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

6 QUANTUM MEASUREMENTTHEORY

When it comes to the measurement of physical observables, quantum mechanics can tell us what measurement results are possible and what the probability is of obtaining each measurement result. It is also important to focus on what the state of the system is after a measurement is made. While measurement generally has no effect on a system in classical mechanics (i.e., macroscopic systems in general), measurement has a profound impact on a quantum mechanical system—altering its state in an irreversible way. Much of the material in this chapter is a review of concepts already introduced. Nevertheless, if this is your first exposure to quantum theory, it is important that you master these topics. Measurement plays a fundamental role in quantum computation because, at some point, we have to be able to get information out of the computational system. In this chapter we will learn the basics about different measurement models used in quantum theory.

DISTINGUISHING QUANTUM STATES AND MEASUREMENT

Measurement plays a central role in quantum mechanics. An act of measurement disturbs a quantum system in a fundamental way. Consider once again a general qubit