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Measure of tripartite quantum entanglement in multi-particle system of two-level atoms

Ram Narayan Deb

Published Apr 4, 2020• 0 citations
OpenAlex

Abstract

We propose a direct measure of tripartite quantum entanglement in an arbitrary symmetric pure state of N correlated two-level atoms (qubits). We compute the third order moments of the collective pseudo-spin operators in terms of the individual atomic operators of the atoms in the assembly and find that all the bipartite quantum correlation terms among the atoms cancel out leaving only the all possible tripartite quantum correlation terms among them. We observe that the third order moments are made up of solely the tripartite quantum correlations among the atoms. This helps to extract out only the tripartite quantum correlations among the N atoms and propose a measure and quantification of these correlations. We also propose the necessary and sufficient condition for the presence of tripartite quantum entanglement in such multi-atomic systems. We conjecture the way of determining the third order moments of the pseudo-spin operators of the two-level atoms in the assembly, experimentally.

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We propose a direct measure of tripartite quantum entanglement in an arbitrary symmetric pure state of N correlated two-level atoms (qubits). We compute the third order moments of the collective pseudo-spin operators in terms of the individual atomic operators of the atoms in the assembly and find that all the bipartite quantum correlation terms among the atoms cancel out leaving only the all possible tripartite quantum correlation terms among them. We observe that the third order moments are made up of solely the tripartite quantum correlations among the atoms. This helps to extract out only the tripartite quantum correlations among the N atoms and propose a measure and quantification of these correlations. We also propose the necessary and sufficient condition for the presence of tripartite quantum entanglement in such multi-atomic systems. We conjecture the way of determining the third order moments of the pseudo-spin operators of the two-level atoms in the assembly, experimentally.
We propose the necessary and sufficient condition for the presence of quantum entanglement in arbitrary symmetric pure states of two-level atomic systems. We introduce a parameter to quantify quantum entanglement in such systems. We express the inherent quantum fluctuations of a composite system of two-level atoms as a sum of the quantum fluctuations of the individual constituent atoms and their correlation terms. This helps to separate out and study solely the quantum correlations among the atoms and obtain the criterion for the presence of entanglement in such multiatomic systems.
Entanglement measures quantify nonclassical correlations present in a quantum system, but can be extremely difficult to calculate, even more so, when information on its state is limited. Here, we consider broad families of entanglement criteria that are based on variances of arbitrary operators and analytically derive the lower bounds these criteria provide for two relevant entanglement measures: the best separable approximation and the generalized robustness. This yields a practical method for quantifying entanglement in realistic experimental situations, in particular, when only few measurements of simple observables are available. As a concrete application of this method, we quantify bipartite and multipartite entanglement in spin-squeezed Bose-Einstein condensates of ∼500 atoms, by lower bounding the best separable approximation and the generalized robustness only from measurements of first and second moments of the collective spin operator.
Entanglement detection is crucial and a necessity in the context of quantum information and quantum computation. This important task has proved to be quite hard for quantum systems of dimensions higher than 2×3, in which case, there exists well established necessary and sufficient conditions like Peres-Horodecki criterion.To tackle this challenge for bipartite systems, we introduce a mathematical framework to reduce the problem to entanglement in a two qubit system. This is done by mapping each subsystem locally into a qubit without increasing entanglement. The mapping is expressed in terms of expectation values of three arbitrary operators in the original state. We give necessary and sufficient conditions for such mapping to be valid from physical point of view, providing thence a versatile tool for dimension reduction in various applications.Our main use of this formalism is as a gate way to derive entanglement criteria for bipartite or multi-partite systemas based on existing ones derived for qubit systems. By mapping each subsystem locally into a qubit, applying entanglement criteria known for qubits on the resulting state automatically gives us entanglement criteria in terms of the chosen operators used to implement the mapping.For the multi-partite case, we focus on spin squeezing inequalities for qubits to derive entanglement criteria for general systems. However, when applying our formalism to this case, an interesting situation arises where one is able to obtain coherent superposition of multi-partite qubit states with different particle number. Hence, to derive better entanglement criteria, we had to consider quantum and/or classical fluctuationsthat may be exhibited by the particle number operator. We derive generalized form of Sørensen-Mølmer’s criterion and of spin squeezing inequalities for fluctuating particle number in terms of arbitrary collective operators. We applied our results to study entanglement in a system of ultra-cold Chromium atoms with spin s = 3 trapped in a bi-dimensional optical lattice incollaboration with Quantum Dipolar Gazes team in Laboratoire de Physique de Laser at Paris Nord 13 university. We showed, in a numerical simulation, that our generalized inequalities are able to detect entanglement in their system using collective operators. Moreover, we show that such observables can be measured using available techniques.
We develop a necessary condition of quantum correlation. It is utilized to construct a ddd-level bipartite Bell-type inequality which is strongly resistant to noise and requires only analyses of O(d)O(d)O(d) measurement outcomes compared to the previous result O(d2)O({d}^{2})O(d2). Remarkably, a connection between the arbitrary high-dimensional bipartite Bell-type inequality and entanglement witnesses is found. Through the necessary condition of quantum correlation, we propose that the witness operators to detect truly multipartite entanglement for a generalized Greenberger-Horne-Zeilinger (GHZ) state with two local measurement settings and a four-qubit singlet state with three settings. Moreover, we also propose a robust entanglement witness to detect a four-level tripartite GHZ state with only two local measurement settings.
This study presents an analytical investigation of entanglement quantification in three-qubit pure states through the Acín canonical representation, which serves as a generalization of the two-qubit Schmidt decomposition. Driven by the intricacies of multipartite entanglement and the shortcomings of current measures, we employ a global concurrence measure derived from the generalized concurrence of various bipartitions within the system. The characteristics of this measure are explored both analytically and numerically across a range of SLOCC entanglement classes, such as product, biseparable, GHZ-type, and W-type states. When compared to the three-tangle and tripartite negativity, our findings indicate that this measure yields complementary insights into multipartite correlations. The outcomes underscore the significance of this measure in elucidating the structure and distribution of entanglement within three-qubit quantum systems.