Experimental generation and characterisation of private states

1. Experimentalgenerationand characterisation of privatestates KonradBanaszekRafał Demkowicz-DobrzańskiMichał Karpiński Wydział Fizyki Uniwersytet Warszawski Krzysztof Dobek Wydział Fizyki, Uniwersytet Adama Mickiewicza w Poznaniu Paweł Horodecki Wydział Fizyki Technicznej i Matematyki Stosowanej, Politechnika Gdańska Karol Horodecki Instytut Informatyki, Uniwersytet Gdański TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAA

2. - - Bell’sinequalities + + B:b = 22.5o A:a = 45o B’:b = 67.5o A’:a = 0o isviolated! Clauser-Horne-Shimony-Holtinequality

3. A. K. Ekert, Phys. Rev. Lett. 67, 661 (1991) For eachphotonpair Alice and Bob selectrandomlymeasurementbases… Quantum cryptography B1:b = 22.5o A1:a = 45o A2:a = 0o B2:b = 67.5o B3:b = 0o …and comparemeasurementsover a public channel afterwards. Perfectcorrelations one-time pad Security test

4. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev. A 60, R773 (1999) Entangledphotonpairs Output state:

5. Correlationmeasurements

6. Entanglementmonogamy Alice Eve Bob • Evenwhen the pairhasbeenprepared by Eve… • …if Alice and Bob verifythat the systems arrived in a maximallyentangledpure state… • …measurementresults will be knownonly to Alice and Bob.

7. Define: Equallyweightedmixture: Statisticalmixture Eve Alice Bob

8. Statisticalmixture Maximallyentangled state Densitymatrix Correlationsbetweenmeasurementoutcomes in the keybasis Security tested by the violation of Bell’sinequalities (Iftrusting quantum theory, could be alsotested by measurements in the basis.)

9. Noisyentanglement How much securekeycan be extractedfrom a noisy state?

10. C. H. Bennett et al., Phys. Rev. Lett. 76, 722 (1996) Distillation M … … N Distillableentanglement:

11. A B Example I B’ A’ Shieldstates: enable Alice and Bob to distinguishlocally and generate the keyusing the standard strategy. Hence

12. A B B’ Example II A’ Whatif • States and cannot be discriminatedunambiguouslyusinglocal operations and classicalcommunication. • Distillableentanglementbounded by log-negativity:

13. K. Horodecki, M. Horodecki, P. Horodecki, and J. Oppenheim,Phys. Rev. Lett. 94, 160502 (2005) Eavesdropping B B’ AA’ E Theworstcasescenario: allthenoiseiscontrolled by Eve

14. Alice measuresanoutcomeawith a probability Alice  Eve channel Eveinfersa from the conditionalstate of hersubsystemE: Holevoquantity:

15. Mutual information B B’ AA’ Keyrate Holevoquantity E Keyrate For Example II, Eve’ssubsystemcontains no informationaboutoutcomes of Alice’smeasurement on herqubit, henceKD = 1.

16. Withouttheshield: Shield key security Thecomplete state: AB   00     01     10   11

17. Doublephotonpairs B’ A B Output state: A’

18. Experimental setup

19. Projectivequbitmeasurements: Four-qubit POVM: Quantum statetomography 34 = 81 measurementbases 34 x 24 = 1296 eventtypes Probability of an outcomei: Densitymatrixestimate : number of eventsi

20. Z. Hradil, Phys. Rev. A 55, R1561 (1997) Probability of an outcomei: Maximumlikelihoodreconstruction – number of eventsi Likelihoodfunction: Maximum-likelihoodestimatemaximizes

21. K. Banaszek, G. M. D’Ariano, M. G. A. Paris, and M. F. Sacchi,Phys. Rev. A61, 010304(R) (1999) Ensuringpositivity: ML: Parametrisation Task: maximize with a constraint PRO: - Guaranteedpositivity CON: • Impracticalinhigherdimensions (>6 qubits) • Underestimateserrors, difficult to includeuncertainty of the measuringdevice (Monte Carlo simulations) • Biasedtowardslow-rankmatrices for undersampled data

22. K. Audenaertand S. Scheel, New J. Phys. 11, 023028 (2009) A priori distribution Bayesianapproach A posteriori: Estimate: • Gaussianapproximation • Truncated to positivedefinitedensityoperators PRO: • Clear statisticalinterpretation • Providesuncertainty • No numericaloptimisation CON: • Difficult to normaliseprobabilitydistribution • A priori distribution not welldefined

23. K. Dobek. M. Karpiński, R. Demkowicz-Dobrzański, K. Banaszek,and P. Horodecki, Phys. Rev. Lett. 106, 030501 (2011) State reconstruction Mahalanobisdistance = 16.8 95% confidenceinterval = 17.1

24. A posteriori ensemble Privacycharacterisation Warning: More conservative Distillableentanglement: ED 0.581(4) Key (cqqscenario): K  0.690(7)

25. Distillationprotocol MeasurequbitsA′B′ inthe same basis. 50% 50% Identicaloutcomes Oppositeoutcomes

26. Reduceddensitymatrix conditional average Single-copydistillation identicaloutcomesA′B′ oppositeoutcomesA′B′ K = 0.354(5) K = 0

27. Distillation-basedapproach Optimalstrategy Cryptographickey Raw key: 3716 bits Raw key: 1859 bits Errorcorrection 2726 bits  1300 bits Privacyamplification Securekey: 2164 bits Securekey:  650 bits

28. Complete densitymatrix for key and shieldsubsystems: Witnessingprivacy where

29. K. Horodecki et al., IEEE Trans. Inf. Theory54, 2621 (2008); ibid. 55, 1898 (2009). In general: Keybound where

30. K. Banaszek, K. Horodecki, and P. Horodecki,Phys. Rev. A 85, 012330 (2012) Suppose we havemeasured Single witness where Thisprovidesanestimate

31. K. Banaszek, K. Horodecki, and P. Horodecki,Phys. Rev. A 85, 012330 (2012) Positivity of impliesthat Single witness Take the worst-casescenario for K w

32. K. Banaszek, K. Horodecki, and P. Horodecki,Phys. Rev. A 85, 012330 (2012) Twoobservables We have:

33. Experimentaldemonstrationof the separationbetweendistillableentanglement and cryptographickeycontents • Practicalcomparison of quantum state reconstructionmethods for a noisymultiqubit state • Fullprivacyanalysisbased on thereconstructed state • Evaluation of highlynon-linearinformationtheoreticquantities • Implementation of a simpleentanglementdistillationprotocol • Witnessingprivacy with fewobservables • Multipledegrees of freedom? Conclusions