Monotonicity of Relative Entropy

• Monotonicity of relative entropy is a principle demonstrating that quantum operations do not increase the distinguishability between states, forming the basis of the data-processing inequality.
• Theoretical proofs by Petz use operator algebra and recovery maps while Uhlmann’s approach employs quadratic interpolation, contrasting practical methodologies in quantum information theory.
• This property impacts quantum thermodynamics, resource theories, and communication by bounding error rates and guiding the design of reversible quantum channels.

The monotonicity of relative entropy is a principle asserting that physical or information-processing transformations cannot increase the distinguishability between quantum states. This property underpins a vast portion of quantum information theory and statistical physics, and possesses deep mathematical foundations and operational consequences. In particular, it equates to the data-processing inequality, ensures the validity of major entropy inequalities, governs information loss under quantum channels, and connects reversibility to recoverability of quantum states.

1. Formal Definition and General Statement

Consider two quantum states (density operators) $\rho$ and $\sigma$ on a Hilbert space. The quantum relative entropy is

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

For any completely positive, trace-preserving (CPTP) map $\mathcal{E}$ (quantum channel), the monotonicity states: $$S(\rho\|\sigma) \geq S(\mathcal{E}(\rho)\| \mathcal{E}(\sigma)).$$ This inequality is known as the data-processing inequality (DPI) and forms the mathematical backbone for the impossibility of increasing information by local, physical transformations (1105.4865, Sagawa, 2012, Pérez-Pardo, 2022, Matheus et al., 14 Sep 2025).

2. Proof Methodologies: Petz vs Uhlmann

There are two principal, structurally distinct proof strategies: Petz's direct operator-algebraic approach and Uhlmann's quadratic form interpolation.

Petz's Operator-Algebraic Approach

Petz's approach reformulates relative entropy in terms of the modular operator $\Delta_{\rho,\sigma}$ on the Hilbert-Schmidt space: $$S(\rho\|\sigma) = -\langle \rho^{1/2}, \log \Delta_{\rho,\sigma} \,\rho^{1/2} \rangle.$$ Upon coarse-graining (e.g., partial trace), Petz introduces an auxiliary isometric lifting operator $V_\rho$ that intertwines the reduced and original modular structures. The initial attempt erroneously applied the contractive Jensen operator inequality, but the correct proof recognizes $V_\rho$ as an isometry in the relevant cases. This enables valid use of the operator-convexity of $- \log x$ for isometries, leading to

$$S(\mathcal{E}(\rho)\| \mathcal{E}(\sigma)) \leq S(\rho\|\sigma),$$

with equality if and only if there is a recovery map—the Petz recovery map—restoring $\rho$ from $\mathcal{E}(\rho)$ (Matheus et al., 14 Sep 2025).

Uhlmann's Interpolation of Sesquilinear Forms

Uhlmann's route is based on Pusz-Woronowicz interpolation of positive sesquilinear forms. Given states $\omega$ and $\nu$ on a unital $C^*$-algebra, Uhlmann constructs an interpolating quadratic form: $$Y_t[\omega_R, \nu_L](a, b) = \langle [a], P^{1-t} Q^t [b] \rangle$$ with $(P,Q)$ commuting positive operators arising from GNS representations. The relative entropy is then defined as a Dini derivative at $t=0$: $$S[\omega, \nu] = -\lim_{t\to 0^+} \frac{Y_t[\omega_R, \nu_L](e, e) - \omega(e, e)}{t}$$ for $e$ the identity. Under a positive, unital (Schwarz) map $\varphi$, Uhlmann's argument shows the pull-back of quadratic forms preserves the ordering, yielding

$$S[\omega, \nu] \geq S[\omega \circ \varphi, \nu \circ \varphi]$$

and, in the operator algebra context, generalizes effortlessly to type III von Neumann algebras and non-invertible states (Pérez-Pardo, 2022, Reible, 8 Jan 2025).

