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Gili Bisker Physics Department, Technion 1
Entanglement: Definition, Purification and measures
Seminar in Quantum Information processing 236823
Gili Bisker
Physics Department Technion
Spring 2006
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 2
Introduction
• Entanglement of pure and mixed states. Identifying Entanglement is easy for a pure state of two systems, but it become extremely difficult when dealing with mixed state in higher dimension.
• Peres necessary criterion that a non entangled mixed state must
satisfy and the proof of its sufficiency in a special case given by the Horodeckis.
• Entanglement is a resource needed for quantum information
(Teleportation), but how can we generate maximally entangles states?
• Quantifying entanglement.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 3
What is Entanglement?
• A pure state of a pair of quantum systems is called entangled if it cannot be written as a product state of the two systems
A Bψ ≠ ⊗ . • A classic example is the singlet state ( )1
2↑↓ − ↓↑ .
• A mixed state is entangled if it cannot be written as the sum of
products of density matrices of the two subsystems A A BA
wρ ρ ρ≠ ⊗∑ . It means that the mixed stated cannot be prepared by two observers in distant labs.
• Example: the state ( )1
2 00 00 11 11ρ = + is not entangled since it can be locally prepared: We choose randomly 0 or 1. If we have 0 we prepare both systems in the state 0 and otherwise 1 .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 4
Separability Criterion for Density Matrices
Asher Peres 1934-2005
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 5
Separability Criterion for Density Matrices
• A given composite quantum system can be prepared by two distant observers if the density matrix of the system is separable:
0, 1A A A A AA Aw w wρ ρ ρ′ ′′= ⊗ > =∑ ∑
• We define the partial transposed: if ( ) ( ),m n A A AmnA
wμ ν μνρ ρ ρ′ ′′=∑ then we denote , ,m n n mμ ν μ νσ ρ= .
• If the state is separable then it means ( )2
TTA A AA
wρ σ ρ ρ′ ′′= = ⊗∑ or:
13 1411 12 11 12 31 32
23 2421 22 21 22 41 42
13 14 33 3433 3431 32
23 24 43 4443 4441 42
ρ ρρ ρ ρ ρ ρ ρρ ρρ ρ ρ ρ ρ ρ
ρ σρ ρ ρ ρρ ρρ ρρ ρ ρ ρρ ρρ ρ
⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟
⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= ⇒ =⎜ ⎟ ⎜ ⎟⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 6
Separability Criterion for Density Matrices
• The transposed matrices ( ) ( )TA Aρ ρ ∗′ ′= are non negative matrices
with a unit trace • ⇒ Density matrices. • ⇒ None of the eigenvalues of σ are negative. • This is a necessary condition for the original density matrix to be
separable. ( )T
A A AAwσ ρ ρ′ ′′= ⊗∑
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 7
Separability Criterion for Density Matrices
• Example:
( )0 0 0 0 1 0 0 00 1 1 0 0 1 0 01 10 1 1 0 0 0 1 04 2 40 0 0 0 0 0 0 1
1 0 0 00 1 2 010 2 1 040 0 0 1
x x xxS I
xx xx x
x
ρ
ρ
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟−− −⎜ ⎟ ⎜ ⎟= + = +⎜ ⎟ ⎜ ⎟−⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠−⎛ ⎞
⎜ ⎟+ −⎜ ⎟=⎜ ⎟− +⎜ ⎟−⎝ ⎠
• After the partial transposition: 1 0 0 2
0 1 0 010 0 1 042 0 0 1
x xx
xx x
σ
− −⎛ ⎞⎜ ⎟+⎜ ⎟=⎜ ⎟+⎜ ⎟− −⎝ ⎠
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 8
Separability Criterion for Density Matrices
• The eigenvalues of σ : ( )11 2 3 4 1 xλ λ λ= = = + and ( )1
4 4 1 3xλ = − . • If the lowest eigenvalues is negative, meaning 1
3x > , the density matrix will not be separable.
• In this case, this criterion is also a sufficient one, meaning that if
13x < the density matrix is indeed separable.
• The Horodeckis proved that this condition is also a sufficient one
for systems with dimension 2 2× and 2 3× . • However, for higher dimension, the partial transpose condition
was shown not to be a sufficient one.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 9
Separability of mixed states: necessary and sufficient conditions
(M. Horodecki, P. Horodecki and R. Horodecki)
• Notations:
o Finite dimensional Hilbert Space 1 2
H H H= ⊗ . o The sets of linear hermitian operators acting on 1 2,H H will be
denoted by 1 2,A A respectively. These sets are Hilbert spaces
with the scalar product ( )†,A B Tr B A= . o The space of linear maps from
1A to
2A is denoted by ( )1 2,L A A .
o A map ( )1 2,L A AΛ∈ is positive if it maps positive operators from 1
A to 2
A , meaning, that if 0A ≥ then ( ) 0AΛ ≥ . o We say that a map Λ is completely positive if the inducted map
1 2:
n n nI A M A MΛ = Λ⊗ ⊗ → ⊗
Is positive for all n , where n n
nM ×∈ and I is the identity map. • We will show that if a state ρ is not separable then there exists a
positive map Λ such that ( )I ρΛ⊗ is not a positive operator. For 2 2× and 2 3× systems, the partial transpose will do the trick.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 10
Separability of mixed states: necessary and sufficient conditions
• An hermitian operator A that acts on 1 2H H H= ⊗ is an
entanglement witness if it fulfills: 1. A is not a positive operator. 2. ( ) 0Tr Aσ ≥ for every separable state σ .
