Math 2.0

Quantum theory Definition

Superdense coding

Builds on QubitBell statePauli matricesTensor product of Hilbert spaces

Definition

Let XX and ZZ be the Pauli matrices σx\sigma_x and σz\sigma_z, and let ∣Φ±⟩,∣Ψ±⟩|\Phi^\pm\rangle, |\Psi^\pm\rangle be the Bell states. Alice and Bob share two qubits AA and BB in the state ∣Φ+⟩AB|\Phi^+\rangle_{AB}, and Alice wants to send two bits (a,b)∈{0,1}2(a, b) \in \set{0,1}^2. Superdense coding is the following protocol.

  1. Alice applies ZaXbZ^a X^b to her qubit AA.
  2. Alice sends the qubit AA to Bob.
  3. Bob measures the pair ABAB in the Bell basis and reads (a,b)(a, b) from the outcome.

Remarks

The check: applying ZaXbZ^a X^b to the first qubit of ∣Φ+⟩|\Phi^+\rangle gives (a,b)=(0,0)↦∣Φ+⟩,(0,1)↦∣Ψ+⟩,(1,0)↦∣Φ−⟩,(1,1)↦∣Ψ−⟩.(a,b) = (0,0) \mapsto |\Phi^+\rangle, \quad (0,1) \mapsto |\Psi^+\rangle, \quad (1,0) \mapsto |\Phi^-\rangle, \quad (1,1) \mapsto |\Psi^-\rangle. For example, (X⊗I)(∣00⟩+∣11⟩)/2=(∣10⟩+∣01⟩)/2=∣Ψ+⟩(X \otimes I)(|00\rangle + |11\rangle)/\sqrt2 = (|10\rangle + |01\rangle)/\sqrt2 = |\Psi^+\rangle. The four results are orthogonal, so the Bell measurement identifies (a,b)(a, b) with certainty.

Non-example: if Alice and Bob start from the product state ∣00⟩|00\rangle instead, the four states (ZaXb⊗I)∣00⟩(Z^a X^b \otimes I)|00\rangle are ±∣00⟩\pm|00\rangle and ±∣10⟩\pm|10\rangle. Only two of them are distinguishable, so Bob recovers just one bit.

The shared pair is consumed, and two classical bits per transmitted qubit is the maximum.

Reading path

To understand this page from the ground up, read these 37 pages in order. The foundations come first.

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Cross product, concept, no evidence levelCross productconceptDeterminant, 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 levelMatrixdefinitionPartial trace, concept, no evidence levelPartial traceconceptQubit, concept, no evidence levelQubitconceptReal projective space, object, no evidence levelReal projective spaceobjectTensor product of Hilbert spaces, concept, no evidence levelTensor product of Hil…conceptTrace 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 levelTransposedefinitionBell state, example, no evidence levelBell stateexampleLinear map, definition, no evidence levelLinear mapdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionOrthogonal matrix, definition, no evidence levelOrthogonal matrixdefinitionMultilinear map, definition, no evidence levelMultilinear mapdefinitionDivision ring, definition, no evidence levelDivision ringdefinitionOrthogonal group, object, no evidence levelOrthogonal groupobjectUnitary matrix, definition, no evidence levelUnitary matrixdefinitionAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionThe quaternions, example, no evidence levelThe quaternionsexampleUnitary group, object, no evidence levelUnitary groupobjectDeterminant, definition, no evidence levelDeterminantdefinitionSpecial 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 matricesobjectSuperdense coding, definition, no evidence levelSuperdense codingdefinition
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. Determinant concept
  3. Group concept
  4. Hermitian matrix concept
  5. Identity matrix object
  6. Lie algebra concept
  7. Lie group concept
  8. Matrix definition
  9. Partial trace concept
  10. Qubit concept
  11. Real projective space object
  12. Tensor product of Hilbert spaces 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. Bell state example
  20. Linear map definition
  21. Unit of a ring definition
  22. Conjugate transpose definition
  23. Orthogonal matrix definition
  24. Multilinear map definition
  25. Division ring definition
  26. Orthogonal group object
  27. Unitary matrix definition
  28. Alternating multilinear map definition
  29. The quaternions example
  30. Unitary group object
  31. Determinant definition
  32. Special orthogonal group object
  33. Special unitary group SU(n) example
  34. Rotation group SO(3) example
  35. SU(2) example
  36. Pauli matrices object
  37. Superdense coding definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Superdense_coding
About this page
Tag466QX46KWR
URIm2://466QX46KWR
Kinddefinition
WritingW0 stub
MSC81P48, 81P45
DefinesSuperdense coding
UsesQubit, Bell state, Pauli matrices, Tensor product of Hilbert spaces
Changed3 November 2026

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