quantum information theory: separability and distillability sfb coherent control €u tmr j. ignacio...
TRANSCRIPT
Quantum information Theory: Separability and distillability
Quantum information Theory: Separability and distillability
SFB Coherent Control€U TMR
J. Ignacio CiracInstitute for Theoretical PhysicsUniversity of Innsbruck
KIAS, November 2001
Entangled statesEntangled states
Superposition principle in Quantum Mechanics:
Two or more systems: entangled states
If the systems can be in
or
then they can also be in
j0i
j1i
c0j0i + c1j1i
j0i A j0i B
j1i A j1i B
c0j0i A j0i B + c1j1i A j1i B
If the systems can be in
or
then they can also be in
A B
Entangled states possess non-local (quantum) correlations:
A BThe outcomes of measurements in A and B are correlated.
In order to explain these correlations classically (with a
realistic theory), we must have non-locality.
Fundamental implications: Bell´s theorem.
Secret communication.
Alice Bob
1. Check that particles are indeed entangled.
Correlations in all directions.
2. Measure in A and B (z direction):
Alice Bob
01110
01110
No eavesdropper present Send secret messages
jÁi = j0; 1i + j1; 0i
Given an entangled pair, secure secret communication is possible
ApplicationsApplications
Computation.
A quantum computer can perform ceratin tasks more efficiently
A quantum computer can do the same as a classical computer ... and more
quantumprocessor
input
ouput
jª in i
jª o u t i
jª o u t i = U jª i n i
- Factorization (Shor).
- Database search (Grover).
- Quantum simulations.
Precission measurements:
Efficient communication:
AliceBob
0
1010 1
1
AliceBob
+We can use less resources
Entangled state
0
1010 1
1
We can measure more precisely
environment
j©iA B jE iE ! jª iA B E
½A B = trE (jª i hª j) 6= j©ih©j
Problem: DecoherenceProblem: Decoherence
A B
The systems get entangled with the environment.
Reduced density operator:
Solution: Entanglement distillationSolution: Entanglement distillation
environment
......
local operation local operation
(classical communication)
Idea:
Distillation:
½
½
½
½
j0; 1i + j1; 0i
Fundamental problems in Quantum Infomation: Separability and distillability
Fundamental problems in Quantum Infomation: Separability and distillability
A B
Are these systems entangled?
½ ...½
½
½
j©i = j0; 1i + j1; 0i
SEPARABILITY DISTILLABILITY
½
Can we distill these systems?
Additional motivations: ExperimentsAdditional motivations: Experiments
jÁi
Long distance Q. communication?
Ion traps
Atomic ensembles
Cavity QED
NMR Quantum dotsJosephson junctions
Optical lattices
Magnetic traps
Distillability: quantum communication.
Separability:
Quantum Information
Th. PhysicsMathematics
Computer Science Th. PhysicsExp. Physics
Physical implementations:Algorithms, etc:
Basic properties:
Q. OpticsCondensed MatterNMR
Separability
Distillability
This talk
OutlineOutline
Separability.
Distillability.
Gaussian states.
Separability.
Distillability.
Multipartite case:
1. Separability1. Separability
1.1 Pure states1.1 Pure states
Product states are those that can be written as:
Otherwise, they are entangled.
Entangled states cannot be created by local operations.
j©i = jai jbi
jai jbi ! ja(t)i jb(t)i
j0i A j0i B
c0j0i A j0i B + c1j1i A j1i B
Examples:Product state:
Entangled state:
Are these systems entangled?
½
Separable states are those that can be prepared by LOCC out of a product state.
Otherwise, they are entangled.
A state is separable iff ½=X
k
pk jak i hak j jbk i hbk j pk ¸ 0where
(Werner 89)
1.2 Mixed states1.2 Mixed states
½
In order to create an entangled state, one needs interactions.
