majorization and entanglement transformations
TRANSCRIPT
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Majorization and Entanglement transformations
Gustavo Martín Bosyk
Instituto de Física La Plata, UNLP, CONICET, La Plata, Argentina
Università degli studi di Cagliari, Cagliari, Italia
In collaboration with:GMB, G. Sergioli, H. Freytes, F. Holik and G. Bellomo, arXiv:1608.04818v1 [quant-ph] (2016)
November 3, 2016
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 1 / 21
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Part I
Majorization
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 2 / 21
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Situation A Situation B
Which situation has less uncertainty?
Game 1: what color is the ball? → A
Game 2: what color is NOT the ball? → B
How to compare probability vectors
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 3 / 21
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Situation A
Situation B
Which situation has less uncertainty?
Game 1: what color is the ball? → A
Game 2: what color is NOT the ball? → B
How to compare probability vectors
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 3 / 21
![Page 5: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/5.jpg)
Situation A Situation B
Which situation has less uncertainty?
Game 1: what color is the ball? → A
Game 2: what color is NOT the ball? → B
How to compare probability vectors
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 3 / 21
![Page 6: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/6.jpg)
Situation A Situation B
Which situation has less uncertainty?
Game 1: what color is the ball? → A
Game 2: what color is NOT the ball? → B
How to compare probability vectors
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 3 / 21
![Page 7: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/7.jpg)
Situation A Situation B
Which situation has less uncertainty?
Game 1: what color is the ball? → A
Game 2: what color is NOT the ball? → B
How to compare probability vectors
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 3 / 21
![Page 8: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/8.jpg)
Situation A Situation B
Which situation has less uncertainty?
Game 1: what color is the ball? → A
Game 2: what color is NOT the ball? → B
How to compare probability vectors
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 3 / 21
![Page 9: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/9.jpg)
Situation A Situation B
Which situation has less uncertainty?
Game 1: what color is the ball? → A
Game 2: what color is NOT the ball? → B
How to compare probability vectors
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 3 / 21
![Page 10: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/10.jpg)
Majorization
De�nition [Marshall, Olkin y Arnold, Inequalities: Theory of Majorization and Its Applications]
Let p = [p1, . . . pN ]t and q = [q1, . . . qN ]t be probability vectors: pi , qi ≥ 0
and∑N
i=1 pi = 1 =∑N
i=1 qi .p is majorized by q, denoted as p ≺ q, if
n∑i=1
p↓i ≤n∑
i=1
q↓i ∀n = 1 . . .N − 1
Example
[1
N. . .
1
N
]t≺ p ≺ [1, 0 . . . 0]t ∀p
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 4 / 21
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Majorization
De�nition [Marshall, Olkin y Arnold, Inequalities: Theory of Majorization and Its Applications]
Let p = [p1, . . . pN ]t and q = [q1, . . . qN ]t be probability vectors: pi , qi ≥ 0
and∑N
i=1 pi = 1 =∑N
i=1 qi .
p is majorized by q, denoted as p ≺ q, if
n∑i=1
p↓i ≤n∑
i=1
q↓i ∀n = 1 . . .N − 1
Example
[1
N. . .
1
N
]t≺ p ≺ [1, 0 . . . 0]t ∀p
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 4 / 21
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Majorization
De�nition [Marshall, Olkin y Arnold, Inequalities: Theory of Majorization and Its Applications]
Let p = [p1, . . . pN ]t and q = [q1, . . . qN ]t be probability vectors: pi , qi ≥ 0
and∑N
i=1 pi = 1 =∑N
i=1 qi .p is majorized by q, denoted as p ≺ q, if
n∑i=1
p↓i ≤n∑
i=1
q↓i ∀n = 1 . . .N − 1
Example
[1
N. . .
1
N
]t≺ p ≺ [1, 0 . . . 0]t ∀p
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 4 / 21
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Majorization
De�nition [Marshall, Olkin y Arnold, Inequalities: Theory of Majorization and Its Applications]
Let p = [p1, . . . pN ]t and q = [q1, . . . qN ]t be probability vectors: pi , qi ≥ 0
and∑N
i=1 pi = 1 =∑N
i=1 qi .p is majorized by q, denoted as p ≺ q, if
n∑i=1
p↓i ≤n∑
i=1
q↓i ∀n = 1 . . .N − 1
Example
[1
N. . .
1
N
]t≺ p ≺ [1, 0 . . . 0]t ∀p
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 4 / 21
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Alternative de�nitions
De�nitions
p ≺ q sii
1 there exist a double stochastic matrix D such that
p = Dq with∑i
Dij =∑j
Dij = 1
2∑N
i=1 φ(pi ) ≤∑N
i=1 φ(qi ) for all concave function φ
Schur-concavity and entropies
Φ : RN 7→ R is Schur-concave if p ≺ q ⇒ Φ(p) ≥ Φ(q)
Shannon entropy: H(p) = −∑
pi ln pi
Tsallis entropy: Tq(p) =∑
pqi −1
1−q
Rényi entropy: Rq(p) =ln
∑pqi
1−q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 5 / 21
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Alternative de�nitions
De�nitions
p ≺ q sii
