Inevitable Negativity: Additivity Commands Negative Quantum Channel Entropy

TL;DR

This work extends the notion of majorization to classical and quantum channels, establishing a robust channel-majorization framework via three equivalent viewpoints (constructive, axiomatic, operational). It develops a standard form, characterization tools, and a consistent entropy concept for channels, and proves a central result: quantum channel entropies must attain negative values under strong additivity. The equivalence across approaches and the link to conditional majorization clarify how uncertainty propagates through channels of varying structure and dimension. The findings reveal fundamental differences between classical and quantum entropy in the channel setting and suggest new operational perspectives on information processing tasks that leverage negative channel entropy as a resource.

Abstract

Quantum channels represent a broad spectrum of operations crucial to quantum information theory, encompassing everything from the transmission of quantum information to the manipulation of various resources. In the domain of states, the concept of majorization serves as a fundamental tool for comparing the uncertainty inherent in both classical and quantum systems. This paper establishes a rigorous framework for assessing the uncertainty in both classical and quantum channels. By employing a specific class of superchannels, we introduce and elucidate three distinct approaches to channel majorization: constructive, axiomatic, and operational. Intriguingly, these methodologies converge to a consistent ordering. This convergence not only provides a robust basis for defining entropy functions for channels but also clarifies the interpretation of entropy in this broader context. Most notably, our findings reveal that any viable entropy function for quantum channels must assume negative values, thereby challenging traditional notions of entropy.

Figures (6)

• Figure 1: A diagram of random permutation superchannel $\Theta$. The classical preprocessing $\mathcal{S}$ sends $z$ to Bob. Bob applies a random permutation channel $\mathcal{D}_z$ corresponding to the received $z$.
• Figure 2: A completely uniformity-preserving superchannel. Sharply squiggly lines represent systems in their maximally mixed (i.e., uniform) state.
• Figure 3: A diagram of a $\mathbf{t}$-gambling game with channel $N^{X \to Y}$. The $\mathbf{t}$ source determines which $k$-game to play. Initially, the player, Alice, learns $k$ partially from $w$, based on which she chooses $x$ to input the channel $\mathcal{N}$. Once the $k$ value is announced, she provides $k$ guesses of the value $y$ based on her chosen $x$.
• Figure 4: The superchannel $\Theta$ maintains the marginal uniformity of the channel $\mathsf{id}^{A\to C}\otimes \mathbf{u}^B$.
• Figure 5: A bijection between mixing superchannels and conditionally mixing channels.

Theorems & Definitions (72)

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