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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
The Teleporters     (Peres ‘96, see also Woronowicz ‘76)
            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.

                                      (see also recent proof by
                                      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 labs paradigm




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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