Projective Equivalence for the Roots of Unity

Let $\mu_{\infty}\subseteq\mathbb{C}$ be the collection of roots of unity and $\mathcal{C}_{n}:=\{(s_{1},\cdots,s_{n})\in\mu_{\infty}^{n}:s_{i}\neq s_{j}\text{ for any }1\leq i<j\leq n\}$. Two elements $(s_{1},\cdots,s_{n})$ and $(t_{1},\cdots,t_{n})$ of $\mathcal{C}_{n}$ are said to be projectively equivalent if there exists $\gamma\in\text{PGL}(2,\mathbb{C})$ such that $\gamma(s_{i})=t_{i}$ for any $1\leq i\leq n$. In this article, we will give a complete classification for the projectively equivalent pairs. As a consequence, we will show that the maximal length for the nontrivial projectively equivalent pairs is $14$

Torsion of elliptic curves and unlikely intersections

We study effective versions of unlikely intersections of images of torsion points of elliptic curves on the projective line.Comment: 19 page

Security proof of differential phase shift quantum key distribution in the noiseless case

Differential phase shift quantum key distribution systems have a high potential for achieving high speed key generation. However, its unconditional security proof is still missing, even though it has been proposed for many years. Here, we prove its security against collective attacks with a weak coherent light source in the noiseless case (i.e. no bit error). The only assumptions are that quantum theory is correct, the devices are perfect and trusted and the key size is infinite. Our proof works on threshold detectors. We compute the lower bound of the secret key generation rate using the information-theoretical security proof method. Our final result shows that the lower bound of the secret key generation rate per pulse is linearly proportional to the channel transmission probability if Bob's detection counts obey the binomial distribution.Comment: Published version, 13 pages, 4 figures, minor changes, references added, acknowledgement adde

Collective quantum phase slips in multiple nanowire junctions

Realization of robust coherent quantum phase slips represents a significant experimental challenge. Here we propose a new design consisting of multiple nanowire junctions to realize a phase-slip flux qubit. It admits good tunability provided by gate voltages applied on superconducting islands separating nanowire junctions. In addition, the gates and junctions can be identical or distinct to each other leading to symmetric and asymmetric setups. We find that the asymmetry can improve the performance of the proposed device, compared with the symmetric case. In particular, it can enhance the effective rate of collective quantum phase slips. Furthermore, we demonstrate how to couple two such devices via a mutual inductance. This is potentially useful for quantum gate operations. Our investigation on how symmetry in multiple nanowire junctions affects the device performance should be useful for the application of phase-slip flux qubits in quantum information processing and quantum metrology.Comment: 12 pages, 6 figure