Based on the theory of quantum mechanics, intrinsic randomness in measurement distinguishes quantum effects from classical ones. From the perspective of states, this quantum feature can be summarized as coherence or superposition in a specific (classical) computational basis. Recently, by regarding coherence as a physical resource, Baumgratz et al. present a comprehensive framework for coherence measures. Here, we propose a quantum coherence measure essentially using the intrinsic randomness of measurement. The proposed coherence measure provides an answer to the open question in completing the resource theory of coherence. Meanwhile, we show that the coherence distillation process can be treated as quantum extraction, which can be regarded as an equivalent process of classical random number extraction. From this viewpoint, the proposed coherence measure also clarifies the operational aspect of quantum coherence. Finally, our results indicate a strong similarity between two types of quantumness -coherence and entanglement.

Quantum key distribution allows remote parties to generate information-theoretic secure keys. The bottleneck throttling its real-life applications lies in the limited communication distance and key generation speed, due to the fact that the information carrier can be easily lost in the channel. For all the current implementations, the key rate is bounded by the channel transmission probability η. Rather surprisingly, by matching the phases of two coherent states and encoding the key information into the common phase, this linear key-rate constraint can be overcome-the secure key rate scales with the square root of the transmission probability, O( √ η), as proposed in twin-field quantum key distribution [Nature (London) 557, 400 (2018)]. To achieve this, we develop an optical-mode-based security proof that is different from the conventional qubit-based security proofs. Furthermore, the proposed scheme is measurement device independent, i.e., it is immune to all possible detection attacks. The simulation result shows that the key rate can even exceed the transmission probability η between two communication parties. In addition, we apply phase postcompensation to devise a practical version of the scheme without phase locking, which makes the proposed scheme feasible with the current technology. This means that quantum key distribution can enjoy both sides of the world-practicality and security.

We study random number generation using a biased source motivated by previous works on this topic, mainly, von Neumman (1951), Elias (1972), Knuth and Yao (1976) and Peres (1992). We study the problem in two cases: first, when the source distribution is unknown, and second, when the source distribution is known. In the first case, we characterize the functions that use a discrete random source of unknown distribution to simulate a target discrete random variable with a given rational distribution. We identify the functions that minimize the ratio of source inputs to target outputs. We show that these optimal functions are efficiently computable. In the second case, we prove that it is impossible to construct an optimal tree algorithm recursively, using algebraic decision procedures. Our model of computation is sufficiently general to encompass previously known algorithms for this problem. RANDOM NUMBER GENERATION USING A BIASED SOURCE BY SUNG-IL PAE ABSTRACT We study random number generation using a biased source motivated by previous works on this topic, mainly, von Neumman (1951), Elias (1972), Knuth and Yao (1976) and Peres (1992). We study the problem in two cases: first, when the source distribution is unknown, and second, when the source distribution is known. In the first case, we characterize the functions that use a discrete random source of unknown distribution to simulate a target discrete random variable with a given rational distribution. We identify the functions that minimize the ratio of source inputs to target outputs. We show that these optimal functions are efficiently computable. In the second case, we prove that it is impossible to construct an optimal tree algorithm recursively, using algebraic decision procedures. Our model of computation is sufficiently general to encompass previously known algorithms for this problem. iii ACKNOWLEDGMENTS I am deeply grateful to my advisor Michael C. Loui. He provided me with advice on almost every aspect of my life as a graduate student, not to mention the countless meetings and comments on my thesis work. I began working with him at an especially difficult time of my graduate study. Without his encouragements and guidance, the complexity of my life would have been even greater.

Quantum random number generators (QRNGs) output genuine random numbers based upon the uncertainty principle. A QRNG contains two parts in general-a randomness source and a readout detector. How to remove detector imperfections has been one of the most important questions in practical randomness generation. We propose a simple solution, measurement-device-independent QRNG, which not only removes all detector side channels but is robust against losses. In contrast to previous fully device-independent QRNGs, our scheme does not require high detector efficiency or nonlocality tests. Simulations show that our protocol can be implemented efficiently with a practical coherent state laser and other standard optical components. The security analysis of our QRNG consists mainly of two parts: measurement tomography and randomness quantification, where several new techniques are developed to characterize the randomness associated with a positive-operator valued measure.

We found an error in the simulation code in our recent work. With the coding error fixed, the simulation result (Fig. 3 in the original paper) is shown below. The longest distance under the same parameters in Table I is 418 km. As is shown in Fig. 1, the key rate line of PM-QKD can still overcome the secret key capacity bound when l > 250 km, and the rate-loss dependence of PM-QKD, R ∼ Oð ffiffi ffi η p Þ, is not affected. The simulation error was brought to our attention when we ran the simulation in a follow-up work, and it was independently pointed out by Ivan Djordjevic. In the Supplemental Material [1], we present the MATLAB code of the PM-QKD key rate for reference. [1] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevX.9.029901 for the MATLAB code of the PM-QKD key-rate evaluation. FIG. 1. Corrected simulation result of Fig. 3 in the original article. The purple dashed line is the original incorrect plot of the PM-QKD key rate result, which is caused by an erroneous extra term ð2πÞ=M when calculating the QBER using Eq. (B22); see lines 27 and 28 in the MATLAB code file in the Supplemental Material [1].