Single-letter Chain Rule for Quantum Relative Entropy

TL;DR

The work addresses the absence of a single-letter quantum chain rule for relative entropy by transporting the classical chain-rule decomposition through POVM-induced ensembles to the quantum setting. It delivers a concrete single-copy chain inequality $D(\rho\|\sigma) - D(\mathcal{M}(\rho)\|\mathcal{N}(\sigma)) \geq - \mathbb{E}_{P_{\rho}^G} D(\mathcal{M}(\rho_j)\|\mathcal{N}(\sigma_j))$, derived via classical-to-quantum reductions and an asymptotic equipartition step, and augments this with a projector-based sufficient condition, a semiclassical variant, and connections to recoverability through twisted recovery maps and universal rotated Petz constructions. The paper further links these single-letter bounds to strengthened data-processing inequalities and measured relative entropy, clarifying how meaningful chain-type inequalities arise already at the one-shot level while acknowledging that tight, fully general quantum chain rules require regularization in the many-copy regime. By introducing a general entropy inequality with a two-reference twisted recovery map and deriving corollaries such as a conditional chain rule and two-channel DPI, the work provides structural insights into quantum information flow and recoverability. Overall, it establishes a notable bridge between classical chain-rule structure and quantum recoverability, offering useful one-shot bounds and highlighting directions for tighter, more general results.

Abstract

Relative entropy is the standard measure of distinguishability in classical and quantum information theory. In the classical case, its loss under channels admits an exact chain rule, while in the quantum case only asymptotic, regularized chain rules are known. We establish new chain rules for quantum relative entropy that apply already in the single-copy regime. The first inequality is obtained via POVM decompositions, extending the point distributions in the classical chain rule to quantum ensemble partitions. The second gives a sufficient condition for the most natural extension of the classical result, which uses projectors as a analog for the classical point distributions. We additionally find a semiclassical chain rule where the point distributions are replaced with the projectors of the initial states, and, finally, we find a relation to previous works on strengthened data processing inequalities and recoverability. These results show that meaningful chain inequalities are possible already at the single-copy level, but they also highlight that tighter bounds remain to be found.

Figures (1)

• Figure 1: Cross section of the $x-z$ plane of the Bloch sphere. Left shows the states $\sigma$ for which \ref{['eq:projBound']} is violated when $\rho=\ketbra{0}$. All the boundaries are not in the set. Right shows the bound for mixed values of $p$. The dots represent the $\rho$ for each value of $p$. Note that in the right picture the cases $p=0$ and $p=\frac{1}{2}$ are calculated for values very close to $0$ and $\frac{1}{2}$ but not exaclty $0$ and $\frac{1}{2}$, since the bound is not continuous at $\frac{1}{2}$ and at $0$ it has a very rapid change.

Theorems & Definitions (28)

• Theorem 1
• proof
• Remark 1
• Remark 2
• Remark 3
• Corollary 1
• proof
• Corollary 2
• proof
• Remark 4