Robust device independent randomness amplification with few devices

1. Robustdevice independent randomnessamplification with few devices 1Department of Computer Science, University College London 2National Quantum Information Centre in Gdańsku, Sopot 3Department of Physics, Adam Mickiewicz University, Poznań 4Institute of Informatics, University of Gdańsk, Gdańsk 5Institute of TheoreticalPhysics and Astrophysic, University of Gdańsk, Gdańsk 6Department of Physics and Applied Math, Technical University of Gdańsk, Gdańsk F.G.S.L Brandao1, R. Ramanathan2 A. Grudka3, K. 4, M. 5,P. 6 Horodeccy arXiv:1310.4544 QIP 2014 Barcelona

2. Motivation or ? Doesanyoneknow the result of throwing a coin ? can one decorrelaterandomness from the third person: eavesdropper ? Example:makeyour choice of product to buy independent of yourknowledge aboutcommercials

3. Device independence • Danger: Evecanselldevice with imprintedbits in advance • We do not trust that devices arebulitdue to specifiation • We onlyrelay on statistics of inputs and outputs of the device • Security proof usesonlythesestatistics

4. Sources of randomness: Santha-Vazirani and type Definition: A sequence X1 ….Xnsatisfiescondition of Santha-Vaziraniif for any i thereis: e – anyvariablethatcouldhaveinfluenced the variable Xi Fact:classics =>„no go” [M. Santha, U.V. Vazirani 1984] Classicalprocessing of SV sourcedoes not lead to randomnessamplfication Weakersource of randomness: A sequence X1 …. Xnislinearsource if (X1 …. Xn) > cn for some c >0 Fact: Non explicitextractor From 2 independent min-entropysources => fullyrandom bit From 3 independent min-entropysources => fullyrandom bit Explicitextractor [Chor and Goldreich ’88, Rao]

5. Quantum mechanicsallowsfor randomnessamplification [R. Colbeck & R. Renner, Nature Physics 2012] Somemeasurements on maximallyenatangledstatesarerandom Idea:results of thesemeasurementsviolatecertain Bell inequality N-th chained Bell inequality Result : For optimal [A.Grudka et al. arXiv:1303.5591] 0.0961 [Gallego et al. arXiv:1210.6514 ] 0 For any Use of 5-partite Mermininequality Thereexistshashfunctionwhichoutputsperfect bit Drawbacks: - hashfunctionis not explicit - asymptoticallymany non-signaling devices - tolerance of noise not included [R. Ramanathan et al. arXiv:1308.4635]

6. The results 0 Starting from any epsilon-Santha-Vaziranisource: obtainbits of randomnesssecure with respect to non-signalingEve 1) Protocol (I) of randomnessamplification with: - single device, but non-explicitfunction - tolerance of noise Maintool: properuse of implicitassumptions 2) Protocol (II) of randomnessamplification with: • 2 devices • explicithashfunction • tolerance of noise Maintool: SV-version of deFinettitheorem

7. The scheme of randomnessamplifier SV source εinsource SV 101011000101110101 Eve Alice Bob P(XZ|U,W) Device: A device Extractor Extractor εout S – randomness of alfabet size Task: 0 εout <εin Quality of randomness: small

8. The protocol I Single Device The protocol : 1) Use Device 1 ntimestaking as inputsbits from SV source 2) Check the level of Bell violationafter n runs 4) Upon goodlevel of Bell violation in 2), applyExtractor to device and SV source

9. Assumptions (I) Assumption1: (fixeddevice) the devicedoes not depend on SV source Assumption 1’: (Markovity) : givenoutput of Eve, the device and SV sourceare product: Cf. [Colbeck and Renner 2012] [Gallego et al. 2012] Note:theseassumptionsare independent

10. Assumptions (II) Assumption2: (conditional non-signaling) Conditionally on theseresults… …theseblocks do not signal Note:itisreasonable, as quantum devices satisfiesit

11. Idea of the proof for protocol I By assumption I: SV sourceservesitself as source independent of the output of the devices Note: we do not imposeindependence of the sources, as thatwould be trivial u time SV source 10100101010111000111010101010 Reason for independence: u decorrelatestwosources y Devices: P(x|u) P(x|y,u) = P(x|u) x Two independent sources => non-explicit extractoryieldssecure bit Note: we have to verifyifdeviceviolates Bell inequality.Ifitdoes not, thereis no way to checkif the deviceis not deterministicfunction to which a SV no-go applies.

