Non-negative subtheories and quasiprobability representations of qubits
Non-negative subtheories and quasiprobability representations of qubits
复制标题
量子位的非负子理论和准概率表示
DOI：
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发表时间：
2012
期刊：
影响因子：
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通讯作者：
S. Bartlett
中科院分区：
文献类型：
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作者：
Joel J. Wallman;S. Bartlett
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。我们试图确定其他几组量子态和量子位的测量在准概率分布中可以是非负的，并确定在这样一组中置换状态的酉变换的非平凡组。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两个以上的基地在任何平面的布洛赫球。此外，在量子比特的任意准概率表示中，有一组唯一的四个基可以是非负的。我们提供了一个详尽的列表的单量子位基的集合是非负的在一些quasiprobibility分布，也关闭下的一组酉变换。这个列表包括两个非平凡的三个基础的家庭，都包括作为特殊情况的单量子比特稳定态。对于qudits，我们证明了在任意的拟概率表示下，在$d$维Hilbert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不同。
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 of a qubit can be non-negative in a quasiprobability distribution, and to identify nontrivial groups of unitary transformations 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 two bases in any plane of the Bloch sphere. Furthermore, there is a unique set of four bases that can be non-negative in an arbitrary quasiprobability representation of a qubit. We provide an exhaustive list of the sets of single qubit bases that are non-negative in some quasiprobability distribution and are also closed under a group of unitary transformations. This list includes two nontrivial families of three bases that both include the single qubit stabilizer states as a special case. For qudits, 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 different from each other.