1. Simulating entanglement without communication
Nicolas Gisin
Nicolas Cerf, Serge Massar, Sandu Popescu
Group of Applied Physics, University of Geneva
Simulating entanglement with a few bits of communication
How strong can no-signaling correlation be?
The PR nonlocal machine
Simulating entanglement with the nonlocal machine
Is Nature sparing with resources?
CHSH-Bell and the PR machines are monogamous
Bell inequality with more settings, combined PR machines
Is partial entanglement more non-local?

2. Simulating entanglement with a few bits of communication (+ shared randomnes)
Bob 
& define measurement bases
The output  &  should
reproduce the Q statistics:
Alice 
Case of singlet: 8 bits, Brassard,Cleve,Tapp, PRL 83,
2 bits, Steiner, Phys.Lett. A270, ,
Gisins Phys.Lett. A260, 323, 1999
1 bit! Toner & Bacon, PRL 91, , 2003
0 bit: impossible (Bell inequality) … but …

3. How strong can no-signaling correlation be?
CHSH-Bell inequality:
1) Q correlation can violate Bell inequalities,
but can’t be used for signaling
2) Q correlation can’t violate the CHSH-Bell inequality
By more than a factor
Are these two facts related? Could there be correlations that:
Do not allow signaling, and
Do violate the CHSH inequality by more than ?
Answer: yes! S. Popescu and D. Rohrlich, quant-ph/

4. The nonlocal Machine Alice Bob x y Non local Machine a b a + b= x.y
E(a=1|x,y) = ½, independent of y  no signaling
E(a,b|0,0) + E(a,b|0,1) + E(a,b|1,0) - E(a,b|1,1) = 4

5. The nonlocal Machine Alice Bob x y Non local Machine a b a + b= x.y
A single bit of communication suffice to simulate the NL Machine
(assuming shared randomness).
But the NL Machine does not allow any communication.
Hence, the NL Machine is a strickly weaker ressource than
communication.

6. Simulating singlets with the NL Machine
Bob
Alice
Non local Machine a b
Given & , the statistics of  &  is that of the singlet state:

7. Is Nature sparing with resources?

8. The CHSH-Bell inequality is monogamous
Theorem:
For all 3-qubit state ABC,
If A-B violates the CHSH-Bell inequality
then neither A-C nor B-C violates it. C B
(see V. Scarani & NG, PRL 87, ,2001,
see also B. Therhal et al., PRL 90,157903,2003)

9. Causal nonlocal machines are monogamousx z
Non local Machine
Non local Machine a y c b
a+b=x.y
a+c=x.z
If then b+c=x(y+z), and Alice can signal to B-C

10. The new inequality for qubits with 3 settings
This is the only new inequality for 3 inputs and binary outputs.
D. Collins et al., J.Phys. A 37, , 2004
I3322=P(a=0,b=0|x=0,y=0)+P(0,0|0,1)+P(0,0|0,2)
+ P(0,0|1,0)+P(0,0|1,1)-P(0,0|1,2)
+ P(0,0|2,0)-P(0,0|2,1)
- P(a=0|x=0) – 2P(b=0|y=0) – P(b=0|y=1)
 0

11. ()= CHSH Pcos()|00>+sin()|11> + (1- CHSH) P|01>
For each , let CHSH be the critical weight such that
()= CHSH Pcos()|00>+sin()|11> + (1- CHSH) P|01>
is at the limit of violating the CHSH inequality
1.03
1.02
1.01
1.00 r )
0.99
trace(B
0.98
0.97
0.96
0.95
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6 q

12. The I3322-Bell inequality is not monogamous
There exists a 3-qubit state ABC, such that
A-B violates the I3322-Bell inequality and
A-C violates it also.
ABC C B
(see D. Collins et al., J.Phys. A 37, , 2004)

13. The nonlocal machine optimal for the I3322 Bell inequalityx y
Alice
Bob
x=x1x2
y=y2y1
Non local Machine
a 1
b 1
Non local Machine
a 2 b2
= 1+2
b=b1 + b2

14. The NL Machine I3322 is not monogamous
Non local Machine
Non local Machine
I3322 > 0
I3322 > 0 C B

15. Simulating partial entanglement
Partially entangled states seem more nonlocal than the max entangled ones !
Nonlocality  entanglement

16. Conclusions
One can simulate the singlet Q correlations with qualitatively less than a bit of communication.
Causal (ie no signaling) correlation can be stronger than the Q correlation.
The NL Machine inspired by the CHSH-Bell inequality suffice to simulate singlet correlation, (one instance of the NL Machine per simulated outcome).
Is Nature sparing with resources?
CHSH-Bell inequality and the NL Machine are monogamous, but the inequality with 3 settings per site is not.
The NL Machine corresponding to the new Bell inequality contains 2 elementary NL machines, “explaining” why it is not monogamous.

17. 1 2

18. x=0
x=1
x=1
x=0

19. (0,0)
(1,0)
(0,1)
(1,1)
(1,1)
(0,1)
(1,0)
(0,0)

20. A=a+1
B=b
(0,0)
(1,0)
(0,1)
(1,1)
(1,1)
(0,1)
(1,0)
(0,0)
B=b+1
A=a