Conditions for the multiplicativity of maximal l_p-norms of channels for fixed integer p
Vittorio Giovannetti, Seth Lloyd, Mary Beth Ruskai
备注：
20 pages: new material added including additional details supporting the conjecture that the Werner-Holevo channel is multiplicative for all p when d > 2^{p-1}; typos corrected. To appear on JMP
We introduce a condition for memoryless quantum channels which, when satisfied guarantees the multiplicativity of the maximal l_p-norm with p a fixed integer. By applying the condition to qubit channels, it can be shown that it is not a necessary condition, although some known results for qubits can be recovered. When applied to the Werner-Holevo channel, which is known to violate multiplicativity when p is large relative to the dimension d, the condition suggests that multiplicativity holds when $d \geqslant 2^{p-1}$. This conjecture is proved explicitly for p=2, 3, 4. Finally, a new class of channels is considered which generalizes the depolarizing channel to maps which are combinations of the identity channel and a noisy one whose image is an arbitrary density matrix. It is shown that these channels are multiplicative for p = 2.
The maximal p-norm multiplicativity conjecture is false
对于所有的1 < p < 2，我们证明了具有非乘法最大p-模的量子信道的存在性。等价地，对于所有1 < p < 2，量子信道的阶p的最小输出Renyi熵不是可加的。发现的违规行为很大。当p接近1时，产品通道的p阶最小输出Renyi熵不需要显著大于其各个因子的最小输出熵。由于p=1对应于冯诺伊曼熵，这些反例表明，如果量子信息理论的可加性猜想是正确的，它不能被证明为最大p-范数乘法的结果。
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Completely Bounded Qusi-Norms, Their Mutiplicativity, and New Additivity Results of Quantum Channels
Institute for Advanced Study in Mathematics, Harbin Institute of Technology（哈尔滨工业大学高等研究院）；Laboratoire de Mathématiques, Université Marie & Louis Pasteur（玛丽·路易·巴斯德大学数学实验室）
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