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

TL;DR

The paper proves a dimension-independent submultiplicativity-type bound for the transposition map composed with a channel difference in the diamond norm: for any CPTP channel $T$ on $\mathsf{L}(\mathcal{H})$ with $\mathcal{H}\cong\mathbb{C}^d$, $\|\Theta\circ(\mathrm{id}-T)\|_{\diamond} \le \alpha\,\|\Theta\|_{\diamond}\,\|\mathrm{id}-T\|_{\diamond}$ with a universal $\alpha=1/\sqrt{2}$. The proof reduces to a matrix inequality for the partial transpose on traceless Hermitian matrices and leverages a chain of norm inequalities, including a Hilbert–Schmidt isometry under $\Theta\otimes\mathrm{id}$ and a traceless-Hilbert–Schmidt bound. The result yields a concrete bound $\|\Theta\circ(\mathrm{id}-T)\|_{\diamond} \le (d/\sqrt{2})\|\mathrm{id}-T\|_{\diamond}$, since $\|\Theta\|_{\diamond}=d$, and proves that equality cannot be attained in finite dimensions unless $T=\mathrm{id}$, suggesting a positive gap. An open question remains whether a strictly better universal constant $\alpha<1/\sqrt{2}$ exists.

Abstract

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

Theorems & Definitions (18)

• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• Remark 1
• Definition 2.1: Column-major vectorization