Quantum Relative Entropy

• Quantum relative entropy is the quantum analogue of KL divergence, defined for density operators with properties like non-negativity, monotonicity, and joint convexity.
• It underpins key quantum protocols by quantifying asymptotic distinguishability in hypothesis testing and establishing optimality in resource theories.
• Its extensions to Rényi divergences and measured entropies enable applications in quantum computing, thermodynamics, and convex optimization.

Quantum relative entropy (QRE), originally introduced by Umegaki, is the principal quantum generalization of the classical Kullback–Leibler (KL) divergence. It is the fundamental information-theoretic measure of state distinguishability, governing error exponents in quantum hypothesis testing, the geometry of quantum state spaces, operational rates in quantum resource theories, and non-asymptotic and asymptotic bounds for quantum protocols. QRE is ubiquitous across quantum information, quantum computing, statistical mechanics, and quantum field theory. Its mathematical structure supports a rich convex-analytic, operational, and algebraic theory, with deep connections to thermodynamics, convex optimization, and quantum channel theory.

1. Definition, Properties, and Operational Meaning

For density operators $\rho$, $\sigma$ on a finite-dimensional Hilbert space $\mathcal{H}$, the Umegaki quantum relative entropy is defined as

$D(\rho\Vert \sigma) = \begin{cases} \Tr\big[\rho(\log\rho - \log\sigma)\big], & \mathrm{supp}(\rho)\subseteq\mathrm{supp}(\sigma),\ +\infty, & \text{otherwise}. \end{cases}$

The function satisfies:

• Nonnegativity: $D(\rho\Vert\sigma)\ge 0$ with equality if and only if $\rho = \sigma$.
• Monotonicity/Data-processing inequality (DPI): For any completely positive trace-preserving (CPTP) map $\mathcal{M}$,

$$D(\rho\Vert \sigma) \ge D(\mathcal{M}(\rho)\Vert \mathcal{M}(\sigma)).$$

• Joint convexity: $D(\cdot\Vert\cdot)$ is convex in $(\rho, \sigma)$.
• Classical limit: For commuting states, $D(\rho\Vert\sigma)$ reduces to the classical KL divergence.

QRE quantifies asymptotic state distinguishability in quantum hypothesis testing (Stein’s Lemma); it is the sharp error exponent for Type-II errors under a fixed Type-I constraint, and thus serves as the rate function for large deviations in quantum statistical mechanics and state estimation (Okamura, 2012).

2. Axiomatic and Operational Characterizations

Matsumoto’s reverse test establishes QRE as the unique monotone, additive, and lower semi-continuous extension of classical relative entropy:

• Axioms: (i) Monotonicity under CPTP, (ii) Normalization to KL for commuting states, (iii) Additivity, and (iv) Asymptotic continuity (Matsumoto, 2010).
• Reverse test: The minimal classical divergence needed to simulate a given quantum pair $(\rho,\sigma)$ via CPTP encoding is

$D^R(\rho\Vert\sigma) = \Tr[\rho\ln(\sqrt{\rho}\,\sigma^{-1}\sqrt{\rho})],$

which universally lower-bounds any monotone quantum relative entropy.

• Asymptotic uniqueness: Under natural operational scenarios, Umegaki’s QRE is the unique (up to constant factors) quantum extension—mirroring resource conversion rates and optimal costs in quantum Shannon theory and resource resource theories (Matsumoto, 2010).

3. Chain Rules, Data Processing, and Recovery

In the classical setting, KL divergence obeys exact chain rules decomposing divergence across components or subsystems. QRE captures an analogous structure only asymptotically:

• Quantum chain rule (asymptotic, regularized) (Fang et al., 2019):

$$D(\mathcal{E}(\rho_{A_1A_2})\Vert \mathcal{F}(\sigma_{A_1A_2})) \le D(\rho_{A_2}\Vert \sigma_{A_2}) + \bar D^{\mathrm{reg}}(\mathcal{E}\Vert\mathcal{F})$$

with $\bar D^{\mathrm{reg}}$ the regularized channel divergence.

• Single-letter bounds (Gasbarri et al., 19 Oct 2025):
  • POVM decomposition:

  $$D(\rho\Vert\sigma) - D(\mathcal{M}(\rho)\Vert\mathcal{N}(\sigma)) \ge - \mathbb{E}_{j\sim P^G_\rho} D(\mathcal{M}(\rho_j)\Vert\mathcal{N}(\sigma_j)),$$

  where $G$ is a POVM and $\rho_j$ are “conditioned” states. - Projector-based and semiclassical decompositions: For spectral projectors $\Pi_j$ of $\rho$,

  $$D(\rho\Vert\sigma) - D(\mathcal{M}(\rho)\Vert\mathcal{N}(\sigma)) \ge - \sum_j p_j D(\mathcal{M}(\Pi_j)\Vert\mathcal{N}(\Pi_j)),$$

  with tightness for commuting states (recovering the classical chain rule).

