Math 2.0

Quantum theory Definition

Quantum teleportation

Builds on QubitBell statePauli matricesLOCCTensor product of Hilbert spacesNo-cloning theoremNo-communication theorem

Definition

Let XX and ZZ be the Pauli matrices σx\sigma_x and σz\sigma_z. Let ∣Φ±⟩,∣Ψ±⟩|\Phi^\pm\rangle, |\Psi^\pm\rangle be the Bell states of two qubits. Alice holds a qubit SS in a state ∣ψ⟩=α∣0⟩+β∣1⟩|\psi\rangle = \alpha|0\rangle + \beta|1\rangle, and Alice and Bob share two qubits AA and BB in the state ∣Φ+⟩AB|\Phi^+\rangle_{AB}. Quantum teleportation is the following LOCC protocol.

  1. Alice measures the pair SASA in the Bell basis ∣Φ+⟩,∣Φ−⟩,∣Ψ+⟩,∣Ψ−⟩|\Phi^+\rangle, |\Phi^-\rangle, |\Psi^+\rangle, |\Psi^-\rangle.
  2. Alice sends the outcome, two classical bits, to Bob.
  3. Bob applies to BB the correction II, ZZ, XX or ZXZX for the outcomes Φ+\Phi^+, Φ−\Phi^-, Ψ+\Psi^+, Ψ−\Psi^- respectively.

After step 3, the qubit BB is in the state ∣ψ⟩|\psi\rangle, up to a global phase.

Remarks

The check rests on the identity ∣ψ⟩S∣Φ+⟩AB=12(∣Φ+⟩SA∣ψ⟩B+∣Φ−⟩SAZ∣ψ⟩B+∣Ψ+⟩SAX∣ψ⟩B+∣Ψ−⟩SAXZ∣ψ⟩B),|\psi\rangle_S |\Phi^+\rangle_{AB} = \tfrac12 \big( |\Phi^+\rangle_{SA} |\psi\rangle_B + |\Phi^-\rangle_{SA} Z|\psi\rangle_B + |\Psi^+\rangle_{SA} X|\psi\rangle_B + |\Psi^-\rangle_{SA} XZ|\psi\rangle_B \big), which follows by expanding both sides in the standard basis. Each outcome has probability 14\tfrac14, and the correction undoes the Pauli factor, since Z2=X2=IZ^2 = X^2 = I and ZX⋅XZ=IZX \cdot XZ = I.

Non-example: if Bob applies no correction, then for the outcomes Φ−\Phi^-, Ψ+\Psi^+, Ψ−\Psi^- he holds Z∣ψ⟩Z|\psi\rangle, X∣ψ⟩X|\psi\rangle or XZ∣ψ⟩XZ|\psi\rangle, which in general differ from ∣ψ⟩|\psi\rangle. Averaged over Alice's outcomes, his state is 12I\tfrac12 I, independent of ψ\psi. This is the no-communication theorem at work: without the classical bits, Bob learns nothing.

At the end, the qubits SS and AA are in a Bell state that carries no information about ψ\psi, in agreement with the no-cloning theorem. The protocol works by linearity for mixed states and for a qubit that is entangled with other systems.

Reading path

To understand this page from the ground up, read these 53 pages in order. The foundations come first.

  1. Completely positive map concept
  2. Cross product concept
  3. Density matrix concept
  4. Determinant concept
  5. Group concept
  6. Hermitian matrix concept
  7. Hilbert space concept
  8. Identity matrix object
  9. Kronecker product concept
  10. LOCC concept
  11. Lie algebra concept
  12. Lie group concept
  13. Matrix definition
  14. Matrix ring object
  15. Partial trace concept
  16. Positive semidefinite matrix concept
  17. Quantum channel concept
  18. Quantum operation concept
  19. Quantum state concept
  20. Qubit concept
  21. Real projective space object
  22. Tensor product of Hilbert spaces concept
  23. Trace of a matrix concept
  24. Unitary matrix concept
  25. Unitary operator concept
  26. Vector space definition
  27. Stinespring dilation theorem statement
  28. Ring definition
  29. Choi's theorem on completely positive maps statement
  30. Square matrix definition
  31. Transpose definition
  32. No-cloning theorem statement
  33. Bell state example
  34. Linear map definition
  35. Unit of a ring definition
  36. Conjugate transpose definition
  37. Orthogonal matrix definition
  38. Kraus representation of a channel statement
  39. Multilinear map definition
  40. Division ring definition
  41. Orthogonal group object
  42. Alternating multilinear map definition
  43. Unitary matrix definition
  44. No-communication theorem statement
  45. The quaternions example
  46. Unitary group object
  47. Determinant definition
  48. Special orthogonal group object
  49. Special unitary group SU(n) example
  50. Rotation group SO(3) example
  51. SU(2) example
  52. Pauli matrices object
  53. Quantum teleportation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Quantum_teleportation
About this page
TagM3MS1YP76J
URIm2://M3MS1YP76J
Kinddefinition
WritingW0 stub
MSC81P48, 81P45
DefinesQuantum teleportation
UsesQubit, Bell state, Pauli matrices, LOCC, Tensor product of Hilbert spaces, No-cloning theorem, No-communication theorem
Changed3 November 2026

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