A quantum analogue of the famous Blackwell Theorem in classical statistics has recently been proposed. Given two quantum channels A and B, a set of payoff functions have been proven to have values for B at least as high as they are for A if and only if there exists a quantum garbling channel E such that A=EB. When such a channel E exists, we can globally compare A and B in terms of their `noisiness'. We show that this method of channel noise comparison is equivalent to one obtained by considering the degradation of the distinguishability of states. Here, the channel A is said to be at least as noisy as the channel B if any ensemble of states, fed into each channel and possibly entangled with ancillae, emerges no more distinguishable from A than it does from channel B, where distinguishability is quantified by the minimum error discrimination probability. We also provide a novel application to eavesdropper detection in quantum cryptography.
We consider the optimal discrimination of bipartite quantum states and provide an upper bound for the maximum success probability of optimal local discrimination. We also provide a necessary and sufficient condition for a measurement to realize the upper bound. We further establish a necessary and sufficient condition for this upper bound to be saturated. Finally, we illustrate our results using an example.
In this work, we provide a complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for mixed four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condi…
We study several variations of the question of minimum-error discrimination of thermal states. Besides of providing the optimal values for the probability of error, we also characterize the optimal measurements. For the case of a fixed Hamiltonian, we show that for a general discrimination problem the optimal measurement is the measurement in the energy basis of the Hamiltonian. We identify a critical temperature, determining whether the given temperature is best distinguishable from thermal state of very high or very low temperatures. Further, we investigate the decision problem of whether th…
Minimum Error Discrimination for an Ensemble of Linearly Independent Pure States
受Belavkin [Belavkin V. P.，Stochastics，1，315（1975）]，独立地由Mochon，[Phys. Rev. A 73，032328，（2006）]，我们制定的问题的最小误差歧视的任何系综的$n$线性无关的纯态剥离的问题，其旋转协方差和保留的问题，只有旋转不变的方面。这是通过将最优条件嵌入矩阵等式和矩阵不等式来完成的。在此条件下，利用隐函数定理，得到一组一阶耦合常非线性微分方程组，该方程组可用于将解从一个已知解的初始点拖到另一个需要求解的点。这种获得解的方式可以通过简单的泰勒级数展开和必要时的解析延拓来完成。因此，我们\emph{complete}所做的工作Belavkin和Mochon最终导致他们的理论解决线性独立的纯态系综的最小误差判别问题。我们还比较了我们的技术的计算复杂性与障碍型内点方法的SDP和表明，我们的技术是计算效率（实际上，比）的SDP算法，与附加的优点是更简单的实现。
Optimal state discrimination with a fixed rate of inconclusive results: Analytical solutions and relation to state discrimination with a fixed error rate
Quantum hypothesis testing is an important tool for quantum information processing. Two main strategies have been widely adopted: in a minimum error discrimination strategy, the average error probability is minimized; while in an unambiguous discrimination strategy, an inconclusive decision (abstention) is allowed to vanish any possibility of errors when a conclusive result is obtained. In both scenarios, the testing between quantum states are relatively well-understood, for example, the ultimate limits of the performance are established decades ago; however, the testing between quantum channe…
A state discrimination problem in an operational probabilistic theory (OPT) is investigated in diagrammatic terms. It is well-known that, in the case of quantum theory, if a state set has a certain symmetry, then there exists a minimum-error measurement having the same type of symmetry. However, to our knowledge, it is not yet clear whether this property also holds in a more general OPT. We show that it also holds in OPTs, i.e., for a symmetric state set, there exists a minimum-error measurement that has the same type of symmetry. It is also shown that this result can be utilized to optimize o…
Quantum state discrimination is a central problem in quantum measurement theory, with applications spanning from quantum communication to computation. Typical measurement paradigms for state discrimination involve a minimum probability of error or unambiguous discrimination with a minimum probability of inconclusive results. Alternatively, an optimal inconclusive measurement, a non-projective measurement, achieves minimal error for a given inconclusive probability. This more general measurement encompasses the standard measurement paradigms for state discrimination and provides a much more pow…
In quantum state discrimination, one aims to identify unknown states from a given ensemble by performing measurements. Different strategies such as minimum-error discrimination or unambiguous state identification find different optimal measurements. Maximum-confidence measurements (MCMs) maximise the confidence with which inputs can be identified given the measurement outcomes. This unifies a range of discrimination strategies including minimum-error and unambiguous state identification, which can be understood as limiting cases of MCM. In this work, we investigate MCMs for general ensembles o…
For the optimal success probability under minimum-error discrimination between $r\geq2$ arbitrary quantum states prepared with any a priori probabilities, we find new general analytical lower and upper bounds and specify the relations between these new general bounds and the general bounds known in the literature. We also present the example where the new general analytical bounds, lower and upper, on the optimal success probability are tighter than most of the general analytical bounds known in the literature. The new upper bound on the optimal success probability explicitly generalizes to $r…