Problem: given , there are infinitely many decompositions
spectral decomposition
need not be orthogonal
Example: two qubits ( )
½
½=X
k
¸ k jª k i hª k j
=X
k
qk j©k i h©k j
= : : :
½=X
k
pk jak i hak j jbk i hbk j
hª k jª j i =±k;j
h©k j©j i 6=±k;j
½=15
(j0; 0i h0; 0j + j1; 1i h1; 1j) +25
j+; +ih+; +j
+110
(j0; ¡ i h0; ¡ j + j1; ¡ i h1; ¡ j)
j§ i =1
p2
(j0i § j1i )
where
½ =120
0
BB@
7 1 2 21 3 2 22 2 3 12 2 1 7
1
CCA
H = C 2 C 2
00 01 10 11
A linear map is called positiveL : A (H ) A (H )½ ¸ 0 ! L (½) ¸ 0
A
B
½
A
½ ½
B
Extensions
state state
state ?
A
B
1.3 Separability: positive maps1.3 Separability: positive maps
: need not be positive, in general
A postive map is completely positive if: ½A B ¸ 0 ! (L 1)(½A B ) ¸ 0
is separable iff for all positive maps ½ ( L 1 ) ( ½) ¸ 0(Horodecki 96)
However, we do not know how to construct all positive maps.
Example: Any physical action.
½ ¸ 0 L (½) ¸ 0
A
B
½
state
A
B
state
Any physical action can be described in terms of a completely positive map.
Example: transposition (in a given basis)
½ =X
i ;j
½i ;j ji i hj j
T (½) =X
i ;j
½i ;j jj i hi j
;
15
µ2 1 + i
1 ¡ i 3
¶!
15
µ2 1 ¡ i
1 + i 3
¶
Extension: partial transposition.
½=X
½i j kl ji ; j i hk; l j (T 1)(½) =X
½i j kl jk; j i hi ; l j
0
BB@
1
CCA
transposes the blocks
Example:
Is called
partial transposition, then
12
1 0 0 10 0 0 00 0 0 01 0 0 1
!12
BB
1 0 0 00 0 1 00 1 0 00 0 0 1
CC
0
@
1
A
0
BB@
1
CCA
Partial transposition is positive, but not completely positive.
A
B
A
B
Is positive
What is known?What is known?
?
SEPARABLE ENTANGLED
PPT NPT
2x2 and 2x3
SEPARABLE ENTANGLED
PPT NPT
(Horodecki and Peres 96)
Gaussian statesSEPAR
ABLE ENTANGLED
PPT NPT
(Giedke, Kraus, Lewenstein, Cirac, 2001)
- Low rank
- Necessary or sufficient conditions
(Horodecki 97)
In general
2. Distillability2. Distillability
...
½
½
½
j©i = j0; 1i + j1; 0i
Can we distill MES using LOCC?
PPT states cannot be distilled. Thus, there are bound entangled states.
There seems to be NPT states that cannot be distilled.
(Horodecki 97)
(DiVincezo et al, Dur et al, 2000)
2.1 NPT states2.1 NPT states
We just have to concentrate on states with non-positive partial transposition.
Idea: If then there exists A and B, such that
Thus, we can concentrate on states of the form:
Physically, this means that
½A B
random the same random
with ~½= (A B )½(A B )y
U U
and still has non-positive partial transposition.
Zd¹ U (U U )~½(U U )y = aP + + bP ¡ where b =
1d¡
tr(P ¡ ~½)
½TA ¸ 0
·Zd¹ U (U U )~½(U U )y
TA
¸ 0
(IBM, Innsbruck 99)
Qubits:
We consider the (unnormalized) family of states:
x3
one can easily find A, B such that (A B )½ N (A B )y ! j©ih©j
Higher dimensions:
x2 3distillable?
there is a strong evidence that they are not distillable: for any finite N, all
projections onto have
Idea: find A, B such that they project
onto with
½(x) = P + + xP ¡
H = C 2 C 2
H = C 3 C 3
½TA ¸ 0
½TA ¸ 0
C 2 C 2 ½TA ¸ 0
C 2 C 2 ½TA ¸ 0
½TA ¸ 0
½TA ¸ 0
NPT
distillable
What is known?What is known?
?
