Quantum state discrimination bounds for finite sample size
Quantum state discrimination bounds for finite sample size
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DOI：
10.1063/1.4768252
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发表时间：
2012-12-01
影响因子：
1.3
通讯作者：
Verstraete, Frank
中科院分区：
文献类型：
--
作者：
Audenaert, Koenraad M. R.;Mosonyi, Milan;Verstraete, Frank
在量子态判别的问题中，我们必须通过测量确定量子系统的状态，基于先验的边信息，即真实状态是两个给定且完全已知的状态之一，或。一般来说，不可能确定真实状态的身份，最优测量策略取决于是否将两种可能的误差（将rho误认为sigma，或相反）视为同等重要。关于量子Chernoff和Hoeffding边界以及量子Stein引理的结果表明，如果系统有多个副本，则最优误差概率随副本数量呈指数衰减，衰减率由rho和sigma之间的一定统计距离（分别为Chernoff距离、Hoeffding距离和相对熵）给出。虽然这些结果为渐近问题提供了一个完整的解，但从实际的角度来看，它们并不完全令人满意。实际上，在现实的场景中，人们只能访问系统的有限多个副本，因此，对于有限的样本量，有错误概率的界限是可取的。本文给出了Stein误差、Chernoff误差、Hoeffding误差以及与Chernoff误差和Hoeffding误差相关的混合误差概率的有限范围。(C) 2012年美国物理研究所。[http://dx.doi.org/10.1063/1.4768252]
In the problem of quantum state discrimination, one has to determine by measurements the state of a quantum system, based on the a priori side information that the true state is one of the two given and completely known states, rho or sigma. In general, it is not possible to decide the identity of the true state with certainty, and the optimal measurement strategy depends on whether the two possible errors (mistaking rho for sigma, or the other way around) are treated as of equal importance or not. Results on the quantum Chernoff and Hoeffding bounds and the quantum Stein's lemma show that, if several copies of the system are available then the optimal error probabilities decay exponentially in the number of copies, and the decay rate is given by a certain statistical distance between rho and sigma (the Chernoff distance, the Hoeffding distances, and the relative entropy, respectively). While these results provide a complete solution to the asymptotic problem, they are not completely satisfying from a practical point of view. Indeed, in realistic scenarios one has access only to finitely many copies of a system, and therefore it is desirable to have bounds on the error probabilities for finite sample size. In this paper we provide finite-size bounds on the so-called Stein errors, the Chernoff errors, the Hoeffding errors, and the mixed error probabilities related to the Chernoff and the Hoeffding errors. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4768252]