Adversarial hypothesis testing and a quantum stein's lemma for restricted measurements

Adversarial hypothesis testing and a quantum stein's lemma for restricted measurements
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对抗性假设检验和受限测量的量子斯坦引理

DOI：
10.1145/2554797.2554816
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发表时间：
2014
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Brandão F
Brandão F
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作者：
Brandão F

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回想一下经典的假设检验设置，其中有两个概率分布PandQ的凸集。一个人接受两种身份证。从分布p ∈ P或分布q ∈ Q中采样，并希望决定从哪个集合中采样点。它是已知的，最佳的指数率，在错误减少可以通过一个简单的最大似然比测试，它不依赖于porq，但只对setsPandQ。我们认为这个模型的自适应推广的选择ofp∈Pandq∈ Q可以改变在每个样本以某种方式，任意依赖于以前的样本。换句话说，在第k轮中，对手在观察了第1轮中的所有先前样本后，k-1，选择sp κ∈Pandqκ∈Q，目的是混淆假设检验。我们证明，即使在这种情况下，最佳的指数误差率可以通过一个简单的最大似然测试，只依赖于pandQ。然后，我们表明，对抗模型在假设检验中的应用forquantum statesususing限制测量。例如，它可以用来研究区分纠缠态从所有可分离状态的集合中的问题，只使用可以通过本地操作和经典通信（LOCC）实现的测量。我们的基本思想是，在我们的设置，纠缠的有害影响可以模拟一个自适应的经典advertisement.We证明了量子斯坦引理在这种设置：在许多情况下，最佳假设检验率是等于一个适当的概念，两个状态之间的量子相对熵。特别是，我们的论点产生了另一种证明李和温特最近加强强次加性量子相对熵。
Recall the classical hypothesis testing setting with two convex sets of probability distributionsPandQ. One receives eitherni.i.d. samples from a distributionp∈Por from a distributionq∈Qand wants to decide from which set the points were sampled. It is known that the optimal exponential rate at which errors decrease can be achieved by a simple maximum-likelihood ratio test which does not depend onporq, but only on the setsPandQ.We consider an adaptive generalization of this model where the choice ofp∈Pandq∈Qcan change in each sample in some way that depends arbitrarily on the previous samples. In other words, in thekth round, an adversary, having observed all the previous samples in rounds 1, ...,κ-1, choosespκ∈Pandqκ∈Q, with the goal of confusing the hypothesis test. We prove that even in this case, the optimal exponential error rate can be achieved by a simple maximum-likelihood test that depends only onPandQ.We then show that the adversarial model has applications in hypothesis testing forquantum statesusing restricted measurements. For example, it can be used to study the problem of distinguishing entangled states from the set of all separable states using only measurements that can be implemented with local operations and classical communication (LOCC). The basic idea is that in our setup, the deleterious effects of entanglement can be simulated by an adaptive classical adversary.We prove a quantum Stein's Lemma in this setting: In many circumstances, the optimal hypothesis testing rate is equal to an appropriate notion of quantum relative entropy between two states. In particular, our arguments yield an alternate proof of Li and Winter's recent strengthening of strong subadditivity for quantum relative entropy.