关于非锐化量子随机变量的中心极限定理
On the central limit theorem for unsharp quantum random variables
摘要：
本文研究了由非锐化量子测量产生的随机变量的弱收敛性质。更准确地说，对于由重复非锐化量子测量产生的随机变量序列，我们研究了相对频率的极限分布。我们提供了所有可分离状态的de Finetti型表示定理，表明该分布可以很好地近似为正态分布的混合。不需要对称性限制，例如置换不变性。此外，我们研究的收敛速度，并表明，相对频率可以稳定到一些常数在最好的速度为$1/\sqrt{N}$阶的所有可分离的输入。另一方面，我们提供了一个严格非尖锐的量子测量的例子，其中更好的速率是通过使用纠缠输入来实现的。这意味着在某些情况下，测量过程产生的噪声可以通过使用纠缠来抑制。我们以量子信息任务的形式提供我们的结果，其中玩家通过使用纠缠输入在极限情况下确定地实现目标，或者通过使用可分离输入确定地失败。
英文摘要：
In this letter we study the weak-convergence properties of random variables generated by unsharp quantum measurements. More precisely, for a sequence of random variables generated by repeated unsharp quantum measurements, we study the limit distribution of relative frequency. We provide the de Finetti-type representation theorem for all separable states, showing that the distribution can be well approximated by mixture of normal distributions. No symmetry restrictions, such as the permutational invariance were needed. Furthermore, we investigate the convergence rates and show that the relative frequency can stabilize to some constant at best at the rate of order $1/\sqrt{N}$ for all separable inputs. On the other hand, we provide an example of a strictly unsharp quantum measurement where the better rates are achieved by using entangled inputs. This means that in certain cases the noise generated by the measurement process can be suppressed by using entanglement. We deliver our result in the form of quantum information task where the player achieves the goal with certainty in the limiting case by using entangled inputs or fails with certainty by using separable inputs.
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