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# Theorem entangled

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```									GEOMETRY OF QUANTUM ENTANGLEMENT
CIRM, Marseille, January 2012
Selected results and open problems of
quantum entanglement theory
Michał Horodecki

Institute of Theoretical Physics and Astrophysics and
National Quantum Information Centre of Gdańsk
University of Gdańsk
PART I

Preliminaries
- states of single and composite systems
- maps and quantum operations

Characterization of entanglement

- in terms of positive maps
- some other criteria

Distillation of entanglement
- LOCC operations
- bound entanglement
- NPT bound entanglement
- extended classes of operations
PART II

Private cryptographic key from entanglement
- inequivalence between private key and maximal entanglement
- drawing key from bound entanglement
- low dimensional PPT states which are key distillable

Entanglement and Local Hidden Variable models

- examples of entangled states with LHV model
- Peres conjecture

Entanglement measures

- locking entanglement
Preliminaries
Quantum system and quantum state
Quantum system and quantum state
Quantum composite system: entanglement
Convex structure

All states

Entangled
Separable               states
states
Entanglement of pure states
Maximally entangled state

Resource for:
- teleportation
- dense coding
- quantum cryptography
- etc.

For d=2
Positive and completely positive maps
(positive maps: Jamiołkowski, Choi, Woronowicz 70-ties,
completely positive maps: Stinespring, Kraus)
Completely positive maps
Examples of positive maps which are not CP
Kraus operators

Preserving of trace
Quantum operations
Characterization of
entanglement
Entanglement of pure bipartite states

How to check that a given state is entangled?
|  a 00  b 01  c 10  d 11

a b 
     X
 c d


U A  U B                 T
U A XU B

We find such U A , U B        that     X '  U A XU B
T

Schmidt
is diagonal, with positive diagonal elements               decomposition

a' 0 
X'         ψ  a' 00  b' 11
 0 b'
Fact: A pure state  is entangled, if its matrix X is of rank greater than 1
Werner ’89
Entanglement of mixed states    (also in mathematical
literature in 70-ties:
Stormer, Osaki)
Two maximally entangled states:

|     1
2
( 00  11 )
|     1
2
( 00  11 )

Their equal mixture:                                Reinhard Werner

1                 1
  |    |  
2                 2
1              1
  | 00 00  | 11 11
2              2

Is not entangled!
Sandu Popescu
Partial transpose

Asher Peres

Stanisław Woronowicz
 11    12    13 14                       11    21 13    23 
                                                                   
  21    22    23  24                     12    22 14    24 
                                            


      32    33 34                              41 33    43 
 31                                          31                    
        42    43  44                           42 34    44 
 41                                          32                    
   (I  T )

Fact: eigenvalues of a quantum state are non-negative
(they are probabilities)
Theorem: Separable (non-entangled) states remain
positive after partial transpose
Example: Bell diagonal states

A    1
2
( 00  11 ),
I
B    1
( 00  11 ),             E 
2                                 4
C    1
2
( 01  10 ),
D    1
2
( 01  10 )
Two-qubit and qubit-qutrit case       [Peres & Horodeccy ’96,
Osaki ’80-ties etc.]

Theorem: For Hilbert space C2C2 and C2C3 a state is separable if and
only if its partial transposition if a positive matrix:

ρ is separable iff                     i.e. ρ is PPT

Sufficiency follows from the
following fact
(Stormer/Woronowicz):
Entangled
Separable            States =           Arbitrary positive map
States = PPT                                      :  2   2 or
= NPT
 :  2  3
is decomposable, i.e.

  CP  CPT
1     2
Dimension greater than 2 x 3
(explicit exmaples for 4x2 and 3x3 given in P. Horodecki 1996)

Separable
PPT entangled       NPT
states
states              entangled
states

There is no easy way to determine whether a given state
is separable or not - it is NP-hard problem (Gurvitz and Barnum)
Separability versus extendability

R. Werner, Lett Math. Phys 17, 359 (1989):

Theorem:
State is separable if and only if
it is n-extendable for every n.

