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The Bloch sphere parametrizes not only pure states but mixed states for 2-level systems. The density operator describing the mixed-state of a 2-level quantum system (qubit) corresponds to a point inside the Bloch sphere with the following coordinates: are the coordinates of the individual states (on the surface of Bloch sphere).
Why is the Bloch sphere named after Felix Bloch?
Bloch sphere. In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system (qubit), named after the physicist Felix Bloch. Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space.
Can a Bloch sphere be generalized to a quantum system?
The Bloch sphere may be generalized to an n-level quantum system, but then the visualization is less useful. For historical reasons, in optics the Bloch sphere is also known as the Poincaré sphere and specifically represents different types of polarizations.
Which is the mapping from the unit 3-sphere to the Bloch sphere?
The mapping from the unit 3-sphere in the two-dimensional state space to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere. , where the coefficient of (or contribution from) each of the two basis vectors is a complex number. This means that the state is described by four real numbers.
Why are wave functions also called Bloch states?
The wave functions in this basis are energy eigenstates (because they are eigenstates of the Hamiltonian), and they are also Bloch states (because they are eigenstates of the translation operators; see Lemma above). and the two operators shall have a common set of eigenfunctions.
How can one choose an arbitrary point on the Bloch sphere?
If the north and south poles are chosen, then both states are on the same line and not orthogonal anymore, so how can one choose an arbitrary point $p$ on the surface of the sphere and possibly decompose it in terms of $0,1$ states in order to find $a$ and $b$?
What kind of vector is the Bloch sphere?
is called the Bloch vector . It is this vector that indicates the point within the sphere that corresponds to a given mixed state. Specifically, as a basic feature of the Pauli vector, the eigenvalues of ρ are . Density operators must be positive-semidefinite, so it follows that
Which is the stereographic projection of the Bloch sphere?
Call this point P. Point u on the plane z = 0 is the stereographic projection of point P on the Bloch sphere. The vector with tail at the origin and tip at P is the direction in 3-D space corresponding to the spinor . The coordinates of P are