Entanglement: Definition, Purification and measuresqip-lab/seminar/gili_bisker.pdf · •...

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


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.



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 .



Separability Criterion for Density Matrices

Asher Peres 1934-2005




• 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

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟= ⇒ =⎜ ⎟ ⎜ ⎟⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠⎝ ⎠




• 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′ ′′= ⊗∑




• 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

− −⎛ ⎞⎜ ⎟+⎜ ⎟=⎜ ⎟+⎜ ⎟− −⎝ ⎠




• 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.



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.




• 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 = − + + =

= − + + = + − − − =

= + − ≥ + − = − ≥




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).




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.



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 .




• 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 .




• 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→ = + → +




• 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.




• 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.




• 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.




• 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= .




• 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⎛ ⎞

= ⎜ ⎟⎝ ⎠




• 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.



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 > .



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+ + − − + + − −+ − + +




• 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

− −

+ −

⊗ = − ⊗ − =

= − − + =

= − − + =

= − + − =

= + ⊗ − = ⊗


BXOR Source Target

Alice 1 3 Flips 3 iff 1 is up Bob 2 4 Flips 4 iff 2 is up



• Summarizing, omitting overall phase:


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



• 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

+ + − − + + − −

+ + − − + + − −

+ + + + + + + +

+ + − − + + − −

− − + + − − + +

− −

⎧ ⎫+ − + − + −⎪ ⎪⎨ ⎬⊗ + − + − + −⎪ ⎪⎩ ⎭

⊗ + − ⊗

+ − − ⊗ + − − ⊗

+ − ⊗ + − − ⊗

+ −=

( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )

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

− − − − − −

+ + + + + + + +

+ + − − + + − −

− − + + − − + +

− − − − − − − −

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⊗ + − ⊗ ⎪⎪ ⎪⎨ ⎬+ − − ⊗ + − ⊗⎪⎪+ − − ⊗ + − − ⊗⎪⎪+ − − ⊗ + − ⊗⎪⎪+ − − ⊗ + − − ⊗⎪⎩ ⎭

⎪⎪⎪⎪⎪⎪⎪




• 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

− −

+ + − − + +

⎧ ⎫+ −⎪ ⎪⎨ ⎬

+ − + − + − + +⎪ ⎪⎩ ⎭




• 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



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):



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)




• 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.




• 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).



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= +




• 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+ + + += +




• 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 .




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⊗




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 = ⊗ ⊗ ⊗ ⊗⎣ ⎦

= ⊗ ⊗ ⊗ ⊗ = =

∑ ∑∑




• 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




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 .



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.




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.




• 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−→

+



Survey of Entanglement Monotones

• We discuss in this section a variety of bipartite entanglement

monotones (meaning they cannot increase under LOCC).



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.



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< < .



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



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 .




• 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∞ = .




• 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

= ⇒ =

⎛ ⎞+ − ⎛ ⎞⎜ ⎟= = = − − = − =⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠




• 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




• 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

∗

⎛ ⎞⎜ ⎟−⎜ ⎟=⎜ ⎟−⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟−⎜ ⎟⊗ ⊗ =⎜ ⎟−⎜ ⎟⎝ ⎠

=

= − − − =

⎛ ⎞+ −⎜ ⎟= = =⎜ ⎟⎝ ⎠



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= = = .




• 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.




• 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 = + + − =



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.



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:



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.



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.



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 .



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=∑



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).



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.



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 = +

.



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.



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.



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!!!



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.



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