Experimentally Generated Random Numbers Certified by the Impossibility of Superluminal Signaling

Transcription

1 Experimentally Generated Random Numbers Certified by the Impossibility of Superluminal Signaling Peter Bierhorst, Lynden K. Shalm, Alan Mink, Stephen Jordan, Yi-Kai Liu, Scott Glancy, Bradley Christensen, Andrea Rommal, Sae Woo Nam, Emanuel Knill National Institute of Standards and Technology, Boulder, CO September 16, / 33

2 First Loophole-Free Bell Experiments Multiple experimental groups have recently reported loophole-free Bell experiments: Hensen et al., Nature , October 2015 Shalm et al., Phys. Rev. Lett , December 2015 Giustina et al., Phys. Rev. Lett , December 2015 Weinfurter et al., QCrypt conference presentation, September / 33

3 First Loophole-Free Bell Experiments Local realism is falsified. Now what? Quantum Information Theory! 3 / 33

4 First Loophole-Free Bell Experiments Local realism is falsified. Now what? Quantum Information Theory! 4 / 33

5 First Loophole-Free Bell Experiments Local realism is falsified. Now what? Quantum Information Theory! 5 / 33

6 Randomness in Bell Experiments Nonlocality + Prohibition of FTL signaling Randomness 6 / 33

7 Experimental Scheme Alice Bob Alice measures a, Bob measures b 7 / 33

8 Experimental Scheme Alice Bob Alice measures a, Bob measures b 8 / 33

9 Experimental Scheme Alice Bob Alice measures a, Bob measures b 9 / 33

10 Experimental Scheme Alice Bob Alice measures a, Bob measures b. 10 / 33

11 Experimental Distributions ab ab a b a b P(+0 ab ) = / 33

12 Experimental Distributions A Local Realist Distribution ab 1/ /2 ab 1/ /2 a b 1/ /2 a b 1/ /2.5 ab ab a b a b ab ab a b a b / 33

13 Experimental Distributions A Local Realist Distribution ab 1/ /2 ab 1/ /2 a b 1/ /2 a b 1/ /2.5 ab ab a b a b ab ab a b a b / 33

14 Experimental Distributions The PR box, a nonlocal distribution: ab 1/ /2 ab 1/ /2 a b 1/ /2 a b 0 1/2 1/ / 33

15 Experimental Distributions A possible decomposition? ab ab a b a b and ab ab a b a b When Bob chooses b, Alice can signal Bob. 15 / 33

16 Experimental Distributions A possible decomposition? ab ab a b a b and ab ab a b a b When Bob chooses b, Alice can signal Bob. 16 / 33

17 Experimental Distributions If one cannot send signals between Alice and Bob, then the randomness in the PR box cannot be eliminated. ab 1/ /2 ab 1/ /2 a b 1/ /2 a b 0 1/2 1/ / 33

18 Data from a photonic loophole-free experiment Raw Counts from a data run: ab ab a b a b Induced empirical distribution (No-signaling adjusted): ab ab a b a b / 33

19 Data from a photonic loophole-free experiment Raw Counts from a data run: ab ab a b a b Induced empirical distribution (No-signaling adjusted): ab ab a b a b / 33

20 Data from a photonic loophole-free experiment Induced empirical distribution (No-signaling adjusted): ab ab a b a b Can be induced by: Local Realist Distribution PR box 20 / 33

21 Data from a photonic loophole-free experiment PR Box Over the course of 132,161,215 trials, this means 3,700 PR boxes. 21 / 33

22 Extracting the Randomness To certify the min-entropy and extract the randomness, we need a protocol that is: Effective in a very low-violation regime (CHSH ) Robust to possible memory effects Handles finite statistics effects (no asymptotic bounds) 22 / 33

23 Extracting the Randomness Existing methods are not effective in our regime Pironio et al., Pironio/Massar, Fehr/Gelles/Schaffner Vidick/Vazirani Chung/Shi/Wu, Miller/Shi Coudron/Yuen Bancal/Sheridan/Scarani, Nieto-Silleras/Pironio/Silman Thinh et al. 23 / 33

