多时间量子通信：有趣但非反事实
Multitime Quantum Communication: Interesting But Not Counterfactual
AI总结：
本文指出SLAZ量子通信协议并非“反事实”，通过引入基于振幅绝对值平方的信道使用量度“Cost”，证明在步骤数趋于无穷的渐近极限中总Cost不趋于零，并利用Gram矩阵性质给出了严格下界。
摘要：
Salih等人（Phys. Rev. Lett. 110 (2013) 170502，以下简称SLAZ）提出的一种在两方之间传输信息的协议，涉及在一系列步骤中将量子振幅通过量子信道来回发送，而不是简单地沿一个方向发送信号。作者声称他们的协议是“反事实的”，其含义是：虽然需要量子信道来连接双方，但随着步骤数趋于无穷大的渐近极限中，其实际使用量变得微乎其微。在此我们表明，这一主张是不正确的，因为它所使用的概率推理在存在量子干涉的中间时刻是无效的。当用一种明确定义的信道使用量度（此处称为“Cost”，等于通过信道发送的振幅的绝对值平方）来取代定义不明确的概率时，在大量步骤的渐近极限中，总Cost不会趋于零，而是受一个严格不等式的下界约束。详细分析表明，SLAZ协议满足该界限。得出该界限的分析利用了这样一个事实：由一组纯量子态内积形成的Gram矩阵在Hilbert子空间上是可加的，并且在酉时间变换下是不变的。其非对角元素（通常不为正数）在形式论证中起着重要作用，同时也提供了一种略显奇怪的可视化信息传输的方式。
英文摘要：
A protocol for transmission of information between two parties introduced by Salih et al., Phys. Rev. Lett. 110 (2013) 170502 (hereafter SLAZ), involves sending quantum amplitude back and forth through a quantum channel in a series of steps, rather than simply sending a signal in one direction. The authors claimed that their protocol was ``counterfactual'' in the sense that while a quantum channel is needed to connect the parties, its actual usage becomes vanishingly small in the asymptotic limit as the number of steps tends to infinity. Here we show that this claim is incorrect because it uses probabilistic reasoning that is not valid at intermediate times in the presence of quantum interference. When ill-defined probabilities are replaced with a well-defined measure of channel usage here called ``Cost'', equal to the absolute square of the amplitude sent through the channel, the total Cost does not go to zero in the asymptotic limit of a large number of steps, but is bounded below by a rigorous inequality. A detailed analysis shows that this bound is satisfied in the SLAZ protocol. The analysis leading to the bound uses the fact that the Gram matrix formed by inner products of a collection of pure quantum states is additive over Hilbert subspaces and invariant under unitary time transformations. Its off-diagonal elements, which in general are not positive, play a significant role in the formal argument as well as providing a somewhat strange way of visualizing the transfer of information.
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