Entanglement and its Quantification

Abolfazl Bayat

白安之

(既来之则安之)

University of Electronic Science and Technology of China

Chengdu, China

1

2

Content of the talk (first part)

Basic concepts of quantum information • Pure states • Mixed states

Quantum operations

• LOCC

Pure state entanglement • Von Neumann entropy

3

Basics of quantum mechanics/information

4

Systems that are fully explained by a unique wave function

BAAB)

2

10(0

2

0100

Pure States

Example 1: A qubit

2

10

Example 2: Bipartite systems

5

Mixed States

Ensemble of pure states:

nnP

P

P

:

:

:

22

11

Density matrix for explaining the state of particles in the box:

i

i

iip

6

Properties of the Density Matrix

1. Hermiticity: 2. Trace one: 3. Positivity:

1)( Tr

0

What positivity means: 0 :

In particular for

0

:

iii

iiii

All eigenvalues of a density matrix should be equal or greater than zero

7

Decomposition of the Density Matrix

Decomposition is not unique

Note that the set of quantum states in each decomposition are not necessarily orthogonal unless for the eigenvectors

j

j

jji

i

ii qp

8

𝜌=1

2 0 0 +

1

2 1 1 =

1

2 + + +

1

2 − −

=1

2

1 00 1

Example:

+ = 0 + 1

2 − =

0 − 1

2 where &

Purity

1)( 22 j

jqTr

j

j

jji

i

ii qp

j

j

jjq 22

)( 2TrP Purity: 11

Pd

Pure states:

dP

d

I 1

1 P

Maximally mixed state:

9

Von Neumann Entropy

j

j

jj

Pure states:

)log()( dSd

I

0)( S

Maximally mixed state:

ijji

Eigenstates

Von Neumann Entropy: j

iiTrS )log()log()(

Note that : 0 log(0)=0

Both purity and von Neumann entropy quantify the purity (or equivalently the mixedness) of the system

10

Evolution

Closed systems: IUUUUUU ,

Open systems:

k k

kkkk ILLLL ,

kL are called Kraus operators

11

Concept of entanglement (pure states)

12

Bipartite Systems

A B

Separable pure states: BAAB

BAAB)

2

10(0

2

0100

Non-separable states are called entangled states

BAAB

2

1100ABAB

AB

13

The most general state:

Schmidt basis:

ji

BAijABji

,

,

Schmidt Decomposition

' ,' '' jjBBiiAA jjii

i

BAi

jiABBAijAB

iiji~

,~

,,

'~~

,'~~

'' iiBBiiAA iiii Properties of Schmidt

decomposition si ' are real and positive (Schmidt coefficients)

14

State of the Subsystem

A B

i

BABBABBA iiTr )(

i

AABAABAB iiTr )(

15

Von Neumann Entropy

A B

i

BBiABABAB iiTr~~

)( 2

BA PP Purity:

Schmidt decomposition:

i

AAiABABBA iiTr~~

)( 2

Subsystems:

Von Neumann Entropy: )()( BA SS If AB is pure:

i

BAiABii~

,~

'~~

,'~~

'' iiBBiiAA iiii

16

A

In separable pure states the subsystems are also pure

Separable Pure States

Separable state: BAAB

1 BA PP

0)()( BA SS B

Subsystems:

17

Entangled states: BAAB

Von-Neumann Entropy of the subsystem quantifies the entanglement

Entangled Pure States

A 1 BA PP

0)()( BA SS B

Subsystems

are not pure

18

Example 1

2

1100ABAB

AB

2 ,

2

IIBA

2

1 BA PP

1)2log()()( BA SS

Maximally entangled

states

1. Subsystems are maximally mixed

2. The entropy of subsystems are maximal

Maximally entangled states:

3. The purity of the subsystems are minimal

19

Non-maximal entangled states:

Entropy of the subsystem can quantify the amount of entanglement

Example 2

ABABAB11

3

200

3

1 11

3

200

3

1 BA

9

5 BA PP

9183.0)3

2log(

3

2)

3

1log(

3

1)()( BA SS

20

Overall state: AB

All entanglement measures are monotonic functions with respect

to the von Neumann entropy

Entanglement of Pure States

A B

)( ABAB Tr

)( ABBA Tr

Entanglement between the two subsystems: )()( BA SSE

)log(0 dE

Separable

states

Maximally entangled

states

21

A 2

1100ABAB

AB

Alice Bob

10'

A

B

A’

BzAAAA

ByAAAA

BxAAAA

BAAAA

BAA

)10(2

1100

)10(2

1001

)10(2

1001

)10(2

1100

2

1

''

''

''

''

'

With 2 classical bits of information Bob gets the exact state of Alice.

