Quantum systems

The Bloch Sphere

what!

• Visualization tool for two level Quantum systems

  • the states are represented as points on the surface of a unit sphere

• A Quantum systems on the Bloch sphere would look like

  • $\theta$: polar angle

    • determines the weight of $|\uparrow ⟩\text{ and }|\downarrow ⟩$ in the superposition

  • $\varphi$: Azimuthal angle

    • sets the phase of the relation ship

  • Note! this also works for MZI setups!

$|\psi (\theta ,\varphi )⟩=\cos \left(\frac{\theta }{2}\right)|\uparrow ⟩+e^{i\varphi }\sin \left(\frac{\theta }{2}\right)|\downarrow ⟩$

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Uncertainty

Stern-Gerlach

• Since there only can be 2 states of it

  • due to the Spin we get these two outputs no matter
    • spin is a vector

The Uncertainty principle

• In QM

  • we calculate probabilities for the physical observable (the output states)
  • We define the average of these states as the Expectation values
  • when quantifiying them some times we want to find the uncertainty of the measurement outcomes