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Simulating entanglement without communication

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

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Simulating entanglement without communication

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  1. 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? Simulating entanglementwithout communication Nicolas Gisin Nicolas Cerf, Serge Massar, Sandu Popescu Group of Applied Physics, University of Geneva

  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, 1874 1999 2 bits, Steiner, Phys.Lett. A270, 239 2000, Gisins Phys.Lett. A260, 323, 1999 1 bit! Toner & Bacon, PRL 91, 187904, 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/979026.

  4. The nonlocal Machine Alice Bob x y Non local Machine b a 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 b a 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 A 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, 117901,2001, see also B. Therhal et al., PRL 90,157903,2003)

  9. Causal nonlocal machines are monogamous x z Non local Machine a Non local Machine 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, 1775-1787, 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. 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 A 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, 1775-1787, 2004)

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

  14. The NL Machine I3322 is not monogamous A 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

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