12. Protocol II Device 2 Device 1 Block 1 Block 2 • Preliminary assumptions: • Devices: • Do not signalbetween • eachother • Areforwardsignaling • (past can influence the future) The protocol : 1) Use Device 1 n timestaking as inputsbits from SV source forming theblock 1 2) Out of n x N2uses of Device 2 choose the block 2 of n uses, by means of SV source 3) Check the level of Bell violation in bothblocks 4) Upon goodlevel of Bell violation in 3), applyExtractor to thesetwoblocks and SV source Security claim: protocol I, upon acceptanceprovidessecure bit up to error thatvanises with uses of devices with high probability

13. Idea of the proof for protocol II 1) By assumption I: SV sourceservesitself as source independent of the output of the devices 2) By assumption II + newtype of deFinettitheorem: The twoblocks of uses of devices (#1 and #2) areproduct with eachother time SV source 10101010111000110101010111000000110 Block 2 Block 1 3 independent sources: good for 3-extractor [Chore,Goldreich ‚88] Securebits (SECRECY AGAINST NO-SIGNALING ADVERSARY )

14. Ingredients of the proof (i) a Bell inequality Q NS P* Entangled 4-qubit in state |, u=0 - measure z , u=1 - measurex x u1, x2 A2 LHV u1 u2 u3 u4 u2,x2 Violatedup to NS value B.P=0 A1 {P(x|u)} A3 u4,x4 x1 x2 x3 x4 A4 u3,x3 Lemma.- SupposeB.{P(x|u)}=  >0then for anymeasurementsettingu and outputx, thereis: Pr(x|u) [1 + ]/3. Interpretation:output of a box ispartiallyrandom, evenwhennoiseisallowed Proof: by Linear Programming (analyticalsolution) [HanggiRenner EUROCRYPT 2010]

15. Ingredients of the proof (ii) Azuma-Hoeffdinginequality for estimation Adaptation of [Pironio et al. 2010,2013Fahr et al., PironioMassar, 2013] – for randomness expansion Azuma-Hoeffdinginequalityenablesestimation With high probability: Thereareatleastg x n of goodboxes (with Bell valueat most δ) with g > 0 Hence for anyu,x: …. Linear in n

16. Ingredients of the proof (iii) de Finettitheorem [BrandaoHarrow STOC’13] + SV correction

17. Ingredients of the proof (iii) de Finettitheorem Device 2 Device 1 Usesprior to Block 2 Block 1 Recall: number of Block 2 ischosen according to bits from SV source Block 2 Averageover the choice of from SV source, and output , Outputsareclose to product

18. Technical part of de Finettibound

19. Putting the piecestogether: We choose a Bell inequality, whichisalgebraicallyviolated on quantum state The protocol : 1) Usedevice #1, N times 2) Out of NxKuses of device #2 choose a block of N uses, by means of SV source 3) Check the level of Bell violation in both the blocks 4) Upon goodlevel of Bell violation in 3), applyExtractor to thesetwoblocks By DeFinetti Block 1 and Block 2 arealmostproduct By Azuma-Hoeffding: enoughgood boxes for linearHmin By assumptions, and the above, thereare 3 Hminsourcesclose to product: The rest of SV source and twoblocks => Extractorswork!

20. Conclusions and Open problems Full randomnessamplification w.r.t. to non-signalingadversaryusing small number of devices ( 1 or 2) ispossible. Noisetolarancedependence show. • Drawback 1 of ourprotocol: to makedeFinettiwork we need to make t large: Largenumber of uses… • Can we obtain a protocol with non-zero rate of amplifiedrandomness and explicit • extractor ? Drawback 2: for single devicenonzerorate but no explicitextractor, for two devices explicitextractor but zero ratedue to error |exp(-cn • Can one achieverandomnessamplification for any epsilon, from bipartite devices ? • (in preparation) • Tolerance of noise – isthere a protocolwhichismorerobustagainstnoise ? • Could we start not from SV-source, but from the one ? Finally : proof applies for 2 devices and 3 devices respectively if we don’tuse the SV apart from setting the inputs.

21. Thankyou for yourattention !