A dimension-independent strict submultiplicativity for the transposition map in diamond norm

TL;DR

This paper establishes a universal bound on the diamond norm of the composition of the transposition map with the difference of quantum channels, demonstrating a dimension-independent strict submultiplicativity property.

Contribution

It proves a universal, explicit constant bound for the diamond norm of the transposition map composed with channel differences, independent of dimension.

Findings

The constant =1/ satisfies the inequality for all dimensions.

The result applies to all quantum channels on finite-dimensional spaces.

It provides a new dimension-independent submultiplicativity property.

Abstract

We prove that there exists an absolute constant $\alpha<1$ such that for every finite dimension $d$ and every quantum channel $T$ on $\mathsf{L}(\mathbb{C}^d)$, $\left\|\Theta\circ(\mathrm{id}-T)\right\|_\diamond \le \alpha\,\left\|\Theta\right\|_\diamond\,\left\|\mathrm{id}-T\right\|_\diamond$, where $\Theta$ is the transposition map. In fact we show the explicit choice $\alpha=1/\sqrt{2}$ works.