We theoretically investigate schemes to discriminate between twononorthogonal quantum states given multiple copies. We consider a number ofstate discrimination schemes as applied to nonorthogonal, mixed states of aqubit. In particular, we examine the difference that local and globaloptimization of local measurements makes to the probability of obtaining anerroneous result, in the regime of finite numbers of copies $N$, and in theasymptotic limit as $N \rightarrow \infty$. Five schemes are considered:optimal collective measurements over all copies, locally optimal localmeasurements in a fixed single-qubit measurement basis, globally optimal fixedlocal measurements, locally optimal adaptive local measurements, and globallyoptimal adaptive local measurements. Here, adaptive measurements are those forwhich the measurement basis can depend on prior measurement results. For eachof these measurement schemes we determine the probability of error (for finite$N$) and scaling of this error in the asymptotic limit. In the asymptoticlimit, adaptive schemes have no advantage over the optimal fixed local scheme,and except for states with less than 2% mixture, the most naive scheme (locallyoptimal fixed local measurements) is as good as any noncollective scheme. Forfinite $N$, however, the most sophisticated local scheme (globally optimaladaptive local measurements) is better than any other noncollective scheme, forany degree of mixture.
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机译：我们从理论上研究了在给定多个副本的情况下区分两个非正交量子态的方案。我们考虑了许多状态判别方案，这些方案适用于aqubit的非正交混合状态。特别地，我们在有限数量的副本$ N $的渐进极限和$ N \ rightarrow \ infty $的渐近极限中研究了局部测量的局部和全局优化对获得错误结果的概率的区别。考虑了五种方案：所有副本上的最佳集体测量，以固定单量子位测量为基础的局部最优局部测量，全局最优固定局部测量，局部最优自适应局部测量和全局最优自适应局部测量。在此，自适应测量是那些测量基础可以依赖于先前的测量结果的测量。对于这些测量方案中的每一个，我们确定误差的概率（对于有限的$ N $）和在渐近极限中的误差缩放。在渐近极限中，自适应方案没有最优固定局部方案的优势，并且除了混合比例小于2％的状态之外，最幼稚的方案（局部最优固定局部度量）与任何非集体方案一样好。但是，对于任何程度的混合，对于无限大的N $，最复杂的局部方案（全局最佳自适应局部度量）要优于任何其他非集体方案。
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