The Bloch sphere maps the state of a single qubit onto a point in three-dimensional coordinates. In the standard computational-basis convention, |0⟩ is the north (+z) pole and |1⟩ is the south (−z) pole. A pure qubit state is represented by a vector from the center to the sphere’s surface; the vector is a mathematical description of the state, not a tiny object moving through ordinary space.
Start with the poles and coordinate axes
Read a Bloch-sphere diagram by identifying its axes and pole labels first. In the standard convention, the z axis runs vertically: |0⟩ is at +z and |1⟩ at −z. The x-y plane cuts through the sphere’s center, forming the equator.
- North pole (+z): |0⟩, the positive eigenstate of the z measurement.
- South pole (−z): |1⟩, the negative eigenstate of the z measurement.
- Equator: states with equal probabilities of |0⟩ and |1⟩ when measured in the computational (z) basis.
- ±x and ±y directions: equatorial states; each pair at opposite ends of an axis represents the two eigenstates for that measurement axis.
These pole labels are the usual abstract-qubit convention. A physical device may encode its basis states in a particular way, but that does not change how the standard Bloch-sphere diagram is read.
Read the state vector using θ and φ
A pure qubit state can be written as:
|ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩
The associated unit Bloch vector is (sinθ cosφ, sinθ sinφ, cosθ). Here θ is the polar angle measured down from +z, and φ is the azimuthal angle around z, commonly measured in the x-y plane from +x toward +y using the chosen right-handed coordinate convention.
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- Find the arrow’s endpoint. For a pure state, it lies on the sphere’s surface.
- Read θ from the +z axis down to the arrow. At θ = 0, the state is |0⟩; at θ = π, it is |1⟩; at θ = π/2, it is on the equator.
- Read φ around the z axis in the x-y plane. Check the diagram’s arrows and handedness: perspective can make the sign or direction of φ hard to infer from the drawing alone.
The half-angle in the ket is important: θ is the angle of the Bloch vector, while the state’s amplitudes use θ/2.
What θ and φ tell you about measurement
In the computational basis, θ determines the outcome probabilities:
- P(0) = cos²(θ/2)
- P(1) = sin²(θ/2)
Changing φ does not change those z-basis probabilities. It does change the state’s position around the equator, distinguishing states that have the same computational-basis outcome probabilities. That distinction matters for measurements along x or y and for interference. So an equatorial point means 50/50 outcomes in the z basis, not that all equatorial states are identical.
Why a two-component qubit is drawn on a sphere
A qubit’s ket has two complex amplitudes. Normalization constrains them, and multiplying the entire ket by a global phase does not change the physical state. After those freedoms are accounted for, a pure qubit state has two independent real parameters, θ and φ. Those parameters specify a point on a two-dimensional surface embedded in three-dimensional space.
The Bloch vector’s three coordinates are the expectation values of the Pauli observables: r = (⟨σx⟩, ⟨σy⟩, ⟨σz⟩). For a pure state, its length is one. Opposite points on any diameter represent orthogonal states, so every measurement axis has a pair of opposite eigenstates.
Distinguish global phase from relative phase
A global phase multiplies the whole ket by the same complex phase factor. It does not move the state to a new point on the Bloch sphere. For example, |ψ⟩ and eiα|ψ⟩ describe the same physical state.
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The relative phase between the |0⟩ and |1⟩ amplitudes is different. In the standard parametrization it is φ, and changing it moves the point around the z axis. This is why different equatorial points can share 50/50 z-basis probabilities yet still represent different states.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Pure states, mixed states, and the Bloch ball
The surface represents pure states. A mixed state—one that represents a statistical mixture rather than a single pure state—has a Bloch vector of length less than one and lies inside the sphere. The center, where the vector length is zero, is the maximally mixed qubit state. Thus, the full set of qubit states is a solid Bloch ball, not only its outer surface.
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Single-qubit unitary operations act as rotations of the Bloch sphere. In the familiar Pauli-gate picture, X, Y, and Z each correspond, up to an overall phase, to a half-turn about the corresponding axis. When interpreting a rotation’s direction or sign, distinguish an operation that rotates the state from a change in the observer’s coordinate frame; diagrams and explanations may use different active or passive conventions.
Check these conventions when diagrams disagree
- Which basis state is labeled at each z pole?
- Which way does φ increase, and what handedness do the axes use?
- Does the arrow end on the surface or inside it?
- Is the arrow labeled as a state vector or a Bloch vector?
Different drawings may orient the axes or angle markings differently. The pole labels and the stated angle convention—not the page’s visual perspective—determine how to interpret a point.
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