非交换Marcus-Ree不等式与Erdős信道
Noncommutative Marcus-Ree inequality and Erdős channels
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中文总结 AI 辅助
本文证明了量子信道中Marcus-Ree不等式的成立条件，定义了Erdős信道并刻画了其结构，推广了经典结果至非交换情形。
中文摘要 AI 辅助
酉量子信道是双随机矩阵的非交换类比。1959年，Marcus和Ree证明了双随机矩阵的Frobenius范数的平方至多等于其最大对角和。我们证明了$\u0005mathbb{M}_d$上混合酉信道的这一不等式的量子版本，并给出了一个例子表明它对于一般酉信道不成立。与经典情形一样，我们将达到等式的信道称为Erdős信道。我们证明了到$\u0005mathbb{M}_d$的任何酉$C^*$-子代数的保迹条件期望是Erdős信道。它也在相同Choi秩的信道中是局部孤立的。接下来，我们引入了量子RCDS信道。这些是支持置换都具有相同对角和的双随机矩阵的非交换类比。我们证明了它们的结构定理，并用它来刻画量子RCDS Erdős信道。我们给出了Choi秩为$2$的Erdős信道的显式形式（在酉等价意义下），并进一步表明存在不可数多个Erdős信道的酉等价类，这与经典情形不同。我们还刻画了任意Choi秩的混合Weyl-酉信道中的Erdős信道。对于量子比特，我们证明了恰好存在四个Erdős信道的酉等价类，它们由恒等信道和Pauli信道的均匀平均表示。最后，限制到对角信道，我们的结果恢复了Marcus-Ree不等式、RCDS双随机矩阵的结构以及Erdős的刻画。
英文摘要
Unital quantum channels are the noncommutative analogues of doubly stochastic matrices. In 1959, Marcus and Ree showed that the squared Frobenius norm of a doubly stochastic matrix is at most its maximal diagonal sum. We prove a quantum version of this inequality for mixed unitary channels on $\mathbb{M}_d$ and give an example showing that it fails for general unital channels. As in the classical case, we call the channels attaining equality as Erdős channels. We show that the trace-preserving conditional expectation onto any unital $C^*$-subalgebra of $\mathbb{M}_d$ is an Erdős channel. It is also locally isolated among channels of the same Choi rank. Next, we introduce quantum RCDS channels. These are the noncommutative analogue of doubly stochastic matrices whose supported permutations all have the same diagonal sum. We prove a structure theorem for them and use it to characterize the quantum RCDS Erdős channels. We give an explicit form, up to unitary equivalence, for the Erdős channels of Choi rank $2$, and further show that there are uncountably many unitary equivalence classes of Erdős channels unlike the classical case. We also characterize the Erdős channels among mixed Weyl-unitary channels of any Choi rank. For qubits, we prove that there are exactly four unitary equivalence classes of Erdős channels and they are represented by the identity channel and uniform averages of Pauli channels. Finally, restricting to the diagonal channels, our results recover the Marcus--Ree inequality, the structure of RCDS doubly stochastic matrices, and the characterization of Erdős.
发表机构
- Indian Institute of Technology, Bombay（印度理工学院孟买分校）
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