A Generalization of Quantum Stein's Lemma
A Generalization of Quantum Stein's Lemma
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DOI：
10.1007/s00220-010-1005-z
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发表时间：
2010-05-01
影响因子：
2.4
通讯作者：
Plenio, Martin B.
中科院分区：
文献类型：
--
作者：
Brandao, Fernando G. S. L.;Plenio, Martin B.
给出许多独立且同分布的(I.I.D.)由态Rho或Sigma描述的量子系统的副本(分别称为零假设和替代假设)，了解真态的身份的最佳测量是什么？在不对称假设检验中，人们感兴趣的是将错误识别Rho而不是sigma的概率降至最低，同时要求在Rho的位置识别sigma的概率由一个小的固定数限定。量子Stein引理将错误概率趋近于零的渐近指数速率定义为Rho和Sigma的量子相对熵，将量子Stein引理推广到另一种假设是由一族态构成的情形，该族可以是非I.I.D.的。我们考虑满足一些自然性质的状态集，其中最重要的是副本排列下的封闭性。然后，我们根据量子相对熵，以非常类似于量子Stein引理的方式来确定误码率函数。我们的结果在纠缠理论中有两个应用。首先，它赋予了一种纠缠度量--纠缠的正则化相对熵--的操作意义。其次，它表明这一措施是忠实的，对每个纠缠态都是严格正的。这特别意味着，只要通过局域操作和经典通信能够将一个多体纠缠态渐近地转换为另一个纠缠态，其转换率必然是非零的。因此，多体纠缠的运算定义与其数学定义是等价的。
Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state rho or sigma (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying rho instead of sigma, while requiring that the probability that sigma is identified in the place of rho is bounded by a small fixed number. Quantum Stein's Lemma identifies the asymptotic exponential rate at which the specified error probability tends to zero as the quantum relative entropy of rho and sigma.We present a generalization of quantum Stein's Lemma to the situation in which the alternative hypothesis is formed by a family of states, which can moreover be non-i.i.d. We consider sets of states which satisfy a few natural properties, the most important being the closedness under permutations of the copies. We then determine the error rate function in a very similar fashion to quantum Stein's Lemma, in terms of the quantum relative entropy.Our result has two applications to entanglement theory. First it gives an operational meaning to an entanglement measure known as regularized relative entropy of entanglement. Second, it shows that this measure is faithful, being strictly positive on every entangled state. This implies, in particular, that whenever a multipartite state can be asymptotically converted into another entangled state by local operations and classical communication, the rate of conversion must be non-zero. Therefore, the operational definition of multipartite entanglement is equivalent to its mathematical definition.