On the violation of Bell’s inequality for all non-product quantum states

We present an explicit reexamination of Gisin’s 1991 original proof concerning the violation of Bell’s inequality for any pure entangled state of two-particle systems. Given the relevance of Gisin’s work, our analysis is motivated by pedagogical reasons and allows the straightening of a few mathematical points in the original proof that in no way change the physical conclusions reached by Gisin.

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Recent results in trapped-ion quantum computing at NIST

Quantum Information and Computation ◽

10.26421/qic1.s-12 ◽

2001 ◽

Vol 1 (Special) ◽

pp. 113-123

Author(s):

D. Kielpinski ◽

A. Ben-Kish ◽

J. Britton ◽

V. Meyer ◽

M.A. Rowe ◽

...

Keyword(s):

The State ◽

Entangled State ◽

Bell’S Inequality ◽

Ambient Conditions ◽

Bell's Inequality ◽

Quantum Register ◽

Sigma Level ◽

Universal Quantum ◽

Encoding Method ◽

First Time

We review recent experiments on entanglement, Bell's inequality, and decoherence-free subspaces in a quantum register of trapped {9Be+} ions. We have demonstrated entanglement of up to four ions using the technique of Molmer and Sorensen. This method produces the state ({|\uparrow\uparrow\rangle}+{|\downarrow\downarrow\rangle})/\sqrt{2} for two ions and the state ({\downarrow}{\downarrow}{\downarrow}{\downarrow} \rangle + | {\uparrow}{\uparrow}{\uparrow}{\uparrow} \rangle)/\sqrt{2} for four ions. We generate the entanglement deterministically in each shot of the experiment. Measurements on the two-ion entangled state violates Bell's inequality at the 8\sigma level. Because of the high detector efficiency of our apparatus, this experiment closes the detector loophole for Bell's inequality measurements for the first time. This measurement is also the first violation of Bell's inequality by massive particles that does not implicitly assume results from quantum mechanics. Finally, we have demonstrated reversible encoding of an arbitrary qubit, originally contained in one ion, into a decoherence-free subspace (DFS) of two ions. The DFS-encoded qubit resists applied collective dephasing noise and retains coherence under ambient conditions 3.6 times longer than does an unencoded qubit. The encoding method, which uses single-ion gates and the two-ion entangling gate, demonstrates all the elements required for two-qubit universal quantum logic.

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BELL'S INEQUALITY AND CONTEXTUAL INFORMATION

International Journal of Quantum Information ◽

10.1142/s0219749906002110 ◽

2006 ◽

Vol 04 (04) ◽

pp. 641-651

Author(s):

F. SAFINEJAD ◽

A. SHAFIEE ◽

R. MALEEH

Keyword(s):

Singlet State ◽

Contextual Information ◽

Classical Model ◽

Entangled State ◽

Bell’S Inequality ◽

Common Cause ◽

Bell's Inequality ◽

The Common ◽

Necessary And Sufficient ◽

General Opinion

The properties of the Shannon entropy (concavity and strong additivity) are satisfied in both classical and quantum experiments, if the context of experiment is considered appropriately. We show that these properties hold true in a classical model with the feature of being contextual, where a correlation is observed between events. Our results show that the classical example is similar to an entangled singlet state for spin-½ particles. But, contrary to general opinion, [Formula: see text] is not a necessary and sufficient condition for violating Bell's inequality, since it has been obtained on the basis of a common cause pattern. In other words, it is possible to reconstruct the amount of information contained in a quantum entangled state according to the common cause criterion satisfying local realism.

