scholarly journals A nonlocal game for witnessing quantum networks

2019 ◽  
Vol 5 (1) ◽  
Author(s):  
Ming-Xing Luo
Keyword(s):  
Computational Complexity ◽  
Main Idea ◽  
Bell Inequalities ◽  
Theoretical Computer Science ◽  
Graph Representation ◽  
Necessary Condition ◽  
Quantum Networks ◽  
Theoretical Computer ◽  
Sufficient And Necessary Condition ◽  
Unified Method

Abstract Nonlocal game as a witness of the nonlocality of entanglement is of fundamental importance in various fields. The well-known nonlocal games or equivalent linear Bell inequalities are only useful for Bell networks consisting of single entanglement. Our goal in this paper is to propose a unified method for constructing cooperating games in network scenarios. We propose an efficient method to construct multipartite nonlocal games from any graphs. The main idea is the graph representation of entanglement-based quantum networks. We further specify these graphic games with quantum advantages by providing a simple sufficient and necessary condition. The graphic games imply a linear Bell testing of the nonlocality of general quantum networks consisting of EPR states. It also allows generating new instances going beyond CHSH game. These results have interesting applications in quantum networks, Bell theory, computational complexity, and theoretical computer science.

scholarly journals THE CHSH-TYPE INEQUALITIES FOR INFINITE-DIMENSIONAL QUANTUM SYSTEMS

2013 ◽  
Vol 27 (21) ◽  
pp. 1350151
Author(s):  
YU GUO
Keyword(s):  
Pure State ◽  
Bell Inequalities ◽  
Quantum Systems ◽  
Necessary Condition ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Pure States ◽  
Sufficient And Necessary Condition

By establishing CHSH operators and CHSH-type inequalities, we show that any entangled pure state in infinite-dimensional systems is entangled in a 2⊗2 subspace. We find that, for infinite-dimensional systems, the corresponding properties are similar to that of the two-qubit case: (i) The CHSH-type inequalities provide a sufficient and necessary condition for separability of pure states; (ii) The CHSH operators satisfy the Cirel'son inequalities; (iii) Any state which violates one of these Bell inequalities is distillable.

Theoretical Computer Science: Computational Complexity

2020 ◽  
pp. 51-89
Author(s):  
Olivier Bournez ◽  
Gilles Dowek ◽  
Rémi Gilleron ◽  
Serge Grigorieff ◽  
Jean-Yves Marion ◽  
...  
Keyword(s):  
Computational Complexity ◽  
Computer Science ◽  
Theoretical Computer Science ◽  
Theoretical Computer

Undecidable complexity statements in -arithmetic

1989 ◽  
Vol 54 (2) ◽  
pp. 415-427
Author(s):  
Ron Sigal
Keyword(s):  
Computational Complexity ◽  
Free Variable ◽  
Theoretical Computer Science ◽  
Programming System ◽  
Theoretical Computer ◽  
Independence Result ◽  
Constructive Version ◽  
Proof Techniques ◽  
Definition Of ◽  
Ordinal Notation

The failure of a large and diverse body of work to settle some of the now-classical questions of computational complexity (notably P =? NP) suggests that they might not, in fact, be resolvable by established proof techniques.Hartmanis and Hopcroft [HH] raised the issue of independent statements about computational complexity in 1976, constructing, for any consistent r.e. theory T capable of expressing statements about Turing machines, a Turing machine MT such that statements which intuitively express the computational complexity of MT are independent of T. Their technique involves a simple diagonalizing search over the theorems of T. In this paper we prove a constructive version of their independence result in the context of a generalization of a hierarchy of free variable logics defined by Rose [R1]. These logics are based on an axiomatized treatment of an extension by Löb and Wainer [LW] of Grzegorczyk's [G] hierarchy into the transfinite. Associated with each extended Grzegorczyk class (relative to an ordinal notation system S satisfying certain conditions) is the logic -arithmetic.Free variable logics are interesting from the perspective of theoretical computer science. We may construe the equational definition of functions as a form of programming system. Each -arithmetic, then, has the nice property of containing both a programming system and a logic for stating and proving facts about programs.

scholarly journals Semantic Domains and Denotational Semantics

1989 ◽  
Vol 18 (276) ◽  
Author(s):  
Carl A. Gunter ◽  
Peter D. Mosses ◽  
Dana S. Scott
Keyword(s):  
Computer Science ◽  
Programming Language ◽  
Main Idea ◽  
Denotational Semantics ◽  
Mathematical Object ◽  
Theoretical Computer Science ◽  
Formal Description ◽  
Theoretical Computer ◽  
Language Semantics ◽  
Semantic Domains

<p>Denotational Semantics is a framework for the formal description of programming language semantics. The main idea of Denotational Semantics is that each phrase of the described language is given a <em>denotation</em>: a mathematical object that represents the contribution of the phrase to the meaning of any program in which it occurs. Moreover, the denotation of each phrase is determined just by the denotations of its subphrases.</p><p>This report consists of two chapters. The first, <em>Semantic Domains</em>, was written by Gunter and Scott. It is concerned with the <em>theory</em> of domains of denotations. The second, <em>Denotational Semantics</em>, was written by Mosses. It explains the formal notation used in denotational descriptions, and illustrates the major standard <em>technigues</em> for finding denotations of programming constructs.</p><p>Both chapters are to appear in the forthcoming <em>Handbook of Theoretical Computer Science</em> (North-Holland).</p>

scholarly journals Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales

2020 ◽  
Vol 18 (1) ◽  
pp. 353-377 ◽  
Author(s):  
Zhien Li ◽  
Chao Wang
Keyword(s):  
Time Scales ◽  
Quaternion Algebra ◽  
Matrix Exponential ◽  
Dynamic Equations ◽  
Necessary Condition ◽  
Quaternion Matrix ◽  
Exponential Functions ◽  
Cauchy Matrix ◽  
Adjoint Matrix ◽  
Sufficient And Necessary Condition

Abstract In this study, we obtain the scalar and matrix exponential functions through a series of quaternion-valued functions on time scales. A sufficient and necessary condition is established to guarantee that the induced matrix is real-valued for the complex adjoint matrix of a quaternion matrix. Moreover, the Cauchy matrices and Liouville formulas for the quaternion homogeneous and nonhomogeneous impulsive dynamic equations are given and proved. Based on it, the existence, uniqueness, and expressions of their solutions are also obtained, including their scalar and matrix forms. Since the quaternion algebra is noncommutative, many concepts and properties of the non-quaternion impulsive dynamic equations are ineffective, we provide several examples and counterexamples on various time scales to illustrate the effectiveness of our results.

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