Lambda(\rho)\Vert\Lambda(\sigma)) \leq D(\rho\Vert\sigma)$$ for any quantum channel $\Lambda$ (CPTP map), for the Umegaki, Rényi ($\beta\leq1$), and Tsallis (for $0<\alpha<1$) entropies.
• Continuity and Limits: The families are continuous in both $\alpha,\beta$, and $\rho,\sigma$; correct limiting behavior interpolates between classical and quantum divergences (Cheng et al., 11 Jun 2025, Wu et al., 11 Jun 2025).
• Convexity and Additivity: Joint convexity in the state arguments; additivity for the logarithmic (Rényi, Umegaki) cases, with pseudo-additivity in the Tsallis regime.
• Infinite Dimensions: Provided the operator monotonicity condition holds, monotonicity and semicontinuity extend to infinite dimensions (Lewin et al., 2013, Androulakis et al., 2022).
• Functional Inequalities: Stochastic control formulations unify Talagrand’s transport, log-Sobolev, Poincaré, and other classical inequalities as minimization problems for a drift generating the target distribution (Lehec, 2010).

4. Physical and Mathematical Applications

Unified entropy formulas underpin a broad class of applications:

• Quantum Statistical Mechanics: Irreversibility, measurement of quantum distinguishability, resource-theoretic quantifiers.
• Continuous Variable & Gaussian Systems: Gaussian relative entropy is decomposed into classical (Shannon-like) and quantum (covariance/mean) terms, with explicit closed forms for $n$-mode states (Parthasarathy, 2021).
• Quantum Field Theory (QFT): The modular Hamiltonian structure, symplectic-flux formulae, and bulk-boundary correspondences yield holographic and conformal field theoretical expressions for relative entropy (Jafferis et al., 2015, Bao et al., 2019).
• Macroscopic Kinetic Theory: In kinetic equations (Vlasov–Fokker–Planck), a modulated relative entropy functional combines microscopic Maxwell–Boltzmann proximity and macroscopic density deviation via nonlocal (Riesz) interaction energies, ensuring control over convergence in both strong and weak topologies (Choi et al., 20 Oct 2025).
• Resource Theories: Unified $(\alpha,\beta)$–relative entropies form the operational basis for measures in coherence, imaginarity, and related resource monotones (Wu et al., 11 Jun 2025).

5. Specialized Structures: Entanglement, Modularity, Boundary Phenomena

The unified formula applies to more structured cases:

• Entanglement: The relative entropy of entanglement is given by minimizing $S(\rho\Vert\sigma)$ over all separable states, with a universal closed-form constructed from the structure of the closest separable state (CSS) and supporting hyperplanes (Friedland et al., 2010).
• Modular Theory (Algebraic QFT, CCR Algebras): On CCR algebras, the relative entropy between quasifree/coherent states is a quadratic form of the difference in displacement vectors, computed using modular data and projectors associated with standard subspaces (Bostelmann et al., 2020).
• Boundary and Left-Right Decompositions in CFT: The left-right relative entropy of boundary states in CFTs is computed via explicit sums over modular S-matrix data, conformal weights, and Ishibashi coefficients, with cutoff-independent universal formulas. The notion of “relative entanglement sectors” organizes states indistinguishable by this entropy according to symmetry/NIM-rep structure (Ghasemi, 14 Nov 2024).

6. Comparison and Synthesis of Unification Principles

All unified formulas share a common functional and structural DNA:

• They interpolate analytically between core entropic quantities (Umegaki, Rényi, Tsallis, and operational monotones).
• Spectral and integral representations turn operator divergences into trace-based, “classicalized” expressions, extending applicability to infinite dimensions and general probabilistic theories (GPTs) (Frenkel, 2022, Androulakis et al., 2022).
• Variational, convex-analytic, and stochastic control frameworks convert operational or resource-theoretic divergence minimization problems into integral or supremum representations (Hollands, 2020, Lehec, 2010).
• The same operator and variational calculus controls quantum coherence/imaginarity measures, resource monotones, and field-theoretic modular flows (Wu et al., 11 Jun 2025, Jafferis et al., 2015).

The core unification thus lies in expressing quantum, classical, and operational distinguishability through a minimal set of analytic, variational, or distribution-theoretic objects that manifestly encode physical, probabilistic, or resource-theoretic monotonicity and convexity principles.