论文标题
高斯州的光子数量矩和累积物
Photon-number moments and cumulants of Gaussian states
论文作者
论文摘要
当以光子数为基础测量时，我们会为高斯状态的矩和累积物开发封闭形式的表达式。我们用循环hafnian表示高斯状态的光子数矩，该功能应用于代表图形的邻接性的$（0,1）$ - 矩阵时，会计算其完美匹配的数量。同样，我们以蒙特利尔人的形式表达了光子数累积物，这是一个新引入的矩阵函数，当应用于$（0,1）$ - 矩阵时，矩阵计数该图的汉密尔顿周期的数量。基于这些图理论连接，我们表明光子数矩和累积物的计算是＃p-hard。此外，我们提供了一种指数时间算法来计算蒙特利尔（以及累积的），与Hafnians的众所周知的结果相匹配。然后，我们证明，当在每个输入中均具有相同的单模高斯状态的均匀有损干涉仪，所有奇数累积液，但第一个均为零。最后，我们采用我们得出的表达方式来研究高斯玻色子采样设置中不同输入状态的累积分布，其中$ k $相同的状态被送入$ \ ell $ - mode干涉仪。我们分析了累积剂与输入状态类型的依赖性，被压缩，有损，挤压或热量的依赖性，以及非vacuum输入数量的函数。我们发现，在模仿有损或无损挤压状态的光子数量累积物时，热状态的表现要比其他经典状态（例如挤压状态）差得多。
We develop closed-form expressions for the moments and cumulants of Gaussian states when measured in the photon-number basis. We express the photon-number moments of a Gaussian state in terms of the loop Hafnian, a function that when applied to a $(0,1)$-matrix representing the adjacency of a graph, counts the number of its perfect matchings. Similarly, we express the photon-number cumulants in terms of the Montrealer, a newly introduced matrix function that when applied to a $(0,1)$-matrix counts the number of Hamiltonian cycles of that graph. Based on these graph-theoretic connections, we show that the calculation of photon-number moments and cumulants are #P-hard. Moreover, we provide an exponential time algorithm to calculate Montrealers (and thus cumulants), matching well-known results for Hafnians. We then demonstrate that when a uniformly lossy interferometer is fed in every input with identical single-mode Gaussian states with zero displacement, all the odd-order cumulants but the first one are zero. Finally, we employ the expressions we derive to study the distribution of cumulants up to the fourth order for different input states in a Gaussian boson sampling setup where $K$ identical states are fed into an $\ell$-mode interferometer. We analyze the dependence of the cumulants as a function of the type of input state, squeezed, lossy squeezed, squashed, or thermal, and as a function of the number of non-vacuum inputs. We find that thermal states perform much worse than other classical states, such as squashed states, at mimicking the photon-number cumulants of lossy or lossless squeezed states.
