Quantum pseudo-telepathy is an intriguing phenomenon which results from the application of quantum information theory to communication complexity. To demonstrate this phenomenon researchers in the field of quantum communication complexity devised a number of quantum non-locality games. The setting of these games is as follows: the players are separated so that no communication between them is possible and are given a certain computational task. When the players have access to a quantum resource called entanglement, they can accomplish the task: something that is impossible in a classical setting. To an observer who is unfamiliar with the laws of quantum mechanics it seems that the players employ some sort of telepathy; that is, they somehow exchange information without sharing a communication channel. This paper provides a formal framework for specifying, implementing, and analysing quantum non-locality games.
Deep Dive into Programming Telepathy: Implementing Quantum Non-Locality Games.
Quantum pseudo-telepathy is an intriguing phenomenon which results from the application of quantum information theory to communication complexity. To demonstrate this phenomenon researchers in the field of quantum communication complexity devised a number of quantum non-locality games. The setting of these games is as follows: the players are separated so that no communication between them is possible and are given a certain computational task. When the players have access to a quantum resource called entanglement, they can accomplish the task: something that is impossible in a classical setting. To an observer who is unfamiliar with the laws of quantum mechanics it seems that the players employ some sort of telepathy; that is, they somehow exchange information without sharing a communication channel. This paper provides a formal framework for specifying, implementing, and analysing quantum non-locality games.
The work develops a formal framework for specifying, implementing, and analysing quantum pseudo-telepathy: an intriguing phenomenon which manifests itself when quantum information theory is applied to communication complexity. To demonstrate this phenomenon researchers in the field of quantum communication complexity devised a number of quantum non-locality games. The setting of these games is as follows: the players are separated so that no communication between them is possible and are given a certain computational task. When the players have access to a quantum resource called entanglement, they can accomplish the task: something that is impossible in a classical setting. To an observer who is unfamiliar with the laws of quantum mechanics it seems that the players employ some sort of telepathy; that is, they somehow exchange information without sharing a communication channel.
Quantum pseudo-telepathy, and quantum non-locality in general, are perhaps the most non-classical and the least understood aspects of quantum information processing. Every effort is made to gain information about the power of these phenomena. Quantum non-locality games in particular have been extensively used to prove separations between quantum and classical communication complexity. The need for a good framework for formal analysis of quantum non-locality is evident.
We look at quantum non-locality in the context of formal methods of program development, or programming methodology. This is the field of computer science concerned with applications of mathematics and logic to software engineering tasks. In particular, the formal methods provide tools to formally express specifications, prove correctness of implementations, and reason about various properties of specifications (e.g. implementability) and implementations (e.g. time and space complexity).
In this work the analysis of quantum non-locality is based on quantum predicative programming ( [33,32]), a recent generalisation of the well-established predicative programming ( [23,24,25]). It supports the style of program development in which each programming step is proved correct as it is made. We inherit the advantages of the theory, such as its generality, simple treatment of recursive programs, and time and space complexity. The theory of quantum programming provides tools to write both classical and quantum specifications, develop quantum programs that implement these specifications, and reason about their comparative time and space complexity all in the same framework.
Presenting new non-locality paradigms or new pseudo-telepathy games is not the subject of this work. Our goal is developing a formal framework that encompasses all aspects of quantum computation and information. Formal analysis of quantum algorithms, including their time complexity, is presented in [33]. Analysis of quantum communication appears in [34]. This paper focuses on formal analysis of non-locality paradigms; we choose known pseudo-telepathy games as illustrative examples of our formalism.
The rest of this work is organised as follows. Section 2 is a brief introduction to quantum predicative programming. The contribution of this work is Section 3 which introduces a formal framework for specifying, implementing, and analysing quan-tum pseudo-telepathy and presents several examples of implementing and analysing non-locality games. Section 5 states conclusions and outlines directions for future research. A brief introduction to quantum computing is included in the Appendix.
This work attempts to bring together two areas of active research: the study of quantum non-locality and applications of formal methods to quantum information and computation. Currently, the two worlds rarely meet.
Quantum non-locality has been studied extensively first by physicists and lately by researchers in the fields of quantum information and quantum communication complexity. Since the work of Bell in 1964 ([8]), researchers have been trying to provide an intuitive explanation of the genuinely non-classical behaviour produced by quantum mechanics. Today, quantum pseudo-telepathy games are considered one of the best and easiest to understand examples of these non-classical phenomena (e.g. [21,14,11,12]).
Formal approaches to quantum programming include the language qGCL [30,38,39], process algebraic approaches developed in [4,27,26], tools developed in the field of category theory by [1,2,3,16,31], functional languages of [6,7,5,35,36], as well as work of [20,19], [17], and [22]. A detailed discussion of the work related to quantum predicative programming is presented in [33]. Some researchers address the subject of formalising quantum non-locality more directly than others (e.g. [38]). To the best of our knowledge, formal approaches to reasoning about quantum pseudo-telepathy games have not been considered.
This section introduces the programming theory of our choice -quantum predicative programming. We briefly intro
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