# 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

Qubit on Bloch Sphere

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!

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

Bloch Sphere

Measurements

AXIOMS OF QUANTUM MECHANICS

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

Summary on Postulates

Postulate 3 in rough form

From last slide

Manin was first compare

You can apply the constant to each Distributive properties Postulate 4

Entanglement

Some convenctions implicit in postulate 4

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

Composed quantum systems – results of Postulate 4

Composite quantum system

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

Covered in 2007, 2011