量子比特的非负子理论和准概率表示
Nonnegative subtheories and quasiprobability representations of qubits
摘要：
准概率表示中的负性通常被解释为非经典行为的指示。然而，这并不排除非负态表现出通常与量子力学相关的现象-单量子比特稳定态具有非负维格纳函数，但在许多量子信息任务中发挥着重要作用。我们试图确定哪些其他的量子态集和测量的量子位可以是非负的准概率表示，并确定非平凡的酉群，置换这样一个集合中的状态。这些状态和测量的集合类似于单个量子位稳定器状态。我们发现，没有准概率表示的量子比特可以是非负的四个以上的基地，在任何准概率表示的非负基地必须满足一定的对称性约束。我们提供了一个详尽的单量子位基的集合列表，这些单量子位基在某些准概率表示中是非负的，并且也被非平凡的酉群置换。这个列表包括两个三个基的家族，它们都包括作为特殊情况的单量子比特稳定态和一个四个基的家族，其对称群是泡利群。对于高维情形，我们证明了在任意拟概率表象下，d维希尔伯特空间的非负基中不可能有超过2^{d^2}个态.此外，这些基必须满足某些对称性约束，对应于要求基彼此充分互补。
英文摘要：
Negativity in a quasiprobability representation is typically interpreted as an indication of nonclassical behavior. However, this does not preclude states that are non-negative from exhibiting phenomena typically associated with quantum mechanics - the single qubit stabilizer states have non-negative Wigner functions and yet play a fundamental role in many quantum information tasks. We seek to determine what other sets of quantum states and measurements for a qubit can be non-negative in a quasiprobability representation, and to identify nontrivial unitary groups that permute the states in such a set. These sets of states and measurements are analogous to the single qubit stabilizer states. We show that no quasiprobability representation of a qubit can be non-negative for more than four bases and that the non-negative bases in any quasiprobability representation must satisfy certain symmetry constraints. We provide an exhaustive list of the sets of single qubit bases that are non-negative in some quasiprobability representation and are also permuted by a nontrivial unitary group. This list includes two families of three bases that both include the single qubit stabilizer states as a special case and a family of four bases whose symmetry group is the Pauli group. For higher dimensions, we prove that there can be no more than 2^{d^2} states in non-negative bases of a d-dimensional Hilbert space in any quasiprobability representation. Furthermore, these bases must satisfy certain symmetry constraints, corresponding to requiring the bases to be sufficiently complementary to each other.
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