QRE monotonicity is intimately linked to the Petz recovery map. The difference $$D(\rho\Vert \sigma) - D(\mathcal{N}(\rho)\Vert \mathcal{N}(\sigma))$$ quantitatively bounds the fidelity between $\rho$ and its Petz recovery $$\mathcal{R}_{\sigma,\mathcal{N}}(\mathcal{N}(\rho))$$: this underpins results in strong subadditivity, convexity, and conditional entropy (Berta et al., 2014).

4. Information Geometry and Metric Responses

QRE generates an information-geometric structure on quantum state space:

• Metric tensor (QRE-susceptibility): For a smooth one-parameter family $\rho(\lambda)$,

$S(\rho(\lambda)\|\rho(\lambda+\delta\lambda)) = \Sigma_{\lambda\lambda} (\delta\lambda)^2 + O(\delta\lambda^3), \quad \Sigma_{ij} = \frac{1}{2}\Tr[\rho^{-1}\partial_i\rho\partial_j\rho].$

• Quantum critical points (QCPs): In quantum many-body systems, the QRE metric susceptibility diverges at QCPs, encoding universal scaling features. For the transverse-field Ising model, the divergence is logarithmic, whereas in non-integrable three-spin chains the divergence is quadratic in system size, reflecting different universality classes (Sarkar et al., 26 Sep 2025).
• Relation to fidelity susceptibility and Rényi divergences: QRE susceptibility generalizes fidelity susceptibility to mixed-state spaces, and preserves connections to the Petz–Rényi spectrum (Sarkar et al., 26 Sep 2025).

5. Quantum Resource Theories and Optimization

Resource measures in quantum information—such as entanglement, coherence, and “magic”—are most generally cast as QRE distances to convex sets of “free” states:

• Relative entropy of entanglement:

$$E_R(\rho) = \min_{\sigma \in \mathrm{Sep}} D(\rho \Vert \sigma).$$

• Resource-theoretic tasks: The regularized QRE, $D^{(\infty)}(\rho\Vert\sigma)$, determines rates in resource conversion, such as entanglement cost and distillable entanglement, and magic-state distillation (Fang et al., 21 Feb 2025).
• QRE programming and convex optimization: Efficient semidefinite programming (SDP) and self-concordant barrier techniques support practical computation of QRE-based quantities in high dimensions (Karimi et al., 2023, Fawzi et al., 2017). Rational approximations of the matrix logarithm enable embedding QRE constraints in optimization frameworks critical for quantum communication, key distribution, and state certification.

6. Extensions: Rényi and Measured Relative Entropies

QRE generalizes to parametrized $\alpha$-divergences:

• Petz–Rényi and sandwiched Rényi relative entropy:

$S_\alpha(\rho\Vert\sigma) = \frac{1}{\alpha-1}\log\Tr\left[\left(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\right)^\alpha\right],$

interpolating between trace distance (for $\alpha = 1/2$), QRE ($\alpha\to 1$), and max-divergence ($\alpha\to\infty$) (Berta et al., 2015, Fröb et al., 2024, Lashkari, 2014).

• Measured relative entropy $D_m$: The maximal classical relative entropy over all POVMs, $D_m(\rho\Vert\sigma)\le D(\rho\Vert\sigma)$, with equality if and only if $[\rho,\sigma]=0$. Sandwiched Rényi divergences dominate measured counterparts for $\alpha > 1/2$ (Berta et al., 2015).
• QFT and modular theory: In algebraic QFT, Petz–Rényi relative entropy is constructed via modular operators and encompasses genuinely quantum fluctuations beyond the classical symplectic structure (Fröb et al., 2024).

7. Applications and Contemporary Algorithms

QRE and its generalizations underpin quantum channel discrimination, state estimation, and statistical inference:

• Large deviation theory and information criteria: QRE is the rate function for quantum large deviation principles, directly justifying information criteria such as AIC and WAIC in the quantum context, with classical accuracies (Okamura, 2012).
• Quantum algorithms: Recent variational quantum algorithms efficiently estimate $D(\rho\Vert\sigma)$ using quadrature techniques and operator ansätze, with polynomial resource scaling and direct applicability to distributed quantum architectures (Lu et al., 13 Jan 2025).
• Thermodynamics and coherence: QRE measures deviation from equilibrium, quantifying quantum coherence consumption and its separation from classical entropy production in thermalization near black holes (Han et al., 2024).

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A plausible implication of this corpus is that QRE serves as the universal non-classical divergence in quantum statistical mechanics, quantum computing, and information processing, bestowing both foundational and practical structure on the geometry and dynamics of quantum information (Gasbarri et al., 19 Oct 2025, Fang et al., 21 Feb 2025, Sarkar et al., 26 Sep 2025, Berta et al., 2014, Karimi et al., 2023). Open problems focus on single-letterization of chain rules, extension to other divergences, modular-theoretic/geometric interpretations, and further computational refinements.