Math 2.0

Quantum theory Definition

Bloch sphere

Builds on QubitDensity matrixPauli matricesSphere

Definition

Let σx,σy,σz\sigma_x, \sigma_y, \sigma_z be the Pauli matrices, and for r=(rx,ry,rz)∈ℝ3r = (r_x, r_y, r_z) \in \R^3 put r⋅σ=rxσx+ryσy+rzσzr \cdot \sigma = r_x \sigma_x + r_y \sigma_y + r_z \sigma_z. Every density matrix ρ\rho of a qubit can be written in exactly one way as ρ=12(I+r⋅σ),r∈ℝ3,∥r∥≤1.\rho = \tfrac12 (I + r \cdot \sigma), \qquad r \in \R^3, \quad \norm{r} \le 1. The vector rr is the Bloch vector of ρ\rho, and the set {r∈ℝ3:∥r∥≤1}\set{r \in \R^3 : \norm{r} \le 1} is the Bloch ball. The Bloch sphere is its boundary, the unit sphere {r∈ℝ3:∥r∥=1}\set{r \in \R^3 : \norm{r} = 1}, whose points are the Bloch vectors of the pure states.

Remarks

In coordinates, the pure state cos⁡(θ/2)∣0⟩+eiφsin⁡(θ/2)∣1⟩\cos(\theta/2)|0\rangle + e^{i\varphi}\sin(\theta/2)|1\rangle, with 0≤θ≤π0 \le \theta \le \pi and 0≤φ<2π0 \le \varphi < 2\pi, has Bloch vector (sin⁡θcos⁡φ,sin⁡θsin⁡φ,cos⁡θ)(\sin\theta\cos\varphi, \sin\theta\sin\varphi, \cos\theta).

Example: ∣+⟩=(∣0⟩+∣1⟩)/2|{+}\rangle = (|0\rangle + |1\rangle)/\sqrt2 has ρ=12(1111)=12(I+σx)\rho = \tfrac12 \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} = \tfrac12(I + \sigma_x), so its Bloch vector is (1,0,0)(1, 0, 0), of norm 11.

Non-example: for r=(0,0,2)r = (0, 0, 2), the matrix 12(I+2σz)=diag(32,−12)\tfrac12(I + 2\sigma_z) = \mathrm{diag}(\tfrac32, -\tfrac12) has a negative eigenvalue, so it is not a state. In general the eigenvalues of 12(I+r⋅σ)\tfrac12(I + r\cdot\sigma) are 12(1±∥r∥)\tfrac12(1 \pm \norm{r}), which explains the condition ∥r∥≤1\norm{r} \le 1.

Orthogonal pure states have opposite Bloch vectors. Conjugation ρ↦UρU∗\rho \mapsto U\rho U^* by a unitary UU acts on Bloch vectors by a rotation, and this gives the covering homomorphism from SU(2)SU(2) onto SO(3)SO(3).

Reading path

To understand this page from the ground up, read these 36 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptDensity matrix, concept, no evidence levelDensity matrixconceptDeterminant, concept, no evidence levelDeterminantconceptGroup, concept, no evidence levelGroupconceptHermitian matrix, concept, no evidence levelHermitian matrixconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLie algebra, concept, no evidence levelLie algebraconceptLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionQubit, concept, no evidence levelQubitconceptReal projective space, object, no evidence levelReal projective spaceobjectSphere, concept, no evidence levelSphereconceptTrace of a matrix, concept, no evidence levelTrace of a matrixconceptUnitary matrix, concept, no evidence levelUnitary matrixconceptVector space, definition, no evidence levelVector spacedefinitionRing, definition, no evidence levelRingdefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionThe quaternions, example, no evidence levelThe quaternionsexampleDeterminant, definition, no evidence levelDeterminantdefinitionUnitary group, object, no evidence levelUnitary groupobjectSpecial orthogonal group, object, no evidence levelSpecial orthogonal gr…objectSpecial unitary group SU(n), example, no evidence levelSpecial unitary group…exampleRotation group SO(3), example, no evidence levelRotation group SO(3)exampleSU(2), example, no evidence levelSU(2)examplePauli matrices, object, no evidence levelPauli matricesobjectBloch sphere, definition, no evidence levelBloch spheredefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Cross product concept
  2. Density matrix concept
  3. Determinant concept
  4. Group concept
  5. Hermitian matrix concept
  6. Identity matrix object
  7. Lie algebra concept
  8. Lie group concept
  9. Matrix definition
  10. Qubit concept
  11. Real projective space object
  12. Sphere concept
  13. Trace of a matrix concept
  14. Unitary matrix concept
  15. Vector space definition
  16. Ring definition
  17. Square matrix definition
  18. Transpose definition
  19. Linear map definition
  20. Unit of a ring definition
  21. Conjugate transpose definition
  22. Multilinear map definition
  23. Orthogonal matrix definition
  24. Division ring definition
  25. Alternating multilinear map definition
  26. Orthogonal group object
  27. Unitary matrix definition
  28. The quaternions example
  29. Determinant definition
  30. Unitary group object
  31. Special orthogonal group object
  32. Special unitary group SU(n) example
  33. Rotation group SO(3) example
  34. SU(2) example
  35. Pauli matrices object
  36. Bloch sphere definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Bloch_sphere
About this page
TagM7QC7XMRPW
URIm2://M7QC7XMRPW
Kinddefinition
WritingW0 stub
MSC81P16, 81P45
DefinesBloch sphere
UsesQubit, Density matrix, Pauli matrices, Sphere
Changed3 November 2026

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