Example: For qubits, the operator 2W I + += − Φ Φ , where
( )12
00 11+Φ = + , is an entanglement witness: 1. ( )2 1 0W I+ + + + + +Φ Φ = Φ − Φ Φ Φ = − < , so this operator is not a positive one. 2. For any pair of states, 0 1a α β= + and 0 1b γ δ= + , we have:
( ) ( )( )
( )( ) ( )( ) ( )
* * * *
2 2 2 2 2 2* * * *
22 2 2 2* * * *
, , , 2 , 1
1 2 Re 2 Re
2Re 2 0
a b W a b a b I a b α γ β δ αγ βδ
αγ αγβ δ βδ α β αγ αγβ δ βδ
αδ βγ αγβ δ αδ βγ αγβ δ αδ βγ
+ += − Φ Φ = − + + =
= − + + = + − − − =
= + − ≥ + − = − ≥
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 11
Separability of mixed states: necessary and sufficient conditions
Lemma: For any inseparable state 1 2A Aρ ∈ ⊗ there exists an entanglement witness such that ( ) 0Tr Aρ < . (The proof uses tools from functional analysis like the Hahn Banach's theorem). Theorem 1: A state
1 2A Aρ ∈ ⊗ is separable if and only if for every positive map
2 1: A AΛ → the operator ( )I ρ⊗Λ is positive.
(Again, the proof uses tools from functional analysis).
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 12
Separability of mixed states: necessary and sufficient conditions
Theorem 2: A state ρ in 2 2⊗ or 2 3⊗ is separable if and only if its partial transposition is a positive operator. Proof: if ρ is separable then its partial transpose 2Tρ is also positive. For the converse statement, we assume that 2Tρ is positive. Every positive map
1 2: A AΛ → , where 2
1 2H H= = or
3 2
1 2,H H= = is of the form
1 2
CP CPTΛ = Λ +Λ , where CP
iΛ is
complete positive and T is the transposition. Since CP
iΛ is complete positive, the map CP
i iIΛ = ⊗Λ is positive, and
if 2Tρ is positive then ( ) ( )( ) ( ) ( ) 2 2
1 2 1 2 1 2
T TCP CP CP CPI I T I Iρ ρ ρ ρ ρ ρ⊗Λ = ⊗ Λ +Λ = ⊗Λ + ⊗Λ = Λ +Λ
Is also positive, and by Theorem 1, ρ is separable.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 13
Concentrating Partial Entanglement by Local Operations
(Bennett, Bernstein, Popescu, Schumacher) • If Alice and Bob are supplied with n pairs of particles in identical
partly entangled pure states, they can concentrate their entanglement into smaller number of maximally entangle pairs (like the singlet), by local actions of each observer.
• The entanglement of partly entangle pure state ( ),A BΨ can be
parameterize by its entropy of entanglement: { } { }2 2log logA A B BH Tr Trρ ρ ρ ρ= − = −
Where: ( ) ( ) ( ){ },A B B ATr A Bρ = Ψ
• The yield of singlets approaches ( )2lognH n−Ο .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 14
Concentrating Partial Entanglement by Local Operations
• Initially, Alice and Bob share n partly entangled particles with
initial entropy of entanglement nH . They can perform only local operations and communicate classically.
• We can write the initial state as:
( )1
, cos 01 sin 10n
AB ABiA B θ θ
=Ψ = ⊗⎡ + ⎤⎣ ⎦
• The entropy of entanglement of one particle:
{ }
2 2
2 2 2 22 2 2
cos 01 01 sin 10 10
log cos log cos sin log sinA AB AB
A AH Tr
ρ θ θ
ρ ρ θ θ θ θ
= +
⎡ ⎤= − = − +⎣ ⎦
• The total entropy Alice and Bob share is nH .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 15
Concentrating Partial Entanglement by Local Operations
• If they have only one particle: ( ), 01 10AB ABA B α βΨ = + , where
we assume α β< , Alice can perform the following measurement:
0
† † †1 1 0 0 1 0 0
0 0 1 1
1 1
M
M M M M M U M M
βλα⎛ ⎞= +⎜ ⎟⎝ ⎠
+ = ⇒ = −
• Whenever she measures 0μ = , the state will be:
( ) ( )0
0
0 0
101 10 01 102AB AB AB ABnormalization
Mp p
μ λβ= ΨΨ→ = + → +
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 16
Concentrating Partial Entanglement by Local Operations
• The probability for getting a maximally entangled state is
20 2p λβ= . In order to increase this probability we would like to
increase λ .
• But we must keep the operator †0 0M M positive:
2*
2 2† *0 0
0 00
0 10 1 0 1M M
β ββλ λα αα
⎛ ⎞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟= = ⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠
• And since we have † †
1 1 0 0 1M M M M+ = , then 22
2 2 2 201 2 2 1pβ αλ λ λβ α
α β≤ ⇒ ≤ ⇒ = ≤ ≤
So we cannot increase λ as much as we want.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 17
Concentrating Partial Entanglement by Local Operations
• For two particles we have
( )( )
2 2
2 2
cos 01 sin 10 cos 01 sin 10
cos 01 01 sin cos 01 10 10 01 sin 10 10
cos 00 11 sin cos 01 10 10 01 sin 11 00AB AB AB AB AB AB AB AB
AA BB AA BB AA BB AA BB
θ θ θ θ
θ θ θ θ
θ θ θ θ
+ + =⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦⊗ + ⊗ + ⊗ + ⊗ =
⊗ + ⊗ + ⊗ + ⊗
• After appropriate measurement done by Alice, they can be left
with the state ( )01 10 10 01
2AA BB AA BB⊗ + ⊗
With probability 2 22sin cosθ θ . • After a CNOT performed locally by Alice and Bob:
( ) ( )1 12 2
01 11 11 01 01 10 11AA BB AA BB AB AB AB⊗ + ⊗ = + ⊗ They have one particle in a maximally entangled state.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 18
Concentrating Partial Entanglement by Local Operations
• Now, let us examine a more general case, with n pairs:
( )1
, cos 01 sin 10n
AB ABi
A B θ θ=
Ψ = ⎡ + ⎤⎣ ⎦∏
• The state has 2n terms with 1n + distinct coefficients:
1cos , cos sin , ,sinn n nθ θ θ θ− … • Alice performs a measurement, projecting the initial state into one
of 1n + orthogonal subspaces corresponding to the power 0, ,k n= … of sinθ in the coefficients, obtaining a result k with
probability:
( ) ( )2 2cos sinn k k
k
np
kθ θ
−⎛ ⎞= ⎜ ⎟⎝ ⎠
• Alice then tells Bob which outcome she obtained.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 19
Concentrating Partial Entanglement by Local Operations
• Alice and Bob will be left with a state kψ which is a maximally
entangled state in a subsystem with 2nk⎛ ⎞⎜ ⎟⎝ ⎠
dimensions, because we
can change the base and get the state:
00 11AB ABAB
n nk k⎛ ⎞⎛ ⎞
+ + + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
• When n →∞ , the probability is maximal for 2sink nθ= .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 20
Concentrating Partial Entanglement by Local Operations
• If we had m identical maximally entangled states ( )00 11
m
AB AB
⊗+ ,
the product has 2m orthogonal elements with the same coefficient.
