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Lower Bound of $l_{1}$ Norm of Coherence of Bipartite Qubit-Qudit System and its Application in the Detection of Entangled Tripartite Qudit-Qubit-Qudit System

Quantum coherence and quantum entanglement are two strong pillars in quantum information theory. We study here for the possibility of any connection between these two important aspects of quantum mechanics while studying the entanglement detection problem for the detection of bipartite higher dimensional entangled states and multipartite entangled states. To achieve our goal, we derive the lower bound $L$ of $l_{1}$ norm of coherence of bipartite qubit-qudit system using the criterion that detect entanglement. Furthermore, we deduce the upper bound $U$ of $l_{1}$ norm of coherence of separable bipartite qubit-qudit system using the separability criterion. Thus, we find that if any $l_{1}$ norm of coherence of bipartite qubit-qudit system is greater than the upper bound $U$ then the given qubit-qudit state is entangled. Finally, we obtained the upper bound $U_{1}$ of $l_{1}$ norm of coherence of separable tripartite state lies either in $2 \otimes d \otimes d$ or $d \otimes 2 \otimes d$ or $d \otimes d \otimes 2$ dimensional Hilbert space using the upper bound $U$. We have shown that if the $l_{1}$ norm of coherence of any tripartite $qubit-qudit-qudit$ or $qudit-qubit-qudit$ or $qudit-qudit-qubit$ system is greater than the derived upper bound $U_{1}$ then the given tripartite system represent an entangled state.