foundations of quantum programming lecture 2: basics of quantum mechanics · 2016-08-15 ·...
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![Page 1: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/1.jpg)
Foundations of Quantum Programming
Lecture 2: Basics of Quantum Mechanics
Mingsheng Ying
University of Technology Sydney, Australia
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Outline
Hilbert Spaces
Linear Operators
Quantum Measurements
Tensor Products
Density Operators
Quantum Operations
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Outline
Hilbert Spaces
Linear Operators
Quantum Measurements
Tensor Products
Density Operators
Quantum Operations
![Page 4: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/4.jpg)
Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → H
I scalar multiplication · : C×H → H
satisfying the conditions:
1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.3. + has the zero element 0, called the zero vector, such that
0 + |ϕ〉 = |ϕ〉.4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.6. λ(µ|ϕ〉) = λµ|ϕ〉.7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
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Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → HI scalar multiplication · : C×H → H
satisfying the conditions:
1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.3. + has the zero element 0, called the zero vector, such that
0 + |ϕ〉 = |ϕ〉.4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.6. λ(µ|ϕ〉) = λµ|ϕ〉.7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
![Page 6: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/6.jpg)
Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → HI scalar multiplication · : C×H → H
satisfying the conditions:1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.
2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.3. + has the zero element 0, called the zero vector, such that
0 + |ϕ〉 = |ϕ〉.4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.6. λ(µ|ϕ〉) = λµ|ϕ〉.7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
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Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → HI scalar multiplication · : C×H → H
satisfying the conditions:1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.
3. + has the zero element 0, called the zero vector, such that0 + |ϕ〉 = |ϕ〉.
4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.6. λ(µ|ϕ〉) = λµ|ϕ〉.7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
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Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → HI scalar multiplication · : C×H → H
satisfying the conditions:1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.3. + has the zero element 0, called the zero vector, such that
0 + |ϕ〉 = |ϕ〉.
4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.6. λ(µ|ϕ〉) = λµ|ϕ〉.7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
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Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → HI scalar multiplication · : C×H → H
satisfying the conditions:1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.3. + has the zero element 0, called the zero vector, such that
0 + |ϕ〉 = |ϕ〉.4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.6. λ(µ|ϕ〉) = λµ|ϕ〉.7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
![Page 10: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/10.jpg)
Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → HI scalar multiplication · : C×H → H
satisfying the conditions:1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.3. + has the zero element 0, called the zero vector, such that
0 + |ϕ〉 = |ϕ〉.4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.
6. λ(µ|ϕ〉) = λµ|ϕ〉.7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
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Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → HI scalar multiplication · : C×H → H
satisfying the conditions:1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.3. + has the zero element 0, called the zero vector, such that
0 + |ϕ〉 = |ϕ〉.4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.6. λ(µ|ϕ〉) = λµ|ϕ〉.
7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
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Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → HI scalar multiplication · : C×H → H
satisfying the conditions:1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.3. + has the zero element 0, called the zero vector, such that
0 + |ϕ〉 = |ϕ〉.4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.6. λ(µ|ϕ〉) = λµ|ϕ〉.7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.
8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
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Vector spacesA (complex) vector space is a nonempty setH with two operations:I vector addition + : H×H → HI scalar multiplication · : C×H → H
satisfying the conditions:1. + is commutative: |ϕ〉+ |ψ〉 = |ψ〉+ |ϕ〉.2. + is associative: |ϕ〉+ (|ψ〉+ |χ〉) = (|ϕ〉+ |ψ〉) + |χ〉.3. + has the zero element 0, called the zero vector, such that
0 + |ϕ〉 = |ϕ〉.4. each |ϕ〉 ∈ H has its negative vector −|ϕ〉 such that|ϕ〉+ (−|ϕ〉) = 0.
5. 1|ϕ〉 = |ϕ〉.6. λ(µ|ϕ〉) = λµ|ϕ〉.7. (λ + µ)|ϕ〉 = λ|ϕ〉+ µ|ϕ〉.8. λ(|ϕ〉+ |ψ〉) = λ|ϕ〉+ λ|ψ〉.
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Inner productAn inner product space is a vector spaceH equipped with an innerproduct:
〈·|·〉 : H×H → C
satisfying the properties:1. 〈ϕ|ϕ〉 ≥ 0 with equality if and only if |ϕ〉 = 0;
2. 〈ϕ|ψ〉 = 〈ψ|ϕ〉∗;3. 〈ϕ|λ1ψ1 + λ2ψ2〉 = λ1〈ϕ|ψ1〉+ λ2〈ϕ|ψ2〉.
I If 〈ϕ|ψ〉 = 0, then |ϕ〉 and |ψ〉 are orthogonal, |ϕ〉 ⊥ |ψ〉.I The length of a vector |ψ〉 ∈ H is
||ψ|| =√〈ψ|ψ〉.
I A vector |ψ〉 is a unit vector if ||ψ|| = 1.
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Inner productAn inner product space is a vector spaceH equipped with an innerproduct:
〈·|·〉 : H×H → C
satisfying the properties:1. 〈ϕ|ϕ〉 ≥ 0 with equality if and only if |ϕ〉 = 0;2. 〈ϕ|ψ〉 = 〈ψ|ϕ〉∗;
3. 〈ϕ|λ1ψ1 + λ2ψ2〉 = λ1〈ϕ|ψ1〉+ λ2〈ϕ|ψ2〉.
I If 〈ϕ|ψ〉 = 0, then |ϕ〉 and |ψ〉 are orthogonal, |ϕ〉 ⊥ |ψ〉.I The length of a vector |ψ〉 ∈ H is
||ψ|| =√〈ψ|ψ〉.
I A vector |ψ〉 is a unit vector if ||ψ|| = 1.
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Inner productAn inner product space is a vector spaceH equipped with an innerproduct:
〈·|·〉 : H×H → C
satisfying the properties:1. 〈ϕ|ϕ〉 ≥ 0 with equality if and only if |ϕ〉 = 0;2. 〈ϕ|ψ〉 = 〈ψ|ϕ〉∗;3. 〈ϕ|λ1ψ1 + λ2ψ2〉 = λ1〈ϕ|ψ1〉+ λ2〈ϕ|ψ2〉.
I If 〈ϕ|ψ〉 = 0, then |ϕ〉 and |ψ〉 are orthogonal, |ϕ〉 ⊥ |ψ〉.I The length of a vector |ψ〉 ∈ H is
||ψ|| =√〈ψ|ψ〉.
I A vector |ψ〉 is a unit vector if ||ψ|| = 1.
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Inner productAn inner product space is a vector spaceH equipped with an innerproduct:
〈·|·〉 : H×H → C
satisfying the properties:1. 〈ϕ|ϕ〉 ≥ 0 with equality if and only if |ϕ〉 = 0;2. 〈ϕ|ψ〉 = 〈ψ|ϕ〉∗;3. 〈ϕ|λ1ψ1 + λ2ψ2〉 = λ1〈ϕ|ψ1〉+ λ2〈ϕ|ψ2〉.
I If 〈ϕ|ψ〉 = 0, then |ϕ〉 and |ψ〉 are orthogonal, |ϕ〉 ⊥ |ψ〉.
I The length of a vector |ψ〉 ∈ H is
||ψ|| =√〈ψ|ψ〉.
I A vector |ψ〉 is a unit vector if ||ψ|| = 1.
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Inner productAn inner product space is a vector spaceH equipped with an innerproduct:
〈·|·〉 : H×H → C
satisfying the properties:1. 〈ϕ|ϕ〉 ≥ 0 with equality if and only if |ϕ〉 = 0;2. 〈ϕ|ψ〉 = 〈ψ|ϕ〉∗;3. 〈ϕ|λ1ψ1 + λ2ψ2〉 = λ1〈ϕ|ψ1〉+ λ2〈ϕ|ψ2〉.
I If 〈ϕ|ψ〉 = 0, then |ϕ〉 and |ψ〉 are orthogonal, |ϕ〉 ⊥ |ψ〉.I The length of a vector |ψ〉 ∈ H is
||ψ|| =√〈ψ|ψ〉.
I A vector |ψ〉 is a unit vector if ||ψ|| = 1.
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Inner productAn inner product space is a vector spaceH equipped with an innerproduct:
〈·|·〉 : H×H → C
satisfying the properties:1. 〈ϕ|ϕ〉 ≥ 0 with equality if and only if |ϕ〉 = 0;2. 〈ϕ|ψ〉 = 〈ψ|ϕ〉∗;3. 〈ϕ|λ1ψ1 + λ2ψ2〉 = λ1〈ϕ|ψ1〉+ λ2〈ϕ|ψ2〉.
I If 〈ϕ|ψ〉 = 0, then |ϕ〉 and |ψ〉 are orthogonal, |ϕ〉 ⊥ |ψ〉.I The length of a vector |ψ〉 ∈ H is
||ψ|| =√〈ψ|ψ〉.
I A vector |ψ〉 is a unit vector if ||ψ|| = 1.
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Cauchy-limitLet {|ψn〉} be a sequence of vectors inH and |ψ〉 ∈ H.
1. If for any ε > 0, there exists a positive integer N such that||ψm − ψn|| < ε for all m, n ≥ N, then {|ψn〉} is a Cauchysequence.
2. If for any ε > 0, there exists a positive integer N such that||ψn − ψ|| < ε for all n ≥ N, then |ψ〉 is a limit of {|ψn〉},|ψ〉 = limn→∞ |ψn〉.
Hilbert spacesA Hilbert space is a complete inner product space; that is, an innerproduct space in which each Cauchy sequence of vectors has a limit.
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Cauchy-limitLet {|ψn〉} be a sequence of vectors inH and |ψ〉 ∈ H.
1. If for any ε > 0, there exists a positive integer N such that||ψm − ψn|| < ε for all m, n ≥ N, then {|ψn〉} is a Cauchysequence.
2. If for any ε > 0, there exists a positive integer N such that||ψn − ψ|| < ε for all n ≥ N, then |ψ〉 is a limit of {|ψn〉},|ψ〉 = limn→∞ |ψn〉.
Hilbert spacesA Hilbert space is a complete inner product space; that is, an innerproduct space in which each Cauchy sequence of vectors has a limit.
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BasesA finite or countably infinite family {|ψi〉} of unit vectors is anorthonormal basis ofH if
1. {|ψi〉} are pairwise orthogonal: |ψi〉 ⊥ |ψj〉 for any i, j with i , j;
2. {|ψi〉} span the whole spaceH: each |ψ〉 ∈ H can be written as alinear combination:
|ψ〉 = ∑i
λi|ψi〉.
I The numbers of vectors in any two orthonormal bases are thesame. It is called the dimension ofH, dimH.
I If an orthonormal basis contains infinitely many vectors, thendimH = ∞.
