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Quantum computing is among the most far-reaching technologies of the 21stcentury, tackling challenges at the cutting edge of physics. This new paradigmin computer science harnesses quantum entanglement, one striking non-intuitivefeature of quantum mechanics and a cornerstone of quantum information, toprovide computation with a quantum speed-up over the best-known classicalalgorithms and to enable encrypted data communication against eavesdropping.The bulk of this article is focused on providing a deep and abidingunderstanding of non-relativistic matrix mechanics by demonstrating thefundamental mathematical identities of the contemporary postulatory approach ofquantum mechanics within the state vector and density operator formalism inmultipartite systems. In addition to that, we derive and analyze the respective1-qubit, 1-qutrit, 2-qubit, and 2-qudit coherent and incoherent densityoperators using Bloch's parametrization for generalized $d$-dimensional$N$-qudit states embedded in the $SU(d)$ Lie group with associate generalizedGell Mann's matrices spanning the $\mathfrak{su}(d)$ Lie algebra. We alsoaddress the fundamental concepts of quantum nondemolition measurements, quantumdecoherence and, particularly, quantum entanglement providing for the latter asystematic view on its historical development and mathematical description inmultipartite systems. We conclude our review by introducing some of theubiquitous quantum quantifiers required to measure degrees of quantumentanglement and quantum coherence, deriving the $p$-norm quantum coherencemeasure for a 1-qubit state.

A thorough introduction to non-relativistic matrix mechanics in multi-qudit systems with a study on quantum entanglement and quantum quantifiers