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
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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:
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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
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Evolution
Closed systems: IUUUUUU ,
Open systems:
k k
kkkk ILLLL ,
kL are called Kraus operators
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Concept of entanglement (pure states)
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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)
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State of the Subsystem
A B
i
BABBABBA iiTr )(
i
AABAABAB iiTr )(
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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
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A
In separable pure states the subsystems are also pure
Separable Pure States
Separable state: BAAB
1 BA PP
0)()( BA SS B
Subsystems:
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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
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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
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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
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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
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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
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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
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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
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Concept of entanglement (mixed states)
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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
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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:
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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
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Classical actions:
Local Operations and Classical Communications
(LOCC)
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LOCC Maps
Local Operations and Classical Communication (LOCC)
Informing the other party about the outcomes of measurements
1. Unitary evolution
2. Measurement
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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
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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
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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
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Extra Properties
Additivity: 6 )()( nEE n
Asymptotic additivity: n
EE
n
n
)()( lim
n
AB
)(
Asymptotic additivity is weaker than additivity 36
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Operational measures
Operational Measures
)()()( : If ABABABAB EELOCC
Any arbitrary state can be made from maximally
entangled states by using LOCC operations
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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 ?
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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)(
)(
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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)(
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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)(
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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
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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
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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.
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Asymptotic measures
Axiomatic Measures
Axiomatic measures:
Distance measures
Entanglement of formation
Negativity
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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.
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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
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Entanglement of Formation
i
iii
i
iiip
F pEpEii
: )( inf,
i
i
iip
There are infinite decompositions:
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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)
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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
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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
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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:
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Negativity
Separable:
i
B
i
A
iip i
B
i
tA
ii
TpA )(
Valid density matrix
0AT
Entangled: 0)( AT
Negativity:
ATN ,2)(
0
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PPT States
Non-distillable entangled states
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0 , AT
i
B
i
A
iip
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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
&
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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
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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
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