GAP Optique Geneva University 1 Quantum Communication Nicolas Gisin  Quantum cryptography: BB84 and uncertainty relations Ekert and entanglement no cloning theorem BB84  Ekert Implementations Eve: optimal individual attack Error correction, privacy amplification, advantage distillation  Quantum Teleportation principle, connection to optimal state estimation and cloning experiments quantum relays and quantum repeaters  Optimal and generalized quantum measurements optimal quantum cloning POVMs (tetrahedron, unambiguous state discrimination) weak measurements Q crypto: RMP 74, , 2002 Q cloning: RMP 77, , 2005

GAP Optique Geneva University 2 Quantum cryptography: a beautiful idea Basic Quantum Mechanics: A quantum measurement perturbs the system QM  limitations However, QM gave us the laser, micro- electronics, superconductivity, etc. New Idea: Let's exploit QM for secure communications

GAP Optique Geneva University 3  If Eve tries to eavesdrop a "quantum communication channel", she has to perform some measurements on individual quanta (single photon pulses).  But, quantum mechanics tells us: every measurement perturbs the quantum system.  Hence the "reading" of the "quantum signal" by a third party reduces the correlation between Alice's and Bob's data.  Alice and Bob can thus detect any undesired third party by comparing (on a public channel) part of their "quantum signal".

GAP Optique Geneva University 4  The "quantum communication channel" is not used to transmit a message (information), only a "key" is transmitted (no information).  If it turns out that the key is corrupted, they simply disregard this key (no information is lost).  If the key passes successfully the control, Alice and Bob can use it safely.  Confidentiality of the key is checked before the message is send.  The safety of Quantum Cryptography is based on the root of Quantum Physics.

GAP Optique Geneva University 5 Modern Cryptology Secrecy is based on: Information theory The key is secrete The key contains the decoding key: Only the two partners have a copy ! The security is proven (Shannon theorem) Example: Message: Key: Coded message: Complexity theory The key is public The public key contains the decoding key, but it is very difficult to find (one way functions) The security is not proven (no one knows whether one way functions exist) Example: 127 x 229 = 29083

GAP Optique Geneva University 6 BB84 protocol: Eve  25% errors

GAP Optique Geneva University 7 Security from Heisenberg uncertainty relations AliceBob Eve P(X, Y, Z) Theorem 1: (I. Csiszàr and J. Körner 1978, U. Maurer 1993) If I(A:B) > min{I(A:E),I(B:E)}, then Alice & Bob can distil a secret key using 1-way communication over an error free authenticated public channel. where I(A:B)= Shannon mutual information = H(A)-H(A|B) = # bits one can save when writing A knowing B

GAP Optique Geneva University 8 Finite-coherent attacks Theorem 2 (Hall, PRL 74,3307,1995) Heisenberg uncertainty relation in Shannon-information terms: I(A:B) + I(A:E) < 2.log(d.c) where c=maximum overlap of eigenvectors and d is the dimension of the Hilbert space. For BB84 with n qubits, d=2 n and c=2^(-n/2). Hence, Theorem 2 reads: I(A:B) + I(A:E) < n It follows from Csiszàr and Körner theorem that the security is guaranteed whenever I(A:B) < 1/2 (per qubit) This corresponds exactly to the bound of the Mayers et al. proofs, i.e. QBER<11% Note: same reasoning valid for 6-state protocols, and for higher dimensions (M. Bourennane et al.).

GAP Optique Geneva University 9 I AE  1-I AB Eve: optimal individual attack

GAP Optique Geneva University 10 Quantum Communication  Quantum Communication is the art of transferring a Q state from one place to another. Example: Q cryptography Q teleportation  Quantum Information is the art of turning a Q paradox into a potentially useful task. Example: Q communication: from no-cloning to Q crypto Q computing: from superpositions to Q parallelism  Note that entanglement and Q nonlocality are always present, at least implicitely. Though their exact power is not yet fully understood

GAP Optique Geneva University 11 Ekert protocol (E91) source aa a=x,z 0,1 bb b=x,z 0,1 Theorem: let If  AB is pure, then

GAP Optique Geneva University 12 Quantum cryptography on noisy channels No cloning theorem: 

GAP Optique Geneva University 13 No cloning theorem and the compatibility with relativity No cloning theorem: It is impossible to copy an unknown quantum state,  / Proof #1: Proof #2: (by contradiction) * Alice Bob } clones M Source of entangled particules Arbitrary fast signaling !

