The Mathematical Language of Quantum Theory From Uncertainty to Entanglement 1st Edition by Teiko Heinosaari, Mario Ziman ISBN 1139210475 9781139210478
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ISBN 10: 1139210475
ISBN 13: 9781139210478
Author: Teiko Heinosaari, Mário Ziman
The Mathematical Language of Quantum Theory From Uncertainty to Entanglement 1st Table of contents:
1: Hilbert space refresher
1.1: Hilbert spaces
1.1.1 Finite- and infinite-dimensional Hilbert spaces
1.1.2 Basis expansion
1.1.3 Example: L2(Ω)
1.2: Operators on Hilbert spaces
1.2.1 The C*-algebra of bounded operators
1.2.2 Partially ordered vector space of selfadjoint operators
1.2.3 Orthocomplemented lattice of projections
1.2.4 Group of unitary operators
1.2.5 Ideal of trace class operators
1.3: Additional useful mathematics
1.3.1 Weak operator topology
1.3.2 Dirac notation and rank-1 operators
1.3.3 Spectral and singular-value decompositions
1.3.4 Linear functionals and dual spaces
1.3.5 Tensor product
2: States and effects
2.1: Duality of states and effects
2.1.1 Basic statistical framework
2.1.2 State space
2.1.3 State space for a finite-dimensional system
2.1.4 From states to effects
2.1.5 From effects to states
2.1.6 Dispersion-free states and Gleason’s theorem
2.2: Superposition structure of pure states
2.2.1 Superposition of two pure states
2.2.2 Interference
2.3: Symmetry
2.3.1 Unitary and antiunitary transformations
2.3.2 State automorphisms
2.3.3 Pure state automorphisms and Wigner’s theorem
2.4: Composite systems
2.4.1 System versus subsystems
2.4.2 State purification
3: Observables
3.1: Observables as positive operator-valued measures
3.1.1 Definition and basic properties of observables
3.1.2 Observables and statistical maps
3.1.3 Discrete observables
3.1.4 Real observables
3.1.5 Mixtures of observables
3.1.6 Coexistence of effects
3.2: Sharp observables
3.2.1 Projection-valued measures
3.2.2 Sharp observables and selfadjoint operators
3.2.3 Complementary observables
3.3: Informationally complete observables
3.3.1 Informational completeness
3.3.2 Symmetric informationally complete observables
3.3.3 State estimation
3.4: Testing quantum systems
3.4.1 Complete versus incomplete information
3.4.2 Unambiguous discrimination of states
3.4.3 How distinct are two states?
3.5: Relations between observables
3.5.1 State distinction and state determination
3.5.2 Coarse-graining
3.6: Example: photon-counting observables
3.6.1 Single-mode electromagnetic field
3.6.2 Nonideal photon-counting observables
4: Operations and channels
4.1: Transforming quantum systems
4.1.1 Operations and complete positivity
4.1.2 Schrödinger versus Heisenberg picture
4.2: Physical model of quantum channels
4.2.1 Isolated versus open systems
4.2.2 Stinespring’s dilation theorem
4.2.3 Operator-sum form of channels
4.3: Elementary properties of quantum channels
4.3.1 Mixtures of channels
4.3.2 Concatenating channels
4.3.3 Disturbance and noise
4.3.4 Conjugate channels
4.4: Parametrizations of quantum channels
4.4.1 Matrix representation
4.4.2 The χ-matrix representation
4.4.3 Choi–Jamiolkowski isomorphism
4.5: Special classes of channels
4.5.1 Strictly contractive channels
4.5.2 Random unitary channels
4.5.3 Phase-damping channels
4.6: Example: qubit channels
5: Measurement models and instruments
5.1: Three levels of description of measurements
5.1.1 Measurement models
5.1.2 Instruments
5.1.3 Compatibility of the three descriptions
5.2: Disturbance caused by a measurement
5.2.1 Conditional output states
5.2.2 No information without disturbance
5.2.3 Disturbance in a rank-1 measurement
5.2.4 Example: BB84 quantum key distribution
5.3: Lüders instruments
5.3.1 Von Neumann’s measurement model
5.3.2 Lüders instrument for a discrete observable
5.3.3 Lüders’ theorem
5.3.4 Example: mean king’s problem
5.4: Repeatable measurements
5.4.1 Repeatability
5.4.2 Wigner–Araki–Yanase theorem
5.4.3 Approximate repeatability
5.5: Programmable quantum processors
5.5.1 Programming of observables and channels
5.5.2 Universal processor for channels
5.5.3 Probabilistic programming
6: Entanglement
6.1: Entangled bipartite systems
6.1.1 Entangled vectors
6.1.2 Entangled positive operators
6.1.3 Nonlocal channels
6.2: Entanglement and LOCC
6.2.1 LOCC ordering and separable states
6.2.2 Maximally entangled states
6.2.3 Majorization criterion for LOCC
6.3: Entanglement detection
6.3.1 Entanglement witnesses
6.3.2 Quantum nonlocal realism
6.3.3 Positive but not completely positive maps
6.3.4 Negative partial transpose (NPT) criterion
6.3.5 Range criterion
6.4: Additional topics in entanglement theory
6.4.1 Entanglement teleportation and distillation
6.4.2 Multipartite entanglement
6.4.3 Evolution of quantum entanglement
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Tags: Teiko Heinosaari, Mario Ziman, Mathematical Language, Entanglement