D. Yang arXiv:quant-ph/0604168)

A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. Phys. Rev. A, 69:022308, 2004.
F. Brandao, M. Christandl and J. Yard arxiv 1011.2751.
Realigment or cross norm criterion

(Chen and Wu, Rudolph 2002)

Realignment: consider the following tranformation

Usually it is not a state anymore.
Permutation criteria

Strong criterion: can detect entanglement of a tripartite state
separable uner any cut!
Necessary and sufficient conditions for more than 2 particles
(M.P. and R Horodecki, quant-ph/0006071)

Problem: provide necessary and sufficient condition of separability
of three qubits
Distillation of entanglement

Distant parties share some joint state

Since the labs are distant, not all operations are easy,
we can introduce

- cheap operations (the easy ones)
- expensive operations, which we count
Restricting class of operations
LOCC operations
Distillability

LOCC operations

Distillable states

All states
Nondistillable
Distillability
Entanglement manipulations: pure states

Suppose we have pair of particles in state
0 – photon in upper path           a0          0 B b1 A 1 B
1 – photon in lower path                     A
T 2  R2  1
T

R

Given that klick was not observed:

normalize
~
  Ta 0   A
0 B b1 A 1 B                           1
2
(0   A
0 B  1 A 1 B)
choose T    b
a
Entanglement manipulations: pure states
Entanglement manipulations: pure states

a 00  b 11        1
2
( 00  11 )

The largest probability of such transition is

pm ax  min{ 2b 2 ,2a 2 }

What if we have TWO pairs?
  a 0    A
0 B b1 A 1 B  a 0        A'
0   B'
 b 1 A' 1 B'   
   AA'
   BB'
 a2 00   AA'
00   BB'

 ab 10   AA'
10   BB'

 01 AA' 01 BB'  b2 11 AA' 11 BB'

Maximally entangled
Many pairs of pure states


n

 a0   A
0   B
b1 A 1 B    
n

Alice and Bob measure NUMBER of 1’s

Example: n  4, k  2                        4 pairs, number of 1’s = 2


n

 a 2b2 1100 A 1100 B  1010 A 1010 B  1001 A 1001 B 
 0110     A
0110 B  0101 A 0101 B  0011 A 0011 B    
After normalization:
1 6
 ( 2)        i
6 i 1    A
i   B       equivalent to   log 6  2.58 e - bits
Many pairs of pure states


n

 a0   A
0    B
b1 A 1 B         
n

Alice and Bob measure NUMBER of 1’s
n
k
 
n
   (k )                1
n  i
 k  i 1
A
i   B
equivalent to        log  e - bits
k 
 
 

Only terms with typical frequency will appear:
k  np where p | b |2                                        H ( x)   x log x  (1  x) log(1  x)

n 
Obtained number of e-bits: log   log(2nH ( p ) )  nH (| a |2 )
 np 
 
Reversible transformation of pure states
(Bennett, Bernstein, Popescu and Schumacher 1996)

 a0   A
0 B b1 A 1 B
 m
   n
 

 m

…
       n                                                                
…

m
 H (| a |2 )              H ( x)   x log x  (1  x) log(1  x)
n

More generally:

m
 E ( ) where E ( ) is entropy of Alice' s (or Bob' s) system
n
Pure states transformations: many copies no error allowed

 m
   n
 
(cf. Nila Datta talk)

 m       m

…
   n                                                           Emin ( )
…

n

 m
    n
 

m
 m      Emax( )
…
   n                                          
n
…
Pure states transformations: single copy case

(Nielsen, 1998)
Theorem: A state  can be transformed into  if and only if 
is majorized by 
 
k                k
i.e.        | a
i 1
k   |   | bk |2 for all k  1,..., d
2

i 1

where    a k , bk       are Schmidt coeffcients of states  ,
Distillability: characterization for two qubit eggs
(HHH1997)

Separable               Entangled
states =                states =
Nondistillable          Distillable

Two qubit entanglement is always useful!
Distillability: Higher dimension
(HHH 1998)

Entangled states

Separable
states            Nondistillable      Distillable
states!           states

There exists a passive type of entanglement which does not allow
for quantum communication: we have called it

Bound entanglement
Partial transpose
NPT bound entanglement
If a state has positive partial transpose (PPT) then it is not distillable

NPT
states

Separable           PPT
states             states                    Distillable
states

Is this set nonempty? Does NPT bound entanglement exist?
Why is this question interesting?

