量子计算中Gleason定理的推广
An Extension of Gleason's Theorem for Quantum Computation
摘要：
我们开发了图灵的计算范式和冯·诺依曼的量子逻辑的合成，作为具有递归的量子计算的模型，使得潜在的非终止计算可以发生，就像在量子图灵机中一样。这个模型是基于冯·诺依曼的量子逻辑扩展到部分状态，这里定义为希尔伯特空间上的子概率测度，配备了自然的逐点偏序。子概率度量允许计算的不终止的一定概率。然后我们得到了Gleason定理的一个推广，证明了对于维数大于2的Hilbert空间，次概率测度的偏序与偏密度算子的集合是序同构的，即，迹类正算子，其迹在0和1之间，具有由正算子导出的通常的偏序。证明了有界可观测量关于部分状态的期望值可以定义为闭有界区间，推广了经典的期望值定义。
英文摘要：
We develop a synthesis of Turing's paradigm of computation and von Neumann's quantum logic to serve as a model for quantum computation with recursion, such that potentially non-terminating computation can take place, as in a quantum Turing machine. This model is based on the extension of von Neumann's quantum logic to partial states, defined here as sub-probability measures on the Hilbert space, equipped with the natural pointwise partial ordering. The sub-probability measures allow a certain probability for the non-termination of the computation. We then derive an extension of Gleason's theorem and show that, for Hilbert spaces of dimension greater than two, the partial order of sub-probability measures is order isomorphic with the collection of partial density operators, i.e., trace class positive operators with trace between zero and one, equipped with the usual partial ordering induced from positive operators. We show that the expected value of a bounded observable with respect to a partial state can be defined as a closed bounded interval, which extends the classical definition of expected value.
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