book

Quantum Computing Explained

by David McMahon

December 2007

Intermediate to advanced

332 pages

7h 28m

English

Wiley-IEEE Press

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Classical InformationInformation Content in a SignalEntropy and Shannon’S Information TheoryProbability BasicsEXERCISES
The QubitVector SpacesLinear Combinations of VectorsUniqueness of a Spanning SetBasis and DimensionInner ProductsOrthonormalityGram-Schmidt OrthogonalizationBra-Ket FormalismThe Cauchy-Schwartz And Triangle InequalitiesSummaryExercises
ObservablesThe Pauli OperatorsOuter ProductsThe Closure RelationRepresentations of Operators Using MatricesOuter Products and Matrix RepresentationsMatrix Representation of Operators in Two-Dimensional SpacesDefinition: The Pauli MatricesHermitian, Unitary, And Normal OperatorsEigenvalues and EigenvectorsSpectral DecompositionThe Trace Of An OperatorImportant Properties of the TraceThe Expectation Value of an OperatorProjection OperatorsPOSITIVE OPERATORSCommutator AlgebraThe Heisenberg Uncertainty PrinciplePolar Decomposition and Singular ValuesThe Postulates of Quantum MechanicsExercises
Representing Composite States in Quantum MechanicsComputing Inner ProductsTensor Products of Column VectorsOperators and Tensor ProductsTensor Products of MatricesExercises

The Density Operator For a Pure StateThe Density Operator For a Mixed StateKey Properties of a Density OperatorCharacterizing Mixed StatesThe Partial Trace And The Reduced Density OperatorThe Density Operator and the Bloch VectorEXERCISES
Distinguishing Quantum States and MeasurementProjective MeasurementsMeasurements on Composite SystemsGeneralized MeasurementsPositive Operator-Valued MeasuresExercises
Bell’S TheoremBipartite Systems and the Bell BasisWhen is a State Entangled?The Pauli RepresentationEntanglement FidelityUsing Bell States For Density Operator RepresentationSchmidt DecompositionPurificationExercises
Classical Logic GatesSingle-Qubit GatesMore Single-Qubit GatesExponentiationThe Z–Y DecompositionBasic Quantum Circuit DiagramsControlled GatesGate DecompositionExercises
Hadamard GatesThe Phase GateMatrix Representation of Serial and Parallel OperationsQuantum InterferenceQuantum Parallelism and Function EvaluationDeutsch-Jozsa AlgorithmQuantum Fourier TransformPhase EstimationShor’S AlgorithmQuantum Searching And Grover’S AlgorithmExercises
TeleportationThe Peres Partial Transposition ConditionEntanglement SwappingSuperdense CodingExercises
A Brief Overview of Rsa EncryptionBasic Quantum CryptographyAn Example Attack: The Controlled Not AttackThe B92 ProtocolThe E91 Protocol (Ekert)Exercises
Single-Qubit ErrorsQuantum Operations And Krauss OperatorsThe Depolarization ChannelThe Bit Flip and Phase Flip ChannelsAmplitude DampingPhase DampingQuantum Error CorrectionExercises
The No-Cloning TheoremTrace DistanceFidelityEntanglement of Formation and ConcurrenceInformation Content and EntropyExercises
Adiabatic ProcessesAdiabatic Quantum ComputingExercises
Cluster StatesAdjacency MatricesStabilizer StatesAside: Entanglement WitnessCluster State ProcessingExercises

Content preview from Quantum Computing Explained

4 TENSOR PRODUCTS

In quantum mechanics we don’t always work with single particles in isolation. In many cases, some of which are seen in the context of quantum information processing, it is necessary to work with multiparticle states. Mathematically, to understand multiparticle systems in quantum mechanics, it is necessary to be able to construct a Hilbert space H that is a composite of the independent Hilbert spaces that are associated with each individual particle. The machinery required to do this goes by the name of the Kronecker or tensor product. We consider the two-particle case.

Suppose that H₁ and H₂ are two Hilbert spaces of dimension N₁ and N₂. We can put these two Hilbert spaces together to construct a larger Hilbert space. We denote this larger space by H and use the tensor product operation symbol ⊗. So we write

The dimension of the larger Hilbert space is the product of the dimensions of H₁and H₂. Once again, we assume that dim(H₁) = N₁ and dim(H₂) = N₂. Then

Next we start getting down to business and learn how to represent state vectors in the composite Hilbert space.

REPRESENTING COMPOSITE STATES IN QUANTUM MECHANICS

A state vector belonging to H is the tensor product of state vectors belonging to H1 and H2. We will show how to represent such vectors explicitly ...