论文标题
旋转频道的混合级别最低
Twirling channels have minimal mixed-unitary rank
论文作者
论文摘要
对于一个正整数$ d $和统一表示$ρ：g \ rightarrow \ mathrm {u}（d）紧凑型组$ g $的$，此表示形式的旋转频道是线性映射$φ：m_d \ rightArrow m_d $ rightarrow m_d $定义为定义$φ（x）= \ int_ {g} \ mathrm {d}μ（g）\，ρ（g）xρ（g^{ - 1}）$ in m_d $中的每个$ x \，其中$μ$是$ g $的$μ$。这样的通道是混合自然通道的示例，因为它们位于固定尺寸的单一通道集的凸壳中。通过Carathéodory的定理，这些通道始终可以表示为单一通道的有限线性组合。我们认为，混合自然的等级旋转频道 - 这是表达频道所需的最小单一共轭数量，作为单一频道的凸组合，并表明每个旋转频道的混合单位级别始终等于其CHOI等级，这两者都等于Vonneumann Algebra Algebra的尺寸。此外，我们展示了如何为这些类型的渠道明确构建最小的混合自然分解并提供一些示例。
For a positive integer $d$ and a unitary representation $ρ:G\rightarrow\mathrm{U}(d)$ of a compact group $G$, the twirling channel for this representation is the linear mapping $Φ: M_d\rightarrow M_d$ defined as $Φ(X)=\int_{G}\mathrm{d}μ(g)\,ρ(g)Xρ(g^{-1})$ for every $X\in M_d$, where $μ$ is the Haar measure on $G$. Such channels are examples of mixed-unitary channels, as they are in the convex hull of the set of unitary channels of a fixed size. By Carathéodory's theorem, these channels can always be expressed as a finite linear combination of unitary channels. We consider the mixed-unitary rank twirling channels---which is the minimum number of distinct unitary conjugations required to express the channel as a convex combination of unitary channels---and show that the mixed-unitary rank of every twirling channel is always equal to its Choi rank, both of which are equal to the dimension of the von Neumann algebra generated by the representation. Moreover, we show how to explicitly construct minimal mixed-unitary decompositions for these types of channels and provide some examples.