I If dimH = n, fix an orthonormal basis {|ψ1〉, ..., |ψn〉}, then avector |ψ〉 = ∑n
i=1 λi|ψi〉 ∈ H is represented by the vector in Cn: λ1...λn
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BasesA finite or countably infinite family {|ψi〉} of unit vectors is anorthonormal basis ofH if
1. {|ψi〉} are pairwise orthogonal: |ψi〉 ⊥ |ψj〉 for any i, j with i , j;2. {|ψi〉} span the whole spaceH: each |ψ〉 ∈ H can be written as a
linear combination:|ψ〉 = ∑
iλi|ψi〉.
I The numbers of vectors in any two orthonormal bases are thesame. It is called the dimension ofH, dimH.
I If an orthonormal basis contains infinitely many vectors, thendimH = ∞.
I If dimH = n, fix an orthonormal basis {|ψ1〉, ..., |ψn〉}, then avector |ψ〉 = ∑n
i=1 λi|ψi〉 ∈ H is represented by the vector in Cn: λ1...λn
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BasesA finite or countably infinite family {|ψi〉} of unit vectors is anorthonormal basis ofH if
1. {|ψi〉} are pairwise orthogonal: |ψi〉 ⊥ |ψj〉 for any i, j with i , j;2. {|ψi〉} span the whole spaceH: each |ψ〉 ∈ H can be written as a
linear combination:|ψ〉 = ∑
iλi|ψi〉.
I The numbers of vectors in any two orthonormal bases are thesame. It is called the dimension ofH, dimH.
I If an orthonormal basis contains infinitely many vectors, thendimH = ∞.
I If dimH = n, fix an orthonormal basis {|ψ1〉, ..., |ψn〉}, then avector |ψ〉 = ∑n
i=1 λi|ψi〉 ∈ H is represented by the vector in Cn: λ1...λn
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BasesA finite or countably infinite family {|ψi〉} of unit vectors is anorthonormal basis ofH if
1. {|ψi〉} are pairwise orthogonal: |ψi〉 ⊥ |ψj〉 for any i, j with i , j;2. {|ψi〉} span the whole spaceH: each |ψ〉 ∈ H can be written as a
linear combination:|ψ〉 = ∑
iλi|ψi〉.
I The numbers of vectors in any two orthonormal bases are thesame. It is called the dimension ofH, dimH.
I If an orthonormal basis contains infinitely many vectors, thendimH = ∞.
I If dimH = n, fix an orthonormal basis {|ψ1〉, ..., |ψn〉}, then avector |ψ〉 = ∑n
i=1 λi|ψi〉 ∈ H is represented by the vector in Cn: λ1...λn
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BasesA finite or countably infinite family {|ψi〉} of unit vectors is anorthonormal basis ofH if
1. {|ψi〉} are pairwise orthogonal: |ψi〉 ⊥ |ψj〉 for any i, j with i , j;2. {|ψi〉} span the whole spaceH: each |ψ〉 ∈ H can be written as a
linear combination:|ψ〉 = ∑
iλi|ψi〉.
I The numbers of vectors in any two orthonormal bases are thesame. It is called the dimension ofH, dimH.
I If an orthonormal basis contains infinitely many vectors, thendimH = ∞.
I If dimH = n, fix an orthonormal basis {|ψ1〉, ..., |ψn〉}, then avector |ψ〉 = ∑n
i=1 λi|ψi〉 ∈ H is represented by the vector in Cn: λ1...λn
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Closed-subspaceLetH be a Hilbert space.
1. If X ⊆ H, and for any |ϕ〉, |ψ〉 ∈ X and λ ∈ C,
1.1 |ϕ〉+ |ψ〉 ∈ X;1.2 λ|ϕ〉 ∈ X,
then X is called a subspace ofH.
2. For each X ⊆ H, its closure X is the set of limits limn→∞ |ψn〉 ofsequences {|ψn〉} in X.
3. A subspace X ofH is closed if X = X.
I For X ⊆ H, the space spanned by X:
spanX =
{n
∑i=1
λi|ψi〉 : n ≥ 0, λi ∈ C and |ψi〉 ∈ X (i = 1, ..., n)
}
I spanX is the closed subspace generated by X.
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Closed-subspaceLetH be a Hilbert space.
1. If X ⊆ H, and for any |ϕ〉, |ψ〉 ∈ X and λ ∈ C,1.1 |ϕ〉+ |ψ〉 ∈ X;
1.2 λ|ϕ〉 ∈ X,
then X is called a subspace ofH.
2. For each X ⊆ H, its closure X is the set of limits limn→∞ |ψn〉 ofsequences {|ψn〉} in X.
3. A subspace X ofH is closed if X = X.
I For X ⊆ H, the space spanned by X:
spanX =
{n
∑i=1
λi|ψi〉 : n ≥ 0, λi ∈ C and |ψi〉 ∈ X (i = 1, ..., n)
}
I spanX is the closed subspace generated by X.
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Closed-subspaceLetH be a Hilbert space.
1. If X ⊆ H, and for any |ϕ〉, |ψ〉 ∈ X and λ ∈ C,1.1 |ϕ〉+ |ψ〉 ∈ X;1.2 λ|ϕ〉 ∈ X,
then X is called a subspace ofH.
2. For each X ⊆ H, its closure X is the set of limits limn→∞ |ψn〉 ofsequences {|ψn〉} in X.
3. A subspace X ofH is closed if X = X.
I For X ⊆ H, the space spanned by X:
spanX =
{n
∑i=1
λi|ψi〉 : n ≥ 0, λi ∈ C and |ψi〉 ∈ X (i = 1, ..., n)
}
I spanX is the closed subspace generated by X.
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Closed-subspaceLetH be a Hilbert space.
1. If X ⊆ H, and for any |ϕ〉, |ψ〉 ∈ X and λ ∈ C,1.1 |ϕ〉+ |ψ〉 ∈ X;1.2 λ|ϕ〉 ∈ X,
then X is called a subspace ofH.2. For each X ⊆ H, its closure X is the set of limits limn→∞ |ψn〉 of
sequences {|ψn〉} in X.
3. A subspace X ofH is closed if X = X.
I For X ⊆ H, the space spanned by X:
spanX =
{n
∑i=1
λi|ψi〉 : n ≥ 0, λi ∈ C and |ψi〉 ∈ X (i = 1, ..., n)
}
I spanX is the closed subspace generated by X.
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Closed-subspaceLetH be a Hilbert space.
1. If X ⊆ H, and for any |ϕ〉, |ψ〉 ∈ X and λ ∈ C,1.1 |ϕ〉+ |ψ〉 ∈ X;1.2 λ|ϕ〉 ∈ X,
then X is called a subspace ofH.2. For each X ⊆ H, its closure X is the set of limits limn→∞ |ψn〉 of
sequences {|ψn〉} in X.3. A subspace X ofH is closed if X = X.
I For X ⊆ H, the space spanned by X:
spanX =
{n
∑i=1
λi|ψi〉 : n ≥ 0, λi ∈ C and |ψi〉 ∈ X (i = 1, ..., n)
}
I spanX is the closed subspace generated by X.
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Closed-subspaceLetH be a Hilbert space.
1. If X ⊆ H, and for any |ϕ〉, |ψ〉 ∈ X and λ ∈ C,1.1 |ϕ〉+ |ψ〉 ∈ X;1.2 λ|ϕ〉 ∈ X,
then X is called a subspace ofH.2. For each X ⊆ H, its closure X is the set of limits limn→∞ |ψn〉 of
sequences {|ψn〉} in X.3. A subspace X ofH is closed if X = X.
I For X ⊆ H, the space spanned by X:
spanX =
{n
∑i=1
λi|ψi〉 : n ≥ 0, λi ∈ C and |ψi〉 ∈ X (i = 1, ..., n)
}
I spanX is the closed subspace generated by X.
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Closed-subspaceLetH be a Hilbert space.
1. If X ⊆ H, and for any |ϕ〉, |ψ〉 ∈ X and λ ∈ C,1.1 |ϕ〉+ |ψ〉 ∈ X;1.2 λ|ϕ〉 ∈ X,
then X is called a subspace ofH.2. For each X ⊆ H, its closure X is the set of limits limn→∞ |ψn〉 of
sequences {|ψn〉} in X.3. A subspace X ofH is closed if X = X.
I For X ⊆ H, the space spanned by X:
spanX =
{n
∑i=1
λi|ψi〉 : n ≥ 0, λi ∈ C and |ψi〉 ∈ X (i = 1, ..., n)
}
I spanX is the closed subspace generated by X.
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1. For any X, Y ⊆ H, X and Y are orthogonal, X ⊥ Y, if |ϕ〉 ⊥ |ψ〉for all |ϕ〉 ∈ X and |ψ〉 ∈ Y.
2. The orthocomplement of a closed subspace X ofH is
X⊥ = {|ϕ〉 ∈ H : |ϕ〉 ⊥ X}.
3. The orthocomplement X⊥ is a closed subspace ofH, (X⊥)⊥ = X.4. Let X, Y be two subspaces ofH. Then
X⊕ Y = {|ϕ〉+ |ψ〉 : |ϕ〉 ∈ X and |ψ〉 ∈ Y}.
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1. For any X, Y ⊆ H, X and Y are orthogonal, X ⊥ Y, if |ϕ〉 ⊥ |ψ〉for all |ϕ〉 ∈ X and |ψ〉 ∈ Y.
2. The orthocomplement of a closed subspace X ofH is
X⊥ = {|ϕ〉 ∈ H : |ϕ〉 ⊥ X}.
3. The orthocomplement X⊥ is a closed subspace ofH, (X⊥)⊥ = X.4. Let X, Y be two subspaces ofH. Then
X⊕ Y = {|ϕ〉+ |ψ〉 : |ϕ〉 ∈ X and |ψ〉 ∈ Y}.
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1. For any X, Y ⊆ H, X and Y are orthogonal, X ⊥ Y, if |ϕ〉 ⊥ |ψ〉for all |ϕ〉 ∈ X and |ψ〉 ∈ Y.
2. The orthocomplement of a closed subspace X ofH is
X⊥ = {|ϕ〉 ∈ H : |ϕ〉 ⊥ X}.
3. The orthocomplement X⊥ is a closed subspace ofH, (X⊥)⊥ = X.
4. Let X, Y be two subspaces ofH. Then
X⊕ Y = {|ϕ〉+ |ψ〉 : |ϕ〉 ∈ X and |ψ〉 ∈ Y}.
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1. For any X, Y ⊆ H, X and Y are orthogonal, X ⊥ Y, if |ϕ〉 ⊥ |ψ〉for all |ϕ〉 ∈ X and |ψ〉 ∈ Y.
2. The orthocomplement of a closed subspace X ofH is
X⊥ = {|ϕ〉 ∈ H : |ϕ〉 ⊥ X}.
3. The orthocomplement X⊥ is a closed subspace ofH, (X⊥)⊥ = X.4. Let X, Y be two subspaces ofH. Then
X⊕ Y = {|ϕ〉+ |ψ〉 : |ϕ〉 ∈ X and |ψ〉 ∈ Y}.