1 there exist a double stochastic matrix D such that
p = Dq with∑i
Dij =∑j
Dij = 1
2∑N
i=1 φ(pi ) ≤∑N
i=1 φ(qi ) for all concave function φ
Schur-concavity and entropies
Φ : RN 7→ R is Schur-concave if p ≺ q ⇒ Φ(p) ≥ Φ(q)
Shannon entropy: H(p) = −∑
pi ln pi
Tsallis entropy: Tq(p) =∑
pqi −1
1−q
Rényi entropy: Rq(p) =ln
∑pqi
1−q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 5 / 21
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Alternative de�nitions
De�nitions
p ≺ q sii
1 there exist a double stochastic matrix D such that
p = Dq with∑i
Dij =∑j
Dij = 1
2∑N
i=1 φ(pi ) ≤∑N
i=1 φ(qi ) for all concave function φ
Schur-concavity and entropies
Φ : RN 7→ R is Schur-concave if p ≺ q ⇒ Φ(p) ≥ Φ(q)
Shannon entropy: H(p) = −∑
pi ln pi
Tsallis entropy: Tq(p) =∑
pqi −1
1−q
Rényi entropy: Rq(p) =ln
∑pqi
1−q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 5 / 21
![Page 17: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/17.jpg)
Alternative de�nitions
De�nitions
p ≺ q sii
1 there exist a double stochastic matrix D such that
p = Dq with∑i
Dij =∑j
Dij = 1
2∑N
i=1 φ(pi ) ≤∑N
i=1 φ(qi ) for all concave function φ
Schur-concavity and entropies
Φ : RN 7→ R is Schur-concave if p ≺ q ⇒ Φ(p) ≥ Φ(q)
Shannon entropy: H(p) = −∑
pi ln pi
Tsallis entropy: Tq(p) =∑
pqi −1
1−q
Rényi entropy: Rq(p) =ln
∑pqi
1−q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 5 / 21
![Page 18: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/18.jpg)
Alternative de�nitions
De�nitions
p ≺ q sii
1 there exist a double stochastic matrix D such that
p = Dq with∑i
Dij =∑j
Dij = 1
2∑N
i=1 φ(pi ) ≤∑N
i=1 φ(qi ) for all concave function φ
Schur-concavity and entropies
Φ : RN 7→ R is Schur-concave if p ≺ q ⇒ Φ(p) ≥ Φ(q)
Shannon entropy: H(p) = −∑
pi ln pi
Tsallis entropy: Tq(p) =∑
pqi −1
1−q
Rényi entropy: Rq(p) =ln
∑pqi
1−q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 5 / 21
![Page 19: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/19.jpg)
Alternative de�nitions
De�nitions
p ≺ q sii
1 there exist a double stochastic matrix D such that
p = Dq with∑i
Dij =∑j
Dij = 1
2∑N
i=1 φ(pi ) ≤∑N
i=1 φ(qi ) for all concave function φ
Schur-concavity and entropies
Φ : RN 7→ R is Schur-concave if p ≺ q ⇒ Φ(p) ≥ Φ(q)
Shannon entropy: H(p) = −∑
pi ln pi
Tsallis entropy: Tq(p) =∑
pqi −1
1−q
Rényi entropy: Rq(p) =ln
∑pqi
1−q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 5 / 21
![Page 20: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/20.jpg)
Alternative de�nitions
De�nitions
p ≺ q sii
1 there exist a double stochastic matrix D such that
p = Dq with∑i
Dij =∑j
Dij = 1
2∑N
i=1 φ(pi ) ≤∑N
i=1 φ(qi ) for all concave function φ
Schur-concavity and entropies
Φ : RN 7→ R is Schur-concave if p ≺ q ⇒ Φ(p) ≥ Φ(q)
Shannon entropy: H(p) = −∑
pi ln pi
Tsallis entropy: Tq(p) =∑
pqi −1
1−q
Rényi entropy: Rq(p) =ln
∑pqi
1−q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 5 / 21
![Page 21: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/21.jpg)
Majorization lattice: POSET + in�mum and supreumum
Set of probability vectors
Let δN ={
[p1, . . . , pN ]t : pi ≥ pi+1 ≥ 0, and∑N
i=1 pi = 1 ≥ pi
}Partially ordered set (POSET)
For all p, q, r ∈ δN one has
re�exivity: p ≺ p
antisymmetry: if p ≺ q and q ≺ p, then p = q
transitivity: if p ≺ q and q ≺ r , then p ≺ r
Majorization is NOT a total order
If p = [0.6, 0.2, 0.2]t and q = [0.5, 0.4, 0.1]t , then p ⊀ q and q ⊀ p.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 6 / 21
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Majorization lattice: POSET + in�mum and supreumum
Set of probability vectors
Let δN ={
[p1, . . . , pN ]t : pi ≥ pi+1 ≥ 0, and∑N
i=1 pi = 1 ≥ pi
}
Partially ordered set (POSET)
For all p, q, r ∈ δN one has
re�exivity: p ≺ p
antisymmetry: if p ≺ q and q ≺ p, then p = q
transitivity: if p ≺ q and q ≺ r , then p ≺ r
Majorization is NOT a total order
If p = [0.6, 0.2, 0.2]t and q = [0.5, 0.4, 0.1]t , then p ⊀ q and q ⊀ p.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 6 / 21
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Majorization lattice: POSET + in�mum and supreumum
Set of probability vectors
Let δN ={
[p1, . . . , pN ]t : pi ≥ pi+1 ≥ 0, and∑N
i=1 pi = 1 ≥ pi
}Partially ordered set (POSET)
For all p, q, r ∈ δN one has
re�exivity: p ≺ p
antisymmetry: if p ≺ q and q ≺ p, then p = q
transitivity: if p ≺ q and q ≺ r , then p ≺ r
Majorization is NOT a total order
If p = [0.6, 0.2, 0.2]t and q = [0.5, 0.4, 0.1]t , then p ⊀ q and q ⊀ p.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 6 / 21
![Page 24: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/24.jpg)
Majorization lattice: POSET + in�mum and supreumum
Set of probability vectors
Let δN ={
[p1, . . . , pN ]t : pi ≥ pi+1 ≥ 0, and∑N
i=1 pi = 1 ≥ pi
}Partially ordered set (POSET)
For all p, q, r ∈ δN one has
re�exivity: p ≺ p
antisymmetry: if p ≺ q and q ≺ p, then p = q
transitivity: if p ≺ q and q ≺ r , then p ≺ r
Majorization is NOT a total order
If p = [0.6, 0.2, 0.2]t and q = [0.5, 0.4, 0.1]t , then p ⊀ q and q ⊀ p.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 6 / 21
![Page 25: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/25.jpg)
Majorization lattice: POSET + in�mum and supreumum
Set of probability vectors
Let δN ={