Non-DISTILLABLE DIS
TILL
ABLE
PPT NPT
2xN
Non-DISTILLABLE
PPT NPT
(Horodecki 97, Dur et al 2000)
Gaussian states(Giedke, Duan, Zoller, Cirac, 2001)
In general
DIS
TILL
ABLE
Non-DISTILLABLE
PPT NPT
DIS
TILL
ABLE
3. Gaussian states3. Gaussian states
Light source: squeezed states:(2-mode approximation)
Decoherence: photon absorption, phase shifts Gaussian state:
jª i = e¸ (ayby¡ ab) jvaci =1X
n=0
¸ n jn; ni
½= e¡ H
where
is at most quadratic in
H = H (X a ; P a; X b; Pb)
X a =ay + a
p2
; P a = iay ¡ a
p2
Atomic ensembles:Internal levels can be approximated
by continuous variables in Gaussian
states
Optical elements:
- Beam splitters:- Lambda plates:- Polarizers, etc.
Gaussian Gaussian
Measurements:
- Homodyne detection:
localoscillator
X, P
A B
n modes m modes
We consider:
½= e¡ H ! ½0= e¡ H 0
½Gaussian
Is separable and/or distillable?½
H = [L 2(R )] n [L 2(R )] m
3.1 What is known?3.1 What is known?
1 mode + 1 mode:
2 modes + 2 modes:
(Duan, Giedke, Cirac and Zoller, 2000; Simon 2000)
is separable iff
There exist PPT entangled states.(Werner and Wolf 2000)
½ ½T A = ( T 1 ) ( ½) ¸ 0
2nX2n
3.2 Separability3.2 Separability
All the information about is contained in:
For valid density operators:
R = (X a1 ; P a1 ; X a2 ; : : : ; X b1 ; P b1 ; : : : )
½
° ®;¯ = 2R e h(R ® ¡ d®)(R ¯ ¡ d¯ )i the „correlation matrix“.
where d®= hR ®i ! 0
° ¸ i J
J = J 2 ©J 2 ©: : : ©J 2where
and J 2 =µ
0 ¡ 11 0
¶
is the „symplectic matrix“
°=µ
A CCT B
¶
2mX2m
CORRELATION MATRIX
Idea: define a map
is a CM of a separable state iff is too.
If is a CM of an entangled state, then either
If is separable, then . This last corresponds to
is no CM
or
is a CM of an entangled state
Given a CM, : does it correspond to a separable state (separable)?°
° 0 ´ ° ° 1 ° 2 ... ° N
° N ° N +1
° N +1
° N +1
° N
° ° N ! ° 1 ½1 = ½A ½B
(for which one can readily see that is separable)
Facts:
A N +1 = B N +1 = A N ¡ R e[C N (B N ¡ i J ) ¡ 1C T ]
C N +1 = ¡ I m[C N (B N ¡ i J ) ¡ 1C T ]
A N +1 = B N +1 = A N ¡ R e[C N (B N ¡ i J ) ¡ 1C T ]
C N +1 = ¡ I m[C N(B N ¡ i J ) ¡ 1C T ]N
A N +1 = B N +1 = A N ¡ R e[C N (B N ¡ i J ) ¡ 1C T ]
C N +1 = ¡ I m[C N (B N ¡ i J ) ¡ 1C T ]
A N +1 = B N +1 = A N ¡ R e[C N (B N ¡ i J ) ¡ 1C T ]
C N +1 = ¡ I m[C N(B N ¡ i J ) ¡ 1C T ]N
N
(G. Giedke, B. Kraus, M. Lewenstein, and Cirac, 2001)
Map for CM‘s:
Map for density operators:
Non-linear
½
½
Gaussian
separable
density operators
° N ! ° N +1
½N = e¡ H N ! ½N +1 = e¡ H N +1
(½N +1)A B = trB ~B f [(½N )A B (½N ) ~A ~B ]X B ~B g
A~A
B~B
A~A
CONNECTION WITH POSITIVE MAPS?
3.3 Distillability3.3 Distillability
Idea: take such that
Two modes: N=M=1:
Symmetric states:
distillable state.
