Optimal Measurement Strategies in Quantum State Discrimination Problems

Abstract

Quantum state discrimination, which discriminates a finite number of quantum states, is a fundamental task with many applications in quantum information processing. The perfect indistinguishability of non-orthogonal quantum states informs us of the importance of optimal measurement strategies in extracting the maximum information of a quantum system. The strategies for quantum state discrimination are classified according to the constraints of measurement.

In chapter 2, we analyze minimum error discrimination of qudit or qubit states and convert it into an optimization problem in Euclidean space. We obtain the solution by considering a quantum state ensemble with special conditions. We also provide the condition for the existence of optimal measurement, which may not need to detect all quantum states. It can provide a way to reduce the number of measurements while maintaining the minimum error. Further, we completely analyze the minimum error discrimination of arbitrary three qubit states with arbitrary prior probabilities.

In chapter 3, we analyze the optimal unambiguous discrimination of three pure quantum states and transform it into an optimization problem of two-dimensional Euclidean space. When one pair of three pure states is orthogonal, we obtain the optimal solution in analytical form. When three pure states are non-orthogonal to each other, we express in analytical form the necessary and sufficient condition for optimal measurement to detect all quantum states and provide a graphical way to find the optimal solution.

In chapter 4, we consider a strategy to minimize the average error probability of conclusive results with a fixed rate of inconclusive results. This optimal discrimination problem can be modified to a minimum error discrimination by adding one appropriate quantum state and a prior probability called an inconclusive degree. By introducing two special inconclusive degrees that determine the beginning and end of proper inconclusive degrees, we analyze the problem of discriminating two arbitrary qubit states with arbitrary prior probabilities. We completely analyze the modified problems corresponding to special and proper inconclusive degrees and in some cases solve the original problem in analytical form. We also provide a numerical method to solve the original problem where it is difficult to analyze in analytical form.