[p1, . . . , pN ]t : pi ≥ pi+1 ≥ 0, and∑N
i=1 pi = 1 ≥ pi
}Partially ordered set (POSET)
For all p, q, r ∈ δN one has
re�exivity: p ≺ p
antisymmetry: if p ≺ q and q ≺ p, then p = q
transitivity: if p ≺ q and q ≺ r , then p ≺ r
Majorization is NOT a total order
If p = [0.6, 0.2, 0.2]t and q = [0.5, 0.4, 0.1]t , then p ⊀ q and q ⊀ p.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 6 / 21
![Page 26: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/26.jpg)
Majorization lattice: POSET + in�mum and supreumum
Set of probability vectors
Let δN ={
[p1, . . . , pN ]t : pi ≥ pi+1 ≥ 0, and∑N
i=1 pi = 1 ≥ pi
}Partially ordered set (POSET)
For all p, q, r ∈ δN one has
re�exivity: p ≺ p
antisymmetry: if p ≺ q and q ≺ p, then p = q
transitivity: if p ≺ q and q ≺ r , then p ≺ r
Majorization is NOT a total order
If p = [0.6, 0.2, 0.2]t and q = [0.5, 0.4, 0.1]t , then p ⊀ q and q ⊀ p.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 6 / 21
![Page 27: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/27.jpg)
Majorization lattice: POSET + in�mum and supreumum
Set of probability vectors
Let δN ={
[p1, . . . , pN ]t : pi ≥ pi+1 ≥ 0, and∑N
i=1 pi = 1 ≥ pi
}Partially ordered set (POSET)
For all p, q, r ∈ δN one has
re�exivity: p ≺ p
antisymmetry: if p ≺ q and q ≺ p, then p = q
transitivity: if p ≺ q and q ≺ r , then p ≺ r
Majorization is NOT a total order
If p = [0.6, 0.2, 0.2]t and q = [0.5, 0.4, 0.1]t , then p ⊀ q and q ⊀ p.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 6 / 21
![Page 28: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/28.jpg)
Majorization lattice: POSET + in�mum and supreumum
Set of probability vectors
Let δN ={
[p1, . . . , pN ]t : pi ≥ pi+1 ≥ 0, and∑N
i=1 pi = 1 ≥ pi
}Partially ordered set (POSET)
For all p, q, r ∈ δN one has
re�exivity: p ≺ p
antisymmetry: if p ≺ q and q ≺ p, then p = q
transitivity: if p ≺ q and q ≺ r , then p ≺ r
Majorization is NOT a total order
If p = [0.6, 0.2, 0.2]t and q = [0.5, 0.4, 0.1]t , then p ⊀ q and q ⊀ p.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 6 / 21
![Page 29: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/29.jpg)
Majorization lattice: POSET + in�mum and supreumum
Set of probability vectors
Let δN ={
[p1, . . . , pN ]t : pi ≥ pi+1 ≥ 0, and∑N
i=1 pi = 1 ≥ pi
}Partially ordered set (POSET)
For all p, q, r ∈ δN one has
re�exivity: p ≺ p
antisymmetry: if p ≺ q and q ≺ p, then p = q
transitivity: if p ≺ q and q ≺ r , then p ≺ r
Majorization is NOT a total order
If p = [0.6, 0.2, 0.2]t and q = [0.5, 0.4, 0.1]t , then p ⊀ q and q ⊀ p.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 6 / 21
![Page 30: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/30.jpg)
Majorization lattice: POSET + in�mum and supremum
Majorization lattice [Cicalese y Vaccaro, IEEE Trans. Inf. Theory 48,933 (2002)]
Let 〈δN ,≺,∧,∨〉, where for all p, q ∈ δN there exists the in�mum p ∧ qand the supremum p ∨ q.
By de�ntion, one has
in�mum: p ∧ q i�p ∧ q ≺ p and p ∧ q ≺ q and s ≺ p ∧ qfor all s such that s ≺ p and s ≺ q
s
p ∧ q
p q
Infimum
supremum: p ∨ q siip ≺ p ∨ q y q ≺ p ∧ q and p ∨ q ≺ sfor all s such that p ≺ s and q ≺ s
s
p ∨ q
p q
Supremum
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 7 / 21
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Majorization lattice: POSET + in�mum and supremum
Majorization lattice [Cicalese y Vaccaro, IEEE Trans. Inf. Theory 48,933 (2002)]
Let 〈δN ,≺,∧,∨〉, where for all p, q ∈ δN there exists the in�mum p ∧ qand the supremum p ∨ q.
By de�ntion, one has
in�mum: p ∧ q i�p ∧ q ≺ p and p ∧ q ≺ q and s ≺ p ∧ qfor all s such that s ≺ p and s ≺ q
s
p ∧ q
p q
Infimum
supremum: p ∨ q siip ≺ p ∨ q y q ≺ p ∧ q and p ∨ q ≺ sfor all s such that p ≺ s and q ≺ s
s
p ∨ q
p q
Supremum
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 7 / 21
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Majorization lattice: POSET + in�mum and supremum
Majorization lattice [Cicalese y Vaccaro, IEEE Trans. Inf. Theory 48,933 (2002)]
Let 〈δN ,≺,∧,∨〉, where for all p, q ∈ δN there exists the in�mum p ∧ qand the supremum p ∨ q.
By de�ntion, one has
in�mum: p ∧ q i�p ∧ q ≺ p and p ∧ q ≺ q and s ≺ p ∧ qfor all s such that s ≺ p and s ≺ q
s
p ∧ q
p q
Infimum
supremum: p ∨ q siip ≺ p ∨ q y q ≺ p ∧ q and p ∨ q ≺ sfor all s such that p ≺ s and q ≺ s
s
p ∨ q
p q
Supremum
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 7 / 21
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Majorization lattice: POSET + in�mum and supremum
Majorization lattice [Cicalese y Vaccaro, IEEE Trans. Inf. Theory 48,933 (2002)]
Let 〈δN ,≺,∧,∨〉, where for all p, q ∈ δN there exists the in�mum p ∧ qand the supremum p ∨ q.
By de�ntion, one has
in�mum: p ∧ q i�p ∧ q ≺ p and p ∧ q ≺ q and s ≺ p ∧ qfor all s such that s ≺ p and s ≺ q
s
p ∧ q
p q
Infimum
supremum: p ∨ q siip ≺ p ∨ q y q ≺ p ∧ q and p ∨ q ≺ sfor all s such that p ≺ s and q ≺ s
s
p ∨ q
p q
Supremum
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 7 / 21
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Majorization lattice: POSET + in�mum and supremum
Majorization lattice [Cicalese y Vaccaro, IEEE Trans. Inf. Theory 48,933 (2002)]
Let 〈δN ,≺,∧,∨〉, where for all p, q ∈ δN there exists the in�mum p ∧ qand the supremum p ∨ q.
