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Quantum conditional entropies from convex trace functionals

Roberto Rubboli, Milad M. Goodarzi, Marco Tomamichel

Abstract

We study geometric properties of trace functionals that generalize those in [Zhang, Adv. Math. 365:107053 (2020)], arising from a novel family of conditional entropies with applications in quantum information. Building on new convexity results for these functionals, we establish data-processing inequalities and additivity properties for our entropies, demonstrating their operational significance. We further prove completeness under duality, chain rules, and various monotonicity properties for this family. Our proofs draw on tools from complex interpolation theory, multivariate Araki--Lieb and Lieb--Thirring inequalities, variational characterizations of trace functionals, and spectral pinching techniques.

Quantum conditional entropies from convex trace functionals

Abstract

We study geometric properties of trace functionals that generalize those in [Zhang, Adv. Math. 365:107053 (2020)], arising from a novel family of conditional entropies with applications in quantum information. Building on new convexity results for these functionals, we establish data-processing inequalities and additivity properties for our entropies, demonstrating their operational significance. We further prove completeness under duality, chain rules, and various monotonicity properties for this family. Our proofs draw on tools from complex interpolation theory, multivariate Araki--Lieb and Lieb--Thirring inequalities, variational characterizations of trace functionals, and spectral pinching techniques.

Paper Structure

This paper contains 35 sections, 45 theorems, 248 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Main results
  3. Relation to multivariate trace functionals
  4. A new concept of conditional entropy
  5. A three-parameter family
  6. Connections to known quantities
  7. Convexity properties of the trivariate trace functional
  8. Joint convexity/concavity/log-concavity
  9. A characterization of optimizers
  10. Additivity via tensorization of optimizers
  11. Data-processing inequality
  12. DPI under channels on Bob's system
  13. DPI under sub-unital channels on Alice's system
  14. Proof of Theorem \ref{['DPItheorem']}
  15. DPI under conditionally mixing channels
  16. ...and 20 more sections

Key Result

Theorem 3.1

The function $(A,B) \mapsto Q_{p,q,r,s}(A,B)$ is

Figures (3)

  • Figure 1: The figure illustrates the DPI region $\mathcal{D}$ in the parameter space $(\alpha, z, \lambda)$. (See Definition \ref{['def:DPIregion']}.) The red region corresponds to $\mathcal{D}_1$, while the blue region corresponds to $\mathcal{D}_2$. The cross-section of $\mathcal{D}$ at $\lambda = t$ is independent of $t$ for $t \in [0,1]$ and corresponds to the DPI region of $D_{\alpha,z}$.
  • Figure 2: The diagram illustrates the connections between the newly introduced conditional entropy $H^\lambda_{\alpha,z}$ and various established quantities explored in the literature.
  • Figure 3: The figure shows the DPI region $\mathcal{D}$ in the coordinates $x_1 = \frac{z}{1-\alpha}$, $x_2 = \frac{1}{1-\alpha} - \lambda$ and $x_3=\frac{z-1}{1-\alpha}-\lambda$. In these coordinates, the DPI region, depicted in yellow, consists of a truncated polyhedral cone and its mirror image. Moreover, the duality relations in Theorem \ref{['Duality']} become $x_1=-\hat{x}_1$, $x_2=-\hat{x}_2$ and $x_3=\hat{x}_3$. Hence, under duality, the points get reflected across the $x_3$ axis. We show the duality relation \ref{['relation 3']} that connects the Petz arrow up with the sandwiched arrow down. The green line corresponds to the Petz arrow up for $\alpha >1$ while the brown line to the sandwiched arrow down for $\alpha<1$.

Theorems & Definitions (89)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4: $\alpha$-$z$ Rényi relative entropy
  • Theorem 3.1
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['log-concavity2']}
  • Lemma 3.3
  • Remark 3.4
  • ...and 79 more