扩展量子操作
Extending quantum operations
摘要：
对于一组给定的量子态或可观测量的输入-输出对，我们提出的问题是，是否存在一个物理上可实现的变换，将每个输入映射到相应的输出。量子态上的物理映射是保迹的完全正映射，但我们也考虑这些要求的变体。我们将完全正性的定义推广到定义在任意子空间上的线性映射，然后将这个概念表述为半定规划，并将它与这个映射的近似扩展的对偶性联系起来。这给出了一个表征的地图，可以近似任意以及限制的地图是完全积极的整个代数，也产生熟悉的扩张定理的运营商空间。对于量子信道的扩展和扩展的概率运算，我们得到了半定的特征，我们也阐明了阿贝尔输入或输出的特殊情况。最后，重新审视一个定理的Alberti和Uhlmann，我们提供了更简单和更广泛适用的条件，某些扩展问题的量子位，并通过使用半定规划制定我们展示反例，看似合理，但错误的概括的Alberti-Uhlmann定理。
英文摘要：
For a given set of input-output pairs of quantum states or observables, we ask the question whether there exists a physically implementable transformation that maps each of the inputs to the corresponding output. The physical maps on quantum states are trace-preserving completely positive maps, but we also consider variants of these requirements. We generalize the definition of complete positivity to linear maps defined on arbitrary subspaces, then formulate this notion as a semidefinite program, and relate it by duality to approximative extensions of this map. This gives a characterization of the maps which can be approximated arbitrarily well as the restriction of a map that is completely positive on the whole algebra, also yielding the familiar extension theorems on operator spaces. For quantum channel extensions and extensions by probabilistic operations we obtain semidefinite characterizations, and we also elucidate the special case of Abelian in- or outputs. Finally, revisiting a theorem by Alberti and Uhlmann, we provide simpler and more widely applicable conditions for certain extension problems on qubits, and by using a semidefinite programming formulation we exhibit counterexamples to seemingly reasonable but false generalizations of the Alberti-Uhlmann theorem.
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