Math 2.0

Quantum theory Foundations and measurement Example

GHZ state

Builds on Tensor product of Hilbert spacesQuantum entanglementPauli matricesHidden-variable theory

Example

Let e0,e1e_0, e_1 be the standard basis of ℂ2\C^2. The GHZ state of three qubits is the unit vector ψ=12(e0⊗e0⊗e0+e1⊗e1⊗e1)\psi = \frac{1}{\sqrt{2}}\big(e_0 \otimes e_0 \otimes e_0 + e_1 \otimes e_1 \otimes e_1\big) in the tensor product ℂ2⊗ℂ2⊗ℂ2\C^2 \otimes \C^2 \otimes \C^2. It is entangled. Indeed, a product vector u⊗v⊗wu \otimes v \otimes w has coefficients cijk=uivjwkc_{ijk} = u_i v_j w_k, so c000c111=c001c110c_{000} c_{111} = c_{001} c_{110}. For ψ\psi the left side is 1/21/2 and the right side is 00. With the Pauli matrices XX and YY, one checks (X⊗X⊗X)ψ=ψ(X \otimes X \otimes X)\psi = \psi. Since Ye0=ie1Y e_0 = i e_1 and Ye1=−ie0Y e_1 = -i e_0, also (X⊗Y⊗Y)ψ=(Y⊗X⊗Y)ψ=(Y⊗Y⊗X)ψ=−ψ(X \otimes Y \otimes Y)\psi = (Y \otimes X \otimes Y)\psi = (Y \otimes Y \otimes X)\psi = -\psi.

Remarks

These four eigenvalue equations give the GHZ argument against local hidden-variable theories. Suppose each qubit kk carries predetermined values xk,yk∈{−1,1}x_k, y_k \in \set{-1, 1} for the measurements of XX and YY. The last three equations force x1y2y3=y1x2y3=y1y2x3=−1x_1 y_2 y_3 = y_1 x_2 y_3 = y_1 y_2 x_3 = -1. Multiplying them gives x1x2x3=−1x_1 x_2 x_3 = -1, since each yk2=1y_k^2 = 1. The first equation forces x1x2x3=1x_1 x_2 x_3 = 1, a contradiction without any inequality. For nn qubits the GHZ state is (e0⊗n+e1⊗n)/2(e_0^{\otimes n} + e_1^{\otimes n})/\sqrt2. Tracing out one qubit of the three-qubit GHZ state leaves a separable mixed state, so its entanglement is destroyed by losing one particle.

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 productconceptDeterminant, concept, no evidence levelDeterminantconceptGroup, concept, no evidence levelGroupconceptHermitian matrix, concept, no evidence levelHermitian matrixconceptHidden-variable theory, concept, no evidence levelHidden-variable theoryconceptIdentity matrix, object, no evidence levelIdentity matrixobjectLie algebra, concept, no evidence levelLie algebraconceptLie group, concept, no evidence levelLie groupconceptMatrix, definition, no evidence levelMatrixdefinitionQuantum entanglement, concept, no evidence levelQuantum entanglementconceptReal 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 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 matricesobjectGHZ state, example, no evidence levelGHZ stateexample
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. Hidden-variable theory concept
  6. Identity matrix object
  7. Lie algebra concept
  8. Lie group concept
  9. Matrix definition
  10. Quantum entanglement 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. 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. GHZ state example

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Greenberger%E2%80%93Horne%E2%80%93Zeilinger_state
About this page
TagA76VV68635
URIm2://A76VV68635
Kindexample
WritingW0 stub
MSC81P40, 81P05
Instance ofQuantum entanglement
UsesTensor product of Hilbert spaces, Quantum entanglement, Pauli matrices, Hidden-variable theory
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Changed3 November 2026

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