Application 1: Teleportation

22

A 2

1100ABAB

AB

Bob B Alice

2

1100

2

1001

2

1001

2

1100

ABAB

ABz

ABAB

ABy

ABAB

ABx

ABAB

AB

I

I

I

II

A single physical 2-level object carries

two classical bits of information

Application 2: Dense coding

23

Content of the talk (second part)

Mixed state entanglement

Operational measures: 1. Entanglement cost 2. Entanglement distillation

Asymptotic measures:

1. Distance measures 2. Entanglement formation 3. Negativity

24

Concept of entanglement (mixed states)

25

Separable Mixed States

Separable states:

i

ii

i

B

i

A

iiAB

pp

p

1 ,0

With local operations and classical communications

Alice and Bob can produce these kind of states 26

Examples for Separable States

Example 1 (Pure states):

AAAAAB BAAB

Example 2: 113

200

3

1AB

Example 3: )(6

200

6

1ZAB II

i

B

i

A

iiAB p

27

Entropy of the subsystem

Separable states: i

B

i

A

iiAB p

i

A

iiABBA pTr )(

i

B

iiABAB pTr )(

Subsystems:

Von Neumann entropy does not quantify the entanglement anymore

as it originates from the initial entropy of the whole system

0)(or )( BA SS System is not entangled but:

28

Examples

Example 1 (Pure states):

AAAAAB

Example 2:

0)( )( BA SS

3

2

6

1IA

113

200

3

1B

113

200

6

1 IAB

6500.0)( AS

9183.0)( BS

Subsystems:

)( )( BA SS 29

Entanglement of Mixed States

Entangled states: i

B

i

A

iiAB p

How to quantify entanglement for a general mixed state?

There is not a unique entanglement measure

30

Classical actions:

Local Operations and Classical Communications

(LOCC)

31

LOCC Maps

Local Operations and Classical Communication (LOCC)

Informing the other party about the outcomes of measurements

1. Unitary evolution

2. Measurement

32

LOCC Kraus Operators

General quantum operation:

k k

kkkk ILLLL ,

LOCC Kraus operators: KKK BAL

k

kkkk

k

kkABkk

final

AB IBBAABABA , init

Trace preserving is equivalent to conservation of probabilities

33

Basic Properties for Entanglement Measures

RE AB )(

i

BAAB iid

,1

maximum is )( ABE

i

B

i

A

iiAB p 0)( ABE

)()( ABAB EE k

kkABkkAB BABA

1

2

3

4

AB

34

Extra Properties

Convexity: 5 )()( AB

i

i

i

i

AB

ii EppE

2

1100 ,

2

110021

ABABExample:

11112

10000

2

1

2

1

2

12211 AB

35

Extra Properties

Additivity: 6 )()( nEE n

Asymptotic additivity: n

EE

n

n

)()( lim

n

AB

)(

Asymptotic additivity is weaker than additivity 36

37

Operational measures

Operational Measures

)()()( : If ABABABAB EELOCC

Any arbitrary state can be made from maximally

entangled states by using LOCC operations

38

Maximally entangled

Ordering the Quantum States (Single Copy)

ABABLOCC

ABABLOCC AB

ABLOCC

)()( ABAB EE )()( ABAB EE )()( ABAB EE

Are LOCC maps enough for giving order to quantum states? NO !!

ABABLOCC )( )( ABAB EE ?

