费米子高斯性可以用与模式无关的样本复杂度来检验

中文摘要 AI 辅助

判断一个未知量子态是否属于某个给定的态族，或者远离该态族，是量子信息论中的一个自然问题。在许多此类情形中，存在一个自然的2-拷贝测试，当态确实属于目标态族时，该测试总是接受；然而，对于远离该态族的态，要界定其被接受的概率通常要困难得多。在此，我们开发了一种简单的新技术，用于对一般态族上界化这一概率，从而也上界化该决策问题本身的样本复杂度。作为该框架的一个特别引人注目的例子，我们证明：判断一个未知的纯$n$模式态是费米子高斯态，或者至少在迹距离上远离所有高斯态$\varepsilon$，可以通过使用最优数量的$\Theta(\varepsilon^{-2})$个拷贝来完成。值得注意的是，我们的分析适用于众所周知的用于检验费米子高斯性的贝尔采样过程，因此我们证明该过程是最优的，即使与进行任意集体和自适应测量的协议相比也是如此。作为第二个例子，我们证明，对于斯莱特行列式态族，类似的决策问题也可以用与模式无关的样本复杂度来解决。

英文摘要

Deciding whether an unknown quantum state either belongs to, or is far from, a given family of states is a natural question of quantum information theory. In many such cases, there is a natural 2-copy test which always accepts whenever the state indeed belongs to the target family; it is often much more difficult, however, to bound the probability of acceptance for states that are far from the family. Here we develop a simple new technique for upper bounding this probability for general families, and therefore upper bounding the sample-complexity of the decision problem itself. As a particularly striking example of this framework, we show that deciding whether an unknown pure $n$-mode state is fermionic Gaussian, or at least $\varepsilon$-far in trace distance from all Gaussian states, can be accomplished using the optimal number $Θ(\varepsilon^{-2})$ of copies. Remarkably, our analysis applies to the well-known Bell sampling procedure for testing fermionic Gaussianity, which we therefore show to be optimal, even compared to protocols making arbitrary collective and adaptive measurements. As a second example, we show that the analogous decision problem for the family of Slater determinant states can also be solved with mode-independent sample-complexity.

发表机构

• Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）

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