Review of Basics and Elementary introduction to quantum postulates.
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Review of Basics and Elementary introduction to quantum postulates
Requirements On Mathematics Apparatus Physical states Mathematic entities Interference phenomena Nondeterministic predictions Model the effects of measurement Distinction between evolution and measurement
What’s Quantum Mechanics A mathematical framework Description of the world known Rather simple rules but counterintuitive applications
Introduction to Linear Algebra Quantum mechanics The basis for quantum computing and quantum information Why Linear Algebra? Prerequisities What is Linear Algebra concerning? Vector spaces Linear operations
Basic linear algebra useful in QM Complex numbers Vector space Linear operators Inner products Unitary operators Tensor products …
Dirac-notation: Bra and Ket For the sake of simplification “ket” stands for a vector in Hilbert “bra” stands for the adjoint of Named after the word “bracket”
Hilbert Space Fundamentals Inner product space: linear space equipped with inner product Hilbert Space (finite dimensional): can be considered as inner product space of a quantum system Orthogonality: Norm: Unit vector parallel to |v :
Hilbert Space (Cont’d) Orthonormal basis: a basis set where Can be found from an arbitrary basis set by Gram-Schmidt Orthogonalization
Inner Products Inner Product is a function combining two vectors It yields a complex number It obeys the following rules
Unitary Operator An operator U is unitary, if Preserves Inner product
Tensor Product Larger vector space formed from two smaller ones Combining elements from each in all possible ways Preserves both linearity and scalar multiplication
Mathematically, what is a qubit ? (1) We can form linear combinations of states A qubit state is a unit vector in a two dimensional complex vector space
Qubits Cont'd We may rewrite as… From a single measurement one obtains only a single bit of information about the state of the qubit There is "hidden" quantum information and this information grows exponentially We can ignore e i as it has no observable effect
Any pair of linearly independent vectors can be a basis!
Postulates in QM Why are postulates important? … they provide the connections between the physical, real, world and the quantum mechanics mathematics used to model these systems - Isaak L. Chuang 2424
Physical Systems - Quantum Mechanics Connections Postulate 1 Isolated physical system Hilbert Space Postulate 2 Evolution of a physical system Unitary transformation Postulate 3 Measurements of a physical system Measurement operators Postulate 4 Composite physical system Tensor product of components entanglement
This was used before CV was invented. You can verify it by multiplying matrices
The Measurement Problem Can we deduce postulate 3 from 1 and 2? Joke. Do not try it. Slides are from MIT.
Quantum Computing Mathematics and Postulates Advanced topic seminar SS02 “Innovative Computer architecture and concepts” Examiner: Prof. Wunderlich Presented by Chensheng Qiu Supervised by Dplm. Ing. Gherman Examiner: Prof. Wunderlich Anuj Dawar, Michael Nielsen Sources