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 vector parallel 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

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

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

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

15. Qubit on Bloch Sphere

16. 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

17. 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

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

19. Measurements of the same qubit in various bases 1/  2

20. Bloch Sphere

21. Measurements

22. AXIOMS OF QUANTUM MECHANICS

23. 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

24. 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

25. Summary on Postulates

26. Postulate 3 in rough form

28. From last slide

29. Manin was first compare

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

31. Entanglement

33. Some convenctions implicit in postulate 4

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

35. Composed quantum systems – results of Postulate 4

36. Composite quantum system

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

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

41. 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

42. Covered in 2007, 2011