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 spacesLinear operations

Basic linear algebra useful in QM

• Complex numbers

• Vector space

• Linear operators

• Inner products

• Unitary operators

• Tensor products

• …

Dirac-notation: Dirac-notation: Bra and Bra and KetKet

• 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 Hilbert Space FundamentalsFundamentals

• 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:

0wv

vvv

v

v

Hilbert Space (Cont’d)

• Orthonormal basis:

a basis set where

• Can be found from an arbitrary basis set by Gram-Schmidt Orthogonalization

nvv ,...,1 ijji vv

Inner Inner ProductsProducts

Inner Products

• Inner Product is a function combining two vectors

• It yields a complex number• It obeys the following rules

• •

C ),(

kkk

kkk wvawav ,,

*),(),( wvwv

0),( vv

Unitary Operator

• An operator U is unitary, if

• Preserves Inner product

IUUτ

Uofadjoint for the stands Uwhere

wvwUvU ,,

Tensor ProductTensor 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 Qubit on Bloch Bloch

SphereSphere

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

0 1

cos 0 sin 12 2

i ie e

cos 0 sin 12 2

ie

We can ignore ei as it has no

observable effect

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

Measurements of the same qubit in Measurements of the same qubit in various basesvarious bases

1/2

Bloch Sphere

MeasurementsMeasurements

AXIOMS OF AXIOMS OF QUANTUM QUANTUM

MECHANICSMECHANICS

Postulates in QMPostulates 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

24242424

Physical Systems -Physical Systems - Quantum Mechanics Connections Quantum Mechanics Connections

Postulate 1Isolated physical

system Hilbert Space

Postulate 2Evolution of a

physical system

Unitary transformation

Postulate 3Measurements of a

physical system

Measurement operators

Postulate 4Composite physical

system

Tensor product of

components

entanglement

Summary on Postulates Summary on Postulates

Postulate 3 in rough Postulate 3 in rough formform

From last slide

Manin was first

compare

You can apply the constant to each

Distributive properties

Postulate 4Postulate 4

Entanglement

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 Composed quantum quantum

systems – results systems – results of Postulate 4of 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 byPresented by

Chensheng QiuChensheng Qiu

Supervised bySupervised by

Dplm. Ing. Gherman Dplm. Ing. Gherman

Examiner: Prof. Wunderlich

Anuj Dawar , Michael Nielsen

Sources

• Covered in 2007, 2011