Extracting classical information from quantum states : fundamental limits, adaptive and finite-length measurements

Abstract

Motivated by the increasing demand for more powerful and efficient optical communication systems, quantum mechanics of information processing has become the key element in determining the fundamental limits of physical channels, and in designing quantum communication systems that approach those fundamental limits. To achieve higher data rates over quantum optical channels, we need to efficiently extract classical information from quantum states. However, peculiar properties of quantum states, such as the no-cloning theorem and the non-reversible measurement process, provide new challenges in the measurement of quantum states; in quantum information science, there is no concept analogous to sufficient statistics in classical information science. Therefore, to extract as much information as possible from quantum states, it is important to choose the right measurement process. In this thesis, we investigate the fundamental question of how to design the measurement process to efficiently extract information from quantum states. First, we consider adaptive measurement, with which we measure each received quantum state one at a time, and then update the next measurement process based on the previous observations. We show that for binary hypothesis testing between two ideal laser light pulses, if we update the adaptive measurement to maximize the communication efficiency at each instant, based on recursively updated knowledge of the receiver, then we can achieve the theoretical lower bound of the detection error probability. Using this viewpoint, we give a natural generalization of the adaptive measurement to general Mary hypothesis testing problems. We also analyze the information capacity with adaptive measurement, and compare the result with that for direct detection receivers and the ultimate capacity of quantum channels (the Holevo limit). We also investigate finite-blocklength joint receivers. The ultimate capacity of quantum channels is calculated under the assumption that an infinite number of quantum states can be collectively measured in one shot. However, this assumption becomes the primary barrier that prevents practical implementations of capacity-achieving joint detection receivers. The maximum number of classical information bits extracted per use of the quantum channel strictly increases with the number of channel outputs jointly measured at the receiver. This phenomenon is called strict superadditivity, and it has been thought of as a unique property that can be observed only in quantum channels, but not in classical discrete memoryless channels (DMCs). In this thesis, we introduce a new aspect of understanding strict superadditivity by comparing the performance of concatenated coding over quantum channels and classical DMCs, for a fixed inner code length. We show that the strict superadditivity in information rate occurs due to a loss of information from hard-decisions at the finite blocklength. We also find a lower bound on the maximum achievable information rate as a function of the length of the quantum measurement. The analysis and new insights into the measurement process of quantum states that we develop in this thesis can be used to improve not only current quantum optical communication systems, but also classical information processing, where the data is too big to be handled with sufficient statistics. Our work would help develop new concepts of efficient statistics that provide systematic ways to choose useful information among big data while discarding the rest.