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Kuantum Bilgi Teorisinde Bell-Türü eşitsizlikler

Seskir, Zeki Can

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

Bell and Bell-like inequalities are essentil tools for identifying and inspecting quantum entanglement, a key phenomenon in quantum information theory. As the development of Quantum Information Theory progresses more and more Bell-like inequalities are formulated. An investigation of these and further inequalities will be given in their correspondences to Bell's and Kochen-Specker theorems. Also Hardy's test will be studied and particular applications of it for bipartitre and tripartite systems are going to be probed.

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Bell and Bell-like inequalities are essentil tools for identifying and inspecting quantum entanglement, a key phenomenon in quantum information theory. As the development of Quantum Information Theory progresses more and more Bell-like inequalities are formulated. An investigation of these and further inequalities will be given in their correspondences to Bell's and Kochen-Specker theorems. Also Hardy's test will be studied and particular applications of it for bipartitre and tripartite systems are going to be probed.
Bell inequalities play a central role in the study of quantum nonlocality and entanglement, with many applications in quantum information. Despite the huge literature on Bell inequalities, it is not easy to find a clear conceptual answer to what a Bell inequality is, or a clear guiding principle as to how they may be derived. In this paper, we introduce a notion of logical Bell inequality which can be used to systematically derive testable inequalities for a very wide variety of situations. There is a single clear conceptual principle, based on purely logical consistency conditions, which underlies our notion of logical Bell inequalities. We show that in a precise sense, all Bell inequalities can be taken to be of this form. Our approach is very general. It applies directly to any family of sets of commuting observables. Thus it covers not only the nnn-partite scenarios to which Bell inequalities are standardly applied, but also Kochen-Specker configurations, and many other examples. There is much current work on experimental tests for contextuality. Our approach directly yields, in a systematic fashion, testable inequalities for a very general notion of contextuality. There has been much work on obtaining proofs of Bell's theorem ``without inequalities'' or ``without probabilities.'' These proofs are seen as being in a sense more definitive and logically robust than the inequality-based proofs. On the hand, they lack the fault-tolerant aspect of inequalities. Our approach reconciles these aspects, and in fact shows how the logical robustness can be converted into systematic, general derivations of inequalities with provable violations. Moreover, the kind of strong non-locality or contextuality exhibited by the GHZ argument or by Kochen-Specker configurations can be shown to lead to maximal violations of the corresponding logical Bell inequalities. Thus the qualitative and the quantitative aspects are combined harmoniously.
Bell nonlocality is a special quantum nonlocality and has emerged as an important resource in quantum information processing tasks. Typically, there are two types of strategies for testing the Bell nonlocality: the Bell inequality method and the “all-versus-nothing proof", the later includes methods such as the GHZ argument and Hardy paradox. The objective of this work is to provide new methods for detecting Bell nonlocality by understanding Hardy-like paradoxes (HLPs) and Hardy-Bell inequalities (HBIs) as well as Hardy inequality. First, the weak HLP (WHLP) is established, and an HBI is obtained for tripartite correlation tensors (CTs) with two inputs and two outcomes. Based on an existing Hardy inequality for an n-qubit pure state, a Hardy inequality is proven for n-partite Bell local CTs. Second, the WHLP and HBI are proven for tripartite states in terms of conditional probabilities. Our WHLP is theoretically and practically easier to construct and more efficient in checking Bell nonlocality compared to the usual HLP.
Recently, an experiment was done in the realm of Quantum Information that won the recent 2022 Nobel prize in Physics. This experiment breaks Bell's inequality, supporting quantum theory over hidden variable theory—the research done is considered one of the pioneering experiments for quantum information science. This research paper documented the theory of the experiment done and experimentally tested for the violation of Bell's inequality, using materials similar to the experiments from the past, in an attempt to provide more support for quantum information. Through repetition of the experiment, it was simple to brainstorm more potential experimentation and design that will further develop entanglement. The mathematical formulation and terms are the same as defined in past works in detail in order to keep notation and inequalities constant throughout the analysis. Using the data points and error propagation, I have shown that Bell's inequality is violated to a great extent. This paper finally addresses potential experimentation regarding quantum entanglement and its applications to quantum sensors and computing technology.
Non-locality and contextuality are key features of quantum mechanics that distinguish it from classical physics. We aim to develop a deeper, more structural understanding of these phenomena, underpinned by robust and elegant mathematical theory with a view to providing clarity and new perspectives on conceptual and foundational issues. A general framework for logical non-locality is introduced and used to prove that 'Hardy's paradox' is complete for logical non-locality in all (2,2,l) and (2,k,2) Bell scenarios, a consequence of which is that Bell states are the only entangled two-qubit states that are not logically non-local, and that Hardy non-locality can be witnessed with certainty in a tripartite quantum system. A number of developments of the unified sheaf-theoretic approach to non-locality and contextuality are considered, including the first application of cohomology as a tool for studying the phenomena: we find cohomological witnesses corresponding to many of the classic no-go results, and completely characterise contextuality for large families of Kochen-Specker-like models. A connection with the problem of the existence of perfect matchings in k-uniform hypergraphs is explored, leading to new results on the complexity of deciding contextuality. A refinement of the sheaf-theoretic approach is found that captures partial approximations to locality/non-contextuality and can allow Bell models to be constructed from models of more general kinds which are equivalent in terms of non-locality/contextuality. Progress is made on bringing recent results on the nature of the wavefunction within the scope of the logical and sheaf-theoretic methods. Computational tools are developed for quantifying contextuality and finding generalised Bell inequalities for any measurement scenario which complement the research programme. This also leads to a proof that local ontological models with `negative probabilities' generate the no-signalling polytopes for all Bell scenarios.
But this conclusion [nonlocality] needs careful discussion in order to clarify what is going on . (Redhead 1987, p. 3) Within the foundations of physics in recent years, Bell's theorem has played the role of what Thomas Kuhn calls a ‘paradigm’: that is, an exemplary piece of work that others learn from, imitate and develop. Following a period of articulation and consolidation, the first generation of developments of the Bell theorem was initiated by Heywood and Redhead (1983). They produced a nonlocality result in the algebraic style of the Bell–Kochen–Specker theorem (Bell 1966; Kochen and Specker 1967), moving away from the probabilistic relations characteristic of the Bell theorems proper. More recently a second generation develops results by Peres (1990), Greenberger–Horne–Zeilinger (1990), and Hardy (1993). In addition to moving away from probabilities, this generation tries to dispense with the limiting inequalities of the Bell theorem to yield socalled ‘Bell theorems without inequalities’. With respect to probabilities, however, Hardy is a half-way house. It requires no inequalities but the result contradicts quantum mechanics under certain locality assumptions only if the statistical predictions of quantum mechanics hold in at least one case. I want to examine the Hardy theorem and its interpretation. Initially, I intend to ignore respects in which it dispenses with probabilities because I want to point out the interesting significance of the theorem in a probabilistic context. We will see that when probabilities are restored, so are inequalities. Then we will see what the theorem has to contribute on the topic of locality.