论文标题
在量子通道的混合级别上
On the mixed-unitary rank of quantum channels
论文作者
论文摘要
在量子信息的理论中，对于任何正整数尺寸$ n $，混合自动量子通道是那些可以用$ n \ times n $复杂单位矩阵表示为凸的线性图。我们考虑任何此类渠道的混合单位等级，这是表达这种形式所需的不同统一共轭的最小数量。我们确定了混合单位排名〜$ n $与混合统一渠道的Choi等级之间的几个新关系，Choi等级等于该渠道的Kraus表示所需的最小非零项数。最值得注意的是，我们证明了每个混合自动频道的不平等$ n \ leq r^2-r+1 $（如$ r = 2 $时的平等$ n = 2 $），并且我们展示了$ n> r $的第一个已知的混合自然频道示例。具体来说，我们证明存在具有Choi排名$ d+1 $的混合渠道，而混合自动排名$ 2D $的无限许多正整数$ D $，包括每个Prime Power $ d $。我们还研究了混合自动沃纳 - 霍尔沃通道的混合自动排名。
In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension $n$, are those linear maps that can be expressed as a convex combination of conjugations by $n\times n$ complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank~$N$ and the Choi rank~$r$ of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a Kraus representation of that channel. Most notably, we prove that the inequality $N\leq r^2-r+1$ is satisfied for every mixed-unitary channel (as is the equality $N=2$ when $r=2$), and we exhibit the first known examples of mixed-unitary channels for which $N>r$. Specifically, we prove that there exist mixed-unitary channels having Choi rank $d+1$ and mixed-unitary rank $2d$ for infinitely many positive integers $d$, including every prime power $d$. We also examine the mixed-unitary ranks of the mixed-unitary Werner--Holevo channels.