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Postulate of quantum mechanics 1
I The state space of a closed (i.e. an isolated) quantum system isrepresented by a Hilbert space.
I A pure state of the system is described by a unit vector in itsstate space.
I A linear combination |ψ〉 = ∑ni=1 λi|ψi〉 of states |ψ1〉, ..., |ψn〉 is
often called their superpositionI Complex coefficients λi are called probability amplitudes.
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Postulate of quantum mechanics 1
I The state space of a closed (i.e. an isolated) quantum system isrepresented by a Hilbert space.
I A pure state of the system is described by a unit vector in itsstate space.
I A linear combination |ψ〉 = ∑ni=1 λi|ψi〉 of states |ψ1〉, ..., |ψn〉 is
often called their superpositionI Complex coefficients λi are called probability amplitudes.
![Page 40: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/40.jpg)
Postulate of quantum mechanics 1
I The state space of a closed (i.e. an isolated) quantum system isrepresented by a Hilbert space.
I A pure state of the system is described by a unit vector in itsstate space.
I A linear combination |ψ〉 = ∑ni=1 λi|ψi〉 of states |ψ1〉, ..., |ψn〉 is
often called their superpositionI Complex coefficients λi are called probability amplitudes.
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Postulate of quantum mechanics 1
I The state space of a closed (i.e. an isolated) quantum system isrepresented by a Hilbert space.
I A pure state of the system is described by a unit vector in itsstate space.
I A linear combination |ψ〉 = ∑ni=1 λi|ψi〉 of states |ψ1〉, ..., |ψn〉 is
often called their superposition
I Complex coefficients λi are called probability amplitudes.
![Page 42: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/42.jpg)
Postulate of quantum mechanics 1
I The state space of a closed (i.e. an isolated) quantum system isrepresented by a Hilbert space.
I A pure state of the system is described by a unit vector in itsstate space.
I A linear combination |ψ〉 = ∑ni=1 λi|ψi〉 of states |ψ1〉, ..., |ψn〉 is
often called their superpositionI Complex coefficients λi are called probability amplitudes.
![Page 43: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/43.jpg)
Example: Qubits
I 2-dimensional Hilbert space:
H2 = C2 = {α|0〉+ β|1〉 : α, β ∈ C}.
I Inner product:
(α|0〉+ β|1〉, α′|0〉+ β′|1〉) = α∗α′ + β∗β′.
I {|0〉, |1〉} is an orthonormal basis ofH2, the computational basis.I A state of a qubit is described by a unit vector |ψ〉 = α|0〉+ β|1〉
with |α|2 + |β|2 = 1.I
|+〉 = |0〉+ |1〉√2
=1√
2
(11
), |−〉 = |0〉 − |1〉√
2=
1√2
(1−1
)
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Example: Qubits
I 2-dimensional Hilbert space:
H2 = C2 = {α|0〉+ β|1〉 : α, β ∈ C}.
I Inner product:
(α|0〉+ β|1〉, α′|0〉+ β′|1〉) = α∗α′ + β∗β′.
I {|0〉, |1〉} is an orthonormal basis ofH2, the computational basis.I A state of a qubit is described by a unit vector |ψ〉 = α|0〉+ β|1〉
with |α|2 + |β|2 = 1.I
|+〉 = |0〉+ |1〉√2
=1√
2
(11
), |−〉 = |0〉 − |1〉√
2=
1√2
(1−1
)
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Example: Qubits
I 2-dimensional Hilbert space:
H2 = C2 = {α|0〉+ β|1〉 : α, β ∈ C}.
I Inner product:
(α|0〉+ β|1〉, α′|0〉+ β′|1〉) = α∗α′ + β∗β′.
I {|0〉, |1〉} is an orthonormal basis ofH2, the computational basis.
I A state of a qubit is described by a unit vector |ψ〉 = α|0〉+ β|1〉with |α|2 + |β|2 = 1.
I
|+〉 = |0〉+ |1〉√2
=1√
2
(11
), |−〉 = |0〉 − |1〉√
2=
1√2
(1−1
)
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Example: Qubits
I 2-dimensional Hilbert space:
H2 = C2 = {α|0〉+ β|1〉 : α, β ∈ C}.
I Inner product:
(α|0〉+ β|1〉, α′|0〉+ β′|1〉) = α∗α′ + β∗β′.
I {|0〉, |1〉} is an orthonormal basis ofH2, the computational basis.I A state of a qubit is described by a unit vector |ψ〉 = α|0〉+ β|1〉
with |α|2 + |β|2 = 1.
I
|+〉 = |0〉+ |1〉√2
=1√
2
(11
), |−〉 = |0〉 − |1〉√
2=
1√2
(1−1
)
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Example: Qubits
I 2-dimensional Hilbert space:
H2 = C2 = {α|0〉+ β|1〉 : α, β ∈ C}.
I Inner product:
(α|0〉+ β|1〉, α′|0〉+ β′|1〉) = α∗α′ + β∗β′.
I {|0〉, |1〉} is an orthonormal basis ofH2, the computational basis.I A state of a qubit is described by a unit vector |ψ〉 = α|0〉+ β|1〉
with |α|2 + |β|2 = 1.I
|+〉 = |0〉+ |1〉√2
=1√
2
(11
), |−〉 = |0〉 − |1〉√
2=
1√2
(1−1
)
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Example: Square summable sequences
I The space of square summable sequences:
H∞ =
{∞
∑n=−∞
αn|n〉 : αn ∈ C for all n ∈ Z and∞
∑n=−∞
|αn|2 < ∞
}.
I Inner product:(∞
∑n=−∞
αn|n〉,∞
∑n=−∞
α′|n〉)
=∞
∑n=−∞
α∗nα′n.
I {|n〉 : n ∈ Z} is an orthonormal basis,H∞ isinfinite-dimensional.
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Example: Square summable sequences
I The space of square summable sequences:
H∞ =
{∞
∑n=−∞
αn|n〉 : αn ∈ C for all n ∈ Z and∞
∑n=−∞
|αn|2 < ∞
}.
I Inner product:(∞
∑n=−∞
αn|n〉,∞
∑n=−∞
α′|n〉)
=∞
∑n=−∞
α∗nα′n.
I {|n〉 : n ∈ Z} is an orthonormal basis,H∞ isinfinite-dimensional.
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Example: Square summable sequences
I The space of square summable sequences:
H∞ =
{∞
∑n=−∞
αn|n〉 : αn ∈ C for all n ∈ Z and∞
∑n=−∞
|αn|2 < ∞
}.
I Inner product:(∞
∑n=−∞
αn|n〉,∞
∑n=−∞
α′|n〉)
=∞
∑n=−∞
α∗nα′n.
I {|n〉 : n ∈ Z} is an orthonormal basis,H∞ isinfinite-dimensional.
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Outline
Hilbert Spaces
Linear Operators
Quantum Measurements
Tensor Products
Density Operators
Quantum Operations
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Linear OperatorsLetH and K be Hilbert spaces. A mapping
A : H → K
is a linear operator if it satisfies the conditions:1. A(|ϕ〉+ |ψ〉) = A|ϕ〉+ A|ψ〉;
2. A(λ|ψ〉) = λA|ψ〉.
Examples
I Identity operator maps each vector inH to itself, denoted IH.I Zero operator maps every vector inH to the zero vector, denoted
0H.I For vectors |ϕ〉, |ψ〉 ∈ H, their outer product is the operator|ϕ〉〈ψ| inH:
(|ϕ〉〈ψ|)|χ〉 = 〈ψ|χ〉|ϕ〉.
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Linear OperatorsLetH and K be Hilbert spaces. A mapping
A : H → K
is a linear operator if it satisfies the conditions:1. A(|ϕ〉+ |ψ〉) = A|ϕ〉+ A|ψ〉;2. A(λ|ψ〉) = λA|ψ〉.
Examples
I Identity operator maps each vector inH to itself, denoted IH.I Zero operator maps every vector inH to the zero vector, denoted
0H.I For vectors |ϕ〉, |ψ〉 ∈ H, their outer product is the operator|ϕ〉〈ψ| inH:
(|ϕ〉〈ψ|)|χ〉 = 〈ψ|χ〉|ϕ〉.
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Linear OperatorsLetH and K be Hilbert spaces. A mapping
A : H → K
is a linear operator if it satisfies the conditions:1. A(|ϕ〉+ |ψ〉) = A|ϕ〉+ A|ψ〉;2. A(λ|ψ〉) = λA|ψ〉.
Examples
I Identity operator maps each vector inH to itself, denoted IH.
I Zero operator maps every vector inH to the zero vector, denoted0H.
I For vectors |ϕ〉, |ψ〉 ∈ H, their outer product is the operator|ϕ〉〈ψ| inH:
(|ϕ〉〈ψ|)|χ〉 = 〈ψ|χ〉|ϕ〉.
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Linear OperatorsLetH and K be Hilbert spaces. A mapping
A : H → K
is a linear operator if it satisfies the conditions:1. A(|ϕ〉+ |ψ〉) = A|ϕ〉+ A|ψ〉;2. A(λ|ψ〉) = λA|ψ〉.
Examples
I Identity operator maps each vector inH to itself, denoted IH.I Zero operator maps every vector inH to the zero vector, denoted
0H.
I For vectors |ϕ〉, |ψ〉 ∈ H, their outer product is the operator|ϕ〉〈ψ| inH:
(|ϕ〉〈ψ|)|χ〉 = 〈ψ|χ〉|ϕ〉.
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Linear OperatorsLetH and K be Hilbert spaces. A mapping
A : H → K
is a linear operator if it satisfies the conditions:1. A(|ϕ〉+ |ψ〉) = A|ϕ〉+ A|ψ〉;2. A(λ|ψ〉) = λA|ψ〉.
Examples
I Identity operator maps each vector inH to itself, denoted IH.I Zero operator maps every vector inH to the zero vector, denoted
0H.I For vectors |ϕ〉, |ψ〉 ∈ H, their outer product is the operator|ϕ〉〈ψ| inH:
(|ϕ〉〈ψ|)|χ〉 = 〈ψ|χ〉|ϕ〉.
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Projection
I Let X be a closed subspace ofH and |ψ〉 ∈ H. Then there existuniquely |ψ0〉 ∈ X and |ψ1〉 ∈ X⊥ such that
|ψ〉 = |ψ0〉+ |ψ1〉.
I Vector |ψ0〉 is called the projection of |ψ〉 onto X, |ψ0〉 = PX|ψ〉.I For closed subspace X ofH, the operator
PX : H → X, |ψ〉 7→ PX|ψ〉
is the projector onto X.
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Projection
I Let X be a closed subspace ofH and |ψ〉 ∈ H. Then there existuniquely |ψ0〉 ∈ X and |ψ1〉 ∈ X⊥ such that
|ψ〉 = |ψ0〉+ |ψ1〉.
I Vector |ψ0〉 is called the projection of |ψ〉 onto X, |ψ0〉 = PX|ψ〉.
