Sandwiched Rényi Relative Entropy

• Sandwiched Rényi Relative Entropy is a one-parameter family of quantum divergences defined for pairs of positive semidefinite operators, generalizing the standard quantum relative entropy.
• It unifies various quantum information measures, including min- and max-relative entropies, while preserving critical properties such as data-processing inequality and unitary invariance.
• Its applications span quantum hypothesis testing, thermodynamic frameworks, and resource theories, making it essential for both theoretical insights and practical quantum information tasks.

The sandwiched Rényi relative entropy, also known as the quantum Rényi divergence, is a one-parameter family of quantum divergences defined for pairs of positive semidefinite operators. It generalizes the quantum relative entropy and unifies several quantum information-theoretic divergences, serving as a parent functional for min- and max-relative entropies, as well as operationally relevant quantities in quantum information theory. Its structural properties, limiting behavior, and data-processing guarantees have made it central in quantum communication theory, matrix analysis, thermodynamics, quantum resource theories, and the study of operator-algebraic structures.

1. Mathematical Definition and Domain

Let $\rho$ be a density operator ($\rho\ge0,\,\mathrm{Tr}\,\rho=1$) and $\sigma$ a positive semidefinite operator on a finite-dimensional Hilbert space $\mathcal{H}$. For $\alpha\in(0,1)\cup(1,\infty)$, the sandwiched Rényi relative entropy is defined by

$$D_{\alpha}(\rho\|\sigma) := \frac{1}{\alpha-1}\;\log\,\mathrm{Tr}\left[\left(\sigma^{\frac{1-\alpha}{2\alpha}}\,\rho\,\sigma^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha}\right].$$

For $\alpha>1$, $D_{\alpha}(\rho\|\sigma)=+\infty$ if $\mathrm{supp}(\rho)\nsubseteq\mathrm{supp}(\sigma)$, ensuring the fractional powers of $\sigma$ act nontrivially only on the support of $\rho$ (Datta et al., 2013).

In the commuting case, this reduces to the Petz Rényi divergence: $$D^{\mathrm{Petz}}_{\alpha}(\rho\|\sigma) = \frac{1}{\alpha-1}\log\,\mathrm{Tr}\big(\rho^{\alpha}\sigma^{1-\alpha}\big).$$

2. Foundational Properties

• Nonnegativity: $D_{\alpha}(\rho\|\sigma) \ge 0$, with equality if and only if $\rho = \sigma$ (Misra et al., 2015).
• Unitary invariance: $$D_{\alpha}(U\rho U^\dagger\|U\sigma U^\dagger)=D_{\alpha}(\rho\|\sigma)$$ for all unitary $U$ (Caginalp, 2022).
• Monotonicity in parameter: For fixed $\rho,\sigma$, $\alpha \mapsto D_{\alpha}(\rho\|\sigma)$ is nondecreasing for $\alpha \ge 1/2$ (Bussandri et al., 2023).
• Data-processing inequality (DPI): For every CPTP map $\Lambda$, $$D_{\alpha}(\Lambda(\rho)\|\Lambda(\sigma)) \le D_{\alpha}(\rho\|\sigma)$$ for $\alpha \ge 1/2$ (Frank et al., 2013, Misra et al., 2015, Bussandri et al., 2023).
• Convexity domain: $D^{*}_{\alpha}$ is jointly convex in $(\rho,\sigma)$ for $\alpha \ge 1/2$ (Mannaï et al., 2024).

3. Limiting and Special Cases

Limit	Sandwiched Rényi (SRD)	Interpretation
$\alpha\to1$	$D(\rho\|\sigma)$	Umegaki (quantum) relative entropy
$\alpha\to\infty$	$\log\|\sigma^{-1/2} \rho \sigma^{-1/2}\|_{\infty}$	Max-relative entropy
$\alpha=1/2$	$-2\log\big\|\sigma^{1/2}\rho^{1/2}\big\|_1$	Min-relative entropy (via fidelity)
$\alpha\to0$	$-\log\mathrm{Tr}[\Pi_\rho\,\sigma]$	Zero-relative Rényi entropy (equal support req'd) (Datta et al., 2013)

For $\alpha=1/2$ this is related to fidelity. The min- and max-relative entropies arise as respective $\alpha=1/2$ and $\alpha\to\infty$ limits, unifying the landscape of non-asymptotic quantum information quantities (Datta et al., 2013, Misra et al., 2015, Nuradha et al., 2023).

