Blochvektors

Bloch vectors are a mathematical construct used in quantum mechanics, particularly in the context of nuclear magnetic resonance (NMR) spectroscopy and quantum information. They represent the state of a two-level quantum system, often referred to as a qubit. A Bloch vector is a vector in a three-dimensional real space. The components of the vector are determined by the probabilities of the system being in each of its two basis states and the relative phase between them.

Specifically, for a qubit in a state represented by the ket |ψ⟩ = α|0⟩ + β|1⟩, where |0⟩ and |1⟩ are the computational basis states and α and β are complex probability amplitudes, the Bloch vector can be expressed in terms of the expectation values of the Pauli matrices. The x, y, and z components of the Bloch vector are given by ⟨σ_x⟩, ⟨σ_y⟩, and ⟨σ_z⟩, respectively, where σ_x, σ_y, and σ_z are the Pauli operators. The magnitude of the Bloch vector is always 1, meaning it lies on the surface of the Bloch sphere.

The Bloch sphere is a unit sphere in this three-dimensional space. Each point on the surface of the Bloch sphere corresponds to a unique pure state of a qubit. The poles of the sphere typically represent the basis states |0⟩ and |1⟩. Rotations of the Bloch vector on the Bloch sphere correspond to unitary transformations applied to the qubit state. This geometric representation provides an intuitive way to visualize and understand the evolution of quantum states, especially under the influence of external fields or control pulses. Mixed states of a qubit are represented by points inside the Bloch sphere.