I For closed subspace X ofH, the operator
PX : H → X, |ψ〉 7→ PX|ψ〉
is the projector onto X.
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Projection
I Let X be a closed subspace ofH and |ψ〉 ∈ H. Then there existuniquely |ψ0〉 ∈ X and |ψ1〉 ∈ X⊥ such that
|ψ〉 = |ψ0〉+ |ψ1〉.
I Vector |ψ0〉 is called the projection of |ψ〉 onto X, |ψ0〉 = PX|ψ〉.I For closed subspace X ofH, the operator
PX : H → X, |ψ〉 7→ PX|ψ〉
is the projector onto X.
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Bounded operators
I An operator A is bounded if there is a constant C ≥ 0 such that
‖A|ψ〉‖ ≤ C · ‖ψ‖
for all |ψ〉 ∈ H.
I The norm of A is
‖A‖ = inf{C ≥ 0 : ||A|ψ〉|| ≤ C · ||ψ|| for all ψ ∈ H}.
I L(H) stands for the set of bounded operators inH.
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Bounded operators
I An operator A is bounded if there is a constant C ≥ 0 such that
‖A|ψ〉‖ ≤ C · ‖ψ‖
for all |ψ〉 ∈ H.I The norm of A is
‖A‖ = inf{C ≥ 0 : ||A|ψ〉|| ≤ C · ||ψ|| for all ψ ∈ H}.
I L(H) stands for the set of bounded operators inH.
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Bounded operators
I An operator A is bounded if there is a constant C ≥ 0 such that
‖A|ψ〉‖ ≤ C · ‖ψ‖
for all |ψ〉 ∈ H.I The norm of A is
‖A‖ = inf{C ≥ 0 : ||A|ψ〉|| ≤ C · ||ψ|| for all ψ ∈ H}.
I L(H) stands for the set of bounded operators inH.
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Operations of operators
(A + B)|ψ〉 = A|ψ〉+ B|ψ〉,(λA)|ψ〉 = λ(A|ψ〉),(BA)|ψ〉 = B(A|ψ〉).
Positive operatorsAn operator A ∈ L(H) is positive if for all states |ψ〉 ∈ H:
〈ψ|A|ψ〉 ≥ 0.
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Operations of operators
(A + B)|ψ〉 = A|ψ〉+ B|ψ〉,(λA)|ψ〉 = λ(A|ψ〉),(BA)|ψ〉 = B(A|ψ〉).
Positive operatorsAn operator A ∈ L(H) is positive if for all states |ψ〉 ∈ H:
〈ψ|A|ψ〉 ≥ 0.
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Löwner order
A v B if and only if B−A = B + (−1)A is positive.
Distance between operators
d(A, B) = sup|ψ〉||A|ψ〉 − B|ψ〉||
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Löwner order
A v B if and only if B−A = B + (−1)A is positive.
Distance between operators
d(A, B) = sup|ψ〉||A|ψ〉 − B|ψ〉||
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Matrix Representation of Operators
I When dimH = n, fix orthonormal basis {|ψ1〉, ..., |ψn〉}, A can berepresented by the n× n complex matrix:
A =(aij)
n×n =
a11 ... a1n...
an1 ... ann
where aij = 〈ψi|A|ψj〉 = (|ψi〉, A|ψj〉).
I If |ψ〉 = ∑ni=1 αi|ψi〉, then
A|ψ〉 = A
α1...αn
=
β1...βn
where βi = ∑n
j=1 aijαj.
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Matrix Representation of Operators
I When dimH = n, fix orthonormal basis {|ψ1〉, ..., |ψn〉}, A can berepresented by the n× n complex matrix:
A =(aij)
n×n =
a11 ... a1n...
an1 ... ann
where aij = 〈ψi|A|ψj〉 = (|ψi〉, A|ψj〉).
I If |ψ〉 = ∑ni=1 αi|ψi〉, then
A|ψ〉 = A
α1...αn
=
β1...βn
where βi = ∑n
j=1 aijαj.
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Unitary Transformations
I For any operator A ∈ L(H), there exists a unique operator A†
such that(A|ϕ〉, |ψ〉) =
(|ϕ〉, A†|ψ〉
).
I Operator A† is called the adjoint of A.I If A =
(aij)
n×n, then
A† =(bij)
n×n
with bij = a∗ji.
I An operator U ∈ L(H) is unitary if U†U = UU† = IH.I All unitary transformations U preserve inner product:
(U|ϕ〉, U|ψ〉) = 〈ϕ|ψ〉.
I If dimH = n, then a unitary operator is represented by an n× nunitary matrix U: U†U = In.
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Unitary Transformations
I For any operator A ∈ L(H), there exists a unique operator A†
such that(A|ϕ〉, |ψ〉) =
(|ϕ〉, A†|ψ〉
).
I Operator A† is called the adjoint of A.
I If A =(aij)
n×n, then
A† =(bij)
n×n
with bij = a∗ji.
I An operator U ∈ L(H) is unitary if U†U = UU† = IH.I All unitary transformations U preserve inner product:
(U|ϕ〉, U|ψ〉) = 〈ϕ|ψ〉.
I If dimH = n, then a unitary operator is represented by an n× nunitary matrix U: U†U = In.
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Unitary Transformations
I For any operator A ∈ L(H), there exists a unique operator A†
such that(A|ϕ〉, |ψ〉) =
(|ϕ〉, A†|ψ〉
).
I Operator A† is called the adjoint of A.I If A =
(aij)
n×n, then
A† =(bij)
n×n
with bij = a∗ji.
I An operator U ∈ L(H) is unitary if U†U = UU† = IH.I All unitary transformations U preserve inner product:
(U|ϕ〉, U|ψ〉) = 〈ϕ|ψ〉.
I If dimH = n, then a unitary operator is represented by an n× nunitary matrix U: U†U = In.
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Unitary Transformations
I For any operator A ∈ L(H), there exists a unique operator A†
such that(A|ϕ〉, |ψ〉) =
(|ϕ〉, A†|ψ〉
).
I Operator A† is called the adjoint of A.I If A =
(aij)
n×n, then
A† =(bij)
n×n
with bij = a∗ji.
I An operator U ∈ L(H) is unitary if U†U = UU† = IH.
I All unitary transformations U preserve inner product:
(U|ϕ〉, U|ψ〉) = 〈ϕ|ψ〉.
I If dimH = n, then a unitary operator is represented by an n× nunitary matrix U: U†U = In.
![Page 73: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/73.jpg)
Unitary Transformations
I For any operator A ∈ L(H), there exists a unique operator A†
such that(A|ϕ〉, |ψ〉) =
(|ϕ〉, A†|ψ〉
).
I Operator A† is called the adjoint of A.I If A =
(aij)
n×n, then
A† =(bij)
n×n
with bij = a∗ji.
I An operator U ∈ L(H) is unitary if U†U = UU† = IH.I All unitary transformations U preserve inner product:
(U|ϕ〉, U|ψ〉) = 〈ϕ|ψ〉.
I If dimH = n, then a unitary operator is represented by an n× nunitary matrix U: U†U = In.
![Page 74: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/74.jpg)
Unitary Transformations
I For any operator A ∈ L(H), there exists a unique operator A†
such that(A|ϕ〉, |ψ〉) =
(|ϕ〉, A†|ψ〉
).
I Operator A† is called the adjoint of A.I If A =
(aij)
n×n, then
A† =(bij)
n×n
with bij = a∗ji.
I An operator U ∈ L(H) is unitary if U†U = UU† = IH.I All unitary transformations U preserve inner product:
(U|ϕ〉, U|ψ〉) = 〈ϕ|ψ〉.
I If dimH = n, then a unitary operator is represented by an n× nunitary matrix U: U†U = In.
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Postulate of quantum mechanics 2
I Suppose that the states of a closed quantum system (i.e. a systemwithout interactions with its environment) at times t0 and t are|ψ0〉 and |ψ〉, respectively.
I Then they are related to each other by a unitary operator Uwhich depends only on the times t0 and t,
|ψ〉 = U|ψ0〉.
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Postulate of quantum mechanics 2
I Suppose that the states of a closed quantum system (i.e. a systemwithout interactions with its environment) at times t0 and t are|ψ0〉 and |ψ〉, respectively.
I Then they are related to each other by a unitary operator Uwhich depends only on the times t0 and t,
|ψ〉 = U|ψ0〉.
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Example: Hadamard transformation
H =1√
2
(1 11 −1
)
H|0〉 = H(
10
)=
1√2
(11
)= |+〉,
H|1〉 = H(
01
)=
1√2
(1−1
)= |−〉.
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Example: Translation
I Let k be an integer. The k-translation operator Tk inH∞ isdefined by
Tk|n〉 = |n + k〉
for all n ∈ Z.
I TL = T−1 and TR = T1. They moves a particle on the line oneposition to the left and to the right, respectively.
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Example: Translation
I Let k be an integer. The k-translation operator Tk inH∞ isdefined by
Tk|n〉 = |n + k〉
for all n ∈ Z.I TL = T−1 and TR = T1. They moves a particle on the line one
position to the left and to the right, respectively.
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Outline
Hilbert Spaces
Linear Operators
Quantum Measurements
Tensor Products
Density Operators
Quantum Operations
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Postulate of quantum mechanics 3
I A quantum measurement on a system with state Hilbert spaceHis described by a collection {Mm} ⊆ L(H) of operators satisfyingthe normalisation condition:
∑m
M†mMm = IH
I Mm are called measurement operators.I The index m stands for the measurement outcomes that may
occur in the experiment.I If the state of a quantum system is |ψ〉 immediately before the
measurement, then for each m,
I the probability that result m occurs in the measurement is
p(m) = ||Mm|ψ〉||2 = 〈ψ|M†mMm|ψ〉 (Born rule)
I the state of the system after the measurement with outcome m is
|ψm〉 =Mm|ψ〉√
p(m).
The normalisation condition implies that the probabilities for alloutcomes sum up to ∑m p(m) = 1.
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Postulate of quantum mechanics 3
I A quantum measurement on a system with state Hilbert spaceHis described by a collection {Mm} ⊆ L(H) of operators satisfyingthe normalisation condition:
∑m
M†mMm = IH
I Mm are called measurement operators.
I The index m stands for the measurement outcomes that mayoccur in the experiment.
I If the state of a quantum system is |ψ〉 immediately before themeasurement, then for each m,
I the probability that result m occurs in the measurement is
p(m) = ||Mm|ψ〉||2 = 〈ψ|M†mMm|ψ〉 (Born rule)
I the state of the system after the measurement with outcome m is
|ψm〉 =Mm|ψ〉√
p(m).
The normalisation condition implies that the probabilities for alloutcomes sum up to ∑m p(m) = 1.
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Postulate of quantum mechanics 3
I A quantum measurement on a system with state Hilbert spaceHis described by a collection {Mm} ⊆ L(H) of operators satisfyingthe normalisation condition:
∑m
M†mMm = IH
I Mm are called measurement operators.I The index m stands for the measurement outcomes that may
occur in the experiment.