Existence of bound entangled states implies that

P. Shor, J. A. Smolin, B. Terhal, arXiv:quant-ph/0010054
Phys. Rev. Lett. 86, 2681--2684 (2001) ,

K. G. H. Vollbrecht, Michael M. Wolf, arXiv:quant-ph/0201103,
Phys. Rev. Lett. 88, 247901 (2002)

NPT
states
PPT
states
Distillable
Separable                            states
states
Papers related to the problem of existence of NPT bound entanglement:

[1] D. P. Divincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and
A. V. Thapliyal, Phys. Rev. A 61, 062312 (2000).
[2] W. Dür, J. I. Cirac, M. Lewenstein, D. Bruss Phys. Rev. A 61,
062313 (2000).
[3] D. Bruß, J. I. Cirac, P. Horodecki, F. Hulpke, B. Kraus, M.
Lewenstein, and A. Sanpera, J. Mod. Opt. 49, 1399 (2002).
[4] B. Kraus, M. Lewenstein, and J. I. Cirac, Phys. Rev. A 65,
042327 (2002).
[5] S. Bandyopadhyay and V. Roychowdhury, Phys. Rev. A 68,
022319 (2003).
[6] J. Watrous, Phys. Rev. Lett. 93, 010502 (2004).
[7] L. Clarisse, Phys. Rev. A 71, 032332 (2005).
[8] L. Clarisse, Quant. Inf. Comput. 6, 539 (2006).
[9] R. O. Vianna and A. C. Doherty, Phys. Rev. A 74, 052306
(2006).
[10] R. Simon, quant-ph/0608250.
[11] I. Chattopadhyay and D. Sarkar, quant-ph/0609050.
[12] F. G. S. L. Brandao and J. Eisert, quant-ph/0709.3835.
[13] Ł. Pankowski, M. Piani, M. Horodecki, and P. Horodecki,
quant-ph/0711.2613.
[14] Ł. Pankowski, F. Brandao, M.Horodecki and G. Smith quant-ph/11….

L. Clarisse, arXiv:quant-ph/0612072, PhD Thesis
Entanglement Distillation; A Discourse on Bound Entanglement
in Quantum Information Theory
What do we know

State is distillable if and only if for some n (HHH1998)

A state for which is it not possible, we call:

Bound entangled                            n-copy nondistillable for all n
Equivalent condition
What do we know

If there exist NPT bound entangled states, then there must
exist Werner states which are NPT bound entangled
NPT
states

Separable
states
PPT
states
Distillable
states

Recall: to find NPT bound, we need to show n-copy nondistillability for ALL n
1-copy non-distilability of Werner states
1-copy non-distilability of Werner states
Numerically: the state on the edge seems to be
2-copy nondistillable
We tried to prove this analytically.
2-copy nondistilability: Equivalent matrix analysis problem
(Ł. Pankowski, M. Piani, MH, P. Horodecki arxiv: 0711.2613)

We have shown it for A, B being diagonalizable matrices (i.e. normal matrices).
Existence of NPT bound entanglement and two-positivity

[D. P. Divincenzo, P. W. Shor, J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, Phys. Rev. A
61, 062312 (2000).]
NPT bound from hiding states?

(DiVincenzo, Leung, Terhal 2001, Werner, Eggeling 2002)
Private cryptographic key
from entanglement
Entanglement and privacy

Recall: the goal of distillation of entanglement was actually
purification – obtaining maximal entanglement in pure form.

A new task: distill from a state private key.

If one can purify entanglement – i.e. obtain maximally
entangled states, then one can also obtain private key.
(Deutsch et al. 1996)

Namely, Alice and Bob distill maximally entangled states,
and then measure it locally:

Gisin and Wolf asked whether this is „if and only if” i.e.
whether key may come only in this way.
Two qubits

Separable           Entangled
states =            states =
Nondistillable=     Distillable=
No key              Private key
Is private key equivalent to maximally entangled state?

For ANY Alice’s basis, there is              Only ONE basis gives key,
Bob’s one which gives key                    other bases are insecure

Thus obtaining private key seems to be less demanding than
obtaining maximally entangled state.
Distillability of private key: mathematical formulation

Recall the formulation of distillability:
Distillability of private key: mathematical formulation
Distillability of private key: mathematical formulation
Which states have perfect bit of key?

Characterization was given in HHH and J.Oppenheim, PRL 2005
Is private key equivalent to maximally entangled state?
Is private key equivalent to maximally entangled state?
Is private key equivalent to maximally entangled state?
Private key from bound entangled states

Can private key be extremely inequivalent with maximally
entangle states ?

