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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
IntroductionBasics of Quantum Mechanics
Outline
1 Introduction
2 Basics of Quantum MechanicsDirac notationLinear algebraOperator relations
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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!
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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!
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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!
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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!
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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!
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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!
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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〉
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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 =
(ab
)
= a(
10
)
︸︷︷︸
|v1〉
+ b(
01
)
︸︷︷︸
|v2〉
|v1〉 and |v2〉 span the complex number vector space.
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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 =
(ab
)
= a(
10
)
︸︷︷︸
|v1〉
+ b(
01
)
︸︷︷︸
|v2〉
|v1〉 and |v2〉 span the complex number vector space.
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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 =
(ab
)
= a(
10
)
︸︷︷︸
|v1〉
+ b(
01
)
︸︷︷︸
|v2〉
|v1〉 and |v2〉 span the complex number vector space.
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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 =
(ab
)
= a(
10
)
︸︷︷︸
|v1〉
+ b(
01
)
︸︷︷︸
|v2〉
|v1〉 and |v2〉 span the complex number vector space.
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
Spanning vectors
A vector space may be spanned by different sets ofspanning vectors.
|v1〉 =1√2
(11
)
|v2〉 =1√2
(1−1
)
Any complex number : |v〉 = (a,b)
|v〉 =a + b√
2|v1〉 +
a − b√2
|v2〉
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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.
A
(∑
i
ai |vi〉)
=∑
i
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〉
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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.
A
(∑
i
ai |vi〉)
=∑
i
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〉
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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.
A
(∑
i
ai |vi〉)
=∑
i
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〉
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
Operators as matrices
If we denote states by vectors, then it’s convenient todenote operators as matrices.
Reason : Matrix algebra is easy!
I =
(1 00 1
)
, |v〉 =
(ab
)
⇒ I|v〉 =
(1 00 1
)(ab
)
=
(ab
)
= |v〉
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
Operators as matrices
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〉 =∑
i
Aij |wi〉
{ Aij } may be interpreted as elements of a matrix.
Note on jargon : Spanning vector set ↔ Basis
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
Operators as matrices
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〉 =∑
i
Aij |wi〉
{ Aij } may be interpreted as elements of a matrix.
Note on jargon : Spanning vector set ↔ Basis
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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
)
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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 ]
w1
w2...
wn
=
n∑
i=1
v∗i wi
Orthonormality : A set of vectors are said to beorthonormal if
〈vi |vj 〉 =
{
0 if i = j
1 if i 6= j
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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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〉〈i |)
|v〉 =∑
i
|i〉〈i |v〉 =∑
i
vi |i〉 = |v〉 ⇒ |i〉〈i | = I
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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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〉〈i |)
|v〉 =∑
i
|i〉〈i |v〉 =∑
i
vi |i〉 = |v〉 ⇒ |i〉〈i | = I
Hande Üstünel Quantum information and quantum computing
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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.
Hande Üstünel Quantum information and quantum computing
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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)
(∑
i
ai |vi 〉 ⊗ |wi〉)
≡∑
i
(aiA|vi〉 ⊗ B|wi〉)
Useful notation : |v〉⊗k means that |v〉 is tensored withitself k times.
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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)
(∑
i
ai |vi 〉 ⊗ |wi〉)
≡∑
i
(aiA|vi〉 ⊗ B|wi〉)
Useful notation : |v〉⊗k means that |v〉 is tensored withitself k times.
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
Dirac notationLinear algebraOperator relations
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)
(∑
i
ai |vi 〉 ⊗ |wi〉)
≡∑
i
(aiA|vi〉 ⊗ B|wi〉)
Useful notation : |v〉⊗k means that |v〉 is tensored withitself k times.
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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Hadamard operator
Defined on a two-dimensional space with spanning vectors
|v0〉 = |0〉 =
(10
)
and |v1〉 = |1〉 =
(01
)
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
∑
x,y
(−1)x·y |x〉〈y |
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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 :
A =
(2 1−1 2
)
|v〉 =
(1−1
)
A|v〉 = λ|v〉 where λ = 1
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Eigenvectors and eigenvalues
Degenaracy
Different eigenvectors having the same eigenvalue are said tobe degenarate.
