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Hilbert spaces and operators
Quantum states are unit vectors in a complex Hilbert space. Observables are Hermitian operators, dynamics are unitary for closed systems, and eigenvalues correspond to possible measurement results. Composite systems use tensor products.
The formalism separates state, transformation, and measurement.
Density operators
A density operator ρ is positive semidefinite with trace one. Pure states satisfy Tr(ρ²)=1; mixed states have lower purity. Density matrices naturally describe uncertainty, noise, and subsystems.
Density matrices are essential for open systems and error analysis.
POVMs and channels
General measurements are represented by positive-operator-valued measures. Physical noise and operations are completely positive trace-preserving maps, often written using Kraus operators.
Quantum channels provide the language for realistic devices and noise.
Applied activity
Advanced exercise: derive or simulate one representative result from this module, document assumptions, and produce a one-page technical interpretation for a non-specialist decision maker.
Read the full lesson text
1. Hilbert spaces and operators
Quantum states are unit vectors in a complex Hilbert space. Observables are Hermitian operators, dynamics are unitary for closed systems, and eigenvalues correspond to possible measurement results. Composite systems use tensor products.
The formalism separates state, transformation, and measurement.
2. Density operators
A density operator ρ is positive semidefinite with trace one. Pure states satisfy Tr(ρ²)=1; mixed states have lower purity. Density matrices naturally describe uncertainty, noise, and subsystems.
Density matrices are essential for open systems and error analysis.
3. POVMs and channels
General measurements are represented by positive-operator-valued measures. Physical noise and operations are completely positive trace-preserving maps, often written using Kraus operators.
Quantum channels provide the language for realistic devices and noise.
4. Applied activity
Advanced exercise: derive or simulate one representative result from this module, document assumptions, and produce a one-page technical interpretation for a non-specialist decision maker.
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