A Tight Bound of Tail Probabilities for a Discrete-time Martingale with Uniformly Bounded Jumps

We investigate the properties of a discrete-time martingale $\{X_m\}_{m\in \mathbb Z_{\geq 0}}$, where all differences between adjacent random variables are limited to be not more than a constant as a promise. In this situation, it is known that the Azuma-Hoeffding inequality holds, which gives an upper bound of a probability for exceptional events. The inequality gives a simple form of the upper bound, and it has been utilized for many investigations. However, the inequality is not tight. We give an explicit expression of a tight upper bound, and we show that it and the bound obtained from the Azuma-Hoeffding inequality have different asymptotic behaviors.Comment: 10pages,2 figure

Aggregating quantum repeaters for the quantum internet

The quantum internet holds promise for performing quantum communication, such as quantum teleportation and quantum key distribution, freely between any parties all over the globe. For such a quantum internet protocol, a general fundamental upper bound on the performance has been derived [K. Azuma, A. Mizutani, and H.-K. Lo, arXiv:1601.02933]. Here we consider its converse problem. In particular, we present a protocol constructible from any given quantum network, which is based on running quantum repeater schemes in parallel over the network. The performance of this protocol and the upper bound restrict the quantum capacity and the private capacity over the network from both sides. The optimality of the protocol is related to fundamental problems such as additivity questions for quantum channels and questions on the existence of a gap between quantum and private capacities.Comment: 5 pages, 2 figure

Next Nearest-Neighbor Correlation Functions of the Spin-1/2 XXZ Chain at Massive Region

The second neighbor correlation functions of the spin-${{1/2}}$ $XXZ$ chain in the ground state are expressed in the form of three dimensional integrals. We show that these integrals can be reduced to one-dimensional ones and thereby evaluate the values of the next nearest-neighbor correlation functions for ${\Delta >1}$.Comment: 12 pages, 1 figur

Unconditional security of coherent-state-based differential phase shift quantum key distribution protocol with block-wise phase randomization

We prove the unconditional security of coherent-state-based differential phase shift quantum key distribution protocol (DPSQKD) with block-wise phase randomization. Our proof is based on the conversion of DPSQKD to an equivalent entanglement-distillation protocol where the estimated phase error rate determines the amount of the privacy amplification. The generated final key has a contribution from events where the sender emits two or more photons, indicating the robustness of DPSQKD against photon-number-splitting attacks.Comment: 15 pages and 12 figure

Algebra and Hilbert space structures induced by quantum probes

In the general setting of quantum controls, it is unrealistic to control all of the degrees of freedom of a quantum system. We consider a scenario where our direct access is restricted to a small subsystem $S$ that is constantly interacting with the rest of the system $E$. What we investigate here is the fundamental structure of the Hilbert space that is caused solely by the restrictedness of the direct control. We clarify the intrinsic space structure of the entire system and that of the operations which could be activated through $S$. The structures hereby revealed would help us make quantum control problems more transparent and provide a guide for understanding what we can implement. They can be deduced by considering an algebraic structure, which is the Jordan algebra formed from Hermitian operators, naturally induced by the setting of limited access. From a few very simple assumptions about direct operations, we elucidate rich structures of the operator algebras and Hilbert spaces that manifest themselves in quantum control scenarios.Comment: Main text is the first 12 pages, and the following 24 pages contain supplementary lemmas and their proofs, including detailed explanations on the Jordan algebra (with hermitian operators

Secure Quantum Network Coding on Butterfly Network

Quantum network coding on the butterfly network has been studied as a typical example of quantum multiple cast network. We propose secure quantum network coding on the butterfly network in the multiple unicast setting based on a secure classical network coding. This protocol certainly transmits quantum states when there is no attack. We also show the secrecy even when the eavesdropper wiretaps one of the channels in the butterfly network.Comment: 11 pages, 2 figure

Concentration inequality using unconfirmed knowledge

We give a concentration inequality based on the premise that random variables take values within a particular region. The concentration inequality guarantees that, for any sequence of correlated random variables, the difference between the sum of conditional expectations and that of the observed values takes a small value with high probability when the expected values are evaluated under the condition that the past values are known. Our inequality outperforms other well-known inequalities, e.g. the Azuma-Hoeffding inequality, especially in terms of the convergence speed when the random variables are highly biased. This high performance of our inequality is provided by the key idea in which we predict some parameters and adopt the predicted values in the inequality.Comment: 16page

Entanglement-assisted classical communication can simulate classical communication without causal order

Phenomena induced by the existence of entanglement, such as nonlocal correlations, exhibit characteristic properties of quantum mechanics distinguishing from classical theories. When entanglement is accompanied by classical communication, it enhances the power of quantum operations jointly performed by two spatially separated parties. Such a power has been analyzed by the gap between the performances of joint quantum operations implementable by local operations at each party connected by classical communication with and without the assistance of entanglement. In this work, we present a new formulation for joint quantum operations connected by classical communication beyond special relativistic causal order but without entanglement and still within quantum mechanics. Using the formulation, we show that entanglement assisting classical communication necessary for implementing a class of joint quantum operations called separable maps can be interpreted to simulate "classical communication" not respecting causal order. Our results reveal a new counter-intuitive aspect of entanglement related to spacetime

Next Nearest-Neighbor Correlation Functions of the Spin-1/2 XXZ Chain at Critical Region

The correlation functions of the spin-1/2 XXZ spin chain in the ground state are expressed in the form of the multiple integrals. For -1< Delta <1, they were obtained by Jimbo and Miwa in 1996. Especially the next nearest-neighbour correlation functions are given as certain three-dimensional integrals. We shall show these integrals can be reduced to one-dimensional ones and thereby evaluate the values of the next nearest-neighbor correlation functions. We have also found that the remaining one-dimensinal integrals can be evaluated analytically, when nu = arccos(Delta)/pi is a rational number.Comment: 10 pages, 2 figure

Secrecy and Robustness for Active Attack in Secure Network Coding and its Application to Network Quantum Key Distribution

In network coding, we discuss the effect of sequential error injection on information leakage. We show that there is no improvement when the operations in the network are linear operations. However, when the operations in the network contains non-linear operations, we find a counterexample to improve Eve's obtained information. Furthermore, we discuss the asymptotic rate in a linear network under the secrecy and robustness conditions as well as under the secrecy condition alone. Finally, we apply our results to network quantum key distribution, which clarifies the type of network that enables us to realize secure long distance communication via short distance quantum key distribution.Comment: We fixed several error