量子降维的局限性
Limitations on quantum dimensionality reduction
摘要：
Johnson-Lindenstrauss引理是一个经典的结果，它意味着任何n个实向量的集合都可以压缩到O（log n）维，而只需要通过一个常数因子来扭曲两两欧几里德距离。在这里，我们考虑这个结果的潜在扩展到量子态的压缩。与经典情形相比，我们证明了在量子信道上不存在任何分布能显著降低量子态的维数，同时以很高的概率保持2-范数距离.我们讨论两个任务，2-范数距离确实是正确的品质因数。在跟踪规范的情况下，我们表明，低秩混合状态的维数可以减少到一个平方根，但基本上没有降维是可能的高度混合状态。
英文摘要：
The Johnson-Lindenstrauss Lemma is a classic result which implies that any set of n real vectors can be compressed to O(log n) dimensions while only distorting pairwise Euclidean distances by a constant factor. Here we consider potential extensions of this result to the compression of quantum states. We show that, by contrast with the classical case, there does not exist any distribution over quantum channels that significantly reduces the dimension of quantum states while preserving the 2-norm distance with high probability. We discuss two tasks for which the 2-norm distance is indeed the correct figure of merit. In the case of the trace norm, we show that the dimension of low-rank mixed states can be reduced by up to a square root, but that essentially no dimensionality reduction is possible for highly mixed states.
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