I If the state of a quantum system is |ψ〉 immediately before themeasurement, then for each m,
I the probability that result m occurs in the measurement is
p(m) = ||Mm|ψ〉||2 = 〈ψ|M†mMm|ψ〉 (Born rule)
I the state of the system after the measurement with outcome m is
|ψm〉 =Mm|ψ〉√
p(m).
The normalisation condition implies that the probabilities for alloutcomes sum up to ∑m p(m) = 1.
![Page 84: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/84.jpg)
Postulate of quantum mechanics 3
I A quantum measurement on a system with state Hilbert spaceHis described by a collection {Mm} ⊆ L(H) of operators satisfyingthe normalisation condition:
∑m
M†mMm = IH
I Mm are called measurement operators.I The index m stands for the measurement outcomes that may
occur in the experiment.I If the state of a quantum system is |ψ〉 immediately before the
measurement, then for each m,
I the probability that result m occurs in the measurement is
p(m) = ||Mm|ψ〉||2 = 〈ψ|M†mMm|ψ〉 (Born rule)
I the state of the system after the measurement with outcome m is
|ψm〉 =Mm|ψ〉√
p(m).
The normalisation condition implies that the probabilities for alloutcomes sum up to ∑m p(m) = 1.
![Page 85: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/85.jpg)
Postulate of quantum mechanics 3
I A quantum measurement on a system with state Hilbert spaceHis described by a collection {Mm} ⊆ L(H) of operators satisfyingthe normalisation condition:
∑m
M†mMm = IH
I Mm are called measurement operators.I The index m stands for the measurement outcomes that may
occur in the experiment.I If the state of a quantum system is |ψ〉 immediately before the
measurement, then for each m,I the probability that result m occurs in the measurement is
p(m) = ||Mm|ψ〉||2 = 〈ψ|M†mMm|ψ〉 (Born rule)
I the state of the system after the measurement with outcome m is
|ψm〉 =Mm|ψ〉√
p(m).
The normalisation condition implies that the probabilities for alloutcomes sum up to ∑m p(m) = 1.
![Page 86: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/86.jpg)
Postulate of quantum mechanics 3
I A quantum measurement on a system with state Hilbert spaceHis described by a collection {Mm} ⊆ L(H) of operators satisfyingthe normalisation condition:
∑m
M†mMm = IH
I Mm are called measurement operators.I The index m stands for the measurement outcomes that may
occur in the experiment.I If the state of a quantum system is |ψ〉 immediately before the
measurement, then for each m,I the probability that result m occurs in the measurement is
p(m) = ||Mm|ψ〉||2 = 〈ψ|M†mMm|ψ〉 (Born rule)
I the state of the system after the measurement with outcome m is
|ψm〉 =Mm|ψ〉√
p(m).
The normalisation condition implies that the probabilities for alloutcomes sum up to ∑m p(m) = 1.
![Page 87: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/87.jpg)
Example
I The measurement of a qubit in the computational basis:
M0 = |0〉〈0|, M1 = |1〉〈1|.
I If the qubit was in state |ψ〉 = α|0〉+ β|1〉 before themeasurement, then:
I the probability of obtaining outcome 0 is
p(0) = 〈ψ|M†0M0|ψ〉 = 〈ψ|M0|ψ〉 = |α|2,
the state after the measurement is
M0|ψ〉√p(0)
= |0〉.
I the probability of outcome 1 is p(1) = |β|2, the state after themeasurement is |1〉.
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Example
I The measurement of a qubit in the computational basis:
M0 = |0〉〈0|, M1 = |1〉〈1|.
I If the qubit was in state |ψ〉 = α|0〉+ β|1〉 before themeasurement, then:
I the probability of obtaining outcome 0 is
p(0) = 〈ψ|M†0M0|ψ〉 = 〈ψ|M0|ψ〉 = |α|2,
the state after the measurement is
M0|ψ〉√p(0)
= |0〉.
I the probability of outcome 1 is p(1) = |β|2, the state after themeasurement is |1〉.
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Example
I The measurement of a qubit in the computational basis:
M0 = |0〉〈0|, M1 = |1〉〈1|.
I If the qubit was in state |ψ〉 = α|0〉+ β|1〉 before themeasurement, then:
I the probability of obtaining outcome 0 is
p(0) = 〈ψ|M†0M0|ψ〉 = 〈ψ|M0|ψ〉 = |α|2,
the state after the measurement is
M0|ψ〉√p(0)
= |0〉.
I the probability of outcome 1 is p(1) = |β|2, the state after themeasurement is |1〉.
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Example
I The measurement of a qubit in the computational basis:
M0 = |0〉〈0|, M1 = |1〉〈1|.
I If the qubit was in state |ψ〉 = α|0〉+ β|1〉 before themeasurement, then:
I the probability of obtaining outcome 0 is
p(0) = 〈ψ|M†0M0|ψ〉 = 〈ψ|M0|ψ〉 = |α|2,
the state after the measurement is
M0|ψ〉√p(0)
= |0〉.
I the probability of outcome 1 is p(1) = |β|2, the state after themeasurement is |1〉.
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Hermitian Operators, Observables
I An operator M ∈ L(H) is Hermitian if it is self-adjoint:
M† = M.
In physics, a Hermitian operator is called an observable.
I An operator P is a projector: P = PX for some closed subspace XofH, if and only if P is Hermitian and P2 = P.
Eigenvectors, Eigenvalues
I An eigenvector of an operator A is a non-zero vector |ψ〉 ∈ H suchthat A|ψ〉 = λ|ψ〉 for some λ ∈ C.
I λ is called the eigenvalue of A corresponding to |ψ〉.I The set of eigenvalues of A is called the (point) spectrum of A
and denoted spec(A).
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Hermitian Operators, Observables
I An operator M ∈ L(H) is Hermitian if it is self-adjoint:
M† = M.
In physics, a Hermitian operator is called an observable.I An operator P is a projector: P = PX for some closed subspace X
ofH, if and only if P is Hermitian and P2 = P.
Eigenvectors, Eigenvalues
I An eigenvector of an operator A is a non-zero vector |ψ〉 ∈ H suchthat A|ψ〉 = λ|ψ〉 for some λ ∈ C.
I λ is called the eigenvalue of A corresponding to |ψ〉.I The set of eigenvalues of A is called the (point) spectrum of A
and denoted spec(A).
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Hermitian Operators, Observables
I An operator M ∈ L(H) is Hermitian if it is self-adjoint:
M† = M.
In physics, a Hermitian operator is called an observable.I An operator P is a projector: P = PX for some closed subspace X
ofH, if and only if P is Hermitian and P2 = P.
Eigenvectors, Eigenvalues
I An eigenvector of an operator A is a non-zero vector |ψ〉 ∈ H suchthat A|ψ〉 = λ|ψ〉 for some λ ∈ C.
I λ is called the eigenvalue of A corresponding to |ψ〉.I The set of eigenvalues of A is called the (point) spectrum of A
and denoted spec(A).
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Hermitian Operators, Observables
I An operator M ∈ L(H) is Hermitian if it is self-adjoint:
M† = M.
In physics, a Hermitian operator is called an observable.I An operator P is a projector: P = PX for some closed subspace X
ofH, if and only if P is Hermitian and P2 = P.
Eigenvectors, Eigenvalues
I An eigenvector of an operator A is a non-zero vector |ψ〉 ∈ H suchthat A|ψ〉 = λ|ψ〉 for some λ ∈ C.
I λ is called the eigenvalue of A corresponding to |ψ〉.
I The set of eigenvalues of A is called the (point) spectrum of Aand denoted spec(A).
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Hermitian Operators, Observables
I An operator M ∈ L(H) is Hermitian if it is self-adjoint:
M† = M.
In physics, a Hermitian operator is called an observable.I An operator P is a projector: P = PX for some closed subspace X
ofH, if and only if P is Hermitian and P2 = P.
Eigenvectors, Eigenvalues
I An eigenvector of an operator A is a non-zero vector |ψ〉 ∈ H suchthat A|ψ〉 = λ|ψ〉 for some λ ∈ C.
I λ is called the eigenvalue of A corresponding to |ψ〉.I The set of eigenvalues of A is called the (point) spectrum of A
and denoted spec(A).
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Eigenspaces
I For each eigenvalue λ ∈ spec(A), the set
{|ψ〉 ∈ H : A|ψ〉 = λ|ψ〉}
is a closed subspace ofH and called the eigenspace of Acorresponding to λ.
I The eigenspaces corresponding to different eigenvalues λ1 , λ2are orthogonal
Spectral Decomposition
I All eigenvalues of an observable (i.e. a Hermitian operator) Mare real numbers.
I
M = ∑λ∈spec(M)
λPλ
where Pλ is the projector onto the eigenspace corresponding to λ.
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Eigenspaces
I For each eigenvalue λ ∈ spec(A), the set
{|ψ〉 ∈ H : A|ψ〉 = λ|ψ〉}
is a closed subspace ofH and called the eigenspace of Acorresponding to λ.
I The eigenspaces corresponding to different eigenvalues λ1 , λ2are orthogonal
Spectral Decomposition
I All eigenvalues of an observable (i.e. a Hermitian operator) Mare real numbers.
I
M = ∑λ∈spec(M)
λPλ
where Pλ is the projector onto the eigenspace corresponding to λ.
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Eigenspaces
I For each eigenvalue λ ∈ spec(A), the set
{|ψ〉 ∈ H : A|ψ〉 = λ|ψ〉}
is a closed subspace ofH and called the eigenspace of Acorresponding to λ.
I The eigenspaces corresponding to different eigenvalues λ1 , λ2are orthogonal
Spectral Decomposition
I All eigenvalues of an observable (i.e. a Hermitian operator) Mare real numbers.
I
M = ∑λ∈spec(M)
λPλ
where Pλ is the projector onto the eigenspace corresponding to λ.
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Eigenspaces
I For each eigenvalue λ ∈ spec(A), the set
{|ψ〉 ∈ H : A|ψ〉 = λ|ψ〉}
is a closed subspace ofH and called the eigenspace of Acorresponding to λ.
I The eigenspaces corresponding to different eigenvalues λ1 , λ2are orthogonal
Spectral Decomposition
I All eigenvalues of an observable (i.e. a Hermitian operator) Mare real numbers.
I
M = ∑λ∈spec(M)
λPλ
where Pλ is the projector onto the eigenspace corresponding to λ.
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Projective Measurements
I An observable M defines a measurement {Pλ : λ ∈ spec(M)},called a projective measurement.
I Upon measuring a system in state |ψ〉, the probability of gettingresult λ is
p(λ) = 〈ψ|Pλ|ψ〉
the state of the system after the measurement is
Pλ|ψ〉√p(λ)
.
I The expectation — average value — of M in state |ψ〉:
〈M〉ψ = ∑λ∈spec(M)
p(λ) · λ = 〈ψ|M|ψ〉.
