The goal of this tutorial is to understand geometric relations to obtain a deeper understand of qubit (quantum-bit) entanglement. We will cover:
-Bloch Sphere Review
-Pure vs Mixed States
-Single Qubit Interactive Visualizations
-Pure States Epistemic Probabilities
-Pure and Mixed States Epistemic Probabilities
-Separable 2-qubit Case
-Full 2-qubit case
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Bloch Sphere Review
The Bloch Sphere is popular among Quantum Mechanics related texts to help visualize and understand the single qubit system. However, in most texts, the Bloch Sphere is introduced to help readers understand only the pure states cases that can be summarized with a ket vector. By setting the radius of the Bloch Sphere equal to 0.5, we demonstrate a geometric Bloch Sphere interpretation that immediately gives off the values of the density matrix, which is the full picture of the quantum state. Another limitation to the Bloch Sphere is that it is hard to generalize this visualization for n-qubit systems because its hard to see the overall picture of the state at a glance. Finding a suitable visualization model for n-qubit is intrinsically hard because the state space grows exponentially while n increases linearly. We establish a torus model for separable 2-qubit systems, which is only a subpicture of the full 2-qubit state space. However, this torus model straightforwardly displays the relation between the 2 qubits, which is not achieved with 2 Bloch Circles. As we shall see in later chapters, the torus model gives of the relative dimensions of the following spaces: the maximally mixed states, partially mixed states, and the pure states. This helps learners build an intuition about these different states a 2-qubit system could be in, and how the dimensions of these sub-states will grow as the size of the n-qubit system grows. Next we'd like to try typing some more text in and see if we can work in an important marginal note to the right. This is the first marginal note. Then after putting in more text we see how it looks.