• Alice and Bob have the state:
00 11AB ABAB
n nk k⎛ ⎞⎛ ⎞
+ + + ⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠
• We'll Define H , such that the above nk⎛ ⎞⎜ ⎟⎝ ⎠
orthogonal states is
equivalent to nH maximally entangled pairs:
2nH nk⎛ ⎞
= ⎜ ⎟⎝ ⎠
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 21
Concentrating Partial Entanglement by Local Operations
• If we take the most likely case, where 2sink nθ= then:
( ) ( )
( ) ( )
2 22 2 2
2 2 2 22 2 22 2
!log logsin sin ! sin !
!log sin log sin cos log cossin ! cos !
n nnHn n n n
n nn n
θ θ θ
θ θ θ θθ θ
⎛ ⎞⎛ ⎞= = =⎜ ⎟⎜ ⎟ ⎜ ⎟−⎝ ⎠ ⎝ ⎠
⎛ ⎞⎡ ⎤= ≈ − +⎜ ⎟ ⎣ ⎦⎜ ⎟
⎝ ⎠
• 2 2 2 2
2 2sin log sin cos log cosH θ θ θ θ⎡ ⎤= − +⎣ ⎦ , which is the entropy of entanglement of the initial pure state.
• n non maximally entangled nH→ maximally entangled.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 22
Purification of Noisy Entanglement (Bennett, Brassard, Popescu, Schumacher, Smolin and Wootters)
• Two observers, who share entangled states, can purify them by
local operations and measurements, using classical communication, and sacrificing some of the pairs to increase the purity of the rest.
• The initial state is:
( ){ }13 1FW F Fψ ψ ψ ψ φ φ φ φ− − + + − − + += + − + +
Where ( )12
01 10ψ ± = ± and ( )12
00 11φ± = ± . • The purity of FW can be express by the Fidelity: FF Wψ ψ− −= • The yield of pure states is shown to be positive if 1
2F > .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 23
Purification of Noisy Entanglement
• First, Bob performs a unilateral yσ rotation. The Unilateral Pauli
rotation iI σ⊗ maps the bell states onto one other, leaving no state unchanged:
maps
maps
maps and
x
y
z
σ ψ φ
σ ψ φ
σ ψ ψ φ φ
± ±
±
± ±
↔
↔
↔ ↔
∓
∓ ∓
• The new state is:
( ){ }13 1F Fφ φ φ φ ψ ψ ψ ψ+ + − − + + − −+ − + +
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 24
Purification of Noisy Entanglement
• The next step is BXOR which is the quantum XOR operating on
two pairs. • The quantum XOR operates on two qubits, flipping the second
(target) spin iff the first (source) spin is up. • For example, we take ψ − as the source and φ− as the target.
( ) ( )( )
( )( ) ( )( )
( ) ( )
1212 34
12
12
12
12
01 10 00 11
0100 0111 1000 1011
0110 0101 1001 1010
01 10 01 10 10 01
01 10 10 01
BXOR BXOR
BXOR
ψ φ
ψ ψ
− −
+ −
⊗ = − ⊗ − =
= − − + =
= − − + =
= − + − =
= + ⊗ − = ⊗
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
BXOR Source Target
Alice 1 3 Flips 3 iff 1 is up Bob 2 4 Flips 4 iff 2 is up
Gili Bisker Physics Department, Technion 25
Purification of Noisy Entanglement
• Summarizing, omitting overall phase:
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Before After BXOR Source Target Source Target φ± φ+ same same ψ ± φ+ same ψ + ψ ± ψ + same φ+ φ± ψ + same same φ± φ− φ ∓ same ψ ± φ− ψ ∓ ψ − ψ ± ψ − ψ ∓ φ− φ± ψ − φ ∓ same
Gili Bisker Physics Department, Technion 26
Purification of Noisy Entanglement
• After the BXOR:
( ) ( ) ( )( )( ) ( ) ( )( )
( )( ) ( ) ( ) ( )( ) ( ) ( )
( )
1 1 13 3 3
1 1 13 3 3
13
1 1 1 13 3 3 3
1 1 13 3 3
13
1 1 1
1 1 1
1
1 1 1 1
1 1 1
1
F F F FBXOR
F F F F
FF F F
F F F F
F F F F
F F
φ φ φ φ ψ ψ ψ ψ
φ φ φ φ ψ ψ ψ ψ
φ φ φ φ φ φ ψ ψ
φ φ φ φ φ φ ψ ψ
φ φ φ φ φ φ ψ ψ
φ φ
+ + − − + + − −
+ + − − + + − −
+ + + + + + + +
+ + − − + + − −
− − + + − − + +
− −
⎧ ⎫+ − + − + −⎪ ⎪⎨ ⎬⊗ + − + − + −⎪ ⎪⎩ ⎭
⊗ + − ⊗
+ − − ⊗ + − − ⊗
+ − ⊗ + − − ⊗
+ −=
( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )
13
1 1 13 3 3
1 1 1 13 3 3 3
1 1 13 3 3
1 1 1 13 3 3 3
1
1 1 1
1 1 1 1
1 1 1
1 1 1 1
F F
F F F F
F F F F
F F F F
F F F F
φ φ φ φ ψ ψ
ψ ψ φ φ ψ ψ ψ ψ
ψ ψ ψ ψ ψ ψ φ φ
ψ ψ φ φ ψ ψ ψ ψ
ψ ψ ψ ψ ψ ψ φ φ
− − − − − −
+ + + + + + + +
+ + − − + + − −
− − + + − − + +
− − − − − − − −
⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⊗ + − ⊗ ⎪⎪ ⎪⎨ ⎬+ − − ⊗ + − ⊗⎪⎪+ − − ⊗ + − − ⊗⎪⎪+ − − ⊗ + − ⊗⎪⎪+ − − ⊗ + − − ⊗⎪⎩ ⎭
⎪⎪⎪⎪⎪⎪⎪
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 27
Purification of Noisy Entanglement
• The target pair is locally measured in the z direction, if the spins
come out parallel (which mean that the target is in the φ φ+ + or the φ φ− − state) the unmeasured source pair is kept, otherwise, it is discarded.