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Projective Measurements
I An observable M defines a measurement {Pλ : λ ∈ spec(M)},called a projective measurement.
I Upon measuring a system in state |ψ〉, the probability of gettingresult λ is
p(λ) = 〈ψ|Pλ|ψ〉
the state of the system after the measurement is
Pλ|ψ〉√p(λ)
.
I The expectation — average value — of M in state |ψ〉:
〈M〉ψ = ∑λ∈spec(M)
p(λ) · λ = 〈ψ|M|ψ〉.
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Projective Measurements
I An observable M defines a measurement {Pλ : λ ∈ spec(M)},called a projective measurement.
I Upon measuring a system in state |ψ〉, the probability of gettingresult λ is
p(λ) = 〈ψ|Pλ|ψ〉
the state of the system after the measurement is
Pλ|ψ〉√p(λ)
.
I The expectation — average value — of M in state |ψ〉:
〈M〉ψ = ∑λ∈spec(M)
p(λ) · λ = 〈ψ|M|ψ〉.
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Outline
Hilbert Spaces
Linear Operators
Quantum Measurements
Tensor Products
Density Operators
Quantum Operations
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Tensor Product of Hilbert Spaces
I LetHi be a Hilbert spaces with {|ψiji〉} as an orthonormal basisfor i = 1, ..., n.
I Write B for the set of the elements in the form:
|ψ1j1 , ..., ψnjn〉 = |ψ1j1 ⊗ ...⊗ ψnjn〉 = |ψ1j1〉 ⊗ ...⊗ |ψnjn〉.
I Then the tensor product ofHi (i = 1, ..., n) is the Hilbert spacewith B as an orthonormal basis:⊗
i
Hi = spanB.
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Tensor Product of Hilbert Spaces
I LetHi be a Hilbert spaces with {|ψiji〉} as an orthonormal basisfor i = 1, ..., n.
I Write B for the set of the elements in the form:
|ψ1j1 , ..., ψnjn〉 = |ψ1j1 ⊗ ...⊗ ψnjn〉 = |ψ1j1〉 ⊗ ...⊗ |ψnjn〉.
I Then the tensor product ofHi (i = 1, ..., n) is the Hilbert spacewith B as an orthonormal basis:⊗
i
Hi = spanB.
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Tensor Product of Hilbert Spaces
I LetHi be a Hilbert spaces with {|ψiji〉} as an orthonormal basisfor i = 1, ..., n.
I Write B for the set of the elements in the form:
|ψ1j1 , ..., ψnjn〉 = |ψ1j1 ⊗ ...⊗ ψnjn〉 = |ψ1j1〉 ⊗ ...⊗ |ψnjn〉.
I Then the tensor product ofHi (i = 1, ..., n) is the Hilbert spacewith B as an orthonormal basis:⊗
i
Hi = spanB.
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Postulate of quantum mechanics 4The state space of a composite quantum system is the tensor productof the state spaces of its components.
Entanglement
I S is a quantum system composed by subsystems S1, ..., Sn withstate Hilbert spaceH1, ...,Hn.
I If for each 1 ≤ i ≤ n, Si is in state |ψi〉 ∈ Hi, then S is in theproduct state |ψ1, ..., ψn〉.
I A state of the composite system is entangled if it is not a productof states of its component systems.
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Postulate of quantum mechanics 4The state space of a composite quantum system is the tensor productof the state spaces of its components.
Entanglement
I S is a quantum system composed by subsystems S1, ..., Sn withstate Hilbert spaceH1, ...,Hn.
I If for each 1 ≤ i ≤ n, Si is in state |ψi〉 ∈ Hi, then S is in theproduct state |ψ1, ..., ψn〉.
I A state of the composite system is entangled if it is not a productof states of its component systems.
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Postulate of quantum mechanics 4The state space of a composite quantum system is the tensor productof the state spaces of its components.
Entanglement
I S is a quantum system composed by subsystems S1, ..., Sn withstate Hilbert spaceH1, ...,Hn.
I If for each 1 ≤ i ≤ n, Si is in state |ψi〉 ∈ Hi, then S is in theproduct state |ψ1, ..., ψn〉.
I A state of the composite system is entangled if it is not a productof states of its component systems.
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Example
I The state space of the system of n qubits:
H⊗n2 = C2n
=
∑x∈{0,1}n
αx|x〉 : αx ∈ C for all x ∈ {0, 1}n
.
I A two-qubit system can be in a product state like |00〉, |1〉|+〉.I It can also be in an entangled state like the Bell states or the EPR
(Einstein-Podolsky-Rosen) pairs:
|β00〉 =1√
2(|00〉+ |11〉), |β01〉 =
1√2(|01〉+ |10〉),
|β10〉 =1√
2(|00〉 − |11〉), |β11〉 =
1√2(|01〉 − |10〉).
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Example
I The state space of the system of n qubits:
H⊗n2 = C2n
=
∑x∈{0,1}n
αx|x〉 : αx ∈ C for all x ∈ {0, 1}n
.
I A two-qubit system can be in a product state like |00〉, |1〉|+〉.
I It can also be in an entangled state like the Bell states or the EPR(Einstein-Podolsky-Rosen) pairs:
|β00〉 =1√
2(|00〉+ |11〉), |β01〉 =
1√2(|01〉+ |10〉),
|β10〉 =1√
2(|00〉 − |11〉), |β11〉 =
1√2(|01〉 − |10〉).
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Example
I The state space of the system of n qubits:
H⊗n2 = C2n
=
∑x∈{0,1}n
αx|x〉 : αx ∈ C for all x ∈ {0, 1}n
.
I A two-qubit system can be in a product state like |00〉, |1〉|+〉.I It can also be in an entangled state like the Bell states or the EPR
(Einstein-Podolsky-Rosen) pairs:
|β00〉 =1√
2(|00〉+ |11〉), |β01〉 =
1√2(|01〉+ |10〉),
|β10〉 =1√
2(|00〉 − |11〉), |β11〉 =
1√2(|01〉 − |10〉).
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Tensor Product of Operators
I Let Ai ∈ L(Hi) for i = 1, ..., n.
I Their tensor product⊗n
i=1 Ai = A1 ⊗ ...⊗An ∈ L (⊗n
i=1Hi):
(A1 ⊗ ...⊗An)|ϕ1, ..., ϕn〉 = A1|ϕ1〉 ⊗ ...⊗An|ϕn〉
Controlled-NOT
I The controlled-NOT or CNOT operator C inH⊗22 = C4:
C|00〉 = |00〉, C|01〉 = |01〉, C|10〉 = |11〉, C|11〉 = |10〉
C =
1 0 0 00 1 0 00 0 0 10 0 1 0
.
I Transform product states into entangled states:
C|+〉|0〉 = β00, C|+〉|1〉 = β01, C|−〉|0〉 = β10, C|−〉|1〉 = β11.
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Tensor Product of Operators
I Let Ai ∈ L(Hi) for i = 1, ..., n.I Their tensor product
⊗ni=1 Ai = A1 ⊗ ...⊗An ∈ L (
⊗ni=1Hi):
(A1 ⊗ ...⊗An)|ϕ1, ..., ϕn〉 = A1|ϕ1〉 ⊗ ...⊗An|ϕn〉
Controlled-NOT
I The controlled-NOT or CNOT operator C inH⊗22 = C4:
C|00〉 = |00〉, C|01〉 = |01〉, C|10〉 = |11〉, C|11〉 = |10〉
C =
1 0 0 00 1 0 00 0 0 10 0 1 0
.
I Transform product states into entangled states:
C|+〉|0〉 = β00, C|+〉|1〉 = β01, C|−〉|0〉 = β10, C|−〉|1〉 = β11.
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Tensor Product of Operators
I Let Ai ∈ L(Hi) for i = 1, ..., n.I Their tensor product
⊗ni=1 Ai = A1 ⊗ ...⊗An ∈ L (
⊗ni=1Hi):
(A1 ⊗ ...⊗An)|ϕ1, ..., ϕn〉 = A1|ϕ1〉 ⊗ ...⊗An|ϕn〉
Controlled-NOTI The controlled-NOT or CNOT operator C inH⊗2
2 = C4:
C|00〉 = |00〉, C|01〉 = |01〉, C|10〉 = |11〉, C|11〉 = |10〉
C =
1 0 0 00 1 0 00 0 0 10 0 1 0
.
I Transform product states into entangled states:
C|+〉|0〉 = β00, C|+〉|1〉 = β01, C|−〉|0〉 = β10, C|−〉|1〉 = β11.
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Tensor Product of Operators
I Let Ai ∈ L(Hi) for i = 1, ..., n.I Their tensor product
⊗ni=1 Ai = A1 ⊗ ...⊗An ∈ L (
⊗ni=1Hi):
(A1 ⊗ ...⊗An)|ϕ1, ..., ϕn〉 = A1|ϕ1〉 ⊗ ...⊗An|ϕn〉
Controlled-NOTI The controlled-NOT or CNOT operator C inH⊗2
2 = C4:
C|00〉 = |00〉, C|01〉 = |01〉, C|10〉 = |11〉, C|11〉 = |10〉
C =
1 0 0 00 1 0 00 0 0 10 0 1 0
.
I Transform product states into entangled states:
C|+〉|0〉 = β00, C|+〉|1〉 = β01, C|−〉|0〉 = β10, C|−〉|1〉 = β11.
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Implementing a General Measurement by a ProjectiveMeasurementI Let M = {Mm} be a quantum measurement in Hilbert spaceH.
I Introduce a new Hilbert spaceHM = span{|m〉} used to recordthe possible outcomes of M.
I Choose a fixed state |0〉 ∈ HM. Define unitary operator inHM ⊗H:
UM(|0〉|ψ〉) = ∑m|m〉Mm|ψ〉
I Define a projective measurement M = {Mm} inHM ⊗H withMm = |m〉〈m| ⊗ IH for every m.
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Implementing a General Measurement by a ProjectiveMeasurementI Let M = {Mm} be a quantum measurement in Hilbert spaceH.I Introduce a new Hilbert spaceHM = span{|m〉} used to record
the possible outcomes of M.
I Choose a fixed state |0〉 ∈ HM. Define unitary operator inHM ⊗H:
UM(|0〉|ψ〉) = ∑m|m〉Mm|ψ〉
I Define a projective measurement M = {Mm} inHM ⊗H withMm = |m〉〈m| ⊗ IH for every m.
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Implementing a General Measurement by a ProjectiveMeasurementI Let M = {Mm} be a quantum measurement in Hilbert spaceH.I Introduce a new Hilbert spaceHM = span{|m〉} used to record
the possible outcomes of M.I Choose a fixed state |0〉 ∈ HM. Define unitary operator inHM ⊗H:
UM(|0〉|ψ〉) = ∑m|m〉Mm|ψ〉
I Define a projective measurement M = {Mm} inHM ⊗H withMm = |m〉〈m| ⊗ IH for every m.
