Bell’s inequality, generalized concurrence and entanglement in qubits

It is well known that the maximal violation of the Bell’s inequality for a two-qubit system is related to the entanglement formation in terms of a concurrence. However, a generalization of this relation to an [Formula: see text]-qubit state has not been found. In this paper, we demonstrate some extensions of the relation between the upper bound of the Bell’s violation and a generalized concurrence in several [Formula: see text]-qubit states. In particular, we show the upper bound of the Bell’s violation can be expressed as a function of the generalized concurrence, if a state can be expressed in terms of two variables. We apply the relation to the Wen-Plaquette model and show that the topological entanglement entropy can be extracted from the maximal Bell’s violation.

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Evaluating the Maximal Violation of the Original Bell Inequality by Two-Qudit States Exhibiting Perfect Correlations/Anticorrelations

Entropy ◽

10.3390/e20110829 ◽

2018 ◽

Vol 20 (11) ◽

pp. 829 ◽

Cited By ~ 5

Author(s):

Andrei Khrennikov ◽

Elena Loubenets

Keyword(s):

Upper Bound ◽

General Class ◽

Bell Inequality ◽

Spin Observable ◽

Mathematical Study ◽

Perfect Correlation ◽

Maximal Violation ◽

Spin Measurements ◽

Qubit States

We introduce the general class of symmetric two-qubit states guaranteeing the perfect correlation or anticorrelation of Alice and Bob outcomes whenever some spin observable is measured at both sites. We prove that, for all states from this class, the maximal violation of the original Bell inequality is upper bounded by 3 2 and specify the two-qubit states where this quantum upper bound is attained. The case of two-qutrit states is more complicated. Here, for all two-qutrit states, we obtain the same upper bound 3 2 for violation of the original Bell inequality under Alice and Bob spin measurements, but we have not yet been able to show that this quantum upper bound is the least one. We discuss experimental consequences of our mathematical study.

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On Bell’s Inequality in PT-Symmetric Quantum Systems

Quantum Reports ◽

10.3390/quantum3030026 ◽

2021 ◽

Vol 3 (3) ◽

pp. 417-424

Author(s):

Sarang S. Bhosale ◽

Biswanath Rath ◽

Prasanta K. Panigrahi

Keyword(s):

Hilbert Space ◽

Quantum Mechanics ◽

Quantum Systems ◽

Inner Product ◽

Bell’S Inequality ◽

Standard Quantum ◽

Bell's Inequality ◽

Qubit System ◽

Symmetric Quantum ◽

No Signaling

Bell’s inequality is investigated in parity-time (PT) symmetric quantum mechanics, using a recently developed form of the inequality by Maccone [Am. J. Phys. 81, 854 (2013) ] , with two PT-qubits in the unbroken phase with real energy spectrum. It is shown that the inequality produces a bound that is consistent with the standard quantum mechanics even after using Hilbert space equipped with CPT inner product and therefore, the entanglement has identical structure with standard quantum mechanics. Consequently, the no-signaling principle for a two-qubit system in PT-symmetric quantum theory is preserved.

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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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A study of quantum correlations in open quantum systems

Quantum Information and Computation ◽

10.26421/qic11.7-8-1 ◽

2011 ◽

Vol 11 (7&8) ◽

pp. 541-562

Author(s):

Indranil Chakrabarty ◽

Subhashish Banerjee ◽

Nana Siddharth

Keyword(s):

Open Quantum Systems ◽

Quantum Correlations ◽

Quantum Systems ◽

Bell’S Inequality ◽

Mixed States ◽

Classical Correlation ◽

Bell's Inequality ◽

Qubit System ◽

Dynamical Parameters ◽

Potential Resource

In this work, we study quantum correlations in mixed states. The states studied are modeled by a two-qubit system interacting with its environment via a quantum non demolition (purely dephasing) as well as dissipative type of interaction. The entanglement dynamics of this two qubit system is analyzed. We make a comparative study of various measures of quantum correlations, like Concurrence, Bell's inequality, Discord and Teleportation fidelity, on these states, generated by the above evolutions. We classify these evoluted states on basis of various dynamical parameters like bath squeezing parameter $r$, inter-qubit spacing $r_{12}$, temperature $T$ and time of system-bath evolution $t$. In this study, in addition we report the existence of entangled states which do not violate Bell's inequality, but can still be useful as a potential resource for teleportation. Moreover we study the dynamics of quantum as well as classical correlation in presence of dissipative coherence.

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