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Review of Basics and Elementary introduction to quantum postulates

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.

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Review of Basics and Elementary introduction to quantum postulates

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  1. Review of Basics and Elementary introduction to quantum postulates

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

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

  4. 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

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

  6. 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”

  7. 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 vectorparallel to |v:

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

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

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

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

  12. Mathematically, what is a qubit ? (1) • We can form linear combinations of states • A qubit state is a unit vector in a two dimensionalcomplex vector space

  13. 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 eia as it has no observable effect

  14. Any pair of linearly independent vectors can be a basis!

  15. 1/2

  16. Bloch Sphere

  17. Measurements

  18. 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 24

  19. Physical Systems - Quantum Mechanics Connections entanglement

  20. Summary on Postulates

  21. Postulate 3 in rough form

  22. From last slide

  23. Manin was first compare

  24. Postulate 4 You can apply the constant to each Distributive properties

  25. Entanglement

  26. Entanglement

  27. Some convenctions implicit in postulate 4

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

  29. Composite quantum system

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

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

  32. Sources Quantum Computing Mathematics and Postulates Presented by Chensheng Qiu Supervised by Dplm. Ing. Gherman Examiner: Prof. Wunderlich Anuj Dawar , Michael Nielsen Advanced topic seminar SS02 “Innovative Computer architecture and concepts” Examiner: Prof. Wunderlich

  33. Covered in 2007

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