  Quantum computing is among the most far-reaching technologies of the 21st
century, tackling challenges at the cutting edge of physics. This new paradigm
in computer science harnesses quantum entanglement, one striking non-intuitive
feature of quantum mechanics and a cornerstone of quantum information, to
provide computation with a quantum speed-up over the best-known classical
algorithms and to enable encrypted data communication against eavesdropping.
The bulk of this article is focused on providing a deep and abiding
understanding of non-relativistic matrix mechanics by demonstrating the
fundamental mathematical identities of the contemporary postulatory approach of
quantum mechanics within the state vector and density operator formalism in
multipartite systems. In addition to that, we derive and analyze the respective
1-qubit, 1-qutrit, 2-qubit, and 2-qudit coherent and incoherent density
operators using Bloch's parametrization for generalized $d$-dimensional
$N$-qudit states embedded in the $SU(d)$ Lie group with associate generalized
Gell Mann's matrices spanning the $\mathfrak{su}(d)$ Lie algebra. We also
address the fundamental concepts of quantum nondemolition measurements, quantum
decoherence and, particularly, quantum entanglement providing for the latter a
systematic view on its historical development and mathematical description in
multipartite systems. We conclude our review by introducing some of the
ubiquitous quantum quantifiers required to measure degrees of quantum
entanglement and quantum coherence, deriving the $p$-norm quantum coherence
measure for a 1-qubit state.
