Introduction Basics of Quantum Mechanics - physics…hande/teaching/talk1.pdf · Introduction...

IntroductionBasics of Quantum Mechanics

Quantum information and quantum computing

Hande Üstünel

Middle East Technical University, Department of Physics

January 6, 2009

Hande Üstünel Quantum information and quantum computing

Outline

1 Introduction

2 Basics of Quantum MechanicsDirac notationLinear algebraOperator relations

The Basic Ideas

The main concerns in computational science andinformation technology:

1 Computational speed/power2 Safe transfer of knowledge

Moore’s Law is about to run up against a size wall.

As size gets smaller, quantum mechanical effects interfere.

Quantum computation has a big speed advantage due toparallel processing.

Quantum cryptography : safe transfer of knowledgebetween parties that might not trust each other

Quantum mechanics is the language!

The Basic Ideas

Dirac notationLinear algebraOperator relations

Dirac notation

The fundamental entity in quantum mechanics is a state. Astate is really a vector in linear algebra.

|ψ〉 : Vector or state, ket in Dirac notation〈ψ| : Dual to the vector or state, bra in Dirac notationA : Matrix, operator in Dirac notation

〈φ|ψ〉 : Inner product|φ〉〈ψ| : Outer product|φ〉|ψ〉 : Tensor product〈φ|A|ψ〉 : Inner product between |φ〉 and A|ψ〉

Basics of linear algebra

Vector space

Vectors in linear algebra live in a space called the vectorspace.

Any vector in a vector space may be written in terms of thespanning vectors {|v1〉, |v2〉 · · · |vn〉}.

Complex numbers ⇒ two-dimensional vectors

a + ib =

︸︷︷︸

|v1〉

︸︷︷︸

|v2〉

|v1〉 and |v2〉 span the complex number vector space.

Vector space

a + ib =

︸︷︷︸

|v1〉

︸︷︷︸

|v2〉

Vector space

a + ib =

︸︷︷︸

|v1〉

︸︷︷︸

|v2〉

Vector space

a + ib =

︸︷︷︸

|v1〉

︸︷︷︸

|v2〉

Spanning vectors

A vector space may be spanned by different sets ofspanning vectors.

|v1〉 =1√2

|v2〉 =1√2

(1−1

Any complex number : |v〉 = (a,b)

|v〉 =a + b√

2|v1〉 +

a − b√2

|v2〉

Operators and Matrices

Linear operator

An operator is a map carrying one vector in one vectorspace to another in the same or a different vector space.

A linear operator acts on the terms of the sum of vectors ina space separately.

ai |vi〉)

aiA(|vi〉)

Let I|v〉 = |v〉 (I is the identity operator)

I (2|v1〉 + 3|v2〉 − 4i |v3〉) = 2I|v1〉 + 3I|v2〉 − 4iI|v3〉= 2|v1〉 + 3|v2〉 − 4i |v3〉

Linear operator

ai |vi〉)

aiA(|vi〉)

Linear operator

ai |vi〉)

aiA(|vi〉)

Operators as matrices

If we denote states by vectors, then it’s convenient todenote operators as matrices.

Reason : Matrix algebra is easy!

(1 00 1

, |v〉 =

⇒ I|v〉 =

(1 00 1

= |v〉

Linear operators as matrices

Linear operator A : V → W between vector spaces V and W .If {|vi 〉} and {|wi〉} → spanning vectors for V and W then thereexists Aij such that

A|vj〉 =∑

Aij |wi〉

{ Aij } may be interpreted as elements of a matrix.

Note on jargon : Spanning vector set ↔ Basis

Linear operators as matrices

Linear operator A : V → W between vector spaces V and W .If {|vi 〉} and {|wi〉} → spanning vectors for V and W then thereexists Aij such that

A|vj〉 =∑

Aij |wi〉

{ Aij } may be interpreted as elements of a matrix.