Comparative Analysis

Feature	Petz	Uhlmann
Conceptual Basis	Modular operator, lifting isometries	Positive form interpolation
Scope	Intuitive, explicit when full rank	Fully general, abstract
Recovery Map Link	Direct to Petz recovery map	General monotonicity
Applicability	Finite-dimensional, invertible states	All normal states
Technical Demand	Operator algebra, isometric lifts	Quadratic form calculus

(Matheus et al., 14 Sep 2025, Reible, 8 Jan 2025)

3. Operational Consequences in Quantum Information

a) Data-Processing and Entropic Uncertainty

Monotonicity clarifies that quantum measurements or noise invariably reduce the distinguishability of states—a fact harnessed for entropic uncertainty relations with quantum side information. It explains the trade-off in knowledge about complementary observables upon sequential or simultaneous measurement, leading to the formal uncertainty relation with quantum side information (UPQSI), where the monotonicity is explicitly used to derive lower bounds on joint conditional entropies (1105.4865): $H(v|c) + H(w|b) \geq -\log r(v,w)$ with $r(v,w) = \max_{j,k} |\langle v_j|w_k \rangle|^2$.

b) Thermodynamics and the Second Law

Monotonicity under CPTP maps underlies second law-like statements such as the Clausius inequality ($W\geq \Delta F$), and the quantum Hatano–Sasa inequality for nonequilibrium steady states, by ensuring non-increase of quantum relative entropy between state sequences under physical evolutions (Sagawa, 2012): $$D(\mathcal{E}(\rho) \| \mathcal{E}(\sigma)) \leq D(\rho \| \sigma)$$ is used to show that the change in entropy is bounded below by excess contributions in driven thermodynamic processes.

c) Resource Theories and Channel Capacities

Monotonicity is essential for the characterization of monotones in resource theories, for error exponents in channel discrimination, and for bounding quantum capacities via sandwiched Rényi divergences. For the sandwiched Rényi divergence $D_\alpha(p\|o)$, monotonicity is proved for all $\alpha \geq 1/2$, confirming its operational legitimacy in contexts like quantum hypothesis testing and one-shot channel coding (Frank et al., 2013).

d) Recovery Maps and Approximate Reversibility

The tightness of monotonicity is realized when the channel action is reversible. Petz's theorem asserts that equality occurs if and only if a Petz recovery map reconstructs $\rho$ from $\mathcal{E}(\rho)$. Quantitative refinements, relating the decrease in relative entropy to the fidelity between $\rho$ and its recovered version, have been established (Berta et al., 2014, Sutter et al., 2015): $$D(\rho\|\sigma) - D(\mathcal{E}(\rho)\| \mathcal{E}(\sigma)) \geq -\log F(\rho, \mathcal{R}_{\sigma, \mathcal{E}}(\mathcal{E}(\rho)))$$ where $F$ denotes the quantum fidelity and $\mathcal{R}_{\sigma, \mathcal{E}}$ is the Petz recovery map.

4. Extensions: Positive Maps, Infinite Dimensions, and Further Generalizations

a) Positive vs Completely Positive

Monotonicity holds under positive trace-preserving maps, not only CPTP (completely positive trace-preserving) maps. This broader validity exposes limitations in the use of relative entropy for detecting non-Markovianity; certain measures become blind if only positivity, not complete positivity, is checked (Müller-Hermes et al., 2015, Sargolzahi et al., 2019).

b) Infinite-Dimensional Operator Algebras

The relative entropy extends to infinite-dimensional von Neumann algebras via modular theory and vector representatives in standard forms. Uhlmann's monotonicity theorem (proven in the Araki–Uhlmann setting) guarantees monotonicity for normal positive functionals under unital Schwarz maps and, hence, covers all physical transformations in algebraic quantum field theory and statistical mechanics (Reible, 8 Jan 2025). Monotonicity is also crucial for defining the two-sided Bogoliubov inequality for KMS (thermal equilibrium) states, governing perturbations and free energy changes in the general setting.

c) Continuity and Stability Aspects

Monotonicity persists for "discontinuity jumps"—infinite sequences of states converging towards a limit. Under quantum operations, the local discontinuity jump of relative entropy cannot increase, implying further stability for limits and approximations in information-theoretic and thermodynamic settings (Shirokov, 2022).