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Implementing a General Measurement by a ProjectiveMeasurementI Let M = {Mm} be a quantum measurement in Hilbert spaceH.I Introduce a new Hilbert spaceHM = span{|m〉} used to record
the possible outcomes of M.I Choose a fixed state |0〉 ∈ HM. Define unitary operator inHM ⊗H:
UM(|0〉|ψ〉) = ∑m|m〉Mm|ψ〉
I Define a projective measurement M = {Mm} inHM ⊗H withMm = |m〉〈m| ⊗ IH for every m.
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Implementing a General Measurement by a ProjectiveMeasurement (Continued)I Then M is realised by the projective measurement M together
with the unitary operator UM.
I For any pure state |ψ〉 ∈ H,
I When we perform measurement M on |ψ〉, the probability ofoutcome m is denoted pM(m), the post-measurement statecorresponding to m is |ψm〉.
I When we perform measurement M on |ψ〉 = UM(|0〉|ψ〉), theprobability of outcome m is denoted pM(m), the post-measurementstate corresponding to m is |ψm〉.
I Then for each m, we have:
pM(m) = pM(m)
|ψm〉 = |m〉|ψm〉
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Implementing a General Measurement by a ProjectiveMeasurement (Continued)I Then M is realised by the projective measurement M together
with the unitary operator UM.I For any pure state |ψ〉 ∈ H,
I When we perform measurement M on |ψ〉, the probability ofoutcome m is denoted pM(m), the post-measurement statecorresponding to m is |ψm〉.
I When we perform measurement M on |ψ〉 = UM(|0〉|ψ〉), theprobability of outcome m is denoted pM(m), the post-measurementstate corresponding to m is |ψm〉.
I Then for each m, we have:
pM(m) = pM(m)
|ψm〉 = |m〉|ψm〉
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Implementing a General Measurement by a ProjectiveMeasurement (Continued)I Then M is realised by the projective measurement M together
with the unitary operator UM.I For any pure state |ψ〉 ∈ H,
I When we perform measurement M on |ψ〉, the probability ofoutcome m is denoted pM(m), the post-measurement statecorresponding to m is |ψm〉.
I When we perform measurement M on |ψ〉 = UM(|0〉|ψ〉), theprobability of outcome m is denoted pM(m), the post-measurementstate corresponding to m is |ψm〉.
I Then for each m, we have:
pM(m) = pM(m)
|ψm〉 = |m〉|ψm〉
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Implementing a General Measurement by a ProjectiveMeasurement (Continued)I Then M is realised by the projective measurement M together
with the unitary operator UM.I For any pure state |ψ〉 ∈ H,
I When we perform measurement M on |ψ〉, the probability ofoutcome m is denoted pM(m), the post-measurement statecorresponding to m is |ψm〉.
I When we perform measurement M on |ψ〉 = UM(|0〉|ψ〉), theprobability of outcome m is denoted pM(m), the post-measurementstate corresponding to m is |ψm〉.
I Then for each m, we have:
pM(m) = pM(m)
|ψm〉 = |m〉|ψm〉
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Implementing a General Measurement by a ProjectiveMeasurement (Continued)I Then M is realised by the projective measurement M together
with the unitary operator UM.I For any pure state |ψ〉 ∈ H,
I When we perform measurement M on |ψ〉, the probability ofoutcome m is denoted pM(m), the post-measurement statecorresponding to m is |ψm〉.
I When we perform measurement M on |ψ〉 = UM(|0〉|ψ〉), theprobability of outcome m is denoted pM(m), the post-measurementstate corresponding to m is |ψm〉.
I Then for each m, we have:
pM(m) = pM(m)
|ψm〉 = |m〉|ψm〉
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Outline
Hilbert Spaces
Linear Operators
Quantum Measurements
Tensor Products
Density Operators
Quantum Operations
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EnsemblesI The state of a quantum system is not completely known: it is in
one of a number of pure states |ψi〉, with respective probabilitiespi, where |ψi〉 ∈ H, pi ≥ 0 for each i, ∑i pi = 1.
I We call {(|ψi〉, pi)} an ensemble of pure states or a mixed state.I The density operator:
ρ = ∑i
pi|ψi〉〈ψi|.
I A pure state |ψ〉may be seen as a special mixed state {(|ψ〉, 1)},its density operator is ρ = |ψ〉〈ψ|.
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EnsemblesI The state of a quantum system is not completely known: it is in
one of a number of pure states |ψi〉, with respective probabilitiespi, where |ψi〉 ∈ H, pi ≥ 0 for each i, ∑i pi = 1.
I We call {(|ψi〉, pi)} an ensemble of pure states or a mixed state.
I The density operator:
ρ = ∑i
pi|ψi〉〈ψi|.
I A pure state |ψ〉may be seen as a special mixed state {(|ψ〉, 1)},its density operator is ρ = |ψ〉〈ψ|.
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EnsemblesI The state of a quantum system is not completely known: it is in
one of a number of pure states |ψi〉, with respective probabilitiespi, where |ψi〉 ∈ H, pi ≥ 0 for each i, ∑i pi = 1.
I We call {(|ψi〉, pi)} an ensemble of pure states or a mixed state.I The density operator:
ρ = ∑i
pi|ψi〉〈ψi|.
I A pure state |ψ〉may be seen as a special mixed state {(|ψ〉, 1)},its density operator is ρ = |ψ〉〈ψ|.
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EnsemblesI The state of a quantum system is not completely known: it is in
one of a number of pure states |ψi〉, with respective probabilitiespi, where |ψi〉 ∈ H, pi ≥ 0 for each i, ∑i pi = 1.
I We call {(|ψi〉, pi)} an ensemble of pure states or a mixed state.I The density operator:
ρ = ∑i
pi|ψi〉〈ψi|.
I A pure state |ψ〉may be seen as a special mixed state {(|ψ〉, 1)},its density operator is ρ = |ψ〉〈ψ|.
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Density Operators
I The trace tr(A) of operator A ∈ L(H):
tr(A) = ∑i〈ψi|A|ψi〉
where {|ψi〉} is an orthonormal basis ofH.
I A density operator ρ is a positive operator with tr(ρ) = 1.I The operator ρ defined by any ensemble {(|ψi〉, pi)} is a density
operator. Conversely, any density operator ρ is defined by an(but not necessarily unique) ensemble {(|ψi〉, pi)}.
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Density Operators
I The trace tr(A) of operator A ∈ L(H):
tr(A) = ∑i〈ψi|A|ψi〉
where {|ψi〉} is an orthonormal basis ofH.I A density operator ρ is a positive operator with tr(ρ) = 1.
I The operator ρ defined by any ensemble {(|ψi〉, pi)} is a densityoperator. Conversely, any density operator ρ is defined by an(but not necessarily unique) ensemble {(|ψi〉, pi)}.
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Density Operators
I The trace tr(A) of operator A ∈ L(H):
tr(A) = ∑i〈ψi|A|ψi〉
where {|ψi〉} is an orthonormal basis ofH.I A density operator ρ is a positive operator with tr(ρ) = 1.I The operator ρ defined by any ensemble {(|ψi〉, pi)} is a density
operator. Conversely, any density operator ρ is defined by an(but not necessarily unique) ensemble {(|ψi〉, pi)}.
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Postulates of Quantum Mechanics in the Language ofDensity Operators
I A closed quantum system from time t0 to t is described byunitary operator U depending on t0 and t:
|ψ〉 = U|ψ0〉
I If the system is in mixed states ρ0, ρ at times t0 and t,respectively, then:
ρ = Uρ0U†.
I If the state of a quantum system was ρ before measurement{Mm} is performed, then the probability that result m occurs:
p(m) = tr(
M†mMmρ
)the system after the measurement:
ρm =MmρM†
mp(m)
.
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Postulates of Quantum Mechanics in the Language ofDensity Operators
I A closed quantum system from time t0 to t is described byunitary operator U depending on t0 and t:
|ψ〉 = U|ψ0〉
I If the system is in mixed states ρ0, ρ at times t0 and t,respectively, then:
ρ = Uρ0U†.
I If the state of a quantum system was ρ before measurement{Mm} is performed, then the probability that result m occurs:
p(m) = tr(
M†mMmρ
)the system after the measurement:
ρm =MmρM†
mp(m)
.
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Postulates of Quantum Mechanics in the Language ofDensity Operators
I A closed quantum system from time t0 to t is described byunitary operator U depending on t0 and t:
|ψ〉 = U|ψ0〉
I If the system is in mixed states ρ0, ρ at times t0 and t,respectively, then:
ρ = Uρ0U†.
I If the state of a quantum system was ρ before measurement{Mm} is performed, then the probability that result m occurs:
p(m) = tr(
M†mMmρ
)the system after the measurement:
ρm =MmρM†
mp(m)
.
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Reduced Density Operators
I We often need to characterise the state of a subsystem of aquantum system.
I It is possible that a composite system is in a pure state, but someof its subsystems must be seen as in a mixed state.
I Let S and T be quantum systems whose state Hilbert spaces areHS andHT, respectively.
I The partial trace over system T:
trT : L(HS ⊗HT)→ L(HS)
trT (|ϕ〉〈ψ| ⊗ |θ〉〈ζ|) = 〈ζ|θ〉 · |ϕ〉〈ψ|I Let ρ be a density operator inHS ⊗HT. Its reduced density
operator for system S:ρS = trT(ρ).
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Reduced Density Operators
I We often need to characterise the state of a subsystem of aquantum system.
I It is possible that a composite system is in a pure state, but someof its subsystems must be seen as in a mixed state.
I Let S and T be quantum systems whose state Hilbert spaces areHS andHT, respectively.
I The partial trace over system T:
trT : L(HS ⊗HT)→ L(HS)
trT (|ϕ〉〈ψ| ⊗ |θ〉〈ζ|) = 〈ζ|θ〉 · |ϕ〉〈ψ|I Let ρ be a density operator inHS ⊗HT. Its reduced density
operator for system S:ρS = trT(ρ).
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Reduced Density Operators
I We often need to characterise the state of a subsystem of aquantum system.
I It is possible that a composite system is in a pure state, but someof its subsystems must be seen as in a mixed state.
I Let S and T be quantum systems whose state Hilbert spaces areHS andHT, respectively.
I The partial trace over system T:
trT : L(HS ⊗HT)→ L(HS)
trT (|ϕ〉〈ψ| ⊗ |θ〉〈ζ|) = 〈ζ|θ〉 · |ϕ〉〈ψ|I Let ρ be a density operator inHS ⊗HT. Its reduced density
operator for system S:ρS = trT(ρ).
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Reduced Density Operators
I We often need to characterise the state of a subsystem of aquantum system.
I It is possible that a composite system is in a pure state, but someof its subsystems must be seen as in a mixed state.
I Let S and T be quantum systems whose state Hilbert spaces areHS andHT, respectively.
