In quantum mechanics and computing, 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.

Mathematically each quantum mechanical system is associated with a separable complex Hilbert space ${\displaystyle H}$. A pure state of a quantum system is represented by a non-zero vector ${\displaystyle \psi }$ in ${\displaystyle H}$. The vectors ${\displaystyle \psi }$ and ${\displaystyle \lambda \psi }$ (with ${\displaystyle \lambda \in \mathbb {C} ^{*}}$) represent the same state. A system with n mutually orthogonal quantum states can be described by a Hilbert space of dimension n. Pure states can be represented as equivalence classes, or, rays in a projective Hilbert space ${\displaystyle \mathbf {P} (H_{n})=\mathbb {C} \mathbf {P} ^{n-1}}$. For a two-dimensional Hilbert space, the space of all such states is the complex projective line ${\displaystyle \mathbb {C} \mathbf {P} ^{1}.}$ This is the Bloch sphere, which can be mapped to the Riemann sphere.

The Bloch sphere is a unit 2-sphere, with antipodal points corresponding to a pair of mutually orthogonal state vectors. The north and south poles of the Bloch sphere are typically chosen to correspond to the standard basis vectors ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$, respectively, which in turn might correspond e.g. to the spin-up and spin-down states of an electron. This choice is arbitrary, however. The points on the surface of the sphere correspond to the pure states of the system, whereas the interior points correspond to the mixed states. The Bloch sphere may be generalized to an n-level quantum system, but then the visualization is less useful.

The natural metric on the Bloch sphere is the Fubini–Study metric. The mapping from the unit 3-sphere in the two-dimensional state space ${\displaystyle \mathbb {C} ^{2}}$ to the Bloch sphere is the Hopf fibration, with each ray of spinors mapping to one point on the Bloch sphere.

Given an orthonormal basis, any pure state ${\displaystyle |\psi \rangle }$ of a two-level quantum system can be written as a superposition of the basis vectors ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$, 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. However, only the relative phase between the coefficients of the two basis vectors has any physical meaning (the phase of the quantum system is not directly measurable), so that there is redundancy in this description. We can take the coefficient of ${\displaystyle |0\rangle }$ to be real and non-negative. This allows the state to be described by only three real numbers, giving rise to the three dimensions of the Bloch sphere.

We also know from quantum mechanics that the total probability of the system has to be one:

Given this constraint, we can write ${\displaystyle |\psi \rangle }$ using the following representation:

The representation is always unique, because, even though the value of ${\displaystyle \phi }$ is not unique when ${\displaystyle |\psi \rangle }$ is one of the states (see Bra-ket notation) ${\displaystyle |0\rangle }$ or ${\displaystyle |1\rangle }$, the point represented by ${\displaystyle \theta }$ and ${\displaystyle \phi }$ is unique.