• If the target pair's z spins parallel, the new source state is:
( ) ( )( )( )
( )( )
22 19
22 52 23 9 3
111 1 1
F F
F F F F F F
φ φ
φ φ ψ ψ ψ ψ
+ +
− − + + − −
⎧ ⎫+ −⎪ ⎪⎨ ⎬
+ − + − + − + +⎪ ⎪⎩ ⎭
• After unilateral yσ rotation, we get:
( ) ( )( )( )
( )( )
22 19
22 52 23 9 3
111 1 1
F F
F F F F F F
ψ ψ
ψ ψ φ φ φ φ
− −
+ + − − + +
⎧ ⎫+ −⎪ ⎪⎨ ⎬
+ − + − + − + +⎪ ⎪⎩ ⎭
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 28
Purification of Noisy Entanglement
• The new Fidelity is:
( )( ) ( )
22 19
22 523 9
11 1
F FF
F F F F+ −
′ =+ − + −
F'(F)
0
0.5
1
0 0.5 1
F
F`F
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 29
Introduction to Entanglement Measures
• Goal: quantifying entanglement and finding magnitudes which
behave monotonically under local transformation. • A typical framework would be two observers who can
communicate classically, sharing a composite system in an entangled state, on which they can only operate locally (LOCC):
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 30
Basic Properties of Entanglement
• Separable state contains no entanglement. A state of many parties A,B,C… is said to be separable if it can be written in the form
... ... 1i i iABC i A B C ii i
p pρ ρ ρ ρ= ⊗ ⊗ ⊗ =∑ ∑ These states can be created by LOCC: Alice samples from the distribution ip , informs all the others of the outcome and then each party X locally prepares i
Xρ . • All non separable states are entangled. Any non separable state ρ can enhance the teleportation power of some other state σ (L. Masanes 2006)
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 31
Basic Properties of Entanglement
• The entanglement of states does not increase under LOCC
transformations. LOCC can only create separable so LOCC cannot create entanglement from a non entangled state. If quantum state ρ can be transformed to σ using LOCC, Then ρ is at least as entangled as σ . • Entanglement does not change under local unitary
transformation. This one follows from the previous one, since local unitary transformations are invertible.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 32
Basic Properties of Entanglement
• There are maximally entangled states. In two party systems a maximally entangled state is:
00 11 1; 1d
d dd
ψ + + + + − −=
Any pure or mixed state can be prepared from such a state with certainty using only LOCC operations. (By the way, an equivalent statement does not exist in multi particle systems, so there is a difficulty in establishing a theory of multi particle entanglement).
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 33
Local Manipulation of Quantum States
• For bi-partite system any pure state can be created using LOCC
from a maximally entangle state, which is a state that is unitary equivalent to ( )1
2 200 11ψ + = + .
