Reverse-type Data Processing Inequality

TL;DR

The paper investigates reverse data-processing behavior by introducing and analyzing relative entropy contraction/expansion across quantum channels. It proves a dimension-based no-go: when $d_A\ge d_B$ and the channel is non-unitary, the expansion coefficient vanishes, while a unitary channel can saturate the bound, and erasure-type channels can violate naive reversibility. It then develops tools based on the BKM metric and CP-order to compare two channels, and applies them to depolarizing, generalized dephasing, and amplitude-damping channels, obtaining nonzero relative expansion coefficients in many cases. As a key application, it constructs a concrete example of a level-1 less noisy channel that is non-degradable, highlighting a nuanced separation within quantum channel classes with implications for channel capacities and algorithmic convergence. The work advances understanding of when and how noise can preserve, rather than merely destroy, distinguishability, and points to open questions about broader divergences and tensorization properties.

Abstract

The quantum data processing inequality asserts that two quantum states become harder to distinguish when a noisy channel is applied. On the other hand, a reverse quantum data processing inequality characterizes whether distinguishability is preserved after the application of a noisy channel. In this work, we explore these concepts through contraction and expansion coefficients of the relative entropy of quantum channels. Our first result is that quantum channels with an input dimension greater than or equal to the output dimension do not have a non-zero expansion coefficient, which means that they cannot admit a reverse data-processing inequality. We propose a comparative approach by introducing a relative expansion coefficient, to assess how one channel expands relative entropy compared to another. We show that this relative expansion coefficient is positive for three important classes of quantum channels: depolarizing channels, generalized dephasing channels, and amplitude damping channels. As an application, we give the first rigorous construction of level-1 less noisy quantum channels that are non-degradable.

Figures (4)

• Figure 1: Illustration of the function defined in \ref{['eqn:auxiliary function f']}.
• Figure 2: A plot of $\widecheck{\eta}_{\mathcal{A}_{\gamma_1},\mathcal{A}_{\gamma_2}}$ for $\gamma_1,\gamma_2 \in (0,1)$.
• Figure 3: Degradable and anti-degradable regions for probabilistic mixture of two amplitude damping channels defined in \ref{['def:target channel']} in the case $p>\frac{1}{2}$. A plot of the corresponding regions for the case of $p<\frac{1}{2}$ can be found in SW24.
• Figure 4: We plot $p_{min}(\gamma_1,\gamma_2)$, an upper bound on the cutoff probability above which $\Psi_{p, \gamma_1,\gamma_2}$ is less noisy by Proposition \ref{['prop:ampdamp regions']}. The highlighted region is the region where $\Psi_{p, \gamma_1,\gamma_2}$ is less noisy but not degradable for any $p\geq p_{min}(\gamma_1,\gamma_2)$. (See the degradability regions in Fig. \ref{['fig:deg region']}.)

Theorems & Definitions (43)

• Theorem 1.1: c.f. Theorem \ref{['main:impossibility']}
• Theorem 1.2
• Theorem 1.3: c.f. Proposition \ref{['prop:ampdamp regions']}
• Theorem 3.1
• Lemma 3.2
• proof : Proof of Lemma \ref{['pure state argument']}
• proof : Proof of Theorem \ref{['main:impossibility']}
• Remark 3.3
• Remark 3.4
• Lemma 4.1