量子协议的范畴语义
A categorical semantics of quantum protocols
摘要：
我们从一个新的角度研究量子信息和计算。我们的方法是基于重铸的标准公理介绍的量子力学，由于冯诺依曼，在一个更抽象的层次上，紧凑的封闭范畴的双产品。我们展示了如何在关键的量子信息协议，如隐形传态，逻辑门隐形传态和纠缠交换中发现的基本结构可以在这个抽象的层次上被捕获。此外，从紧闭包和双积的--显然是纯定性的--结构的组合中，出现了“标量”和“玻恩规则”。这种抽象和结构的观点为描述和推理量子系统开辟了新的可能性。它还显示了公理自由度：我们可以显示对标量C（I，I）的（半）环的要求，其中C是范畴，I是张量单位，以执行各种协议，如隐形传态。我们的形式主义捕捉了协议的信息流方面（见quant-ph/0402014），以及由于量子不确定性而产生的分支。这与标准解释形成对比，在标准解释中，经典信息流“在”通常的量子力学形式主义之外。
英文摘要：
We study quantum information and computation from a novel point of view. Our approach is based on recasting the standard axiomatic presentation of quantum mechanics, due to von Neumann, at a more abstract level, of compact closed categories with biproducts. We show how the essential structures found in key quantum information protocols such as teleportation, logic-gate teleportation, and entanglement-swapping can be captured at this abstract level. Moreover, from the combination of the --apparently purely qualitative-- structures of compact closure and biproducts there emerge `scalars` and a `Born rule'. This abstract and structural point of view opens up new possibilities for describing and reasoning about quantum systems. It also shows the degrees of axiomatic freedom: we can show what requirements are placed on the (semi)ring of scalars C(I,I), where C is the category and I is the tensor unit, in order to perform various protocols such as teleportation. Our formalism captures both the information-flow aspect of the protocols (see quant-ph/0402014), and the branching due to quantum indeterminism. This contrasts with the standard accounts, in which the classical information flows are `outside' the usual quantum-mechanical formalism.
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