Calculation of quantum discord for qubit-qudit or N qubits
Calculation of quantum discord for qubit-qudit or N qubits
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DOI：
10.1088/1751-8113/45/9/095303
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发表时间：
2011-06
期刊：
影响因子：
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通讯作者：
Sai Vinjanampathy;A. Rau
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文献类型：
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作者：
Sai Vinjanampathy;A. Rau
量子不和谐是量子相关的一种，它被定义为二部系统中量子互信息与经典相关之间的差异。到目前为止，讨论的都是只有几个独立参数的小系统。我们在这里扩展到一个更广泛的状态类别，当第二方是任意维d时，只要第一个被测量的方是一个量子位。我们提出了计算量子不和谐的两个公式，第一个与原始的熵定义有关，第二个与最近提出的几何距离度量有关，它导致了一个解析公式。在熵计算中对量子位的跟踪被简化为一个非常简单的处方。当d维系统为所谓的X态时，密度矩阵只沿对角线和反对角线具有非零元素，从而在视觉上看起来像字母X，则可以解析地进行熵计算。全二部量子位-量子位系统的这种状态可以命名为“扩展X状态”，其密度矩阵由四个块矩阵组成，每个块矩阵在视觉上表现为一个X。熵计算中涉及的优化通常超过两个参数，在许多情况下减少到一个，并且像我们的数值研究证明的那样，对于绝大多数密度矩阵集合完全避免。我们的结果也适用于N量子位系统的状态，其中“扩展X状态”由（2−1）个状态组成，其数量大于N量子位的X状态的（2−1）个。虽然这些仍然小于N个量子位的状态总数（2−1），但涉及的参数数量仍然很大。在N = 2的情况下，它们包含了整个15维参数空间，即N = 2的扩展X态代表了完整的量子位-量子位系统。
Quantum discord, a kind of quantum correlation, is defined as the difference between quantum mutual information and classical correlation in a bipartite system. It has been discussed so far for small systems with only a few independent parameters. We extend here to a much broader class of states when the second party is of arbitrary dimension d, so long as the first, measured, party is a qubit. We present two formulæ to calculate quantum discord, the first relating to the original entropic definition and the second to a recently proposed geometric distance measure which leads to an analytical formulation. The tracing over the qubit in the entropic calculation is reduced to a very simple prescription. And, when the d-dimensional system is a so-called X state, the density matrix having non-zero elements only along the diagonal and anti-diagonal so as to appear visually like the letter X, the entropic calculation can be carried out analytically. Such states of the full bipartite qubit-qudit system may be named “extended X states”, whose density matrix is built of four block matrices, each visually appearing as an X. The optimization involved in the entropic calculation is generally over two parameters, reducing to one for many cases, and avoided altogether for an overwhelmingly large set of density matrices as our numerical investigations demonstrate. Our results also apply to states of a N-qubit system, where “extended X states” consist of (2 − 1) states, larger in number than the (2 − 1) of X states of N qubits. While these are still smaller than the total number (2 − 1) of states of N qubits, the number of parameters involved is nevertheless large. In the case of N = 2, they encompass the entire 15-dimensional parameter space, that is, the extended X states for N = 2 represent the full qubit-qubit system.