By de�ntion, one has
in�mum: p ∧ q i�p ∧ q ≺ p and p ∧ q ≺ q and s ≺ p ∧ qfor all s such that s ≺ p and s ≺ q
s
p ∧ q
p q
Infimum
supremum: p ∨ q siip ≺ p ∨ q y q ≺ p ∧ q and p ∨ q ≺ sfor all s such that p ≺ s and q ≺ s
s
p ∨ q
p q
Supremum
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 7 / 21
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Calculating the in�mum
In�mum
Let p, q ∈ δN , the in�mum s inf ≡ p ∧ q is such that
s infi = min
{i∑
l=1
pl ,i∑
l=1
ql
}−
i−1∑l=1
s infi ,
with s inf0 ≡ 0.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 8 / 21
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Calculating the in�mum
In�mum
Let p, q ∈ δN , the in�mum s inf ≡ p ∧ q is such that
s infi = min
{i∑
l=1
pl ,i∑
l=1
ql
}−
i−1∑l=1
s infi ,
with s inf0 ≡ 0.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 8 / 21
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Calculating the supremum
Supremum
Let p, q ∈ δN , the supremum, ssup ≡ p ∨ q, is obtained as follows
1 s = [s1, . . . , sN ]t : s1 = max {p1, q1} andsi = max
{∑il=1 pl ,
∑il=1 ql
}−∑i−1
l=1 sl with i ∈ [2,N]
2 r = [r1, . . . , rN ]t :
(a) let j be the smallest integer in [2,N] such that rj > rj−1
(b) let k be the greatest integer in [1, j − 1] such that rk−1 ≥∑j
l=k rlj−k+1
= a
with r0 > 1(c) let t the probability vector given by
tl ≡{
a for l = k , k + 1, . . . , jrl otherwise.
3 Applying transformations 2.(a)− 2.(c) with the input probability
vector s, one obtains the supremum in no more than N − 1 iterations.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 9 / 21
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Calculating the supremum
Supremum
Let p, q ∈ δN , the supremum, ssup ≡ p ∨ q, is obtained as follows
1 s = [s1, . . . , sN ]t : s1 = max {p1, q1} andsi = max
{∑il=1 pl ,
∑il=1 ql
}−∑i−1
l=1 sl with i ∈ [2,N]
2 r = [r1, . . . , rN ]t :
(a) let j be the smallest integer in [2,N] such that rj > rj−1
(b) let k be the greatest integer in [1, j − 1] such that rk−1 ≥∑j
l=k rlj−k+1
= a
with r0 > 1(c) let t the probability vector given by
tl ≡{
a for l = k , k + 1, . . . , jrl otherwise.
3 Applying transformations 2.(a)− 2.(c) with the input probability
vector s, one obtains the supremum in no more than N − 1 iterations.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 9 / 21
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Calculating the supremum
Supremum
Let p, q ∈ δN , the supremum, ssup ≡ p ∨ q, is obtained as follows
1 s = [s1, . . . , sN ]t : s1 = max {p1, q1} andsi = max
{∑il=1 pl ,
∑il=1 ql
}−∑i−1
l=1 sl with i ∈ [2,N]
2 r = [r1, . . . , rN ]t :
(a) let j be the smallest integer in [2,N] such that rj > rj−1
(b) let k be the greatest integer in [1, j − 1] such that rk−1 ≥∑j
l=k rlj−k+1
= a
with r0 > 1(c) let t the probability vector given by
tl ≡{
a for l = k, k + 1, . . . , jrl otherwise.
3 Applying transformations 2.(a)− 2.(c) with the input probability
vector s, one obtains the supremum in no more than N − 1 iterations.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 9 / 21
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Calculating the supremum
Supremum
Let p, q ∈ δN , the supremum, ssup ≡ p ∨ q, is obtained as follows
1 s = [s1, . . . , sN ]t : s1 = max {p1, q1} andsi = max
{∑il=1 pl ,
∑il=1 ql
}−∑i−1
l=1 sl with i ∈ [2,N]
2 r = [r1, . . . , rN ]t :
(a) let j be the smallest integer in [2,N] such that rj > rj−1
(b) let k be the greatest integer in [1, j − 1] such that rk−1 ≥∑j
l=k rlj−k+1
= a
with r0 > 1(c) let t the probability vector given by
tl ≡{
a for l = k, k + 1, . . . , jrl otherwise.
3 Applying transformations 2.(a)− 2.(c) with the input probability
vector s, one obtains the supremum in no more than N − 1 iterations.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 9 / 21
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Calculating the supremum
Supremum
Let p, q ∈ δN , the supremum, ssup ≡ p ∨ q, is obtained as follows
1 s = [s1, . . . , sN ]t : s1 = max {p1, q1} andsi = max
{∑il=1 pl ,
∑il=1 ql
}−∑i−1
l=1 sl with i ∈ [2,N]
2 r = [r1, . . . , rN ]t :
(a) let j be the smallest integer in [2,N] such that rj > rj−1
(b) let k be the greatest integer in [1, j − 1] such that rk−1 ≥∑j
l=k rlj−k+1
= a
with r0 > 1(c) let t the probability vector given by
tl ≡{
a for l = k, k + 1, . . . , jrl otherwise.
3 Applying transformations 2.(a)− 2.(c) with the input probability
vector s, one obtains the supremum in no more than N − 1 iterations.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 9 / 21
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Calculating the supremum
Supremum
Let p, q ∈ δN , the supremum, ssup ≡ p ∨ q, is obtained as follows
1 s = [s1, . . . , sN ]t : s1 = max {p1, q1} andsi = max
{∑il=1 pl ,
∑il=1 ql
}−∑i−1
l=1 sl with i ∈ [2,N]
2 r = [r1, . . . , rN ]t :
(a) let j be the smallest integer in [2,N] such that rj > rj−1
(b) let k be the greatest integer in [1, j − 1] such that rk−1 ≥∑j
l=k rlj−k+1
= a
with r0 > 1
(c) let t the probability vector given by
tl ≡{
a for l = k, k + 1, . . . , jrl otherwise.