The matrix
A =
2 0 10 2 01 0 1
has two eigenvectors |v1〉 = [1 0 0] and |v2〉 = [0 1 0]corresponding to the eigenvalue 2.
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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 :
Z =
(1 00 −1
)
= |0〉〈0| − |1〉〈1|
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More on linear operators
Adjoints
For a linear operator A, there exists an operator A† whichsatisfies
〈v |A|w〉 = 〈w |A†|v〉A† is known as the adjoint or Hermitian conjugate of A.
In matrix notation, the adjoint is the transpose of the complexconjugate.
(1 + 3i 2i1 + i 1 − 4i
)†
=
(1 − 3i 1 − i−2i 1 + 4i
)
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Properties of adjoints :1 Action on vectors : (A|v〉)† = 〈v |A†
2 Adjoints of outer product : (|w〉〈v |)† = |v〉〈w |3 (A†)† = A
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More on linear operators
Hermitian operators
An operator is Hermitian if it is equal to its adjoint : A = A†
Unitary operators
An operator is unitary if its inverse equals its adjoint : UU† = I
Unitary operators preserve inner products :
(U|v〉,U|w〉) = 〈v |U†U|w〉 = 〈v |w〉
Hande Üstünel Quantum information and quantum computing
IntroductionBasics of Quantum Mechanics
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More on linear operators
Hermitian operators
An operator is Hermitian if it is equal to its adjoint : A = A†
Unitary operators
An operator is unitary if its inverse equals its adjoint : UU† = I
Unitary operators preserve inner products :
(U|v〉,U|w〉) = 〈v |U†U|w〉 = 〈v |w〉
Hande Üstünel Quantum information and quantum computing
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Trace of a matrix
Trace of a matrix is the sum of its diagonal elements.
tr(A) ≡∑
i
Aii
Properties :
Trace is cyclic
tr(ABC) = tr(BCA) = tr(CAB)
Unitary transforms preserve trace
tr(UAU†) = tr(U†UA) = tr(A)
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Projectors
Suppose that
W is a k-dimensional subspace of the d -dimensionalvector space V .
W has an orthonormal basis |1〉, |2〉, · · · , |d〉V has an orthonormal basis |1〉, |2〉, · · · , |k〉
The following operator
P ≡k∑
i=1
|i〉〈i |
is then a projector onto the subspace of W .
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Projectors (properties)
P is Hermitian :
P† =
(k∑
i=1
|i〉〈i |)†
=
k∑
i=1
|i〉〈i | = P
P is idempotent (its higher powers is equal to itself)
P2 = P
We can define a complimentary operator to P, which wedenote by Q
Q = I − P
which projects on the subspace of V spanned by thevectors |k + 1〉, |k + 2〉, · · · , |d〉
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P is Hermitian :
P† =
(k∑
i=1
|i〉〈i |)†
=
k∑
i=1
|i〉〈i | = P
P is idempotent (its higher powers is equal to itself)
P2 = P
We can define a complimentary operator to P, which wedenote by Q
Q = I − P
which projects on the subspace of V spanned by thevectors |k + 1〉, |k + 2〉, · · · , |d〉
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Projectors (properties)
P is Hermitian :
P† =
(k∑
i=1
|i〉〈i |)†
=
k∑
i=1
|i〉〈i | = P
P is idempotent (its higher powers is equal to itself)
P2 = P
We can define a complimentary operator to P, which wedenote by Q
Q = I − P
which projects on the subspace of V spanned by thevectors |k + 1〉, |k + 2〉, · · · , |d〉
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Commutator and anti-commutator
The commutator and the anti-commutator are new operatorsdefined through applying two operators in succession asfollows :
Commutator : [A,B] ≡ AB − BA
Anti-commutator : {A,B} ≡ AB + BA
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Commutator and anti-commutator
Commutation relations for Pauli matrices
If X ,Y and Z are the Pauli matrices as defined previously then
Commutator :
[X ,Y ] = 2iZ , [Y ,Z ] = 2iX , [Z ,X ] = 2iX
Anti-commutator :
{X ,Y} = {Y ,Z} = {Z ,X} = 0
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