Math 2.0

Quantum theory Entanglement Theorem

Purification of a quantum state

Builds on Tensor product of Hilbert spacesDensity matrixPartial traceSpectral theorem for Hermitian matricesUnitary operatorSchmidt decomposition

Statement

Let HAH_A be a complex Hilbert space of finite dimension, and let ρ\rho be a density matrix on HAH_A of rank rr. Let HBH_B be a complex Hilbert space with dim⁡HB≥r\dim H_B \ge r.

  1. There is a unit vector ψ∈HA⊗HB\psi \in H_A \otimes H_B with tr⁡B∣ψ⟩⟨ψ∣=ρ\tr_B |\psi\rangle\langle\psi| = \rho, where tr⁡B\tr_B is the partial trace over HBH_B.
  2. If ψ\psi and φ\varphi are two unit vectors in HA⊗HBH_A \otimes H_B with tr⁡B∣ψ⟩⟨ψ∣=tr⁡B∣φ⟩⟨φ∣=ρ\tr_B |\psi\rangle\langle\psi| = \tr_B |\varphi\rangle\langle\varphi| = \rho, then φ=(I⊗U)ψ\varphi = (I \otimes U)\psi for some unitary operator UU on HBH_B.

Such a vector ψ\psi is called a purification of ρ\rho.

Remarks

By the spectral theorem, ρ=∑i=1rpi∣ei⟩⟨ei∣\rho = \sum_{i=1}^{r} p_i |e_i\rangle\langle e_i| with pi>0p_i > 0 and eie_i orthonormal. Choose orthonormal f1,…,frf_1, \dots, f_r in HBH_B and put ψ=∑ipi ei⊗fi\psi = \sum_i \sqrt{p_i} \, e_i \otimes f_i; a direct computation gives tr⁡B∣ψ⟩⟨ψ∣=ρ\tr_B |\psi\rangle\langle\psi| = \rho. For uniqueness, the Schmidt decomposition of any purification has the form ∑ipi ei⊗fi′\sum_i \sqrt{p_i} \, e_i \otimes f_i', up to a choice of eigenbasis. A unitary on HBH_B that maps the family fif_i to the family fi′f_i' then relates the two purifications.

Example: the maximally mixed qubit state 12I\tfrac12 I on ℂ2\C^2 is purified by (∣00⟩+∣11⟩)/2(|00\rangle + |11\rangle)/\sqrt{2} in ℂ2⊗ℂ2\C^2 \otimes \C^2.

The bound dim⁡HB≥r\dim H_B \ge r is necessary. If dim⁡HB<r\dim H_B < r, every vector of HA⊗HBH_A \otimes H_B has Schmidt rank at most dim⁡HB\dim H_B, so its reduced state has rank less than rr.

Going deeper

For a student

Purification lets one replace mixed states by pure states on a larger space, which often simplifies proofs. It underlies the Stinespring picture of quantum channels and Uhlmann's formula for the fidelity. The uniqueness clause is also the reason why a party holding the purifying system can steer the other party's state.

Reading path

To understand this page from the ground up, read these 40 pages in order. The foundations come first.

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Density matrix, concept, no evidence levelDensity matrixconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionOrthonormal basis, concept, no evidence levelOrthonormal basisconceptPartial trace, concept, no evidence levelPartial traceconceptQuantum entanglement, concept, no evidence levelQuantum entanglementconceptSingular value, concept, no evidence levelSingular valueconceptTensor product of Hilbert spaces, concept, no evidence levelTensor product of Hil…conceptUnitary operator, concept, no evidence levelUnitary operatorconceptVector space, definition, no evidence levelVector spacedefinitionSquare matrix, definition, no evidence levelSquare matrixdefinitionTranspose, definition, no evidence levelTransposedefinitionInner product, definition, no evidence levelInner productdefinitionLinear combination, definition, no evidence levelLinear combinationdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionLinear independence, definition, no evidence levelLinear independencedefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionLinear span, definition, no evidence levelLinear spandefinitionHermitian matrix, definition, no evidence levelHermitian matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionOrthonormal family, definition, no evidence levelOrthonormal familydefinitionAdjoint of a linear map, definition, no evidence levelAdjoint of a linear m…definitionEigenvalue, definition, no evidence levelEigenvaluedefinitionSpanning set, definition, no evidence levelSpanning setdefinitionPositive semidefinite matrix, definition, no evidence levelPositive semidefinite…definitionSelf-adjoint operator, definition, no evidence levelSelf-adjoint operatordefinitionBasis, definition, no evidence levelBasisdefinitionEigenvector, definition, no evidence levelEigenvectordefinitionGram matrices are positive semidefinite, statement, no evidence levelGram matrices are pos…statementOrthonormal basis, definition, no evidence levelOrthonormal basisdefinitionSpectral theorem, statement, no evidence levelSpectral theoremstatementSingular value, definition, no evidence levelSingular valuedefinitionSpectral theorem for Hermitian matrices, statement, no evidence levelSpectral theorem for …statementSingular value decomposition, statement, no evidence levelSingular value decomp…statementSchmidt decomposition, statement, no evidence levelSchmidt decompositionstatementPurification of a quantum state, statement, no evidence levelPurification of a qua…statement
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. Density matrix concept
  2. Identity matrix object
  3. Matrix definition
  4. Orthonormal basis concept
  5. Partial trace concept
  6. Quantum entanglement concept
  7. Singular value concept
  8. Tensor product of Hilbert spaces concept
  9. Unitary operator concept
  10. Vector space definition
  11. Square matrix definition
  12. Transpose definition
  13. Inner product definition
  14. Linear combination definition
  15. Linear map definition
  16. Linear subspace definition
  17. Conjugate transpose definition
  18. Inner product space definition
  19. Orthogonal vectors definition
  20. Linear independence definition
  21. Endomorphism definition
  22. Linear span definition
  23. Hermitian matrix definition
  24. Unitary matrix definition
  25. Orthonormal family definition
  26. Adjoint of a linear map definition
  27. Eigenvalue definition
  28. Spanning set definition
  29. Positive semidefinite matrix definition
  30. Self-adjoint operator definition
  31. Basis definition
  32. Eigenvector definition
  33. Gram matrices are positive semidefinite statement
  34. Orthonormal basis definition
  35. Spectral theorem statement
  36. Singular value definition
  37. Spectral theorem for Hermitian matrices statement
  38. Singular value decomposition statement
  39. Schmidt decomposition statement
  40. Purification of a quantum state statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantum_state_purification
About this page
TagCDXS11AEW9
URIm2://CDXS11AEW9
Kindstatement, theorem
WritingW0 stub
MSC81P16, 81P40
StatusThis copy of the site has no build data. Run m2 status CDXS11AEW9.
AboutDensity matrix, Partial trace
UsesTensor product of Hilbert spaces, Density matrix, Partial trace, Spectral theorem for Hermitian matrices, Unitary operator, Schmidt decomposition
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Changed3 November 2026

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