39

Asymptotic Approach

n

n copy

LOCC

Alice Bob m

m copy

Alice Bob

Having many copies allows for more complex operations

n

m

E

E

nLOCCAB

AB Sup lim)(

)(

40

Entanglement Cost

n

ME

)(

n copy of ME states

LOCC

Alice Bob m

m copy (m>n)

Alice Bob

m

nE

nLOCC

C Sup lim)(

41

Entanglement Distillation (Concentration)

n

n copy

LOCC

Alice Bob m

ME

m copy of ME states (m<n)

Alice Bob

n

mE

nLOCC

D Sup lim)(

42

Non-Distillable Entanglement

AB

0)(

0)(

D

C

i

B

i

A

ii

E

E

p

Non-distillable or bound entanglement:

Bound entanglement (or PPT states) cannot be used

for teleportation or dense coding

43

Entanglement Cost vs. Entanglement Distillation

1)(2ABTr

For pure states:

)()()()( BAABDABC SSEE

C. H. Bennett, H. Bernstein, S. Popescu and B. Schumacher, Phys. Rev. A 53,

2046 (1996).

That is why von Neumann entropy of the subsystem is considered

as the unique measure of entanglement for pure states

44

Entanglement Cost vs. Entanglement Distillation

For an arbitrary entanglement measure L which is asymptotically

additive we have:

)()(

)( lim ABD

n

AB

nABC E

n

LE

M.J. Donald, M. Horodecki, and O. Rudolph, J. Math. Phys. 43, 4252 (2002).

Entanglement cost and entanglement distillations are extremal measures.

Entanglement cost and entanglement distillations are extremal measures.

45

46

Asymptotic measures

Axiomatic Measures

Axiomatic measures:

Distance measures

Entanglement of formation

Negativity

47

Distance Based Measures

Separable Entangled

The set of all states

),( )( inf

DESEP

D is a distance function and can be considered

as relative entropy or trace norm distance.

48

Entanglement of Formation

i

i

iip

No! Because decomposition of a density matrix is not unique

Pure state

i

iiiEpE )(Is it possible to quantify entanglement as:

Example: 2

1100 ,

2

110021

ABAB

11112

10000

2

1

2

1

2

12211 AB

11 E 12 E 000 E 011 E

49

Entanglement of Formation

i

iii

i

iiip

F pEpEii

: )( inf,

i

i

iip

There are infinite decompositions:

50

Concurrence Entanglement of formation has been computed for two qubits:

yyyy *~ ~R

iji ji :for : 4321,0max)( C

2

11log

2

11

2

11log

2

11)(

2222 CCCCE

C is called concurrence and can be computed as:

C is a monotonic function of C and thus usually concurrence

is also used as an entanglement measure

W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998)

51

Asymptotic Definition

)()(

)( lim

C

n

F

nF E

n

EE

P. Hayden, M. Horodecki, and B.M. Terhal, J. Phys. A 34, 6891 (2001).

Entanglement of formation is not additive and thus:

)()( FF EE

52

M. B. Hastings, Nature Physics 5, 255 (2009)

Transpose of a Density Matrix

1)( Tr

0

Density matrix: jiji

ji,

.

ijji

ji

t ,

.

The transpose of the density matrix is also a valid density matrix:

)( tt

1)( tTr

0t

53

Partial Transpose of a Density Matrix

Density matrix:

The partial transpose of the density matrix is defined as:

ji

BBAAklij ljki,

,

ji

BBAAklij

TljikA

,

,

Bipartite System:

ji

BBAAklij

TjlkiB

,

,

)( AA TT

1)( ATTr

0AT

Properties:

54

Negativity

Separable:

i

B

i

A

iip i

B

i

tA

ii

TpA )(

Valid density matrix

0AT

Entangled: 0)( AT

Negativity:

ATN ,2)(

0

55

PPT States

Non-distillable entangled states

56

0 , AT

i

B

i

A

iip

57

Entanglement Witness

Entanglement Witness

A Hermitian operator W is

entanglement witness if:

Separable Entangled

The set of all states

0)( WTr

0)( WTr0)( : WTrSEP

0)( : WTrSEP

Example: CHSH Bell inequality is a well known entanglement witness

&

58

Summary

Pure states: von Neumann entropy Mixed states: There is not a unique measure

• Entanglement cost (upper bound) • Entanglement distillation (lower bound) • Entanglement of formation (for qubits concurrence) • Distance entanglement • Negativity (easily computable)

Except negativity the rest of measures are often notoriously difficult to compute

59

Good Reviews

1- An introduction to entanglement measures M. B. Plenio, S. Virmani Quantum Information and Computation Vol:7, 1-51, 2007 2- Entanglement measures M. Horodecki Quantum Information and Computation Vol:1, 3-26, 2001

60