3 Applying transformations 2.(a)− 2.(c) with the input probability
vector s, one obtains the supremum in no more than N − 1 iterations.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 9 / 21
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Calculating the supremum
Supremum
Let p, q ∈ δN , the supremum, ssup ≡ p ∨ q, is obtained as follows
1 s = [s1, . . . , sN ]t : s1 = max {p1, q1} andsi = max
{∑il=1 pl ,
∑il=1 ql
}−∑i−1
l=1 sl with i ∈ [2,N]
2 r = [r1, . . . , rN ]t :
(a) let j be the smallest integer in [2,N] such that rj > rj−1
(b) let k be the greatest integer in [1, j − 1] such that rk−1 ≥∑j
l=k rlj−k+1
= a
with r0 > 1(c) let t the probability vector given by
tl ≡{
a for l = k , k + 1, . . . , jrl otherwise.
3 Applying transformations 2.(a)− 2.(c) with the input probability
vector s, one obtains the supremum in no more than N − 1 iterations.
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 9 / 21
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Calculating the supremum
Supremum
Let p, q ∈ δN , the supremum, ssup ≡ p ∨ q, is obtained as follows
1 s = [s1, . . . , sN ]t : s1 = max {p1, q1} andsi = max
{∑il=1 pl ,
∑il=1 ql
}−∑i−1
l=1 sl with i ∈ [2,N]
2 r = [r1, . . . , rN ]t :
(a) let j be the smallest integer in [2,N] such that rj > rj−1
(b) let k be the greatest integer in [1, j − 1] such that rk−1 ≥∑j
l=k rlj−k+1
= a
with r0 > 1(c) let t the probability vector given by
tl ≡{
a for l = k , k + 1, . . . , jrl otherwise.
3 Applying transformations 2.(a)− 2.(c) with the input probability
vector s, one obtains the supremum in no more than N − 1 iterations.GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 9 / 21
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Continuation
Example
If p = [0.6, 0.15, 0.15, 0.1]t y q = [0.5, 0.25, 0.20, 0.05]t , then
p ∧ q = [0.5, 0.25, 0.15, 0.1]t
p ∨ q = [0.6, 0.175, 0.175, 0.05]t
Properties
bottom element: s0 ≡[1N . . .
1N
]ttop element: s1 ≡ [1, 0 . . . 0]t
majorization lattice is NOT modular:
if r ≺ q ; r ∨ (p ∧ q) = (r ∨ p) ∧ qp q
r
p ∧ q
p ∨ q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 10 / 21
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Continuation
Example
If p = [0.6, 0.15, 0.15, 0.1]t y q = [0.5, 0.25, 0.20, 0.05]t , then
p ∧ q = [0.5, 0.25, 0.15, 0.1]t
p ∨ q = [0.6, 0.175, 0.175, 0.05]t
Properties
bottom element: s0 ≡[1N . . .
1N
]ttop element: s1 ≡ [1, 0 . . . 0]t
majorization lattice is NOT modular:
if r ≺ q ; r ∨ (p ∧ q) = (r ∨ p) ∧ qp q
r
p ∧ q
p ∨ q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 10 / 21
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Continuation
Example
If p = [0.6, 0.15, 0.15, 0.1]t y q = [0.5, 0.25, 0.20, 0.05]t , then
p ∧ q = [0.5, 0.25, 0.15, 0.1]t
p ∨ q = [0.6, 0.175, 0.175, 0.05]t
Properties
bottom element: s0 ≡[1N . . .
1N
]ttop element: s1 ≡ [1, 0 . . . 0]t
majorization lattice is NOT modular:
if r ≺ q ; r ∨ (p ∧ q) = (r ∨ p) ∧ qp q
r
p ∧ q
p ∨ q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 10 / 21
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Continuation
Example
If p = [0.6, 0.15, 0.15, 0.1]t y q = [0.5, 0.25, 0.20, 0.05]t , then
p ∧ q = [0.5, 0.25, 0.15, 0.1]t
p ∨ q = [0.6, 0.175, 0.175, 0.05]t
Properties
bottom element: s0 ≡[1N . . .
1N
]ttop element: s1 ≡ [1, 0 . . . 0]t
majorization lattice is NOT modular:
if r ≺ q ; r ∨ (p ∧ q) = (r ∨ p) ∧ qp q
r
p ∧ q
p ∨ q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 10 / 21
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Continuation
Example
If p = [0.6, 0.15, 0.15, 0.1]t y q = [0.5, 0.25, 0.20, 0.05]t , then
p ∧ q = [0.5, 0.25, 0.15, 0.1]t
p ∨ q = [0.6, 0.175, 0.175, 0.05]t
Properties
bottom element: s0 ≡[1N . . .
1N
]ttop element: s1 ≡ [1, 0 . . . 0]t
majorization lattice is NOT modular:
if r ≺ q ; r ∨ (p ∧ q) = (r ∨ p) ∧ qp q
r
p ∧ q
p ∨ q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 10 / 21
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Continuation
Example
If p = [0.6, 0.15, 0.15, 0.1]t y q = [0.5, 0.25, 0.20, 0.05]t , then
p ∧ q = [0.5, 0.25, 0.15, 0.1]t
p ∨ q = [0.6, 0.175, 0.175, 0.05]t
Properties
bottom element: s0 ≡[1N . . .
1N
]ttop element: s1 ≡ [1, 0 . . . 0]t
majorization lattice is NOT modular:
if r ≺ q ; r ∨ (p ∧ q) = (r ∨ p) ∧ qp q
r
p ∧ q
p ∨ q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 10 / 21
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Continuation
Example
If p = [0.6, 0.15, 0.15, 0.1]t y q = [0.5, 0.25, 0.20, 0.05]t , then
p ∧ q = [0.5, 0.25, 0.15, 0.1]t
p ∨ q = [0.6, 0.175, 0.175, 0.05]t
Properties
bottom element: s0 ≡[1N . . .
1N
]t
top element: s1 ≡ [1, 0 . . . 0]t
majorization lattice is NOT modular:
if r ≺ q ; r ∨ (p ∧ q) = (r ∨ p) ∧ qp q
r
p ∧ q
p ∨ q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 10 / 21
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Continuation
Example
If p = [0.6, 0.15, 0.15, 0.1]t y q = [0.5, 0.25, 0.20, 0.05]t , then
p ∧ q = [0.5, 0.25, 0.15, 0.1]t
p ∨ q = [0.6, 0.175, 0.175, 0.05]t
Properties
bottom element: s0 ≡[1N . . .
