Comparing quantum channels using Hermitian-preserving trace-preserving linear maps: A physically meaningful approach

TL;DR

This work proposes a physically meaningful framework to compare quantum channels using Hermitian-preserving trace-preserving (HP-TP) linear maps. It formalizes an asymptotic power preorder $\Lambda_1 \succeq_{asymp} \Lambda_2$ based on informationally complete measurements and shows this is equivalent to the existence of a HP-TP map $\Theta$ with $\Lambda_2=\Theta\circ\Lambda_1$, thereby linking output statistics to HP post-processing. The authors establish a hierarchy among preorders, with CPTP post-processing, positive maps, and HP post-processing forming a nested structure, and they discuss implications for quantum device incompatibility. They also illustrate that asymptotic power does not imply post-processing compatibility, underscoring the nuanced landscape of channel comparisons and suggesting directions for future work in entanglement witnesses, thermodynamics, and information-theoretic tasks.

Abstract

In quantum technologies, quantum channels are essential elements for the transmission of quantum states. The action of a quantum channel usually introduces noise in the quantum state and thereby reduces the information contained in it. Concatenating a quantum channel with another quantum channel makes it more noisy and degrades its information and resource preservability. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems. These are special cases of Hermitian-preserving trace-preserving linear maps. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. More precisely, given a pair of quantum channels and an arbitrary unknown input state, we show that if the output state of one quantum channel from the pair can be obtained from the output statistics of the other channel from the pair using some quantum measurement, then the latter channel from the pair can be obtained from the former channel by concatenating it with a Hermitian-preserving trace-preserving linear map. This relation between these two channels is a preorder, and we try to study its characterization in this work. We also illustrate the implications of our results for the incompatibility of quantum devices through an example.

Figures (1)

• Figure 1: For a given channel $\Lambda\in\mathscr{C}(\mathcal{H},\mathcal{K})$, $\mathscr{C}^{HP}_{\mathcal{H}\rightarrow\overline{\mathcal{H}}}(\Lambda)$ represents set of all quantum channels obtained by concatenating it with Hermitian-preserving trace-preserving linear maps, $\mathscr{C}^{P}_{\mathcal{H}\rightarrow\overline{\mathcal{H}}}(\Lambda)$ represents set of all quantum channels obtained by concatenating it with positive maps and $\mathscr{C}^{CP}_{\mathcal{H}\rightarrow\overline{\mathcal{H}}}(\Lambda)$ represents set of all quantum channels obtained by post-processing it with a quantum channels. They follow the subset relation:$\mathscr{C}^{CP}_{\mathcal{H}\rightarrow\overline{\mathcal{H}}}(\Lambda)\subseteq\mathscr{C}^{P}_{\mathcal{H}\rightarrow\overline{\mathcal{H}}}(\Lambda)\subseteq\mathscr{C}^{asymp}_{\mathcal{H}\rightarrow\overline{\mathcal{H}}}(\Lambda)=\mathscr{C}^{HP}_{\mathcal{H}\rightarrow\overline{\mathcal{H}}}(\Lambda)$

Theorems & Definitions (18)

• Proposition 1
• Proposition 2
• Definition 1
• Remark 1: Justification of Definition \ref{['Def:chan_power_IC']}
• Proposition 3
• proof
• Theorem 1
• proof
• Remark 2
• Example 1