Invariants of the quantum graph of the partial trace

TL;DR

The paper interprets quantum channels through their non-commutative confusability graphs and computes four invariants for the partial-trace channel $\mathrm{Tr}_n\otimes\mathrm{id}_k$. It identifies the associated operator system as $S=\mathrm{B}(\mathbb{C}^n)\otimes\mathds{1}_k$ and determines its orthogonal complement, enabling exact results: the independence number $\alpha(S)=k$ and zero-error capacity $C_0(S)=\log k$; the Lovász function satisfies $\vartheta(S)=k^2$ if $n\ge k$ and $\vartheta(S)=nk$ if $n\le k$, while the quantum Lovász function is $\tilde{\vartheta}(S)=k^2$. The results clarify how these invariants depend on the dimension parameters $n$ and $k$ and illustrate the effectiveness of operator-system techniques and Choi-type inequalities in analyzing quantum confusability graphs.

Abstract

We compute the independence number, zero-error capacity, and the values of the Lovász function and the quantum Lovász function for the quantum graph associated to the partial trace quantum channel $\operatorname{Tr}_n\otimes\mathrm{id}_k\colon\operatorname{B}(\mathbb{C}^n\otimes\mathbb{C}^k)\to\operatorname{B}(\mathbb{C}^k)$.

Theorems & Definitions (11)

• Definition 1.1
• Proposition 2.1
• proof
• Corollary 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.4
• proof
• Corollary 2.5