• We'll consider a general states written in the Schmidt
decomposition form: 00 11φ α β= +
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 34
Local Manipulation of Quantum States
• Let us define the operators:
( )( ) ( )
0
1
0 0 1 1
0 1 1 0 1 0 0 1
A I
A
α β
α β
+ ⊗
+ ⊗ +
• They satisfy:
† †0 0 1 1A A A A I I+ = ⊗
And:
1 10 2 1 22 2
A Aψ φ ψ φ+ += = So:
† †0 2 2 0 1 2 2 1A A A Aφ φ ψ ψ ψ ψ+ + + += +
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 35
Local Manipulation of Quantum States
• Let us now add an ancilla 0 to Alice's system:
( )12 2
0 00 0 11A AB A ABψ + → + • Alice applies to her system the unitary transformation:
00 00 11
01 10 01
α β
α β
→ +
→ +
• We get: ( ) ( )( )
( ) ( )( )12
12
00 11 0 10 01 1
0 00 11 1 01 10B BA A
A AB AB A AB AB
α β α β
α β α β
+ + + =
+ + +
• Alice measures her ancilla, if the outcome is 0 , then they are left
with φ . If the outcome is 1 , Bob performs xσ on his particle, and they are left with φ .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 36
Local Manipulation of Quantum States
These two procedures are equivalent: if we want to perform a measurement described by the measurement operators mM : We add an ancilla with an orthogonal basis m corresponding to the possible measurement outcomes. If the state of the ancilla at the beginning is 0 , then we define an operator U on the product 0ψ by: 0 mm
U M mψ ψ=∑ . If we do a projective measurement of the ancilla mP I m m= ⊗ , the outcome m occurs with probability:
( ) † †0 0m m mp m U P U M Mψ ψ ψ ψ= = And the state after the measurement will be:
†m
m m
Mm
M Mψ
ψ ψ⊗
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 37
Local Manipulation of Quantum States
Let A BH Hρ ∈ ⊗ , where A BH H H= = . Let A B
A BH Hφ φ φ= ⊗ ∈ ⊗ be a product state, then there is a LOCC operation Λ such that ( )ρ φ φΛ = :
( ) ( ) ( ) ( ) ( ){ }( ) ( ) ( )
1 1
, 1
1 1 1 1d dA B B A A B A A B A B Bj i
d A B A B A B A Bi j
j i i j
i j i j Tr
ρ φ φ ρ φ φ
φ φ ρ φ φ φ ρ φ φ φ
= =
=
⎡ ⎤Λ = ⊗ ⊗ ⊗ ⊗⎣ ⎦
= ⊗ ⊗ ⊗ ⊗ = =
∑ ∑∑
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 38
Local Manipulation of Quantum States
• We want to transform 1ψ to 2ψ , (general pure state of two
parties) using LOCC. • We can write the state in their Schmidt decomposition:
1 21 1
n n
i A B i A Bi i
i i i iψ α ψ α= =
′ ′ ′= =∑ ∑
Where the ,i iα α ′ are real and given in a decreasing order: 1 2
1 2
n
n
α α α
α α α
> > >
′ ′ ′> > >
• It has been shown (M.A Nielsen, 1999) that the task is possible if
and only if the { }iα are majorized by { }iα ′ , meaning, for every
1 l n≤ < we have 1 1
l li ii i
α α= =
′≤∑ ∑ and 1 1
n ni ii i
α α= =
′=∑ ∑ . • For example, consider the distribution: { } { }3 1 1 2 1 1 1, , and , , ,
5 5 5 5 5 5 5
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 39
Local Manipulation of Quantum States
Consequences: • There are pairs of states that neither of them can be converted into
the other with certainty. For example, consider the distribution: { } { }5 1 1 1 4 3 1, , , and , ,
8 8 8 8 8 8 8
• Given a pure state d
+Ψ , where d+Ψ is a maximally entangled
state, there is a LOCC operation Λ such that ( )d σ+Λ Ψ = , for
any state A BH Hσ ∈ ⊗ , because the distribution ( )11
dd i= is majorized
by any probability distribution on { }1,...,d .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 40
Definition of Entanglement Measures and Entanglement Monotones
• Entanglement monotone ( )E ρ needs to satisfy to following: 1. ( )E ρ is a mapping from density matrices into positive real
numbers: ( )Eρ ρ +→ ∈ . 2. For normalization, we set ( ) 2logdE dψ + = . 3. ( ) 0E ρ = If the state is separable. 4. ( )E ρ Doesn't increase under LOCC.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 41
Definition of Entanglement Measures and Entanglement Monotones
Frequently, some may also impose additional requirements: • For pure states the measure reduces to the entropy of
entanglement: ( ) ( )E Hψ ψ ψ ψ= . • Convexity (mixing of state does not increase entanglement):
( ) ( )i i i ii iE p p Eρ ρ≤∑ ∑
• Additivity: because there are entanglement measures that do not
satisfy the condition ( ) ( )nE nEσ σ⊗ = or ( ) ( ) ( )E E Eσ ρ σ ρ⊗ = + , we define an asymptotic version:
• ( ) ( )lim
n
n
EE
nσ
σ⊗
∞
→∞
Which automatically satisfies additivity.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 42
Definition of Entanglement Measures and Entanglement Monotones
• Continuity: any entanglement monotone that is additive on pure
state and "asymptotically continuous" must equal ( )H ρ on all pure states.
• Asymptotically continuous: for two sequences of states ,n n nHρ σ ∈
such that 0n nρ σ− → , the following property holds: ( ) ( )
( )2
01 log dim
n n
n
E EH
ρ σ−→
+
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 43
Survey of Entanglement Monotones
• We discuss in this section a variety of bipartite entanglement
monotones (meaning they cannot increase under LOCC).
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 44
Entanglement Cost
• Say we want to transform n mρ σ⊗ ⊗→ , for some large integers n
and m . The larger ratio r m n= we may achieve would give an indication about the relative entanglement
• The entanglement cost ( )CE ρ quantifies the maximal possible
rate r at which one can convert blocks of two qubits maximally entangled states into many copies of ρ .
• We denote a general LOCC operation by Ψ , we can define the
entanglement cost as:
( ) ( ){ }{ }2 2inf : lim inf 0rnn
C nE r Trρ ρ ψ ψ
⊗⊗ + +
→∞ Ψ
⎡ ⎤= −Ψ =⎢ ⎥⎣ ⎦
• The ( )CE ρ measures how many maximally entangles states are
required to create copies of ρ by LOCC alone.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 45
The Distillable Entanglement
• We can ask about the reverse process: at what rate may we obtain
maximally entangled states (of two qubits) from an input supply of states of the form ρ ? (distillation or concentration)
• The distillable entanglement ( )DE ρ provide us the rate at which
noisy mixed state ρ may be converted into singlet state by LOCC:
( ) ( ) ( ){ }2 2sup : lim inf 0rnn
D nE r Trρ ρ ψ ψ
⊗⊗ + +
→∞ Ψ
⎡ ⎤= Ψ − =⎢ ⎥⎣ ⎦
• Two of the Horodeckis showed (Phys. Rev. Lett. 84, 2000) that
any entanglement measure E with the properties: Normalized ( ) 2logdE dψ + = , Monotonic under LOCC, Continuous and
Additive ( ( ) ( )nE nEσ σ⊗ = ) satisfies ( ) ( ) ( )D CE E Eρ ρ ρ< < .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 46
Entropy of Entanglement
• For pure states ( ) ( )D CE Eρ ρ= and they are equal to the entropy of
entanglement, which is defined by: ( ) ( ) ( )A BH S tr S trψ ψ ψ ψ ψ ψ=
Where S is the von Neumann entropy ( ) { }2logS trρ ρ ρ= − . • For a general mixed state σ , ( )AS tr σ may not equal ( )BS tr σ • Now we can say: Given a large number N of copies of 1 1ψ ψ ,
we can distill ( )1 1N H ψ ψ⋅ singlet states and then create from those singlets ( ) ( )1 1 2 2M N H Hψ ψ ψ ψ≈ copies of 2 2ψ ψ .