5. Generalized and Axiomatic Contexts

Beyond the quantum case, the monotonicity of relative entropy arises axiomatically from the postulate that distinguishability cannot increase under noise (data-processing). For any operationally meaningful divergence $D$, this property and additivity force $D$ to interpolate between minimal and maximal divergences (min- and max-relative entropies), and to possess continuity in the interior of the probability simplex. This establishes a bijection between entropy and relative entropy, and operationalizes why quantities like the quantum (or classical) relative entropy and Rényi divergences occupy their central role (Gour et al., 2020).

6. Geometric and Physical Analogues

In geometric contexts, such as the paper of hypersurfaces in hyperbolic space, a notion of “relative entropy” defined via renormalized areas satisfies a monotonicity property along mean curvature flow: the difference in renormalized area (relative entropy) between two hypersurfaces is non-increasing in time, quantifying geometric “closeness” in analogy with information theory (Yao, 2022).

Key Formulas and Conceptual Summary

• Umegaki quantum relative entropy: $$D(\rho\|\sigma) = \operatorname{Tr}[\rho (\log \rho - \log \sigma)]$$
• Monotonicity / Data Processing Inequality: $$D(\mathcal{E}(\rho)\| \mathcal{E}(\sigma)) \leq D(\rho\| \sigma)$$ for CPTP $\mathcal{E}$
• Petz recovery map: $$\mathcal{R}_{\sigma,\mathcal{E}}(Y) = \sigma^{1/2} \mathcal{E}^*(\mathcal{E}(\sigma)^{-1/2} Y \mathcal{E}(\sigma)^{-1/2}) \sigma^{1/2}$$
• Characterization of equality: $$D(\rho\|\sigma) = D(\mathcal{E}(\rho)\| \mathcal{E}(\sigma))$$ if and only if a Petz map recovers $\rho$ from $\mathcal{E}(\rho)$.
• Bounds via divergence families: Any data-processing and additive divergence $D$ satisfies $D_{\min} \leq D \leq D_{\max}$, with $$D_{\min}(p\|q) = -\log \sum_{i\in \text{supp}(p)}q_i$$ and $D_{\max}(p\|q) = \log \max_i (p_i/q_i)$ (Gour et al., 2020).
• Von Neumann algebra generalization: For a unital Schwarz map $\alpha: M_1 \to M_2$ and positive normal functionals $\psi_{1,2}$, $\phi_{1,2}$ satisfying $\psi_2 \circ \alpha \leq \psi_1$ and $\phi_2 \circ \alpha \leq \phi_1$,

$$S_{M_1}(\psi_1, \phi_1) \leq S_{M_2}(\psi_2, \phi_2)$$

(Reible, 8 Jan 2025).

References (arXiv ids)

• (1105.4865)
• (Sagawa, 2012)
• (Frank et al., 2013)
• (Lewin et al., 2013)
• (Berta et al., 2014)
• (Deuchert et al., 2015)
• (Sutter et al., 2015)
• (Müller-Hermes et al., 2015)
• (Lin et al., 2016)
• (Casini et al., 2016)
• (Zhao et al., 2017)
• (Sargolzahi et al., 2019)
• (Gour et al., 2020)
• (Carlen et al., 2022)
• (Yao, 2022)
• (Pérez-Pardo, 2022)
• (Shirokov, 2022)
• (Reible, 8 Jan 2025)
• (Matheus et al., 14 Sep 2025)

These results collectively place the monotonicity of relative entropy at the heart of quantum information theory, statistical mechanics, and mathematical physics, providing an essential constraint on physical, information-theoretic, and even geometric processes.