Optimal measurements for symmetric quantum states with applications to optical communication
Optimal measurements for symmetric quantum states with applications to optical communication
复制标题
对称量子态的最佳测量及其在光通信中的应用
DOI：
10.1103/physreva.92.062333
复制
发表时间：
2015
影响因子：
2.9
通讯作者：
M. Silva
中科院分区：
文献类型：
--
作者：
H. Krovi;S. Guha;Z. Dutton;M. Silva
最小误差概率（MPE）测量区分具有量子力学所允许的最小平均误差概率的一组候选量子态。测量值为MPE的条件由Yuen、Kennedy和Lax（YKL）推导。MPE测量已经被发现用于在群作用下形成单个轨道的状态，即，在集合中的状态上存在传递的群作用。对于这样的状态集，称为几何均匀（GU）的福尼，它表明，“相当好的测量”（PGM）达到MPE。即便如此，评估PGM在GU集合上获得的实际错误概率（和其他性能指标）涉及对大矩阵进行求逆，并且通常不容易。我们的第一个贡献是一个公式的MPE和条件概率的GU集，使用组表示理论。接下来，我们考虑在群作用下具有多个轨道的纯态集合。这种状态被称为复合几何均匀（CGU）。一般CGU集的MPE测量值未知。在本文中，我们将展示如何我们的代表性理论描述的GU集的最佳测量自然推广到CGU的情况下。我们展示了如何通过将问题简化为求解几个联立方程来计算CGU集的MPE测量。方程的数目取决于不可约表示的重数空间的大小。对于许多常见的群表示（如几个实用的好的线性代码），这是更容易处理的解决大型半定程序-这是需要解决的YKL条件数值任意状态集。我们展示了如何评估与量子限制经典光通信相关的一些例子的CGU状态的MPE测量。
The minimum probability of error (MPE) measurement discriminates between a set of candidate quantum states with the minimum average error probability allowed by quantum mechanics. Conditions for a measurement to be MPE were derived by Yuen, Kennedy and Lax (YKL). MPE measurements have been found for states that form a single orbit under a group action, i.e., there is a transitive group action on the states in the set. For such state sets, termed geometrically uniform (GU) by Forney, it was shown that the `pretty good measurement' (PGM) attains the MPE. Even so, evaluating the actual probability of error (and other performance metrics) attained by the PGM on a GU set involves inverting large matrices, and is not easy in general. Our first contribution is a formula for the MPE and conditional probabilities of GU sets, using group representation theory. Next, we consider sets of pure states that have multiple orbits under the group action. Such states are termed compound geometrically uniform (CGU). MPE measurements for general CGU sets are not known. In this paper, we show how our representation-theoretic description of optimal measurements for GU sets naturally generalizes to the CGU case. We show how to compute the MPE measurement for CGU sets by reducing the problem to solving a few simultaneous equations. The number of equations depends on the sizes of the multiplicity space of irreducible representations. For many common group representations (such as those of several practical good linear codes), this is much more tractable than solving large semi-definite programs---which is what is needed to solve the YKL conditions numerically for arbitrary state sets. We show how to evaluate MPE measurements for CGU states for some examples relevant to quantum-limited classical optical communication.