Mixture of Gaussians in the open quantum random walks
- PDF / 1,034,224 Bytes
- 31 Pages / 439.37 x 666.142 pts Page_size
- 72 Downloads / 244 Views
Mixture of Gaussians in the open quantum random walks Chul Ki Ko1 · Hyun Jae Yoo2 Received: 26 January 2020 / Accepted: 4 July 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We discuss the Gaussian and the mixture of Gaussians in the limit of open quantum random walks. The central limit theorems for the open quantum random walks under certain conditions were proven by Attal et al (Ann Henri Poincaré 16(1):15–43, 2015) on the integer lattices and by Ko et al (Quantum Inf Process 17(7):167, 2018) on the crystal lattices. The purpose of this paper is to investigate the general situation. We see that the Gaussian and the mixture of Gaussians in the limit depend on the structure of the invariant states of the intrinsic quantum Markov semigroup whose generator is given by the Kraus operators which generate the open quantum random walks. Some concrete models are considered for the open quantum random walks on the crystal lattices. Due to the intrinsic structure of the crystal lattices, we can conveniently construct the dynamics as we like. Here, we consider the crystal lattices of Z2 with intrinsic two points, hexagonal, triangular, and Kagome lattices. We also discuss Fourier analysis on the crystal lattices which gives another method to get the limit theorems. Keywords Open quantum random walks · Quantum dynamical semigroup · Invariant states · Crystal lattices · Central limit theorem · Mixture of Gaussians Mathematics Subject Classification 82B41 · 82C41
B
Hyun Jae Yoo [email protected] Chul Ki Ko [email protected]
1
University College, Yonsei University, 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, Korea
2
School of Computer Engineering and Applied Mathematics, and Institute for Integrated Mathematical Sciences, Hankyong National University, 327 Jungang-ro, Anseong-si, Gyeonggi-do 17579, Korea 0123456789().: V,-vol
123
244
Page 2 of 31
C. K. Ko, H. J. Yoo
1 Introduction In this paper, we investigate the limit distributions of the open quantum random walks (OQRWs hereafter) [1–3,8]. The limit distributions of OQRWs have been investigated in several papers. See, for instance, references [1,2,7,9]. The central limit theorem (CLT in short) was firstly shown in [1] for OQRWs on the integer lattices, and then, it was shown for models on the crystal lattices in [8]. In both cases, in order that the CLT holds, a certain condition must be provided. Recall that the OQRWs consist of repeated two dynamics: an intrinsic change of states followed by space movements. Considering solely the intrinsic dynamics, it defines a quantum Markov semigroup (QMS shortly). The condition for the CLT is that the QMS should be irreducible and thereby there exists only one invariant state for the QMS. We refer to Sect. 3 for the details. One naturally then asks what would be the limit distribution of the OQRWs if the irreducibility fails to hold. It is the main motivation of this paper to answer this question. After demonstrating some known results on the structure of invariant states for QMSs [4
Data Loading...