• In the infinite limit ( ) ( )1 1 2 2H Hψ ψ ψ ψ is the optimal
conversion rate from 1 1ψ ψ to 2 2ψ ψ
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 47
Entanglement of Formation
• The Entanglement of formation ( )FE ρ of a mixed state ρ is the
least expected entanglement of any ensemble of pure states realizing ρ :
( ) ( ){ }min :F i i i i i ii iE p H pρ ψ ψ ρ ψ ψ= =∑ ∑
Where ( ) ( ) ( ) { } { }2 2log logA B A A B BH S S Tr Trψ ψ ρ ρ ρ ρ ρ ρ= = = − = − And { } { },A B B ATr Trρ ψ ψ ρ ψ ψ= = • In order for Alice and Bob to create the system ρ without
transferring quantum states between them, they must already share the amount of entanglement equivalent of ( )FE ρ .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 48
Entanglement of Formation
• A closed formula for ( )FE ρ is known for bi-partite qubit systems: • We define ( ) ( )y y y yρ σ σ ρ σ σ∗= ⊗ ⊗ . • We denote ( ) { }1 2 3 4max 0,C ρ λ λ λ λ= − − − , where the iλ are the
square roots of the eigenvalues of the matrix ρρ , in decreasing order, and
( ) ( )21 12F
CE s
ρρ
⎛ ⎞+ −⎜ ⎟=⎜ ⎟⎝ ⎠
Where ( ) ( ) ( )2 2log 1 log 1s x x x x x= − − − − . • The asymptotic version defined as:
( ) ( )lim F
F n
EE
nρ
ρ⊗
∞
→∞=
is equal the entanglement cost ( ) ( )F CE Eρ ρ∞ = .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 49
Entanglement of Formation
• Example for maximally entangled state:
( )( )
( ) ( )
12
2
0 0 0 00 1 1 0101 10 01 100 1 1 020 0 0 0
0 10
0 0 1 00 0 0 1
01 0
0 0 0 0 0 0 0 00 1 1 0 0 1 1 01 10 1 1 0 0 1 1 02 20 0 0 0 0 0 0 0
y y
y y y y
s s
i ii i
ρ
σ σ
ρ σ σ ρ σ σ ρρ ρ ρ∗
⎛ ⎞⎜ ⎟−⎜ ⎟= = − − =⎜ ⎟−⎜ ⎟⎝ ⎠
−⎛ ⎞⎜ ⎟− −⎛ ⎞ ⎛ ⎞ ⎜ ⎟⊗ = ⊗ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟−⎝ ⎠
⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟− −⎜ ⎟ ⎜ ⎟= ⊗ ⊗ = ⇒ = = =⎜ ⎟ ⎜ ⎟− −⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
{ } { } ( )
( ) ( )2
2 2 2
1,0,0,0 1
1 1 1 1 1 1 1 1log log log 12 2 2 2 2 2 2F
eigenvalues C
CE s s
ρρ ρ
ρρ
= ⇒ =
⎛ ⎞+ − ⎛ ⎞⎜ ⎟= = = − − = − =⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 50
Entanglement of Formation
• Example for a separable state:
( ) ( )
{ } { }( )
( ) ( ) ( ) ( )2
2 2
1 0 0 00 0 0 00 0 0 00 0 0 0
0 10
0 0 1 00 0 0 1
01 0
0 0 0 00 0 0 00 0 0 00 0 0 1
00,0,0,0
0
1 11 log 1 log 1
2
y y
y y y y
F
i ii i
eigenvalues
C
CE C s s
ρ
σ σ
ρ σ σ ρ σ σ
ρρρρ
ρ
ρρ
∗
⎛ ⎞⎜ ⎟⎜ ⎟=⎜ ⎟⎜ ⎟⎝ ⎠
−⎛ ⎞⎜ ⎟− −⎛ ⎞ ⎛ ⎞ ⎜ ⎟⊗ = ⊗ =⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎜ ⎟−⎝ ⎠⎛ ⎞⎜ ⎟⎜ ⎟= ⊗ ⊗ =⎜ ⎟⎜ ⎟⎝ ⎠
=
=
=
⎛ ⎞+ −⎜ ⎟= = = = − − =⎜ ⎟⎝ ⎠
E 0
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 51
Entanglement of Formation
• Examples for Inseparable state:
( ) ( )
( ) { }( )
( ) ( ) ( )
18
3 18 4
314 8
18
25 1 1 164 64 64 64
5 1 1 1 18 8 8 8 4
2
0 0 00 00 00 0 0
1 0 0 00 13 3 01
64 0 3 13 00 0 0 1
, , ,
1 10.9841 0.1178
2
y y y y
F
eigenvalue
C
CE s s
ρ
ρ σ σ ρ σ σ
ρρ
ρ
ρρ
∗
⎛ ⎞⎜ ⎟−⎜ ⎟=⎜ ⎟−⎜ ⎟⎝ ⎠
⎛ ⎞⎜ ⎟−⎜ ⎟⊗ ⊗ =⎜ ⎟−⎜ ⎟⎝ ⎠
=
= − − − =
⎛ ⎞+ −⎜ ⎟= = =⎜ ⎟⎝ ⎠
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 52
Negativity and Logarithmic Negativity
• The Negativity, ( )N ρ , is an entanglement monotone that attempts
to quantify the negativity in the spectrum of the partial transposed matrix. We will define it as:
( )2 12
T
Nρ
ρ−
=
Where †X Tr X X= is the trace norm. • The trace norm of any Hermitian operator A is equal to the sum of
the absolute values of the eigenvalues of A. • For density matrices: † 1Tr Trρ ρ ρ ρ= = = .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 53
Negativity and Logarithmic Negativity
• Another entanglement monotone is the logarithmic negativity:
( ) 22log T
NE ρ ρ= Which is an additive one ( ) ( ) ( )1 2 1 2N N NE E Eρ ρ ρ ρ⊗ = + :
( ) ( )
( ) ( ) ( )( )( )( ) ( )
( ) ( ) ( )
2
1 2 2 1 2 2 1 2
† ††2 1 2 1 2 2 1 1 2 2
† †† †2 1 1 2 2 2 1 1 2 2
††2 1 1 2 2 2 1 2
log log
log log
log log
log log
T TN
T T T T
T T T T
T TN N
E
Tr Tr
