How well can we approximate a quantum channel output state using a random codebook with a certain size? In this work, we study the quantum soft covering problem. Namely, we use a random codebook with codewords independently sampled from a prior distribution and send it through a classical-quantum channel to approximate the target state. When using a random codebook sampled from an independent and identically distributed prior with a rate above the quantum mutual information, we show that the expected trace distance between the codebook-induced state and the target state decays with exponent given by the sandwiched R\'enyi information. On the other hand, when the rate of the codebook size is below the quantum mutual information, the trace distance converges to one exponentially fast. We obtain similar results when using a random constant composition codebook, whereas the sandwiched Augustin information expresses the error exponent. In addition to the above large deviation analysis, our results also hold in the moderate deviation regime. That is, we show that even when the rate of the codebook size approaches the quantum mutual information moderately quickly, the trace distance still vanishes asymptotically.
翻译:我们如何利用一个具有一定尺寸的随机代码簿来估计量子频道输出状态? 在这项工作中,我们研究量子软覆盖问题。 也就是说, 我们使用一个随机代码簿, 由先前分发的样本进行独立抽样, 并通过古典量子频道发送, 以接近目标状态。 当我们使用一个独立且同样分布的代码簿, 其比例高于量子相互信息时, 我们显示, 代码簿引发的状态和目标状态之间的预期痕量距离会随着三明治R\' enyi信息给出的提示而衰减。 另一方面, 当代码簿的速率低于量子共享信息时, 追踪距离会迅速趋近到一个。 当使用随机常数代码簿时, 我们获得类似的结果, 而 三明治的奥古斯丁信息会显示错误。 除了上述大偏差分析外, 我们的结果也会维持在中度偏差机制中。 也就是说, 即使代码簿的速率接近量子相互信息, 追踪距离仍然会迅速消失 。