I.e. can we get private key from bound entangled states?

There do not exist bound entangled states which have perfect
bit of key,

But:              they may be arbitrary good bit of key.

HHH + J. Oppenheim, arXiv:quant-
ph/0309110, PRL 94, 160502 (2005)

unconditional security:

HHH + D. Leung + J. Oppenheim,
arXiv:quant-ph/0702077,
arXiv:quant-ph/0608195
PRL, IEEE accepted
Basic open question: can we distill private key from all
entangled states?

Separable
states

PPT
states
Other question

The smallest PPT state, from which we can obtain key
acts on 4x4 system

Does there exist PPT state on 3x3 system, from which
one can obtain key?

(Note that for 2x2 and 2x3 there is no such problem: all PPT states
do not give private key, as they are separable)
Entanglement versus
Local Hidden Variable models
Detecting entanglement via Bell inequalities

Bell 60-ties, Clauser, Horne, Shimony and Holt 60-ties, Werner 89

A, A'                                                          B, B'

A, A' , B, B'   are observables with two outcomes +1 and -1
| AB  A' B  AB'  A' B' | 2

Proof of Bell-CHSH inequality:
( ') B  ( ') B '
AA         AA                  ( ') B  ( ') B '
AA         AA
2          0                      0          2

For maximally etangled state, there exist obervables such that

AB  A' B  AB'  A' B'  2 2  2
Example: Detecting entanglement via CHSH inequality

Fact: if for a given state there exist observables that
violate Bell inequalities, then the state is entangled
Example: Max. entangled state mixed with white noise
I
  p |      (1  p)
4

Max. entangled state:          Max. chaotic state:
CHSH value = 2 2               CHSH value =0
CHSH (  )  p 2 2
Violation of CHSH,
must be entangled
I
4                                                                | 
p0                                      1/ 2               p 1
Local Hidden Variable model
Werner states vs Bell inequalities

I
  p |      (1  p)
4
CHSH (  )  p 2 2                                0.7056      0.7071
Vertesi, 465   CHSH, 2
observables    observables

Existence of local-realistic model             Violation of CHSH
I                                            ?
4                                                                       | 
p0                    1/ 3     0.416                             p 1
proj. meas. 0.6595
gen. meas. 1 / 2  0.7071

separable                         entangled states
states
Peres conjecture

Bound entangled states satisfy all Bell inequalities

Progress:

Acin et al. and Toth:
multipartite bound entangled states which violate Bell inequalities

Augusiak and P. Horodecki:
Four-partite states that violate Bell inequalities

Vertesi:
Tripartite bound entangled state which is separable under any bipartite
cut that violates Bell inequalities

Open: does there exist PPT bipartite state, which violates Bell
Inequalies?
Entanglement measures
Entanglement measures

Main division:

1) operational measures – which come from some task

2) axiomatic measures – functions which satisfy some
natural postulates

Examples of operational measures:

- distillable entanglement
- distillable key

Basic postulates for axiomatic measures:

- nonnegative function defined on states
- nonincreasing under LOCC operations
- zero on separable states
LOCC operations
Digression

No fixed domain and co-domain.

Given entanglement measure is usually defined on
all possible tensor products

Hence more precisely, entanglement measure is a collection
of functions:

satisfying compatibility conditions:
Digression

The monotonicity condition writes then as follows:

Usually we use shorthand notation:
Just two examples of axiomatic entanglement measures

Entanglement of formation
(Bennett, DiVincenzo, Smolin and Wooters 1996)

Relative entropy of entanglement
(Plenio, Vedral, Rippin and Knight 1997)
Locking

If a quantity drops by a lot when we remove single qubit,
we say it is lockable.
(DiVincenzo, M.H., Leung, Smolin and Terhal 2002)

Entanglement can be locked
(HHH and Oppenheim 2005)
Lockable entanglement measures include:

entanglement of formation / entanglement cost
squashed entanglement

Nonlockable entanglement measure:

Relative entropy of entanglement. It can drop at most by 2
(proved in HHHO, example that 2 is possible - Andrzej Grudka)

Open problem: is distillable entanglement lockable?
Open problem: locking down to product state

Typically, in examples of locking, removal of a qubit causes
the state to be separable (or PPT).

Suppose that after removing a qubit, state becomes product.

Is it possible to lock entanglement in such case?
(J. Oppenheim, A. Winter, MH)

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