The Bloch sphere is a map of a single qubit’s state. In the standard computational-basis convention, the north pole is |0⟩ and the south pole is |1⟩. A pure state is represented by a vector from the center to the sphere’s surface; its direction encodes the state, not a tiny object moving through ordinary space.
Start with the poles and coordinate axes
Read a Bloch-sphere diagram by finding its axes and checking which states it assigns to the poles. In the usual convention, the vertical z axis passes through |0⟩ at the north (+z) pole and |1⟩ at the south (−z) pole. The x-y plane cuts through the sphere at the equator.
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- The z-axis eigenstates are the two poles: |0⟩ and |1⟩.
- The ±x and ±y eigenstates lie around the equator.
- Opposite points on any diameter represent orthogonal states, so each axis picks out a pair of opposite eigenstates.
These are standard abstract-qubit labels. A physical system can encode its two basis states differently, so use the labels shown on a particular diagram rather than assuming every device uses the same physical encoding.
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A common parametrization of a pure qubit state is:
|ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩
Here θ is the polar angle measured down from +z, and φ is the azimuthal angle around the z axis, conventionally read in the x-y plane from +x toward +y in a right-handed coordinate system. The corresponding unit Bloch vector is:
r = (sin θ cos φ, sin θ sin φ, cos θ)
- At θ = 0, the vector points north and the state is |0⟩.
- At θ = π, it points south and the state is |1⟩.
- At θ = π/2, it lies on the equator.
Check the angle arrows in the diagram: illustrations may draw φ differently or use a different viewing perspective. If the convention is not specified, the apparent direction of φ alone is not enough to determine its sign.
What the vector represents—and why a sphere?
The arrow is the Bloch vector associated with a physical state ray; it is not the ket itself. A normalized two-component complex state has two independent real parameters once normalization and physically irrelevant global phase are removed. Those parameters, θ and φ, locate a point on a two-dimensional surface embedded in three-dimensional space.
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Multiplying the whole ket by the same phase does not change the physical state or its Bloch-sphere point. By contrast, the relative phase between the |0⟩ and |1⟩ amplitudes changes φ and moves the point around the sphere. This is why equatorial states can all give equal probabilities in a z-basis measurement yet still represent different states.
Use the sphere to understand measurement probabilities
For the parametrization above, measurement in the computational (z) basis gives:
- P(0) = cos²(θ/2)
- P(1) = sin²(θ/2)
Thus θ determines the probabilities of the two z-basis outcomes. The azimuth φ does not change those probabilities, but it distinguishes states around the equator and matters for measurements along x or y and for interference. A qubit state is more than this pair of probabilities.
Tell pure states from mixed states
A pure state lies on the sphere’s surface, with a Bloch vector of length one. A mixed state lies inside the sphere, with vector length less than one; the center represents the maximally mixed qubit state. The full set of pure and mixed single-qubit states is therefore often called the Bloch ball.
In expectation-value form, the Bloch coordinates are r = (⟨σx⟩, ⟨σy⟩, ⟨σz⟩), where the σ terms are the Pauli observables. A pure state has |r| = 1, while a mixed state has |r| ≤ 1. Carnegie Mellon University’s Quantum Computation and Quantum Information course notes describe the sphere representation in terms of the direction for which the corresponding spin component is positive.
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What single-qubit gates do to the picture
Single-qubit unitary operations act as rotations of the Bloch sphere. In the familiar Pauli-gate picture, X, Y, and Z each produce a half-turn about the corresponding axis, up to a global phase. The sphere can also be described from a changed coordinate frame; that is distinct from actively rotating the state. Rotation signs depend on whether a diagram uses an active or passive convention.
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A quick diagram-reading checklist
- Find the axes and confirm which pole is labeled |0⟩.
- Read θ from +z down to the state vector.
- Read φ around the x-y plane, checking the diagram’s handedness and angle arrows.
- Check whether the vector ends at the surface or inside the sphere: surface means pure, interior means mixed.
The University of Illinois Urbana-Champaign’s PHYS 523 Fall 2025 slide on the Bloch sphere shows the standard pole, axis, and angle labels. If a diagram differs, inspect its conventions before interpreting the point.
Further reading
For a wider treatment of quantum information, Cambridge University Press lists Michael A. Nielsen and Isaac L. Chuang’s Quantum Computation and Quantum Information, 10th Anniversary Edition. It covers substantially more than reading this diagram.
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