1N
]ttop element: s1 ≡ [1, 0 . . . 0]t
majorization lattice is NOT modular:
if r ≺ q ; r ∨ (p ∧ q) = (r ∨ p) ∧ qp q
r
p ∧ q
p ∨ q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 10 / 21
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Continuation
Example
If p = [0.6, 0.15, 0.15, 0.1]t y q = [0.5, 0.25, 0.20, 0.05]t , then
p ∧ q = [0.5, 0.25, 0.15, 0.1]t
p ∨ q = [0.6, 0.175, 0.175, 0.05]t
Properties
bottom element: s0 ≡[1N . . .
1N
]ttop element: s1 ≡ [1, 0 . . . 0]t
majorization lattice is NOT modular:
if r ≺ q ; r ∨ (p ∧ q) = (r ∨ p) ∧ qp q
r
p ∧ q
p ∨ q
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 10 / 21
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Part II
Entanglement transformations
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 11 / 21
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Local operations and classical communication (LOCC)
Alice y Bob share an initial entangled pure state |ψ〉Goal: obtain the target entangled pure state |φ〉 by using
local operations and classical communications (LOCC)
which is the condition for this process of entanglementtransformation to be possible?
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 12 / 21
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Local operations and classical communication (LOCC)
Alice y Bob share an initial entangled pure state |ψ〉
Goal: obtain the target entangled pure state |φ〉 by using
local operations and classical communications (LOCC)
which is the condition for this process of entanglementtransformation to be possible?
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 12 / 21
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Local operations and classical communication (LOCC)
Alice y Bob share an initial entangled pure state |ψ〉Goal: obtain the target entangled pure state |φ〉 by using
local operations and classical communications (LOCC)
which is the condition for this process of entanglementtransformation to be possible?
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 12 / 21
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Local operations and classical communication (LOCC)
Alice y Bob share an initial entangled pure state |ψ〉Goal: obtain the target entangled pure state |φ〉 by using
local operations and classical communications (LOCC)
which is the condition for this process of entanglementtransformation to be possible?
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 12 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉
if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if
ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Entanglement transformation and majorization
Nielsen Theorem [Phys. Rev. Lett. 83, 436 (1999)]
Let consider the Schmidt decomposition of the states:
initial: |ψ〉 =∑N
i=1
√ψi |iA〉 |iB〉 con ψ = [ψ1, . . . , ψN ] ∈ δN
target: |φ〉 =∑N
j=1
√φj |jA〉 |jB〉 con φ = [φ1, . . . , φN ] ∈ δN
|ψ〉 →LOCC
|φ〉 if and only if ψ ≺ φ
This condition does not depend on the Schmidt basis
As it is expected, the Nielsen condition is not satis�ed in general
Example
|ψ〉 =√0.6 |00〉+
√0.15 |11〉+
√0.15 |22〉+
√0.1 |33〉
|φ〉 =√0.5 |00〉+
√0.25 |11〉+
√0.2 |22〉+
√0.05 |33〉
one has |ψ〉 =LOCC
|φ〉 due to ψ ⊀ φ and φ ⊀ ψ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 13 / 21
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Approximate entanglement transformations
initial state|ψ〉
target state
|φ〉
\
|χ1〉...|χM〉
Goal: �nd |χ〉 closest to |φ〉
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 14 / 21
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Approximate entanglement transformations
initial state|ψ〉
target state
|φ〉\
|χ1〉...|χM〉
Goal: �nd |χ〉 closest to |φ〉
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 14 / 21
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Approximate entanglement transformations
initial state|ψ〉
target state
|φ〉\
|χ1〉
...|χM〉
Goal: �nd |χ〉 closest to |φ〉
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 14 / 21
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Approximate entanglement transformations
initial state|ψ〉
target state
|φ〉\
|χ1〉
...
|χM〉
Goal: �nd |χ〉 closest to |φ〉
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 14 / 21
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Approximate entanglement transformations
initial state|ψ〉
target state
|φ〉\
|χ1〉...|χM〉
Goal: �nd |χ〉 closest to |φ〉
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 14 / 21
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Approximate entanglement transformations
initial state|ψ〉
target state
|φ〉\
|χ1〉...|χM〉
Goal: �nd |χ〉 closest to |φ〉
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 14 / 21
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Approximate entanglement transformations with Fidelity
Vidal et. al Criterion [Phys. Rev. A 62, 012304 (2000)]
Let |ψ〉 and |φ〉 be the initial and target such that |ψ〉 9LOCC
|φ〉.
They de�ne |χopt〉 as the closest to the target in the sense of maximal
�delity:
|χopt〉 = argmax|χ〉:|ψ〉 →
LOCC|χ〉F (|φ〉 , |χ〉),
where F (|φ〉 , |χ〉) = | 〈φ|χ〉 |2 is the �delity between the states |φ〉 and |χ〉
Equivalent problem
χopt = arg maxχ:ψ≺χ
F (φ, χ)
where F (φ, χ) =(∑
i
√φiχi
)2is the �delity between the vectors φ and χ
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Approximate entanglement transformations with Fidelity
Vidal et. al Criterion [Phys. Rev. A 62, 012304 (2000)]
Let |ψ〉 and |φ〉 be the initial and target such that |ψ〉 9LOCC
|φ〉.They de�ne |χopt〉 as the closest to the target in the sense of maximal
�delity:
|χopt〉 = argmax|χ〉:|ψ〉 →
LOCC|χ〉F (|φ〉 , |χ〉),
where F (|φ〉 , |χ〉) = | 〈φ|χ〉 |2 is the �delity between the states |φ〉 and |χ〉
Equivalent problem
χopt = arg maxχ:ψ≺χ
F (φ, χ)
where F (φ, χ) =(∑
i
√φiχi
)2is the �delity between the vectors φ and χ
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Approximate entanglement transformations with Fidelity
Vidal et. al Criterion [Phys. Rev. A 62, 012304 (2000)]
Let |ψ〉 and |φ〉 be the initial and target such that |ψ〉 9LOCC
|φ〉.They de�ne |χopt〉 as the closest to the target in the sense of maximal
�delity:
|χopt〉 = argmax|χ〉:|ψ〉 →
LOCC|χ〉F (|φ〉 , |χ〉),
where F (|φ〉 , |χ〉) = | 〈φ|χ〉 |2 is the �delity between the states |φ〉 and |χ〉
Equivalent problem
χopt = arg maxχ:ψ≺χ
F (φ, χ)
where F (φ, χ) =(∑
i
√φiχi
)2is the �delity between the vectors φ and χ
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Approximate entanglement transformations with Fidelity
Vidal et. al Criterion [Phys. Rev. A 62, 012304 (2000)]
Let |ψ〉 and |φ〉 be the initial and target such that |ψ〉 9LOCC
|φ〉.They de�ne |χopt〉 as the closest to the target in the sense of maximal
�delity:
|χopt〉 = argmax|χ〉:|ψ〉 →
LOCC|χ〉F (|φ〉 , |χ〉),
where F (|φ〉 , |χ〉) = | 〈φ|χ〉 |2 is the �delity between the states |φ〉 and |χ〉
Equivalent problem
χopt = arg maxχ:ψ≺χ
F (φ, χ)
where F (φ, χ) =(∑
i
√φiχi
)2is the �delity between the vectors φ and χ
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Expression of the optimum: χopt
χopt =
rk
φlk = φ1...
