Simulating quantum correlations as a sampling problem Julien Degorre | L.R.I. Université Paris Sud > + | L.I.T.Q. Université Montréal > (joint work with : Sophie Laplante* and Jérémie Roland*˚) * L.R.I. Université Paris Sud ˚Université Libre de Bruxelles Physical Review A, 72:062314, 2005
The problem of simulating quantum correlations L.H.V. model L.H.V. model + + Non Local Box Communication (worst case) 1 nlbit 1bit L.H.V. model [CGMP05] + [TB03] Post-Selection (detection loophole) Efficiency 2/3 [GG99] L.H.V. model Distributed + Sampling Problem Communication (Average) [DLR05] [S99], [CGM00] 1/10
Local Hidden Variable Model λ Doesn't depend on the inputs! Input: Input: a b Alice Bob B A a , ∈ {− 1, 1 } b , ∈ {− 1, 1 } Output: Output: BelI's theorem: impossible to reproduce quantum correlations 2 /10
Biased Hidden Variable Model Infinite biased Shared Randomness s ~ ∣ a ⋅ s ∣ s ∈ S 2 2 a s b s Alice Bob B s = sign b , b ⋅ s a , a ⋅ A s =− sign s a ⋅ P A, B = 1 − AB b 4 3/ 10
a ⋅ s ~ ∣ s ∣ Step 1: Local Sampling of the biased distribution: 2 0 , 1 , 2 , ... , k , ... ~ U S 2 Shared randomness independent on the inputs : The rejection method: Set k=0 k ~ U S 2 1. Alice picks u k ~ U [ 0,1 ] 2. Alice picks u k ≤ ∣ a ⋅ k ∣ 3. Test whether s = k k If test succeeds, Alice ACCEPTS and sets k Go back to 1 with k=k+1 Otherwise, Alice REJECTS s ~ ∣ a ⋅ s ∣ When the process terminates, Alice has 2 4 /10
Step 2: Distributed Sampling problem With communication 0 , 1 , ... k ~ U S 2 b a Bob Alice Rejection method : Index k Set k= 0 k ~ U S 2 s = 1. Alice picks k Communication 2 bits on average u k ~ U [ 0,1 ] 2. Alice picks [Steiner99 + Gisin² 00] u k ≤ ∣ k ∣ a ⋅ 3. Test If s = k s ~ ∣ a ⋅ s ∣ s s Go to 1. k=k+1 2 5 /10
Step 2: Distributed Sampling problem With post selection 0 , 1 , ... k ~ U S 2 b a Bob Alice Rejection method : 0 ~ U S 2 s = 1. Alice picks Post selection 0 Communication [Gisin Gisin 00] u 0 ~ U [ 0,1 ] 2. Alice picks u 0 ≤ ∣ a ⋅ 0 ∣ 3. Test If s = 0 s ~ ∣ a ⋅ s ∣ s s Abort 2 P(output)= 1/2 6 /10
But we can be more clever... 0 , 1 , ... Used only by Alice ! k ~ U S 2 b a Bob Alice Rejection method : 0 ~ U S 2 s = 1. Alice picks Post selection 0 Communication [Gisin Gisin 00] u 0 ~ U [ 0,1 ] 2. Alice picks u 0 ≤ ∣ a ⋅ 0 ∣ 3. Test If s = 0 s ~ ∣ a ⋅ s ∣ s s Abort 2 P(output)= 1/2 6 /10
Local Sampling of the biased distribution: a new method Recall the rejection method: 0 ~ U S 2 1. Alice picks u 0 ~ U [ 0,1 ] 2. Alice picks u 0 ≤ ∣ a ⋅ 0 ∣ 3. Test whether s = 0 0 If Test OK Alice ACCEPTS and sets 0 Otherwise Alice REJECTS go back to 1 with another s ~ ∣ a ⋅ s ∣ / 2 So, Alice has 7 /10
Local Sampling of the biased distribution: a new method Recall the rejection method: ∣ a ⋅ ∣ ~ U [ 0,1 ] 0 ~ U S 2 1. Alice picks when ~ U S 2 u 0 = ∣ a ⋅ 1 ∣ ~ U [ 0,1 ] u 0 ~ U [ 0,1 ] 2. Alice picks u 0 = ∣ a ⋅ 1 ∣ ≤ ∣ a ⋅ 0 ∣ 3. Test whether s = 0 0 If Test OK Alice ACCEPTS and sets 0 Otherwise Alice REJECTS go back to 1 with another s ~ ∣ a ⋅ s ∣ / 2 So, Alice has 7 /10
Local Sampling of the biased distribution: a new method The Choice method: ∣ a ⋅ ∣ ~ U [ 0,1 ] 0 ~ U S 2 1. Alice picks when ~ U S 2 u 0 = ∣ a ⋅ 1 ∣ ~ U [ 0,1 ] u 0 ~ U [ 0,1 ] 2. Alice picks u 0 = ∣ a ⋅ 1 ∣ ≤ ∣ a ⋅ 0 ∣ 3. Test whether s = 0 0 If Test OK Alice ACCEPTS and sets s = 1 1 Otherwise Alice ACCEPTS and sets s ~ ∣ a ⋅ s ∣ / 2 So, Alice has 7 /10
Step 2: Distributed Sampling problem With communication 0 , 1 ~ U S 2 b a Bob Alice Choice method : x = 0 or 1 0 ~ U S 2 Bob sets : 1. Alice picks Communication 1 ~ U S 2 s = 2. Alice picks x 1 bit Worst Case [TB03] 3. Tests whether ∣ a ⋅ 1 ∣ ≤ ∣ a ⋅ 0 ∣ s = 0 If yes s = 1 s ~ ∣ a ⋅ s ∣ If no s s With 2 8 /10
Without resource 0 , 1 ~ U S 2 a b Choice method : Bob always s = 0 , sets: 0 1 Bob Alice picks Alice Tests whether ∣ a ⋅ 1 ∣ ≤ ∣ a ⋅ 0 ∣ s = 0 If yes x=0 s = 1 If no x=1 Simulation of non separable Werner State. − ∣ 1 − p 1 − × W = p ∣ With p=1/2 4 Alice's output : Bob's output : a , a ⋅ B sign b , b ⋅ A s = − sign s s = s 9 /10
With a non local box 0 , 1 ~ U S 2 a b Choice method : Bob always s = 0 , sets: 0 1 Bob Alice picks Alice Tests whether Bob Tests whether ∣ a ⋅ 1 ∣ ≤ ∣ a ⋅ 0 ∣ sign 0 = sign b ⋅ b ⋅ 1 s = 0 If yes If yes: cool ! y=0 x=0 s = 1 If no: Aïe ! y=1 If no x=1 y x x ∧ y = a ⊕ b Non-Local Box: a b Alice's output : Bob's output : −− 1 a − 1 b a , a , a ⋅ a ⋅ B sign b , b ⋅ A A s = s = − sign sign s s s = s 9 /10
Conclusion ● The distributed sampling problem give us a unified framework for the problem of simulating quantum correlations. ● Related results: POVMs : Post Selection( Efficiency 1/3), Communication and Non-local Boxes (2 nl-Bits + 4 bits on average) Some preliminary results for higher dimensions. ● Open problems: Multipartite states and non maximally entangled states. 10 /10
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