A B
A B
Non-symmetric states:
A B A B
General case: N,M
A B
A B
symmetric state.
two modes
° A B = ° B A
½ ½T ¸ 0
½Ts ¸ 0 ½T ¸ 0
~½Ts ¸ 0 ½T
s ¸ 0
½TN ;M ¸ 0 ½T
1;1 ¸ 0
is distillable if and only if ½ ½T ¸ 0
There are no NPT Gaussian states.
(Giedke, Duan, Zoller, and Cirac, 2001)
4. Multipartite case.4. Multipartite case.
A B Are these systems entangled?
Fully separable states are those that can be prepared by LOCC out of a product state.
½
C
½=NX
k=1
pk jak i A hak j jbk i B hbk j jck i C hck j
We can also consider partitions:
Separable A-(BC) Separable B-(AC) Separable C-(AB)
A B
C
A B
C
A B
C
NX
k=1
pk jak i A hak j j' k i B C h' k jNX
k=1
pk jbk i B hbk j j' k i A C h' k jNX
k=1
pk jck i C hck j j' k i A B h' k j
4.1 Bound entangled states.4.1 Bound entangled states.
Consider
A B
C
A B
C
½=NX
k= 1
pk jak i A hak j j' k i B C h' k j =NX
k= 1
pk jbk i B hbk j j' k i A C h' k j
but such that it is not separable C-(AB).
Is B entangled with A or C?
Is A entangled with B or C?
Is C entangled with A or B?
Consequence: Nothing can be distilled out of it. It is a bound entangled state.
QUESTIONS:
4.2 Activation of BES.4.2 Activation of BES.
A B
C
A B
C
but A and B can act jointly
A B
C
singlets
Consider
(Dür and Cirac, 1999)
Then they may be able to distill GHZ states.
N o t d is t illa b le
D is ti lla b le
N o t-d is t illa b le
N o t-d is t illa b le
42
85
137
6
42
85
137
6
4
2 85
137
6
42
85
137
6
N o t-d is t il la b le
D is ti lla b le
N o t-d is t il la b le
42
8
1 2
13
7
6
115
91 0
4
8
1 2
7
6
11
91 0
42
8
1 2
13
7
6
11
59
1 0
2
13
5
2
1
4
3
2
1
4
3
Distillable iff two groups3 and 5 particles
Distillable iff two groups35-45% and 65-55%
Distillable iff two groups
have more than 2 particles.
Two parties can distill iff the
other join
If two particles remain
separated not distillable.
Superactivation
(Shor and Smolin, 2000)
A B
C
Two copies
ACTIVATION OF BOUND ENTANGLED STATES
4.3 Family of states4.3 Family of states
where
There are parameters.2N ¡ 1
Define:
Any state can be depolarized to this form.
5. Conclusions5. Conclusions
Maybe we can use the methods developed here to attack the general problem.
The separability problem is one of the most challanging problems in quantum
Information theory. It is relevant from the theoretical and experimental point of view.
Multipartite systems:
New behavior regarding separability and bound entanglement.
Family of states which display new activation properties.
Gaussian states:
Solved the separability and distillability problem for two systems.
Solved the separability problem for three (1-mode) systems
SFB Coherent Control€U TMR
Geza Giedke
Wolfgang Dür
Guifré Vidal
Barbara Kraus
J.I.C.
Innsbruck:
Collaborations:
M. Lewenstein
R. Tarrach (Barcelona)
P. Horodecki (Gdansk)
L.M. Duan (Innsbruck)
P. Zoller (Innsbruck)
Hannover
EQUIP
KIAS, November 2001
Institute for Theoretical PhysicsInstitute for Theoretical Physics
€
FWF SFB F015:„Control and Measurement of Coherent Quantum Systems“
EU networks:„Coherent Matter Waves“, „Quantum Information“
EU (IST):„EQUIP“
Austrian Industry:Institute for Quantum Information Ges.m.b.H.
P. ZollerJ. I. Cirac
Postdocs: - L.M. Duan (*) - P. Fedichev - D. Jaksch - C. Menotti (*) - B. Paredes - G. Vidal - T. Calarco
Ph D: - W. Dur (*) - G. Giedke (*) - B. Kraus - K. Schulze