φlk−1−1
...
r2
φl2...
φl2−1−1
r1
φl1...
φl1−1−1 = φN
with lk the least integer in [1, lk − 1]such that
rk = minl∈[1,lk−1−1]
El(ψ)− Elk−1(ψ)
El(φ)− Elk−1(φ)
where El(ψ) =∑N
l ′=l ψl ′ for all
l = 1, . . . ,N
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Expression of the optimum: χopt
χopt =
rk
φlk = φ1...
φlk−1−1
...
r2
φl2...
φl2−1−1
r1
φl1...
φl1−1−1 = φN
with lk the least integer in [1, lk − 1]such that
rk = minl∈[1,lk−1−1]
El(ψ)− Elk−1(ψ)
El(φ)− Elk−1(φ)
where El(ψ) =∑N
l ′=l ψl ′ for all
l = 1, . . . ,N
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Expression of the optimum: χopt
χopt =
rk
φlk = φ1...
φlk−1−1
...
r2
φl2...
φl2−1−1
r1
φl1...
φl1−1−1 = φN
with lk the least integer in [1, lk − 1]such that
rk = minl∈[1,lk−1−1]
El(ψ)− Elk−1(ψ)
El(φ)− Elk−1(φ)
where El(ψ) =∑N
l ′=l ψl ′ for all
l = 1, . . . ,N
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Relationship with the supremum
Does the lattice structure of majorization play some role?
Theorem [GMB, G. Sergioli, H. Freytes, F. Holik and G. Bellomo]
Let |ψ〉 and |φ〉 the initial and target states, one has
φ ≺ χsup ≺ χopt
where χsup ≡ ψ ∨ φ
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Relationship with the supremum
Does the lattice structure of majorization play some role?
Theorem [GMB, G. Sergioli, H. Freytes, F. Holik and G. Bellomo]
Let |ψ〉 and |φ〉 the initial and target states, one has
φ ≺ χsup ≺ χopt
where χsup ≡ ψ ∨ φ
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Relationship with the supremum
Does the lattice structure of majorization play some role?
Theorem [GMB, G. Sergioli, H. Freytes, F. Holik and G. Bellomo]
Let |ψ〉 and |φ〉 the initial and target states, one has
φ ≺ χsup ≺ χopt
where χsup ≡ ψ ∨ φ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 17 / 21
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Relationship with the supremum
Does the lattice structure of majorization play some role?
Theorem [GMB, G. Sergioli, H. Freytes, F. Holik and G. Bellomo]
Let |ψ〉 and |φ〉 the initial and target states, one has
φ ≺ χsup ≺ χopt
where χsup ≡ ψ ∨ φ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 17 / 21
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Relationship with the supremum
Theorem [GMB, G. Sergioli, H. Freytes, F. Holik and G. Bellomo]
Let |ψ〉 and |φ〉 the initial and target states, one has
φ ≺ χsup ≺ χopt
where χsup ≡ ψ ∨ φ
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 18 / 21
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Relationship with the supremum
Theorem [GMB, G. Sergioli, H. Freytes, F. Holik and G. Bellomo]
Let |ψ〉 and |φ〉 the initial and target states, one has
φ ≺ χsup ≺ χopt
where χsup ≡ ψ ∨ φ
ψ
φ
Case 1: if |ψ〉 →LOCC
|φ〉
= χsup = χopt
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 18 / 21
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Relationship with the supremum
Theorem [GMB, G. Sergioli, H. Freytes, F. Holik and G. Bellomo]
Let |ψ〉 and |φ〉 the initial and target states, one has
φ ≺ χsup ≺ χopt
where χsup ≡ ψ ∨ φ
φ
ψ
χopt
Case 2: if |ψ〉 9LOCC
|φ〉 and |ψ〉 ←LOCC
|φ〉
= χsup
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 18 / 21
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Relationship with the supremum
Theorem [GMB, G. Sergioli, H. Freytes, F. Holik and G. Bellomo]
Let |ψ〉 and |φ〉 the initial and target states, one has
φ ≺ χsup ≺ χopt
where χsup ≡ ψ ∨ φ
χsup χopt
φ ψ
(a) χsup ⊀ χopt and χopt ⊀ χsup
χsup
χopt
φ ψ
(b) χopt ≺ χsup
χopt
χsup
φ ψ
(c) χsup ≺ χopt
Case 3: |ψ〉 =LOCC
|φ〉
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 18 / 21
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Fidelity vs. majorization
Example
Let:
ψ = [0.6, 0.15, 0.15, 0.1]t
φ = [0.5, 0.25, 0.2, 0.5]t
one has
χopt = [0.6, 0.2, 0.16, 0.4]t with �delity F (φ, χopt) ≈ 0.989
χsup = [0.6, 0.175, 0.175, 0.05]t with �delity F (φ, χsup) ≈ 0.987
But, we have seen that:
φ ≺ χsup ≺ χopt
Fidelity does not respect the majorization order in general
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 19 / 21
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Fidelity vs. majorization
Example
Let:
ψ = [0.6, 0.15, 0.15, 0.1]t
φ = [0.5, 0.25, 0.2, 0.5]t
one has
χopt = [0.6, 0.2, 0.16, 0.4]t with �delity F (φ, χopt) ≈ 0.989
χsup = [0.6, 0.175, 0.175, 0.05]t with �delity F (φ, χsup) ≈ 0.987
But, we have seen that:
φ ≺ χsup ≺ χopt
Fidelity does not respect the majorization order in general
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 19 / 21
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Fidelity vs. majorization
Example
Let:
ψ = [0.6, 0.15, 0.15, 0.1]t
φ = [0.5, 0.25, 0.2, 0.5]t
one has
χopt = [0.6, 0.2, 0.16, 0.4]t with �delity F (φ, χopt) ≈ 0.989
χsup = [0.6, 0.175, 0.175, 0.05]t with �delity F (φ, χsup) ≈ 0.987
But, we have seen that:
φ ≺ χsup ≺ χopt
Fidelity does not respect the majorization order in general
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 19 / 21
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Fidelity vs. majorization
