Quantifying Entanglement

As a relevant concept in Physics, the role played by entanglement in quantum information processing is paramount, as it is the main resource enabling considerable technological achievements, such as quantum communication, quantum cryptography, quantum computational speed-up and so on. Mathematically, Quantum entanglement is the most unbelievable non-classical property of compound states that cannot be decomposed as a statistical mixtures of product states over subsystems, and it has a very complex structure, encasing many features described by just as much entanglement measures. Among all of them, the Quantum Conditional Mutual Information is a particularly interesting one and represents the object this thesis is devoted to. The exact quantification of many information measures is generally a daunting problem because it would involve a huge amount of computational resources that outstrip the capability on any existing computer: since the eigenvalues and relevant entropies of a density matrix operator and its subsystems are expected to be known, this task becomes quickly computationally demanding for large enough systems, setting itself as seemingly unsolvable. For this reason, it behooves us to understand how we can keep using them while endeavoring to devise some meaningful approximations. The upshot is that we try to handle with the aforementioned quantity in terms of lower and upper bounds, which can be obtained analytically in a simple yet effective way. This purpose can be generalized to other important information measures without spectrum reconstruction. In particular, we emphasize how our lower bound is tighter than the Carlen-Lieb's one, a well-known scientific result establishing a sharper refinement on the strong subadditivity inequality of the Von Neumann Entropy. Moreover, we run simulations over a suitable quantum state whose outcomes ostensibly validate our proposal. However, hitting the bullseye comes up after drawing a parallel between the classical and quantum worlds concerning some preliminary notions of Information Theory, to which we will dedicate the first and second chapter of the thesis, respectively. We will then proceed to the analytical derivation of the bounds and subsequently apply them to a mixed state in order to check their actual correctness.