24 Extracting the Randomness The following papers develop statistical analysis techniques that are effective for falsifying local realism in low-violation photonic experiments, and suggest a way forward for certifying min-entropy. A robust mathematical model for a loophole-free ClauserHorne experiment, P. Bierhorst, J. Phys. A 2015 Requirements for a loophole-free photonic Bell test using imperfect setting generators, J. Kofler, M. Giustina, J.-A. Larsson, M. W. Mitchell, Phys. Rev. A 2016 Asymptotically optimal data analysis for rejecting local realism, Y. Zhang, S. Glancy, E. Knill, Phys. Rev. A / 33

25 Extracting the Randomness The following papers develop statistical analysis techniques that are effective for falsifying local realism in low-violation photonic experiments, and suggest a way forward for certifying min-entropy. A robust mathematical model for a loophole-free ClauserHorne experiment, P. Bierhorst, J. Phys. A 2015 Requirements for a loophole-free photonic Bell test using imperfect setting generators, J. Kofler, M. Giustina, J.-A. Larsson, M. W. Mitchell, Phys. Rev. A 2016 Asymptotically optimal data analysis for rejecting local realism, Y. Zhang, S. Glancy, E. Knill, Phys. Rev. A / 33

26 The Protocol 1 Choose a Bell function T from experimental outcomes to R +, satisfying T > 0, E LR (T ) 1, E PR (T ) = 1 + m, m > 0 2 In an experiment of n trials, compute n i=1 T i = V. 3 For values of V >> 1, our result puts a lower bound on the amount of min-entropy σ s present in the data 4 Extract randomness to ɛ-uniform bits using Trevisan extractor 26 / 33

27 The Protocol 1 Choose a Bell function T from experimental outcomes to R +, satisfying T > 0, E LR (T ) 1, E PR (T ) = 1 + m, m > 0 2 In an experiment of n trials, compute n i=1 T i = V. 3 For values of V >> 1, our result puts a lower bound on the amount of min-entropy σ s present in the data 4 Extract randomness to ɛ-uniform bits using Trevisan extractor ab ab a b a b m = Generate T with method of Zhang, Glancy, Knill, Phys. Rev. A / 33

28 The Protocol 1 Choose a Bell function T from experimental outcomes to R +, satisfying T > 0, E LR (T ) 1, E PR (T ) = 1 + m, m > 0 2 In an experiment of n trials, compute n i=1 T i = V. 3 For values of V >> 1, our result puts a lower bound on the amount of min-entropy σ s present in the data 4 Extract randomness to ɛ-uniform bits using Trevisan extractor n = 132, 161, 215 V = / 33

29 The Protocol 1 Choose a Bell function T from experimental outcomes to R +, satisfying T > 0, E LR (T ) 1, E PR (T ) = 1 + m, m > 0 2 In an experiment of n trials, compute n i=1 T i = V. 3 For values of V >> 1, our result puts a lower bound on the amount of min-entropy σ s present in the data 4 Extract randomness to ɛ-uniform bits using Trevisan extractor Key result: [ ] 2m + 1 n V ɛ s σ s n log 2 2m When the protocol is passed, σ s is the average min entropy of the output string, conditioned on the setting string. 29 / 33

30 The Protocol 1 Choose a Bell function T from experimental outcomes to R +, satisfying T > 0, E LR (T ) 1, E PR (T ) = 1 + m, m > 0 2 In an experiment of n trials, compute n i=1 T i = V. 3 For values of V >> 1, our result puts a lower bound on the amount of min-entropy σ s present in the data 4 Extract randomness to ɛ-uniform bits using Trevisan extractor We use updated software we wrote based on the Mauerer, Portmann, Scholz (arxiv:1212:.0520) implementation of Trevisan s extractor. We can extract up to t ɛ-uniform bits where t obeys the following formula: t + 4 log 2 t σ s log 2 ɛ ext 30 / 33

31 Extractable Bits Results 1400 Error vs. Extractable Bits Log 10 Error Probability 31 / 33

32 Extractable Bits Results 2500 Blind 1 Blind 2 Error vs. Extractable Bits: New Data Log 10 Error Probability 32 / 33

33 Conclusion 256 Extracted Bits: Thank you! Peter Bierhorst National Institute of Standards and Technology, Boulder, CO 33 / 33