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Nov 21, 2013 (3 years and 6 months ago)

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Recent Progress in Many
-
Body Theories

Barcelona, 20 July 2007

Antonio Acín
1,2

J. Ignacio Cirac
3

Maciej Lewenstein
1,2


1
ICFO
-
Institut de Ciències Fotòniques (Barcelona)

2
ICREA
-
Institució Catalana de Recerca i Estudis Avançats

3
Max
-
Planck Institute for Quantum Optics


Cryptographic properties of
nonlocal correlations

Entanglement Percolation
in Quantum Networks


Quantum Information Theory (QIT) studies how information
can be transmitted and processed when encoded on
quantum states.


New information applications are possible because of
quantum features
: communication complexity and
computational speed
-
up, secure information transmission
and quantum teleportation.


The key resource for all these applications is quantum
correlations, or entanglement.


A pure state is entangled whenever it cannot be written in a
product form:



A mixed state is entangled whenever it cannot be obtained
by mixing product states:


Quantum Information Theory

b
a
AB



i
i
i
i
i
i
AB
b
b
a
a
p




Quantum Communication

Distant parties aim at establishing maximally entangled two
-
qubit states.



11
00
2
1


AB

Crypto:

If the parties share this state, they know they have no correlations
with any third party. By measuring the state they obtain a perfect secret key.

More in general, if the parties have this state they can teleport any qubit. Thus,
a maximally entangled state is equivalent to a perfect quantum channel
.

M
CORR
CORR
Entanglement Theory

Given a quantum state:


Is it entangled?


If yes, can the parties transform many copies of it into fewer
maximally entangled states?


What are the optimal procedures?

Entanglement Swapping:

A

B

A

B

By local operations and classical communication (LOCC) at the repeater, the
distant parties are able to establish a maximally entangled state between them.

Quantum Networks

Quantum Network:
N

distant nodes share a quantum state
ρ
.

A
B
ρ

The goal is to establish an entangled state between two distant nodes,
A

and
B
, by local operations and classical communication (LOCC).

A
B
Quantum Networks

...
A
B
R1
R2
RN
1D Structures: the nodes are connected by a series of quantum repeaters.

One of our main goals is to consider geometries of larger dimension.

There exist several possible figures of merit:





i
AB
i
E
i
E
LOCC




Pr
max
The maximum probability such that
A

and
B

share a two
-
qubit
maximally entangled state,



11
00
2
1



Briegel, Dür, Cirac

and Zoller, PRL’98



i
AB
i
E
E
LOCC

min
max




The averaged concurrence.



The worst
-
case entanglement.



The singlet
-
conversion probability, SCP.



The averaged concurrence.



The worst
-
case entanglement.



The singlet
-
conversion probability, SCP.

Quantum Networks

We focus on a simple version of the problem where (i) the network has a
well
-
defined geometry and (ii) the state connecting the nodes are pure.

...
...
...
.
.
.
.
.
.
.
.
.
.
.
.
...
...
φ




d
i
i
ii
1


0
2
1




d




Despite their apparent
simplicity, these networks
already contain rich and
intriguing features
.

Example:

2
/
1
2
1






Classical Entanglement Percolation

A

B

A

B

φ

Φ

OK
p
Majorization Theory:





1
1
2
,
1
min



OK
p
Bond Percolation

Lattice

Percolation Threshold

Square

Triangular

Honeycomb


1/2


0.3473


0.6527




18
/
sin
2





18
/
sin
2
1

The classical entanglement percolation strategy
(CEP) defines some bounds for the minimal amount
of entanglement for non
-
exponential SCP.

Nielsen & Vidal

Entanglement Percolation


Is Classical Entanglement Percolation always
optimal?


If not, does it predict the right asymptotic
behaviour?

NO

NO

The distribution of entanglement though a
quantum network defines a new type of phase
transition, an entanglement phase transition
that we call
entanglement percolation.


1D Geometries

1 Repeater

A

B

2
1
2


11
00
2
1





A

A

B

2
1
2
1



2
2
2
1
2
1
2




A

B

One has , which is better than the CEP strategy.



2
1
2
,
1
min
OK
OK
p
p
SCP




B

ES
(zz)

Worst
-
case strategy: the goal is to maximize the minimum of the
entanglement over the measurement outcomes.



i
AB
i
E
E
LOCC

min
max


The optimal strategy is ES (zx basis) and
gives the same entanglement for all
i
.

The intermediate repeater does not imply any loss of SCP!

(this property of course does not scale with the number of repeaters)

1D Geometries

Asymptotic regime

A

11
00
2
1





R1

R2

RN

B





k
k
r
N
r
r
N
N
M
M
C




det
2
det
2
sup
1
2
1
1







1.
The exponential decay of the
SCP

whenever
automatically follows from this result.

2.
Most of these results can be translated to arbitrary dimension,
especially for one
-
way communication LOCC strategies.



1
det
2

k

An exponential decay of the entanglement is observed whenever the
connection between the repeater does not majorize the singlet.

The same result is obtained by CEP.

Verstraete, Martín
-
Delgado and Cirac, PRL’04

2D Geometries

A

B



CEP:




Previous strategy:





2
2
1
1
OK
p




2
1
1
OK
p
SCP



Not surprisingly, CEP is not optimal for finite lattices.

Finite
-
size entanglement percolation

A

A

B

B

A

B

6498
.
0
2
/
1
1



A singlet can be established
with probability one whenever

i

j


2D Geometries

Using the previous measurement strategy, we already see some
differences with the classical case.

...
...
...
.
.
.
.
.
.
.
.
.
.
.
.
...
...
...
...
...
.
.
.
.
.
.
.
.
.
.
.
.
...
...
Many end points can
be connected with
probability one!

2D Geometries



2
2
1
2
2
11
00











2

OK
p


th
h
a
OK
p
p



2
1
)
(
1
2



th
t
b
OK
p
p



1
)
(
1
2

CEP

Combining entanglement swapping and
CEP, long
-
distance entanglement can be
established in a network where CEP fails.


The distribution of entanglement through
quantum networks defines a framework where
statistical methods and concepts naturally apply.


It leads to a novel type of critical phenomenon,
an entanglement phase transition that we call
entanglement percolation.


Is any amount of pure
-
state entanglement
between the nodes sufficient for entanglement
percolation?


More examples beyond CEP.


Mixed states?

Conclusions

Raussendorf, Bravyi and Harrington, PRA’05

Mixed states

...
...
...
.
.
.
.
.
.
.
.
.
.
.
.
...
...
In this case, it is much easier to obtain lower
bounds for long
-
distance entanglement.

ρ

Given a mixed state, there exist
many different ensembles:

i
i
i
p




E
E
S
S
p
p







E
E
p
p
min


If
p
E
(
ρ
) is smaller than the
percolation threshold
probability
→ long
-
distance
entanglement is impossible.

Conclusions

Quantum Information
Theory

Many
-
Body Systems

Antonio Acín, J. Ignacio Cirac and Maciej Lewenstein, Entanglement
Percolation in Quantum Networks, Nature Phys.
3
, 256 (2007).