标题： 矩形正映射的Sinkhorn-Knopp定理 显示英文标题

标题： Sinkhorn-Knopp theorem for rectangular positive maps

摘要： 在本工作中，我们将Sinkhorn-Knopp定理应用于矩形正映射$(T:M_k\rightarrow M_m)$。我们将其支持和全支持的概念扩展到这些映射。我们证明，正映射$T:M_k\rightarrow M_m$与双随机映射等价当且仅当$T:M_k\rightarrow M_m$与具有全支持的正映射等价。此外，如果$k$和$m$互质，则$T:M_k\rightarrow M_m$与双随机映射等价当且仅当$T:M_k\rightarrow M_m$具有支持。此结果为滤波正规形式提供了必要且充分条件，该形式常用于量子信息理论中以简化检测纠缠的任务。 设$A=\sum_{i=1}^nA_i\otimes B_i\in M_k\otimes M_m$为一个状态，$G_A: M_k\rightarrow M_m$为正映射$G_A(X)=\sum_{i=1}^nB_itr(A_iX)$。 我们证明当且仅当$A$可以转化为滤波标准形式时，$G_A: M_k\rightarrow M_m$与具有全支撑的正映射等价。 我们证明了任何状态$A\in M_k\otimes M_m\simeq M_{km}$，如果$\dim(\ker(A))<k-1$，当$k=m$，以及$\dim(\ker(A))<\min\{k,m\}$，当$k\neq m$，都可以转化为滤波标准形式。 最近，注意到矩形正映射$T:M_k\rightarrow M_m$的容量与某种平方正映射$\widetilde{T}:M_{mk}\rightarrow M_{mk}$的容量之间的联系。 在这里，我们得到了这些映射之间更深入的联系。 作为我们主要结果的推论，我们证明了当且仅当$\widetilde{T}:M_{mk}\rightarrow M_{mk}$与双随机映射等价时，$T:M_k\rightarrow M_m$与双随机映射等价。 显示英文摘要

摘要： In this work, we adapt Sinkhorn-Knopp theorem for rectangular positive maps $(T:M_k\rightarrow M_m)$. We extend their concepts of support and total support to these maps. We show that a positive map $T:M_k\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $T:M_k\rightarrow M_m$ is equivalent to a positive map with total support. Moreover, if $k$ and $m$ are coprime then $T:M_k\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $T:M_k\rightarrow M_m$ has support. This result provides a necessary and sufficient condition for the filter normal form, which is commonly used in Quantum Information Theory in order to simplify the task of detecting entanglement. Let $A=\sum_{i=1}^nA_i\otimes B_i\in M_k\otimes M_m$ be a state and $G_A: M_k\rightarrow M_m$ be the positive map $G_A(X)=\sum_{i=1}^nB_itr(A_iX)$. We show that $A$ can be put in the filter normal form if and only if $G_A: M_k\rightarrow M_m$ is equivalent to a positive map with total support. We prove that any state $A\in M_k\otimes M_m\simeq M_{km}$ such that $\dim(\ker(A))<k-1$, if $k=m$, and $\dim(\ker(A))<\min\{k,m\}$, if $k\neq m$, can be put in the filter normal form. Recently, a connection between the capacity of a rectangular positive map $T:M_k\rightarrow M_m$ and the capacity of a certain square positive map $\widetilde{T}:M_{mk}\rightarrow M_{mk}$ was noticed. Here, we obtain a deeper connection between these maps. As a corollary of our main results, we prove that $T:M_k\rightarrow M_m$ is equivalent to a doubly stochastic map if and only if $\widetilde{T}:M_{mk}\rightarrow M_{mk}$ is equivalent to a doubly stochastic map.