Quantum Cryptography

1
Entanglement Percolation in Quantum Networks
Cryptographic properties of nonlocal correlations
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
Recent Progress in Many-Body Theories Barcelona,
20 July 2007
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Quantum Information Theory

• 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

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Quantum Communication
Distant parties aim at establishing maximally
entangled two-qubit states.
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.
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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
A
B
B
By local operations and classical communication
(LOCC) at the repeater, the distant parties are
able to establish a maximally entangled state
between them.
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Quantum Networks
Quantum Network N distant nodes share a quantum
state ?.
?
The goal is to establish an entangled state
between two distant nodes, A and B, by local
operations and classical communication (LOCC).
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Quantum Networks
1D Structures the nodes are connected by a
series of quantum repeaters.
Briegel, Dür, Cirac and Zoller, PRL98
One of our main goals is to consider geometries
of larger dimension.
There exist several possible figures of merit

• The averaged concurrence.
• The worst-case entanglement.
• The singlet-conversion probability, SCP.

• The averaged concurrence.
• The worst-case entanglement.
• The singlet-conversion probability, SCP.

The maximum probability such that A and B share a
two-qubit maximally entangled state,
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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.
f
Example
Despite their apparent simplicity, these networks
already contain rich and intriguing features.
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Classical Entanglement Percolation
f
F
A
B
A
B
Nielsen Vidal
Majorization Theory
Bond Percolation
Lattice Percolation Threshold
Square Triangular Honeycomb 1/2 0.3473 0.6527
The classical entanglement percolation strategy
(CEP) defines some bounds for the minimal amount
of entanglement for non-exponential SCP.
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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.
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1D Geometries
1 Repeater
A
B
ES (zz)
A
B
A
B
A
B
One has
, which is better than the CEP strategy.
The intermediate repeater does not imply any loss
of SCP! (this property of course does not scale
with the number of repeaters)
Worst-case strategy the goal is to maximize the
minimum of the entanglement over the measurement
outcomes.
The optimal strategy is ES (zx basis) and gives
the same entanglement for all i.
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1D Geometries
R1
R2
RN
Asymptotic regime
A
B
Verstraete, Martín-Delgado and Cirac, PRL04

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.

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.
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2D Geometries

• CEP
• Previous strategy

A
B
Not surprisingly, CEP is not optimal for finite
lattices.
Finite-size entanglement percolation
A singlet can be established with probability one
whenever
A
A
B
B
A
B
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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!
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2D Geometries
CEP
Combining entanglement swapping and CEP,
long-distance entanglement can be established in
a network where CEP fails.
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Conclusions

• 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?

Raussendorf, Bravyi and Harrington, PRA05
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Mixed states
In this case, it is much easier to obtain lower
bounds for long-distance entanglement.
Given a mixed state, there exist many different
ensembles
?
If pE(?) is smaller than the percolation
threshold probability ? long-distance
entanglement is impossible.
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Conclusions
Quantum Information Theory
Many-Body Systems
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Thanks for your attention!
Antonio Acín, J. Ignacio Cirac and Maciej
Lewenstein, Entanglement Percolation in Quantum
Networks, Nature Phys. 3, 256 (2007).