Note on jargon : Spanning vector set ↔ Basis

Pauli matrices

Four useful matrices

σ0 = I =

(1 00 1

σ1 = X =

(0 11 0

σ2 = Y =

(0 −ii 0

σ3 = Z =

(1 00 −1

Products

Inner product

Inner product between vectors (kets) |v〉 = [v1v2 · · · vn] and|w〉 = [w1w2 · · ·wn]

(|v〉, |w〉) = 〈v |w〉 = [v∗1 v∗

2 · · · v∗n ]

v∗i wi

Orthonormality : A set of vectors are said to beorthonormal if

〈vi |vj 〉 =

0 if i = j

1 if i 6= j

Products

Outer product

Representation of a linear operator. Define an operator A suchthat

A|v ′〉 ≡ (|w〉〈v |) |v ′〉 = 〈v |v ′〉|w〉.A is an outer product operator.

Completeness relation

Consider an arbitrary vector |v〉 =∑

i vi |v〉 where 〈i |v〉 = vi .(∑

|i〉〈i |)

|v〉 =∑

|i〉〈i |v〉 =∑

vi |i〉 = |v〉 ⇒ |i〉〈i | = I

Products

Outer product

Representation of a linear operator. Define an operator A suchthat

A|v ′〉 ≡ (|w〉〈v |) |v ′〉 = 〈v |v ′〉|w〉.A is an outer product operator.

Completeness relation

Consider an arbitrary vector |v〉 =∑

i vi |v〉 where 〈i |v〉 = vi .(∑

|i〉〈i |)

|v〉 =∑

|i〉〈i |v〉 =∑

vi |i〉 = |v〉 ⇒ |i〉〈i | = I

Products

Tensor product

A tensor product is a way of putting vector spaces together toform larger vector spaces.

In matrix notation, let Am×n and Bp×q be two matrices. Then

A ⊗ B =

A11B A12B · · · A1nBA21B A22B · · · A2nB

...... · · · ...

Am1B An2B · · · AmnB

where AijB are p × q submatrices.

Products

The tensor product operator A ⊗ B acts on the tensorproduct space V ⊗ W as follows :

A ⊗ B(|v〉 ⊗ |w〉) = A|v〉 ⊗ B|w〉

A ⊗ B is a linear operator

(A ⊗ B)

ai |vi 〉 ⊗ |wi〉)

≡∑

(aiA|vi〉 ⊗ B|wi〉)

Useful notation : |v〉⊗k means that |v〉 is tensored withitself k times.

Products

A ⊗ B(|v〉 ⊗ |w〉) = A|v〉 ⊗ B|w〉

(A ⊗ B)

ai |vi 〉 ⊗ |wi〉)

≡∑

Products

A ⊗ B(|v〉 ⊗ |w〉) = A|v〉 ⊗ B|w〉

(A ⊗ B)

ai |vi 〉 ⊗ |wi〉)

≡∑

Products

Hadamard operator

Defined on a two-dimensional space with spanning vectors

|v0〉 = |0〉 =

and |v1〉 = |1〉 =

Remember that 〈1|0〉 = 〈0|1〉 = 0

On a single state it can be written as

H =1√2[(|0〉 + |1〉)〈0| + (|0〉 − |1〉)〈1|)]

The Hadamard operator on n tensored states

H⊗n =1√2

(−1)x·y |x〉〈y |

Eigenvectors and eigenvalues

An eigenvector of a linear operator A is a nonzero vectorsatisfying the relation

A|v〉 = λ|v〉where λ is a complex number and is defined as the eigenvaluecorresponding to the eigenvector.

An example from matrix algebra :

(2 1−1 2

|v〉 =

(1−1

A|v〉 = λ|v〉 where λ = 1

Eigenvectors and eigenvalues

Degenaracy

Different eigenvectors having the same eigenvalue are said tobe degenarate.

The matrix

2 0 10 2 01 0 1

has two eigenvectors |v1〉 = [1 0 0] and |v2〉 = [0 1 0]corresponding to the eigenvalue 2.

Diagonalizability

A diagonal representation for an operator A on a vector spaceV is a representation A =

i λi |i〉〈i | where the vectors form anorthonormal set of eigenvectors for A.

Example :

(1 00 −1

= |0〉〈0| − |1〉〈1|

Introduction Basics of Quantum Mechanics - physics…hande/teaching/talk1.pdf · Introduction...

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