I The partial trace over system T:
trT : L(HS ⊗HT)→ L(HS)
trT (|ϕ〉〈ψ| ⊗ |θ〉〈ζ|) = 〈ζ|θ〉 · |ϕ〉〈ψ|
I Let ρ be a density operator inHS ⊗HT. Its reduced densityoperator for system S:
ρS = trT(ρ).
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Reduced Density Operators
I We often need to characterise the state of a subsystem of aquantum system.
I It is possible that a composite system is in a pure state, but someof its subsystems must be seen as in a mixed state.
I Let S and T be quantum systems whose state Hilbert spaces areHS andHT, respectively.
I The partial trace over system T:
trT : L(HS ⊗HT)→ L(HS)
trT (|ϕ〉〈ψ| ⊗ |θ〉〈ζ|) = 〈ζ|θ〉 · |ϕ〉〈ψ|I Let ρ be a density operator inHS ⊗HT. Its reduced density
operator for system S:ρS = trT(ρ).
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Outline
Hilbert Spaces
Linear Operators
Quantum Measurements
Tensor Products
Density Operators
Quantum Operations
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Super-Operators
I Unitary transformations are suited to describe the dynamics ofclosed quantum systems.
I For open quantum systems that interact with the outside, weneed a more general notion of quantum operation.
I A linear operator in vector space L(H) is called a super-operatorinH.
I LetH and K be Hilbert spaces. For any super-operator E inHand super-operator F in K, their tensor product E ⊗ F is thesuper-operator inH⊗K: for each C ∈ L(H⊗K),
(E ⊗ F )(C) = ∑k
αk(E(Ak)⊗F (Bk))
where C = ∑k αk(Ak ⊗ Bk), Ak ∈ L(H), Bk ∈ L(K) for all k.
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Super-Operators
I Unitary transformations are suited to describe the dynamics ofclosed quantum systems.
I For open quantum systems that interact with the outside, weneed a more general notion of quantum operation.
I A linear operator in vector space L(H) is called a super-operatorinH.
I LetH and K be Hilbert spaces. For any super-operator E inHand super-operator F in K, their tensor product E ⊗ F is thesuper-operator inH⊗K: for each C ∈ L(H⊗K),
(E ⊗ F )(C) = ∑k
αk(E(Ak)⊗F (Bk))
where C = ∑k αk(Ak ⊗ Bk), Ak ∈ L(H), Bk ∈ L(K) for all k.
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Super-Operators
I Unitary transformations are suited to describe the dynamics ofclosed quantum systems.
I For open quantum systems that interact with the outside, weneed a more general notion of quantum operation.
I A linear operator in vector space L(H) is called a super-operatorinH.
I LetH and K be Hilbert spaces. For any super-operator E inHand super-operator F in K, their tensor product E ⊗ F is thesuper-operator inH⊗K: for each C ∈ L(H⊗K),
(E ⊗ F )(C) = ∑k
αk(E(Ak)⊗F (Bk))
where C = ∑k αk(Ak ⊗ Bk), Ak ∈ L(H), Bk ∈ L(K) for all k.
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Super-Operators
I Unitary transformations are suited to describe the dynamics ofclosed quantum systems.
I For open quantum systems that interact with the outside, weneed a more general notion of quantum operation.
I A linear operator in vector space L(H) is called a super-operatorinH.
I LetH and K be Hilbert spaces. For any super-operator E inHand super-operator F in K, their tensor product E ⊗ F is thesuper-operator inH⊗K: for each C ∈ L(H⊗K),
(E ⊗ F )(C) = ∑k
αk(E(Ak)⊗F (Bk))
where C = ∑k αk(Ak ⊗ Bk), Ak ∈ L(H), Bk ∈ L(K) for all k.
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Quantum Operations
I Let the states of a system at times t0 and t are ρ and ρ′,respectively. Then they must be related to each other by asuper-operator E depending only on the times t0 and t:
ρ = E(ρ0).
I A quantum operation in a Hilbert spaceH is a super-operator inH satisfying:
1. tr[E(ρ)] ≤ tr(ρ) = 1 for each density operator ρ inH;2. (Complete positivity) For any extra Hilbert spaceHR, (IR ⊗ E)(A)
is positive provided A is a positive operator inHR ⊗H, where IRis the identity operator in L(HR).
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Quantum Operations
I Let the states of a system at times t0 and t are ρ and ρ′,respectively. Then they must be related to each other by asuper-operator E depending only on the times t0 and t:
ρ = E(ρ0).
I A quantum operation in a Hilbert spaceH is a super-operator inH satisfying:
1. tr[E(ρ)] ≤ tr(ρ) = 1 for each density operator ρ inH;2. (Complete positivity) For any extra Hilbert spaceHR, (IR ⊗ E)(A)
is positive provided A is a positive operator inHR ⊗H, where IRis the identity operator in L(HR).
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Quantum Operations
I Let the states of a system at times t0 and t are ρ and ρ′,respectively. Then they must be related to each other by asuper-operator E depending only on the times t0 and t:
ρ = E(ρ0).
I A quantum operation in a Hilbert spaceH is a super-operator inH satisfying:
1. tr[E(ρ)] ≤ tr(ρ) = 1 for each density operator ρ inH;
2. (Complete positivity) For any extra Hilbert spaceHR, (IR ⊗ E)(A)is positive provided A is a positive operator inHR ⊗H, where IRis the identity operator in L(HR).
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Quantum Operations
I Let the states of a system at times t0 and t are ρ and ρ′,respectively. Then they must be related to each other by asuper-operator E depending only on the times t0 and t:
ρ = E(ρ0).
I A quantum operation in a Hilbert spaceH is a super-operator inH satisfying:
1. tr[E(ρ)] ≤ tr(ρ) = 1 for each density operator ρ inH;2. (Complete positivity) For any extra Hilbert spaceHR, (IR ⊗ E)(A)
is positive provided A is a positive operator inHR ⊗H, where IRis the identity operator in L(HR).
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Examples
I Let U be a unitary transformation in a Hilbert spaceH. Define:
E(ρ) = UρU†
for every density operator ρ. Then E is a quantum operation.
I Let M = {Mm} be a quantum measurement inH.
1. For each m, if for any system state ρ before measurement, define
Em(ρ) = pmρm = MmρM†
where pm is the probability of outcome m and ρm is thepost-measurement state corresponding to m, then Em is a quantumoperation.
2. For any system state ρ before measurement, the post-measurementstate is
E(ρ) = ∑mEm(ρ) = ∑
mMmρM†
m
whenever the measurement outcomes are ignored. Then E is aquantum operation.
![Page 152: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/152.jpg)
Examples
I Let U be a unitary transformation in a Hilbert spaceH. Define:
E(ρ) = UρU†
for every density operator ρ. Then E is a quantum operation.I Let M = {Mm} be a quantum measurement inH.
1. For each m, if for any system state ρ before measurement, define
Em(ρ) = pmρm = MmρM†
where pm is the probability of outcome m and ρm is thepost-measurement state corresponding to m, then Em is a quantumoperation.
2. For any system state ρ before measurement, the post-measurementstate is
E(ρ) = ∑mEm(ρ) = ∑
mMmρM†
m
whenever the measurement outcomes are ignored. Then E is aquantum operation.
![Page 153: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/153.jpg)
Examples
I Let U be a unitary transformation in a Hilbert spaceH. Define:
E(ρ) = UρU†
for every density operator ρ. Then E is a quantum operation.I Let M = {Mm} be a quantum measurement inH.
1. For each m, if for any system state ρ before measurement, define
Em(ρ) = pmρm = MmρM†
where pm is the probability of outcome m and ρm is thepost-measurement state corresponding to m, then Em is a quantumoperation.
2. For any system state ρ before measurement, the post-measurementstate is
E(ρ) = ∑mEm(ρ) = ∑
mMmρM†
m
whenever the measurement outcomes are ignored. Then E is aquantum operation.
![Page 154: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/154.jpg)
Examples
I Let U be a unitary transformation in a Hilbert spaceH. Define:
E(ρ) = UρU†
for every density operator ρ. Then E is a quantum operation.I Let M = {Mm} be a quantum measurement inH.
1. For each m, if for any system state ρ before measurement, define
Em(ρ) = pmρm = MmρM†
where pm is the probability of outcome m and ρm is thepost-measurement state corresponding to m, then Em is a quantumoperation.
2. For any system state ρ before measurement, the post-measurementstate is
E(ρ) = ∑mEm(ρ) = ∑
mMmρM†
m
whenever the measurement outcomes are ignored. Then E is aquantum operation.
![Page 155: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/155.jpg)
Kraus TheoremThe following statements are equivalent:
1. E is a quantum operation in a Hilbert spaceH;
2. (System-environment model) There are an environment system Ewith state Hilbert spaceHE, and a unitary transformation U inHE ⊗H and a projector P onto some closed subspace ofHE ⊗Hsuch that
E(ρ) = trE
[PU(|e0〉〈e0| ⊗ ρ)U†P
]for all density operator ρ inH, where |e0〉 is a fixed state inHE;
3. (Kraus operator-sum representation) There exists a finite orcountably infinite set of operators {Ei} inH such that ∑i E†
i Ei v Iand
E(ρ) = ∑i
EiρE†i
for all density operators ρ inH. We write: E = ∑i Ei ◦ E†i .
![Page 156: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/156.jpg)
Kraus TheoremThe following statements are equivalent:
1. E is a quantum operation in a Hilbert spaceH;2. (System-environment model) There are an environment system E
with state Hilbert spaceHE, and a unitary transformation U inHE ⊗H and a projector P onto some closed subspace ofHE ⊗Hsuch that
E(ρ) = trE
[PU(|e0〉〈e0| ⊗ ρ)U†P
]for all density operator ρ inH, where |e0〉 is a fixed state inHE;
3. (Kraus operator-sum representation) There exists a finite orcountably infinite set of operators {Ei} inH such that ∑i E†
i Ei v Iand
E(ρ) = ∑i
EiρE†i
for all density operators ρ inH. We write: E = ∑i Ei ◦ E†i .
![Page 157: Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics · 2016-08-15 · Foundations of Quantum Programming Lecture 2: Basics of Quantum Mechanics Mingsheng Ying](https://reader034.vdocument.in/reader034/viewer/2022042306/5ed223ec5e0ec842bd789803/html5/thumbnails/157.jpg)
Kraus TheoremThe following statements are equivalent:
1. E is a quantum operation in a Hilbert spaceH;2. (System-environment model) There are an environment system E
with state Hilbert spaceHE, and a unitary transformation U inHE ⊗H and a projector P onto some closed subspace ofHE ⊗Hsuch that
E(ρ) = trE
[PU(|e0〉〈e0| ⊗ ρ)U†P
]for all density operator ρ inH, where |e0〉 is a fixed state inHE;
3. (Kraus operator-sum representation) There exists a finite orcountably infinite set of operators {Ei} inH such that ∑i E†
i Ei v Iand
E(ρ) = ∑i
EiρE†i
for all density operators ρ inH. We write: E = ∑i Ei ◦ E†i .