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Maximally entangled state and Bell’s inequality in qubits

Annals of Physics ◽

10.1016/j.aop.2018.05.016 ◽

2018 ◽

Vol 395 ◽

pp. 183-195

Author(s):

Su-Kuan Chu ◽

Chen-Te Ma ◽

Rong-Xin Miao ◽

Chih-Hung Wu

Keyword(s):

Entangled State ◽

Bell’S Inequality ◽

Maximally Entangled State ◽

Bell's Inequality

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Tripartite entanglement and quantum correlation

Journal of High Energy Physics ◽

10.1007/jhep05(2021)185 ◽

2021 ◽

Vol 2021 (5) ◽

Author(s):

Xingyu Guo ◽

Chen-Te Ma

Keyword(s):

Quantum Entanglement ◽

Upper Bound ◽

Pure State ◽

Quantum Correlation ◽

Quantum States ◽

Bell’S Inequality ◽

Tripartite Entanglement ◽

Bell's Inequality ◽

Unitary Transformations ◽

Proper Diagnosis

Abstract We provide an analytical tripartite-study from the generalized R-matrix. It provides the upper bound of the maximum violation of Mermin’s inequality. For a generic 2-qubit pure state, the concurrence or R-matrix characterizes the maximum violation of Bell’s inequality. Therefore, people expect that the maximum violation should be proper to quantify Quantum Entanglement. The R-matrix gives the maximum violation of Bell’s inequality. For a general 3-qubit state, we have five invariant entanglement quantities up to local unitary transformations. We show that the five invariant quantities describe the correlation in the generalized R-matrix. The violation of Mermin’s inequality is not a proper diagnosis due to the non-monotonic behavior. We then classify 3-qubit quantum states. Each classification quantifies Quantum Entanglement by the total concurrence. In the end, we relate the experiment correlators to Quantum Entanglement.

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SOME DERIVATIONS OF BELL’S INEQUALITY

Danish Yearbook of Philosophy ◽

10.1163/24689300_0290104 ◽

1994 ◽

Vol 29 (1) ◽

pp. 63-105

Author(s):

Kaj Børge Hansen

Keyword(s):

Quantum Physics ◽

Mathematical Problem ◽

Hidden Variables ◽

Quantum States ◽

Valuable Insight ◽

Bell’S Inequality ◽

Bell's Inequality ◽

Fixed Direction ◽

Insight Into ◽

Classical Picture

Eight derivations of Bell’s inequality are given. First a simple and concrete derivation is given drawing on the full strength of the two hypotheses of locality and hidden variables. Two attempts at deriving Bell’s inequality without locality are made. They fail, but give valuable insight into the form which nonlocality must take in quantum physics. Arguments are given against this form of nonlocality. Ontological ideas which allow separation, realism, and locality in quantum mechanics (QM) are indicated. In the following five derivations, a number of variants of the assumption of hidden variables are tried. Among the insights which these derivations give rise to are: (1) Particles cannot be assumed to have spin or polarisation values for as much as one fixed direction. (2) The classical picture of electromagnetic radiation is incompatible with QM. (3) Heisenberg’s idea of potentiality in elementary particles is incompatible with QM. (4) There is a weakest form of hidden variables, called realisability, which is sufficient to yield Bell’s inequality. (5) Quantum states cannot be identified with physical states. The mathematical problem of the integration of quantum states into physical states is nontrivial.

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Review on investigating the possibility of local hidden variable theories-quantum teleportation

International Journal of Advanced Mathematical Sciences ◽

10.14419/ijams.v6i1.8755 ◽

2018 ◽

Vol 6 (1) ◽

pp. 1

Author(s):

Dorcas Attuabea Addo ◽

Steven Abel ◽

Richard Kwame Ansah ◽

Isaac Nkrumah

Keyword(s):

Single Particle ◽

Quantum Teleportation ◽

Entangled State ◽

Bell’S Inequality ◽

Variable Theory ◽

Bell's Inequality ◽

Hidden Variable ◽

Hidden Variable Theories ◽

Nonlocal Nature ◽

Local Hidden Variable

The core of the paper was to investigate the possibility of local hidden variable theory and its application in quantum teleportation. We reviewed literature on the Bell's inequality which is necessary for quantum teleportation. Quantum teleportation utilises a single-particle entangled state which can be successfully achieved by the application of the locality assumption which leads to Bell's inequality. A violation of the Bell's inequality signifies the nonlocal nature of a single particle useful for quantum teleportation.

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