多维连续时间量子随机游动的Itô公式

数学物理学报 ›› 2016, Vol. 36 ›› Issue (4): 771-782.

多维连续时间量子随机游动的Itô公式

康元宝

重庆师范大学数学科学学院重庆 401331

收稿日期:2015-11-03 修回日期:2016-04-12 出版日期:2016-08-26 发布日期:2016-08-26
作者简介:康元宝,kangyuanb@163.com
基金资助:
国家自然科学基金（11461061）和重庆师范大学博士启动基金项目（15XLB013）资助

Itô Formula for Multidimensional Continuous-Time Quantum Random Walk

Kang Yuanbao

School of Mathematics Science, Chongqing Normal University, Chongqing 401331

Received:2015-11-03 Revised:2016-04-12 Online:2016-08-26 Published:2016-08-26
Supported by:
Support by the NSFC(11461061)and the Dr Start-up Founds Project of Chongqing Normal University(15XLB013)

摘要/Abstract

摘要：

基于文献[5-6]和[18]的思想，该文提出了关于高维连续时间量子随机游动（简记为CQRW）的Itô公式. 作为应用，随后建立了一个关于高维CQRW的Tanaka公式.

关键词: CQRW, Itô, 公式, Tanaka公式

Abstract:

In the paper, following [5-6] we present an Itô formula for the multidimensional continuous-time quantum random walk (CQRW, for short) based on [18]. As an application we then establish a Tanaka's formula for the high-dimensional CQRW.

Key words: CQRW, Itô, formula, Tanaka's formula

中图分类号:

O211.6

康元宝. 多维连续时间量子随机游动的Itô公式[J]. 数学物理学报, 2016, 36(4): 771-782.

Kang Yuanbao. Itô Formula for Multidimensional Continuous-Time Quantum Random Walk[J]. Acta mathematica scientia,Series A, 2016, 36(4): 771-782.

参考文献

[1] Kunita H. Itô's stochastic calculus: its surprising power for applications. Stochastic Process Appl, 2010, 120: 622-652
[2] Pelo P, Aurel A, Stan I. An Itô formula for a family of stochastic integrals and related Wong-Zakai theorems. Stochastic Processes and their Applications, 2013, 123: 3183-3200
[3] Fujita T. Stochastic Calculus for Finance (in Japanese). Tokyo: Kodansha, 2002
[4] Fujita T. Random Walks and Stochastic Calculus (in Japanese). Tokyo: Nippon-Hyoron-Sha, 2008
[5] Konno N. A note on Itô's formula for discrete-time quantum walk. 2011, arXiv:1112.4335[quant-ph]
[6] Ampadu C. Itô's formula for the discrete-time quantum walk in two dimensions. Journal of Quantum Information Science, 2012, 2: 40-46
[7] Ko C. Interacting Fock spaces and the moments of the limit distribution for quantum random walk. Inf Dim Anal Quantum Probab Rel Topics, 2013, 16(1): 1350003
[8] Mackay T D, Bartlett S D, Stephenson L T, Sanders B C. Quantum walks in higher dimensions. J Phys A: Math Gen, 2002, 35: 2745-2753
[9] Obata N. A note on Konno's paper on quantum walk. Inf Dim Anal Quantum Probab Rel Topics, 2006, 9: 299-304
[10] Konishi N, Konno N. Localization of two-dimensional quantum walks. Phys Rev A, 2004, 69: 052323
[11] Ko C K, Yoo H J. The generator and quantum Markov semigroup for quantum random walks. Kodai Math J, 2013, 36: 363-385
[12] Grimmett G, Janson S, Scudo P F. Weak limits for quantum random walks. Phys Rev E, 2004, 69: 026119
[13] Muelken A, Blumen A. Quantum transport on two-dimensional regular graphs. J Phys A: Math Gen, 2006, 39: 14997-15012
[14] Yang W, Liu C, Zhang K. A path integral formula with applications to quantum random walks in Z^d. J Phys A: Math Theor, 2007, 40: 8487-8516
[15] Gudder S. Quantum measures and integrals. 2011, arXiv:1105.3781
[16] Hudson R L, Parthasarathy K R. Quantum Itô's formula and stochastic calculus. Comm Math Phys, 1984, 93: 301-323
[17] Biane P. Itô's stochastic calculus and Heisenberg commutation relations. Stochastic Process Appl, 2010, 120: 698-720
[18] Kang Y B, Wang C S. Itô formula for one-dimensional continuous-time quantum random walk. Physica A: Statistical Mechanics and its Applications, 2014, 414: 154-162
[19] Kang Y B, Wang C S. Quantum random walk polynomial and quantum random walk measure. Quantum Information Processing, 2014, 5: 1191-1209