Some Generalised Fixed Point Theorems Applied to Quantum Operations

Yusuf, Umar Batsari; Kumam, Poom; Yoo-Kong, Sikarin

doi:10.3390/sym12050759

Open AccessArticle

Some Generalised Fixed Point Theorems Applied to Quantum Operations

by

Umar Batsari Yusuf

^1,2,†

,

Poom Kumam

^2,3,4,*,†

and

Sikarin Yoo-Kong

^5,†

¹

Department of Mathematics and Statistics, College of Science and Technology, Hassan Usman Katsina Polytechnic, Katsina P.M.B 2052, Nigeria

²

Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand

³

Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand

⁴

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

⁵

The Institute for Fundamental Study(IF), Naresuan University, Phitsanulok 65000, Thailand

^*

Author to whom correspondence should be addressed.

^†

These authors contributed equally to this work.

Symmetry 2020, 12(5), 759; https://doi.org/10.3390/sym12050759

Submission received: 21 March 2020 / Revised: 15 April 2020 / Accepted: 15 April 2020 / Published: 6 May 2020

(This article belongs to the Special Issue Symmetry in Ordinary and Partial Differential Equations and Applications)

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Abstract

In this paper, we consider an order-preserving mapping T on a complete partial b-metric space satisfying some contractive condition. We were able to show the existence and uniqueness of the fixed point of T. In the application aspect, the fidelity of quantum states was used to establish the existence of a fixed quantum state associated to an order-preserving quantum operation. The method we presented is an alternative in showing the existence of a fixed quantum state associated to quantum operations. Our method does not capitalise on the commutativity of the quantum effects with the fixed quantum state(s) (Luders’s compatibility criteria). The Luders’s compatibility criteria in higher finite dimensional spaces is rather difficult to check for any prospective fixed quantum state. Some part of our results cover the famous contractive fixed point results of Banach, Kannan and Chatterjea.

Keywords: partial b-metric space; order-preserving; fixed point; quantum operation; fidelity of quantum state; quantum state

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MDPI and ACS Style

Yusuf, U.B.; Kumam, P.; Yoo-Kong, S. Some Generalised Fixed Point Theorems Applied to Quantum Operations. Symmetry 2020, 12, 759. https://doi.org/10.3390/sym12050759

AMA Style

Yusuf UB, Kumam P, Yoo-Kong S. Some Generalised Fixed Point Theorems Applied to Quantum Operations. Symmetry. 2020; 12(5):759. https://doi.org/10.3390/sym12050759

Chicago/Turabian Style

Yusuf, Umar Batsari, Poom Kumam, and Sikarin Yoo-Kong. 2020. "Some Generalised Fixed Point Theorems Applied to Quantum Operations" Symmetry 12, no. 5: 759. https://doi.org/10.3390/sym12050759

APA Style

Yusuf, U. B., Kumam, P., & Yoo-Kong, S. (2020). Some Generalised Fixed Point Theorems Applied to Quantum Operations. Symmetry, 12(5), 759. https://doi.org/10.3390/sym12050759

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