Quantum channels can be described via a unitary coupling of system and environment, followed by a trace over the environment state space. Taking the trace instead over the system state space produces a different mapping which we call the conjugate channel. We explore the properties of conjugate channels and describe several different methods of construction. In general, conjugate channels map M_d to M_d' with d < d', and different constructions may differ by conjugation with a partial isometry.
We show that a channel and its conjugate have the same minimal output entropy and maximal output p-norm. It then follows that the additivity and multiplicativity conjectures for these measures of optimal output purity hold for a product of channels if and only if they also hold for the product of their conjugates. This allows us to reduce these conjectures to the special case of maps taking M_d to M_d' with a minimal representation of dimension at most d.
We find explicit expressions for the conjugates for a number of well-known examples, including entanglement-breaking channels, unital qubit channels, the depolarizing channel, and a subclass of random unitary channels. For the entanglement-breaking channels, channels this yields a new class of channels for which additivity and multiplicativity of optimal output purity can be established. For random unitary channels using the generalized Pauli matrices, we obtain a new formulation of the multiplicativity conjecture. The conjugate of the completely noisy channel plays a special role and suggests a mechanism for using noise to transmit information.
Non-Multiplicativity of the Holevo Barycenter of Quantum Channels
Universität zu Köln（科隆大学）；University of Camerino（卡梅里诺大学）；Istituto Nazionale di Fisica Nucleare, Sezione di Perugia（意大利国家核物理研究所佩鲁吉亚分部）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）
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。
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）
From Permutation Symmetry to Communication Bounds and Additivity
Perimeter Institute for Theoretical Physics（Perimeter理论物理研究所）；University of Waterloo（滑铁卢大学）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）
