Quantum entanglement can be simulated without communication Nicolas - PowerPoint PPT Presentation

Quantum entanglement can be simulated without communication Nicolas J. Cerf Centre for Quantum Information and Communication Université Libre de Bruxelles (joint work with Nicolas Gisin, Serge Massar, and Sandu Popescu) Physical Review Letters 94, 220403 (2005) QIP |2006>, Paris, January 2006 1

Simulation of E.P.R. experiment | Ψ‾ > Input: Input:  a  b Alice Bob A ∈ {− 1,  1 } B ∈ {− 1,  1 } Output: Output: a ⋅ P  A, B = 1 − AB  b  a ⋅ E  A B =− b 4 P  A = 1 / 2 ∀ P  B = 1 / 2 ∀  a b 2 CAUSALITY

Local Hidden Variable (LHV) Model λ Shared randomness: Input: Input: a   b Alice Bob B  A  a ,  ∈ {− 1,  1 } b ,  ∈ {− 1,  1 } Output: Output: b = ∫ ∈ p  A  a ,  a ,  B  E  A B ∣ b ,  ? a ⋅ 3 =− b BUT...

Bell's Theorem: No Local Hidden Variable model can simulate the quantum correlations of the EPR experiment Indeed, any LHV model must satisfy the CHSH inequality: a 1 ,  b 0 ,  a 1 ,  b 0 ,  ∣ C   a 0 ,  b 1 ∣≤ 2 ∀  a 0 ,  b 1 ∈ S 2 a 1 ,  b 0 ,  a 0 ,  a 0 ,  a 1 ,  a 1 ,  C   a 0 ,  b 1 = E  AB ∣ b 0  E  AB ∣ b 1  E  AB ∣ b 0 − E  AB ∣ b 1  with In quantum mechanics: a 1 ,  b 0 ,  a 1 ,  b 0 ,  b 1 = 2  2 ∃  a 0 ,  b 1 ∈ S 2 C   a 0 ,  such that So we need extra resources, in addition to those allowed by any Local Hidden Variable model The amount of extra resources that is needed gives us some measure of the non-locality of QM 4 (Maudlin 92; Brassard, Cleve, Tapp 99)

Additional resources Classical communication : in number of bits (on average or in worst case) Classical communication Allows for superluminal communication Freedom to post-select (detection loophole) : the parties are given Freedom to post-select (detection loophole) the possibility to output “no result”, simulating an imperfect detector Does not allow for superluminal communication but probabilistic Non-Local Box : in number of uses Non-Local Box Remains causal : strictly weaker resource than 1 bit of communication x y Popescu and Rohrlich 94 a ⊕ b = x ∧ y van Dam 00 x , y , a ,b ∈ { 0,1 } a b 5

Outline of the known protocols a ,   b Resource Amount Reference Communication 1.17 bit on Average Equator Maudlin 92 Brassard, Cleve, Tapp 99 Communication 8 bits in Worst Case Sphere Steiner 99 Communication Equator 1.48 bit on Average Gisin, Gisin 99 Post-Selection P(A_output)= P(B_output)= 2/3 Sphere Communication NJC, Gisin, Massar 00 Sphere 1.19 bit on Average Communication Sphere Toner, Bacon 03 1 bit in Worst Case Non-Local Box ( this talk ) ( this talk ) 1 use in Worst Case Sphere but no communication 6

Non-Local Box C=4 ● Maximally non-local : maximally violates CHSH inequality C=4 ● Causal +1 +1 +1 –1 a 1 ,  b 0 ,  a 0 ,  a 0 ,  a 1 ,  a 1 ,  C   a 0 ,  b 1 = E  AB ∣ b 0  E  AB ∣ b 1  E  AB ∣ b 0 − E  AB ∣ b 1  y=0 x=1 y=1 x=0 a and b are anticorrelated when x = 1 and y = 1, x y x , y , a ,b ∈ { 0,1 } otherwise they are correlated p  a = 0 ∣ x , y = p  a = 0 ∣ x = 1 x ∧ y = a ⊕ b 2 p  b = 0 ∣ x , y = p  b = 0 ∣ y = 1 2 A =1-2a B =1-2b 7 { 0,1 }{ 1, − 1 }

Is it a sufficient resource to simulate any VN measurement on an EPR state? It is sufficiently nonlocal (more than QM !) It is causal (just like QM !) : does not “spoil” resources It admits binary inputs, while there are infinitely many possible VN measurements HOW DOES IT WORK ? Next slide HOW DOES IT WORK ? Next slide WHY DOES IT WORK ? Next talk WHY DOES IT WORK ? Next talk 8

 1 ,    2 a   b Bob Alice y = sgn     sgn  b ⋅  b ⋅  a ⋅ a ⋅  −  x = sgn   1  sgn   2  sgn  t = 0 t  0  ± =    1 ± with  2 with = 1 t ≤ 0 x ∧ y = a ⊕ b a ,   1 ,  a ⋅ B   2 =− 1  2 [ b  sgn  b ,   1 ,  b ⋅ A   2 = 1 − 2 [ a  sgn   1 ]   ] a ⋅ RESULT: E  A B =− RESULT: b 9

Monogamy : Non-Local Box cannot be shared 0 1 x y z x ∧ y = a ⊕ b x ∧ z = a ⊕ c a b c x ∧ y ⊕ z = a ⊕ b ⊕ a ⊕ c = b ⊕ c → Non causal ! y = 0 ∧ z = 1 b ⊕ c = x Non causal ! ● Exploit monogamy to do QKD (talk by N. Gisin, A. Acin, L. Masanes) 10 ● Characterize monogamy in general (talk by B. Toner)

Conclusion & Perspectives ● Extend to non-maximally entangled states 1 use of Non-Local Box is not sufficient N. Brunner, N. Gisin, V. Scarani, 05 Non-maximally entangled state is “more non-local” Non-maximally entangled state is “more non-local” ● Extend to POVM measurements (related) ● Extend to multipartite states and/or higher dimensions Non-Local Box appears to be useful conceptual tool Non-Local Box (non-locality characterization, secret key distribution, communication complexity, bit commitment,...) 11