4. Connections to Other Divergences

• Petz Rényi divergence: For commuting density matrices or in the classical case, the sandwiched form reduces to the Petz version; otherwise, it "sandwiches" $\rho$ between fractional powers of $\sigma$, more accurately capturing noncommutative regimes (Datta et al., 2013, Misra et al., 2015).
• Measured Rényi entropy: For $\alpha > 1/2$, the sandwiched Rényi is strictly larger than the measured Rényi entropy, and for $\alpha < 1/2$, it is strictly smaller, providing counterexamples to DPI below $\alpha=1/2$ (Berta et al., 2015).

5. Variational and Operator-Algebraic Formulations

• Variational formula: For $\alpha>1$,

$$\mathrm{Tr}\left(\sigma^{\frac{1-\alpha}{2\alpha}}\rho\sigma^{\frac{1-\alpha}{2\alpha}}\right)^{\alpha} = \sup_{\omega>0}\left\{ \alpha\,\mathrm{Tr}[\rho\,\omega^{1-1/\alpha}] + (1-\alpha)\mathrm{Tr}[\sigma\,\omega] \right\}.$$

For $0<\alpha<1$, the supremum is replaced by an infimum (Berta et al., 2015, Frank et al., 2013).

• Noncommutative $L_p$-space extension: The sandwiched Rényi relative entropy extends to states on arbitrary von Neumann algebras via Kosaki's interpolation and coincides with the Araki–Masuda divergences (Jencova, 2016, Jenčová, 2017).

6. Operational and Resource-Theoretic Applications

• Quantum hypothesis testing & strong converse: The sandwiched Rényi divergence determines strong-converse exponents for quantum channel coding, hypothesis testing, and one-shot capacities, especially for $\alpha>1$ (Wilde et al., 2013, Seshadreesan et al., 2017).
• Thermodynamical universality: In the Rényi-thermodynamic framework, the sandwiched Rényi relative entropy yields Rényi generalizations of the Clausius inequality and free energy, with the "form invariance" underlying the universality of the second law for all $\alpha$ (Misra et al., 2015).
• Quantum resource theories: Measures constructed from the sandwiched Rényi relative entropy (e.g., coherence, entanglement) satisfy the requirements of monotonicity and convexity for $\alpha \ge 1/2$, as shown in frameworks such as the Baumgratz–Cramer–Plenio (BCP) coherence theory (Xu, 2018, Mannaï et al., 2024).

7. Generalizations, Extensions, and Physical Implications

• Operator-algebraic (modular) generalization: The modular-operator framework allows the algebraic definition of Rényi divergences in QFT and holographic error correction codes, where the equality of bulk and boundary sandwiched Rényi entropy is equivalent to bulk-boundary subregion duality and Ryu–Takayanagi formula in AdS/CFT (Caginalp, 2022).
• Chain and decomposition rules: Sandwiched Rényi divergences generate valid quantum generalizations of chain, decomposition, and uncertainty relations via interpolation methods, recovering Shannon and von Neumann entropy equalities in the $\alpha \to 1$ limit (McKinlay, 2021).
• Gradient flows and dissipative dynamics: The primitive Lindblad equation with GNS-detailed balance is a gradient flow of $D_{\alpha}$ for any $\alpha$, with $D_{\alpha}$ decaying exponentially fast under such dynamics, extending contractivity properties beyond von Neumann relative entropy (Cao et al., 2018).

8. Notable Examples and Sector-Specific Applications

• Entanglement monogamy/polygamy: Sandwiched Rényi–based entanglement monotones remain valid and convex for all $\alpha\ge1/2$ and capture the full range of monogamy/polygamy transitions in tripartite scenarios. For prototypical states and thermal spin models, $$M_{\alpha}^* = E_{\alpha}^*(\rho_{A:BC}) - E_{\alpha}^*(\rho_{AB}) - E_{\alpha}^*(\rho_{AC})$$ can switch sign, and only the sandwiched form remains consistent in the convexity window (Mannaï et al., 2024).
• Coherence quantification: Two full families of coherence monotones based on the sandwiched Rényi divergence satisfy the full BCP axiom set for $\alpha\ge1/2$, and coincide with known geometric coherence at $\alpha = 1/2$ (Xu, 2018).
• Boundary CFTs: The left–right sandwiched Rényi entropy in boundary CFTs exactly classifies relative entanglement sectors using only modular $S$-matrix data and is UV-finite by construction (Ghasemi, 2024).

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• Ghasemi, “Left-Right Relative Entropy” (Ghasemi, 2024)
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• Leditzky et al., “Rényi-Holevo inequality...” (Bussandri et al., 2023)
• Anshu et al., “Fidelity-Based Smooth Min-Relative Entropy” (Nuradha et al., 2023)
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