Tr Tr Tr
Tr Tr E E
ρ ρ ρ ρ ρ ρ
ρ ρ ρ ρ ρ ρ ρ ρ
ρ ρ ρ ρ ρ ρ ρ ρ
ρ ρ ρ ρ ρ ρ
⊗ = ⊗ = ⊗ =
= ⊗ ⊗ = ⊗ =
⎡ ⎤= ⊗ = =⎢ ⎥⎣ ⎦⎡ ⎤⎡ ⎤= + = +⎢ ⎥⎣ ⎦ ⎣ ⎦
• If the state is separable then 2 1Tρ = and the negativity and the
logarithmic negativity vanish.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 54
Negativity and Logarithmic Negativity
• Consider the example we used for Peres criterion:
( )14
xxS Iρ
−= +
• The eigenvalues of 2Tρ : ( )11 2 3 4 1 xλ λ λ= = = + and ( )1
4 4 1 3xλ = − . • If the lowest eigenvalues is negative, meaning 1
3x > , the density matrix will not be separable:
( ) ( )2 3 31 1 1 14 4 2 2 2 21 1 3 1T x x xρ = + − − = + > + =
• If 1
3x < the density matrix is indeed separable: ( ) ( )2 3 1
4 41 1 3 1T x xρ = + + − =
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 55
Relative Entropy of Entanglement
• The quantum relative entropy is ( ) { }log logS Trρ σ ρ ρ σ ρ= − and
it measures the distinguish-ability between quantum states. • The relative entropy of entanglement with respect to a set X will
be defined as: ( ) ( )infX
R XE S
σρ ρ σ
∈=
• The set X can be taken as the set of separable states, or any other
set of states, depending upon what you are regarding as "worthless" states.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 56
Relative Entropy of Entanglement
• We can also define this measure using another distance measure to
quantify how far a particular state is from a chosen set of disentangled states:
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 57
Entanglement Witness Monotone
• An hermitian operator A is defined as an entanglement witness if:
( )
( )
0and
such that 0
SEP Tr A
Tr A
ρ ρ
ρ ρ
∀ ∈ ≥
∃ <
So A acts as a linear hyperplane separating some entangled states from the convex set of separable ones.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 58
Entanglement Witness Monotone • The amount of violation of an Entanglement Witness is a measure
of the non Separability of a given state: ( ) ( ){ }max 0,witE A Tr Aρ= −
• We can choose sets of entanglement witnesses and define a
monotone as the minimal violation over all witnesses of the set.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 59
Entanglement Measure which is not a Monotone (G. Gour and R. W. Spekkens)
• We define the "Entanglement of Assistance" (EoA): Suppose that
Alice and Bob have a bipartite system in the mixed state ABρ and Charlie holds a purification of this state.
• The tripartite system held by Alice, Bob and Charlie is in a pure
state ABC
Ψ , such that { }AB C ABCTrρ = Ψ Ψ .
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 60
Entanglement Measure which is not a Monotone • The EoA quantifies the maximum of pure state entanglement
(quantified by the entropy of entanglement H ) that Alice and Bob can extract from ABρ by Charlie performing a measurement on his system and reporting the outcome to Alice and Bob:
( ){ }
( ),
maxi i AB
Ast AB i i ABp i
E p Hψ
ρ ψ= ∑
• Where ( ) ( ) ( ) ( ), , , ,log ,i A i A i A i A i B i iAB ABH S Tr Trψ ρ ρ ρ ρ ψ ψ= = − = .
The maximization is over all the ensembles { },i i ABp ψ such that: AB i i iABi
pρ ψ ψ=∑
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 61
Entanglement Measure which is not a Monotone • The EoA is a measure of entanglement for tripartite states and for
it to be an entanglement monotone; it must be non increasing under LOCC operations between all three parties.
• "Entanglement of Collaboration" (EoC): Same as the EoA, only
now, the parties can use arbitrary tripartite LOCC operation. • Here Alice and Bob are allowed to assist Charlie in assisting them. • If there exists a state for which the EoC is greater then the EoA,
then the EoA is not an entanglement monotone, because in this state it would be possible to increase the EoA by an LOCC operation (Alice and Bob communicating with Charlie).
