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Review of Basics and Elementary introduction to quantum postulates

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Requirements On Mathematics Apparatus Physical states Mathematic entities Interference phenomena Nondeterministic predictions Model the effects of measurement Distinction between evolution and measurement

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What’s Quantum Mechanics A mathematical framework Description of the world known Rather simple rules but counterintuitive applications

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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

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Basic linear algebra useful in QM Complex numbers Vector space Linear operators Inner products Unitary operators Tensor products …

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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”

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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 :

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Hilbert Space (Cont’d) Orthonormal basis: a basis set where Can be found from an arbitrary basis set by Gram-Schmidt Orthogonalization

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Inner Products Inner Product is a function combining two vectors It yields a complex number It obeys the following rules

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Unitary Operator An operator U is unitary, if Preserves Inner product

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Tensor Product Larger vector space formed from two smaller ones Combining elements from each in all possible ways Preserves both linearity and scalar multiplication

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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

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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

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Any pair of linearly independent vectors can be a basis!

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1/ 2

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Bloch Sphere

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Measurements

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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

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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

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Summary on Postulates

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Postulate 3 in rough form

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From last slide

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Manin was first compare

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You can apply the constant to each Distributive properties Postulate 4

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Entanglement

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Some convenctions implicit in postulate 4

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We assume the opposite Leads to contradiction, so we cannot decompose as this Entangled state as opposed to separable states

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Composite quantum system

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This was used before CV was invented. You can verify it by multiplying matrices

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The Measurement Problem Can we deduce postulate 3 from 1 and 2? Joke. Do not try it. Slides are from MIT.

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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

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Covered in 2007

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