Example
Let:
ψ = [0.6, 0.15, 0.15, 0.1]t
φ = [0.5, 0.25, 0.2, 0.5]t
one has
χopt = [0.6, 0.2, 0.16, 0.4]t with �delity F (φ, χopt) ≈ 0.989
χsup = [0.6, 0.175, 0.175, 0.05]t with �delity F (φ, χsup) ≈ 0.987
But, we have seen that:
φ ≺ χsup ≺ χopt
Fidelity does not respect the majorization order in general
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 19 / 21
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Fidelity vs. majorization
Example
Let:
ψ = [0.6, 0.15, 0.15, 0.1]t
φ = [0.5, 0.25, 0.2, 0.5]t
one has
χopt = [0.6, 0.2, 0.16, 0.4]t with �delity F (φ, χopt) ≈ 0.989
χsup = [0.6, 0.175, 0.175, 0.05]t with �delity F (φ, χsup) ≈ 0.987
But, we have seen that:
φ ≺ χsup ≺ χopt
Fidelity does not respect the majorization order in general
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 19 / 21
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Fidelity vs. majorization
Example
Let:
ψ = [0.6, 0.15, 0.15, 0.1]t
φ = [0.5, 0.25, 0.2, 0.5]t
one has
χopt = [0.6, 0.2, 0.16, 0.4]t with �delity F (φ, χopt) ≈ 0.989
χsup = [0.6, 0.175, 0.175, 0.05]t with �delity F (φ, χsup) ≈ 0.987
But, we have seen that:
φ ≺ χsup ≺ χopt
Fidelity does not respect the majorization order in general
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 19 / 21
![Page 99: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/99.jpg)
Fidelity vs. majorization
Example
Let:
ψ = [0.6, 0.15, 0.15, 0.1]t
φ = [0.5, 0.25, 0.2, 0.5]t
one has
χopt = [0.6, 0.2, 0.16, 0.4]t with �delity F (φ, χopt) ≈ 0.989
χsup = [0.6, 0.175, 0.175, 0.05]t with �delity F (φ, χsup) ≈ 0.987
But, we have seen that:
φ ≺ χsup ≺ χopt
Fidelity does not respect the majorization order in general
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 19 / 21
![Page 100: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/100.jpg)
Fidelity vs. majorization
Example
Let:
ψ = [0.6, 0.15, 0.15, 0.1]t
φ = [0.5, 0.25, 0.2, 0.5]t
one has
χopt = [0.6, 0.2, 0.16, 0.4]t with �delity F (φ, χopt) ≈ 0.989
χsup = [0.6, 0.175, 0.175, 0.05]t with �delity F (φ, χsup) ≈ 0.987
But, we have seen that:
φ ≺ χsup ≺ χopt
Fidelity does not respect the majorization order in general
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 19 / 21
![Page 101: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/101.jpg)
Fidelity vs. majorization
Example
Let:
ψ = [0.6, 0.15, 0.15, 0.1]t
φ = [0.5, 0.25, 0.2, 0.5]t
one has
χopt = [0.6, 0.2, 0.16, 0.4]t with �delity F (φ, χopt) ≈ 0.989
χsup = [0.6, 0.175, 0.175, 0.05]t with �delity F (φ, χsup) ≈ 0.987
But, we have seen that:
φ ≺ χsup ≺ χopt
Fidelity does not respect the majorization order in general
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 19 / 21
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Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
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Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
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Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
![Page 105: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/105.jpg)
Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
![Page 106: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/106.jpg)
Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
![Page 107: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/107.jpg)
Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
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Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
![Page 109: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/109.jpg)
Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
![Page 110: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/110.jpg)
Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
![Page 111: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/111.jpg)
Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
![Page 112: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/112.jpg)
Intepreting the supreum
Distance on the majorization lattice
Let two proability vectors p, q ∈ δN . A distance d is de�ned as:
d(p, q) = H(p) + H(q)− 2H(p ∨ q) con H(p) = −∑i
pi ln pi
positivity: d(p, q) ≥ 0 with d(p, q) = 0 i� p = q
symmetry: d(p, q) = d(q, p)
triangle inequality: d(p, r) + d(r , q) ≥ d(p, q)
compatible with the lattice: if p ≺ q ≺ r ⇒ d(p, r) = d(p, q) + d(q, r)
Supremum state
|ψ〉 →LOCC
|χsup〉 ≡∑j
√χsupj |jA〉 |jB〉 with χsup ≡ ψ ∨ φ
is the closest to target in the sense of minimal distance d :
|χsup〉 = argmin|χ〉:|ψ〉 →
LOCC|χ〉
d(|φ〉 , |χ〉)
it has more entanglement entropy than the optimumRecall: Entropy of Schmidt coe�cients is the entanglement entropy
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 20 / 21
![Page 113: Majorization and Entanglement transformations](https://reader035.vdocuments.us/reader035/viewer/2022071519/613c6590f237e1331c514e8d/html5/thumbnails/113.jpg)
Grazie mille!!!
Questions, comments...
GM Bosyk (IFLP&UNICA) AMQI 2016 03/11/16 21 / 21