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 62
Entanglement Measure which is not a Monotone • A demonstration for which this is the case, in a tripartite system of
dimensions 8 4 2× × : ( ) ( )( ) ( )
00 11 22 33 0 00 11 22 33 114 40 51 62 73 40 51 62 73ABC
i i
i i
⎡ ⎤+ + + + + − − +Φ = ⎢ ⎥
+ + + + + + + + −⎢ ⎥⎣ ⎦
Where ( )12
0 1i± = ± . • Alice performs a measurement with two possible outcomes:
0 0 1 1 2 2 3 3+ + + and 4 4 5 5 6 6 7 7+ + + and she sends the outcome to Charlie.
• If the first outcome occurred, Charlie performs a measurement in
the { }0 , 1 basis and if the second outcome occurred, he measures in the { },+ − basis. In each case, he leaves Alice and Bob with a maximally entangled 8 4× state.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 63
Entanglement Measure which is not a Monotone • If no communication from Alice and Bob to Charlie is allowed,
then Charlie cannot always create a maximally entangled state for Alice and Bob.
• First note that the initial state can be written as follows:
0 10 1ABC C AB CABu uΦ = + Where:
( ) ( )( ) ( )( ) ( )
3 10 0, 40
3 11 1, 40
0 4 0 1 2 5 1
2 6 2 3 2 7 3
0 4 0 1 1 2 6 2 3 3
kAB k
kAB k
zu c k
z
u c k i z i z
=
=
⎡ ⎤+ + + +⎢ ⎥= =⎢ ⎥+ + +⎣ ⎦
⎡ ⎤= = + + + − + −⎣ ⎦
∑
∑
And ( )12
1z i≡ + . • The reduced density matrix is:
0 0 1 1AB C ABC ABABTr u u u uρ = Φ Φ = +
.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 64
Entanglement Measure which is not a Monotone • Charlie cannot even create a maximally entangled state with some
probability. • Given the Hughston-Jozsa-Wootters theorem, it is enough to show
that no convex decomposition of ABρ contains a maximally entangled states.
(HJW gave A complete classification of quantum ensembles having a given density matrix) • Any decomposition of ABρ is proportional to 0 1 , ,ABABx u y u x y+ ∈
(why?) • So it is enough to show that any state of that form is not a
maximally entangled state of a 8 4× system.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 65
Entanglement Measure which is not a Monotone • If 0 1 ABABx u y u+ is maximally entangled then all of its Schmidt
coefficient must be equal. • The coefficient are:
( )( )( )( )
2 21 11 16 2
2 212 16
2 21 13 16 2
2 214 16
2
2
x iy x y
x y x
x iy x y
x y x
λ
λ
λ
λ
= + + +
= + +
= − + +
= − +
• Thus, 1 2 3 4λ λ λ λ= = = if and only if 0x y= = . ⇒ Any state of the form 0 1 ABABx u y u+ cannot by maximally entangled. ⇒ Entanglement of Assistance is not an entanglement monotone.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 66
Summary
• We saw some properties of entanglement, ways to increase it and
ways to quantify it. • The idea of entanglement monotone. • This is just the tip of the iceberg!!!
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 67
Reference 1. A. Peres. "Separable Criterion for Density Matrices", Physical Review Letters, volume 77, August 1996. 2. M. Horodecki, P. Horodecki and R. Horodecki. "Separability of mixed states: necessary and sufficient
conditions" Physics Letters A, Volume 223, November 1996. 3. J. E. Avron and Netanel H. Lindner, Private conversations. 4. C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher, J. A. Smolin and W. K. Wootters. "Purification of
Noisy Entanglement and Faithful Teleportation via Noisy Channels", Physical Review Letters, volume 76, January 1996.
5. C.H. Bennett, H. J. Bernstein, S. Popescu and B. Schumacher. "Concentrating Partial Entanglement by local Operations", Physical Review A, volume 53, April 1996.
6. C. H. Bennett, D. P. DiVincenzo, J.A. Smolin and W. K. Wootters. "Mixed State Entanglement and Quantum Error Correction", Physical Review A, volume 54, November 1996.
7. M. B. Plenio and S. Virmani. "An introduction to Entanglement Measures" arxiv: quant-ph/0504163, v2, February 2006.
8. V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knight. "Quantifying Entanglement". Physical Review Letters, volume 78, March 1997.
9. G. Vidal, "Entanglement Monotones", Journal of Modern Optics, Volume 47, February 2000. 10. M. A. Nielsen and I. L. Chang, "Quantum Computation and Quantum Information", Cambridge, 2001. 11. G. Vidal and R. F. Werner. "Computable measure of Entanglement", Physical Review A, volume 65,
February 2002. 12. W. K. Wootters. "Entanglement of Formation of an Arbitrary State of Two Qubits", Physical Review
Letters, volume 80, March 1998. 13. M. J. Donald, M. Horodecki and O. Rudolph. "The Uniqueness Theorem For Entanglement Measures",
Journal of Mathematical Physics, volume 43, September 2002. 14. Dieter Heiss "Fundamentals of Quantum Information: Quantum Computation, Communication,
Decoherence and All That", Springer 2002. 15. Lecture Notes for the course "Quantum information" given by Dr. Benni Reznik, Tel Aviv University,
March 2005. 16. G. Gour and R. W. Spekkens. "Entanglement of Assistance is not an entanglement monotone" arxiv: quant-
ph/0512139, December 2005. 17. G. Gour, Private conversations.
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary
Gili Bisker Physics Department, Technion 68
The End
1. Introduction 2. What is entanglement? 3. Separability Criterion 4. Necessary and sufficient conditions • Entanglement witness 5. Concentrating partial entanglement 6. Purification 7. Entanglement Measures • Properties of Entanglement • Local Manipulation • Definitions • Entanglement Monotones: Cost, Distillable, Entropy, Formation, Negativity, Relative Entropy, Witness • Entanglement Measure which is not a Monotone 8. Summary