Quantum information Theory: Separability and distillability

1
Quantum information Theory Separability
and distillability
J. Ignacio Cirac Institute for Theoretical
Physics University of Innsbruck
KIAS, November 2001
SFB Coherent ControlU TMR
2
Entangled states
Superposition principle in Quantum Mechanics
If the systems can be in
or
then they can also be in
Two or more systems entangled states
If the systems can be in
A
B
or
then they can also be in
Entangled states possess non-local (quantum)
correlations
The outcomes of measurements in A and B are
correlated.
A
B
In order to explain these correlations
classically (with a realistic theory), we must
have non-locality.
Fundamental implications Bells theorem.
3
Applications
Secret communication.
Alice
Bob
1. Check that particles are indeed entangled.
2. Measure in A and B (z direction)
Alice
Bob
Correlations in all directions.
0 1 1 1 0
0 1 1 1 0
No eavesdropper present
Send secret messages
Given an entangled pair, secure secret
communication is possible
4
Computation.
ouput
quantumprocessor
input
A quantum computer can perform ceratin tasks
more efficiently
A quantum computer can do the same as a classical
computer ... and more
- Factorization (Shor). - Database search
(Grover). - Quantum simulations.
5
Precission measurements
We can measure more precisely
Efficient communication
Bob
Alice
Bob
Alice
We can use less resources

Entangled state
6
Problem Decoherence
environment
A
B
The systems get entangled with the environment.
Reduced density operator
7
Solution Entanglement distillation
Idea
...
local operation
local operation
environment
(classical communication)
Distillation
...
8
Fundamental problems in Quantum Infomation
Separability and distillability
SEPARABILITY
DISTILLABILITY
A
B
...
Can we distill these systems?
Are these systems entangled?
9
Additional motivations Experiments
Separability
Ion traps
Cavity QED
Optical lattices
Magnetic traps
NMR
Quantum dots
Josephson junctions
Atomic ensembles
Distillability quantum communication.
Long distance Q. communication?
10
Basic properties
This talk
Separability
Th. Physics
Mathematics
Distillability
Quantum Information
Algorithms, etc
Physical implementations
Computer Science
Th. Physics
Exp. Physics
Q. Optics
Condensed Matter
NMR
11
Outline
Separability.
Distillability.
Gaussian states.
Separability.
Distillability.
Multipartite case
12
1. Separability
Are these systems entangled?
1.1 Pure states
Product states are those that can be written as
Otherwise, they are entangled.
Examples
Product state
Entangled state
Entangled states cannot be created by local
operations.
13
1.2 Mixed states
In order to create an entangled state, one needs
interactions.
Separable states are those that can be prepared
by LOCC out of a product state. Otherwise, they
are entangled.
where
A state is separable iff
(Werner 89)
14
Problem given , there are infinitely many
decompositions
spectral decomposition
need not be orthogonal
Example two qubits ( )
00 01 10 11
where
15
1.3 Separability positive maps
A linear map is
called positive
A
B
state
state
Extensions
A
A
B
B
state
?
need not be positive, in general
A postive map is completely positive if
is separable iff for all
positive maps
(Horodecki 96)
However, we do not know how to construct all
positive maps.
16
Example Any physical action.
A
A
B
B
state
state
Any physical action can be described in terms of
a completely positive map.
17
Example transposition (in a given basis)
Is positive
A
A
Extension partial transposition.
B
B
Is called partial transposition
, then
Example
transposes the blocks
Partial transposition is positive, but not
completely positive.
18
What is known?
PPT
NPT
In general
- Low rank
SEPARABLE
ENTANGLED
- Necessary or sufficient conditions
?
(Horodecki 97)
2x2 and 2x3
Gaussian states
(Horodecki and Peres 96)
(Giedke, Kraus, Lewenstein, Cirac, 2001)
PPT
NPT
PPT
NPT
ENTANGLED
ENTANGLED
SEPARABLE
SEPARABLE
19
2. Distillability
...
Can we distill MES using LOCC?
PPT states cannot be distilled. Thus, there are
bound entangled states.
(Horodecki 97)
There seems to be NPT states that cannot be
distilled.
(DiVincezo et al, Dur et al, 2000)
20
2.1 NPT states
(IBM, Innsbruck 99)
We just have to concentrate on states with
non-positive partial transposition.
Idea If then there exists A and B,
such that
with
Physically, this means that
A
B
random
the same random
and still has non-positive partial transposition.
Thus, we can concentrate on states of the form
where
21
We consider the (unnormalized) family of states
Qubits
x
3
distillable
one can easily find A, B such that
Higher dimensions
x
2
3
?
distillable
NPT
Idea find A, B such that they project onto
with
there is a strong evidence that they are not
distillable for any finite N, all projections
onto have
22
What is known?
PPT
NPT
In general
DISTILLABLE
Non-DISTILLABLE
?
2xN
Gaussian states
(Horodecki 97, Dur et al 2000)
(Giedke, Duan, Zoller, Cirac, 2001)
PPT
NPT
PPT
NPT
DISTILLABLE
DISTILLABLE
Non-DISTILLABLE
Non-DISTILLABLE
23
3. Gaussian states
Light source
squeezed states (2-mode approximation)
Gaussian state
Decoherence photon absorption, phase shifts
where
is at most quadratic in
Internal levels can be approximated by continuous
variables in Gaussian states
Atomic ensembles
24
Optical elements
Measurements
X, P

