Bloch sphere ( $S^{2}$ )

The Bloch sphere is the unit sphere of Bloch vectors, corresponding to pure qubit states through

ρ = (I + r \cdot σ) /2

. A local unitary change of qubit basis induces a rotation of this sphere, with its common phase physically irrelevant. Mixed states occupy the interior of the unit ball.

Pure-qubit overlap identity ( $∣ ⟨ u ∣ v ⟩ ∣^{2} = \frac{1}{2} (1 + u \cdot v)$ )

(I + r \cdot σ) /2

. The trace identities of the Pauli matrices,

Tr σ_{i} = 0

and

Tr (σ_{i} σ_{j}) = 2 δ_{ij}

, make the trace of their product equal to the displayed overlap. Antipodal unit vectors represent orthogonal states; vectors at right angles correspond to squared overlap one half. For mixed states the same trace formula gives Hilbert–Schmidt overlap, not in general quantum fidelity.

The rotation-invariant average of pure qubit states is uniform area measure on the Bloch sphere, induced by Haar measure on unitary transformations. For a uniformly distributed unit Bloch vector

r

E r_{i} = 0

and

E r_{i} r_{j} = δ_{ij} /3

. Uniform polar angle without the factor

sin θ

is a different distribution.

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The Bloch sphere is a geometrical representation of the state space of a two-level quantum mechanical system, commonly referred to as a qubit. In quantum mechanics, qubits are the fundamental units of quantum information, analogous to classical bits, but they can exist in superpositions of 0 and 1 states. The Bloch sphere provides a visualization of the pure states of a qubit as points on the surface of a sphere.