recent progress in many-body theories barcelona, 20 july 2007 antonio acín 1,2 j. ignacio cirac 3...
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Recent Progress in Many-Body Theories
Barcelona, 20 July 2007
Antonio Acín1,2
J. Ignacio Cirac3
Maciej Lewenstein1,2
1ICFO-Institut de Ciències Fotòniques (Barcelona)2ICREA-Institució Catalana de Recerca i Estudis Avançats
3Max-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
baAB
iiiii
iAB bbaap
Quantum CommunicationDistant parties aim at establishing maximally entangled two-qubit states.
11002
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 BR1 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:
iAB
i
EiELOCC
Prmax
The maximum probability such that A and B share a two-qubit maximally entangled state, 1100
2
1
Briegel, Dür, Ciracand Zoller, PRL’98
iAB
iEE
LOCC
minmax
• 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.
...
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...
... ... ... ......
...
φ
d
ii ii
1
021 d
Despite their apparent simplicity, these networks already contain rich and intriguing features.
Example: 2/121
Classical Entanglement Percolation
A B A B
φ Φ
OKp
Majorization Theory: 112,1min OKp
Bond Percolation
Lattice Percolation Threshold
Square
Triangular
Honeycomb
1/2
0.3473
0.6527
18/sin2 18/sin21
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 Geometries1 Repeater
A B
212
1100 21
A
A B2121
22
21
212
A B
One has , which is better than the CEP strategy. 212,1min OKOK ppSCP
BES (zz)
Worst-case strategy: the goal is to maximize the minimum of the entanglement over the measurement outcomes.
iAB
iEE
LOCC
minmax
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
A1100 21
R1 R2 RN
B
kkr
NrrN NMMC det2det2sup 121 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.
1det2 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:
2211 OKp
211 OKpSCP
Not surprisingly, CEP is not optimal for finite lattices.
Finite-size entanglement percolation
A AB B A B
6498.02/1 1
A singlet can be established with probability one whenever
i
j
2D GeometriesUsing the previous measurement strategy, we already see some
differences with the classical case.
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...
Many end points can be connected with
probability one!
2D Geometries 2
21
2
2 1100
2OKp
thh
aOK pp 2
1)( 12
tht
bOK pp 1)( 12
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
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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:
ii
ip
EESS pp
EE pp min
If pE(ρ) is smaller than the percolation threshold
probability → long-distance entanglement is impossible.
Conclusions
Quantum Information Theory
Many-Body Systems
Thanks for
your attention!
Antonio Acín, J. Ignacio Cirac and Maciej Lewenstein, Entanglement Percolation in Quantum Networks, Nature Phys. 3, 256 (2007).