• Beam splitters
• Lambda plates
• Polarizers, etc.

- Homodyne detection
Gaussian
Gaussian
local oscillator
B
We consider
A
m modes
n modes
Gaussian
Is separable and/or distillable?
25
3.1 What is known?
1 mode 1 mode
is separable iff
(Duan, Giedke, Cirac and Zoller, 2000 Simon 2000)
2 modes 2 modes
There exist PPT entangled states.
(Werner and Wolf 2000)
26
3.2 Separability
CORRELATION MATRIX
All the information about is contained in
the correlation matrix.
where
2nX2n
2mX2m
For valid density operators
is the symplectic matrix
where
and
27
Given a CM, does it correspond to a
separable state (separable)?
(G. Giedke, B. Kraus, M. Lewenstein, and Cirac,
2001)
Idea define a map
...
Facts
is a CM of a separable state iff is
too.
is no CM
If is a CM of an entangled state, then
either
or
is a CM of an entangled state
If is separable, then . This
last corresponds to
(for which one can readily see that is separable)
28
CONNECTION WITH POSITIVE MAPS?
Map for CMs
Map for density operators
Non-linear
29
3.3 Distillability
(Giedke, Duan, Zoller, and Cirac, 2001)
Idea take such that
Two modes NM1
Symmetric states
Non-symmetric states
distillable state.
symmetric state.
General case N,M
two modes
is distillable if and only if
There are no NPT Gaussian states.
30
4. Multipartite case.
C
A
Are these systems entangled?
B
Fully separable states are those that can be
prepared by LOCC out of a product state.
We can also consider partitions
Separable A-(BC)
Separable B-(AC)
Separable C-(AB)
C
C
C
A
A
A
B
B
B
31
4.1 Bound entangled states.
Consider
C
C
A
A
B
B
but such that it is not separable C-(AB).
QUESTIONS
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.
32
4.2 Activation of BES.
(Dür and Cirac, 1999)
Consider
C
C
A
B
A
B
but A and B can act jointly
C
A
B
singlets
Then they may be able to distill GHZ states.
33
ACTIVATION OF BOUND ENTANGLED STATES
Distillable iff two groups 3 and 5 particles
Distillable iff two groups 35-45 and 65-55
Distillable iff two groups have more than 2
particles.
If two particles remain separated not distillable.
Superactivation
Two parties can distill iff the other join
(Shor and Smolin, 2000)
C
A
B
Two copies
34
4.3 Family of states
Define
where
There are parameters.
Any state can be depolarized to this form.
35
5. Conclusions
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.
Gaussian states
Solved the separability and distillability
problem for two systems.
Solved the separability problem for three
(1-mode) systems
Maybe we can use the methods developed here to
attack the general problem.
Multipartite systems
New behavior regarding separability and bound
entanglement.
Family of states which display new activation
properties.
36
Hannover
Innsbruck
M. Lewenstein
Geza Giedke
Barbara Kraus
Collaborations
Wolfgang Dür
Guifré Vidal
R. Tarrach (Barcelona)
P. Horodecki (Gdansk)
J.I.C.
L.M. Duan (Innsbruck)
P. Zoller (Innsbruck)
SFB Coherent ControlU TMR
EQUIP
KIAS, November 2001
37
Institute for Theoretical Physics
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
P. Zoller J. I. Cirac
FWF SFB F015 Control and Measurement of
Coherent Quantum Systems EU networksCoherent
Matter Waves, Quantum Information EU
(IST) EQUIP Austrian Industry Institute for
Quantum Information Ges.m.b.H.