---
title: Sandwiched Rényi Relative Entropy
url: https://www.emergentmind.com/topics/sandwiched-renyi-relative-entropy
type: topic
---

# Sandwiched Rényi Relative Entropy

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 $\rho$ [1308.5961]. 

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 $\rho = \sigma$ [1505.06980].
- **Unitary invariance:** $D_{\alpha}(U\rho U^\dagger\|U\sigma U^\dagger)=D_{\alpha}(\rho\|\sigma)$ for all unitary $U$ [2204.07694].
- **Monotonicity in parameter:** For fixed $\rho,\sigma$, $\alpha \mapsto D_{\alpha}(\rho\|\sigma)$ is nondecreasing for $\alpha \ge 1/2$ [2309.04539].
- **Data-processing inequality (DPI):** For every CPTP map $\Lambda$, $D_{\alpha}(\Lambda(\rho)\|\Lambda(\sigma)) \le D_{\alpha}(\rho\|\sigma)$ for $\alpha \ge 1/2$ [1306.5358, 1505.06980, 2309.04539].
- **Convexity domain:** $D^{*}_{\alpha}$ is jointly convex in $(\rho,\sigma)$ for $\alpha \ge 1/2$ [2411.01995].

## 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) [1308.5961] |

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 [1308.5961, 1505.06980, 2305.05859].

## 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 [1308.5961, 1505.06980].
- **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 $\alpha=1/2$ [1512.02615].

## 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 [1512.02615, 1306.5358].

- **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 [1609.08462, 1707.00047].

## 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 $\alpha>1$ [1306.1586, 1706.09885].
- **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 $\alpha$ [1505.06980].
- **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 [1808.04662, 2411.01995].

## 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 [2204.07694].
- **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 [2106.10415].
- **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 [1810.00906].

## 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 [2411.01995].
- **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 $\alpha = 1/2$ [1808.04662].
- **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 [2411.09406].

## References

- Datta & Leditzky, “A limit of the quantum Renyi divergence” [1308.5961]
- Beigi, “Sandwiched Rényi divergence satisfies data processing inequality” [1306.5920]
- Frank & Lieb, “Monotonicity of a relative Rényi entropy” [1306.5358]
- Müller-Lennert et al., “On quantum Rényi entropies: a new definition and some properties” [1306.3142]
- Wilde et al., “Strong converse for the classical capacity...” [1306.1586]
- Xu, “Coherence measures based on sandwiched Rényi relative entropy” [1808.04662]
- Jenčová, “Rényi relative entropies and noncommutative $L_p$-spaces II” [1707.00047]
- Cao, Lu, Lu, “Gradient flow...for primitive Lindblad equations...” [1810.00906]
- Gao, Junge, LaRacuente, “Relative entropy for von Neumann subalgebras” [1909.01906]
- Seshadreesan, Lami, Wilde, “Renyi relative entropies of quantum Gaussian states” [1706.09885]
- Ghasemi, “Left-Right Relative Entropy” [2411.09406]
- Majidy et al., “Rényi relative entropy based monogamy of entanglement in tripartite systems” [2411.01995]
- Jenčová, “Rényi relative entropies and noncommutative $L_p$-spaces” [1609.08462]
- Leditzky et al., “Rényi-Holevo inequality...” [2309.04539]
- Anshu et al., “Fidelity-Based Smooth Min-Relative Entropy” [2305.05859]
- Knott, “Rényi divergence inequalities via interpolation...” [2106.10415]
- Engelhardt et al., “Sandwiched Renyi Relative Entropy in AdS/CFT” [2204.07694]

Source: https://www.emergentmind.com/topics/sandwiched-renyi-relative-entropy