Simulating quantum correlations as a sampling problem

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Simulating quantum correlations as a sampling
problem
Julien Degorre L.R.I. Université Paris Sud -
L.I.T.Q. Université de Montréal (joint work
with Sophie Laplante and Jérémie Roland)
L.R.I. Université de Paris Sud Université
Libre de Bruxelles
Physical Review A, 72062314, 2005
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The problem of simulating quantum correlations
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Local Hidden Variable Model
?
Doesn't depend on the inputs!
Input
Input
Alice
Bob
Output
Output
BelI's theorem impossible to reproduce quantum
correlations
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Biased Hidden Variable Model
Infinite biased Shared Randomness
Alice
Bob
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Step 1 Local Sampling of the biased distribution
Shared randomness independent on the inputs
The rejection method
Set k0
1. Alice picks 2. Alice
picks 3. Test whether

If test succeeds, Alice ACCEPTS and
sets
Go back to 1 with kk1
Otherwise, Alice REJECTS
When the process terminates, Alice has
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Step 2 Distributed Sampling problem
With communication
With post selection
Bob
Alice
Rejection method 1. Alice picks
2. Alice picks 3. Test If

Index k
Set k 0
Post selection
Communication
2 bits on average Steiner99 Gisin² 00
Gisin Gisin 00
Abort
Go to 1. kk1
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But we can be more clever...
Used only by Alice !
Bob
Alice
Rejection method 1. Alice picks
2. Alice picks 3. Test If

Post selection
Communication
Gisin Gisin 00
Abort
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Local Sampling of the biased distribution a new
method
Recall the rejection method
The Choice method
1. Alice picks
3. Test whether

2. Alice pick
So, Alice has
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Step 2 Distributed Sampling problem
With communication
Bob
Alice
Choice method 1. Alice picks
2. Alice picks 3. Tests
whether
x 0 or 1
Bob sets
Communication
1 bit Worst Case TB03
If yes
If no
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Without Resources !
Bob
Choice method
Alice
Bob always sets
Alice picks
Test If
If yes
If no
Bob's output
Alice's output
Simulation of non separable Werner State.
With p1/2
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With non local boxes

• A non-local box doesn't allow Alice and Bob to
  share

the biased random variable
the mutual information

• To simulate the singlet correlations with a
  shared biased random
• variable

Bob only needs to generate
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With a non local box
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With a non local box
Choice method
Bob always sets
Bob
Alice
Alice picks
Tests whether
Bob Tests whether
If yes
y0
If yes cool !
x0
y1
If no
If no Aïe !
x1
y
x
Non-Local Box

Simulation of non separable Werner State.
ß
With p1/2
a
He sets
She sets
Bob's output
Alice's output
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Other results further work

• Results on POVMs

• Post Selection Efficiency 1/3

• Communication and Non-local Boxes 2 nl-Bits and
  4 bits on average

• Multiparty states and non maximally entangled
  states ?

• Some preliminary results for higher dimensions