GAP Optique Geneva University 14 Optimal Universal non-signaling Quantum Cloning symmetric and universal universal no.signaling achievable by the Hillery-Buzek UQCM N.Gisin, Phys. Lett.A 242, 1-3, 1998

GAP Optique Geneva University 15 BB84  E91 source aa a=x,z 0,1 bb b=x,z 0,1 source Alice Indistinguishable from a single photon source. The qubit is coded in the a-basis And holds the bit value given by Alice results.

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GAP Optique Geneva University 17 Experimental Realization  Single photon source  Polarization or phase control during the single photon propagation  Single photon detection avalanche photodiode (Germanium or InGaAs) in Geiger mode  dark counts based on supraconductors  requires cryostats laser pulses strongly attenuated (  0.1 photon/pulse) photon pair source (parametric downconversion) true single-photon source parallel transport of the polarization state (Berry topological phase)  no vibrations fluctuations of the birefringence  thermal and mechanical stability depolarization  polarization mode dispersion smaller than the source coherence Stability of the interferometers coding for the phase

GAP Optique Geneva University 18 Telecommunication wavelengths  Attenuation (  transparency)  Chromatic dispersion  Components available [  m]  [dB/km]T 10km 0.821% % %  Two windows

GAP Optique Geneva University 19 Single Photon Generation (1) Attenuated Laser Pulse Simple, handy, uses reliable technology  today’s best solution Poissonian Distribution 0% 20% 40% 60% 80% 100% Number of photons per pulse Probability Mean = 1 Mean = 0.1 Attenuating Medium 1nrather tha or..." 2or 1 0"

GAP Optique Geneva University 20 Avalanche photodiodes  Single-photon detection with avalanches in Geiger mode  macroscopic avalanche triggered by single-photon Silicon:1000 nm Germanium:1450 nm InGaAs/InP:1600 nm

GAP Optique Geneva University 21 Noise sources  Charge tunneling across the junction  not significant  Band to band thermal excitation  reduce temperature  Afterpulses  release of charges trapped during a previous avalanche  increase temperature Optimization !!!

GAP Optique Geneva University 22 Efficiency and Dark Counts

GAP Optique Geneva University 23 experimental Q communication for theorists tomorrow: Bell inequalities and nonlocal boxes

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GAP Optique Geneva University 25 Polarization effects in optical fibers:  Polarization encoding is a bad choice !

GAP Optique Geneva University 26 Phase Coding  Single-photon interference Basis 1:  A = 0;  Basis 2:  A =  Basis:  B = 0;  Compatible:Alice  A  D i BobD i  A (  A -  B = n  ) Incompatible:Alice and Bob ?? (  A -  B =  ) Bases

GAP Optique Geneva University 27 Difficulties with Phase Coding  Stability of a 20 km long interferometer? Problems: stabilization of the path difference  active feedback control stability of the interfering polarization states Time (ns) Coincidences long -long 0 Time Window short -short short - long + long - short 

GAP Optique Geneva University 28 The Plug-&-Play configuration  Simplicity, self-stabilization J.Mod.Opt. 47, 517, 2000

GAP Optique Geneva University 29 Faraday mirrors Faraday rotator standard mirror (  incidence) Faraday rotator FM Independent of 

GAP Optique Geneva University 30 QC over 67 km, QBER  5% + aerial cable (in Ste Croix, Jura) ! D. Stucki et al., New Journal of Physics 4, , Quant-ph/ RMP 74, , 2002, Quant-ph/

GAP Optique Geneva University 31  Company established in 2001 Spin-off from the University of Geneva  Products Quantum Cryptography (optical fiber system) Quantum Random Number Generator Single-photon detector module (1.3  m and 1.55  m)  Contact information web:

GAP Optique Geneva University 32 Quantum Random Number Generator to be announced next week at CEBIT  Physical randomness source  Commercially available  Applications Cryptography Numerical simulations Statistics

GAP Optique Geneva University 33 Photon pairs source laser nonlinear birefringent crystal filtre l s,i lplp  Parametric fluorescence  Energy and momentum conservation  Phase matching determines the wavelengths and propagation directions of the down-converted photons

GAP Optique Geneva University 34 2-photon Q cryptography: Franson interferometer Two unbalanced interferometers  no first order interferences One can not distinguish between "long-long" and "short-short" Hence, according to QM, one should add the probability amplitudes  interferences (of second order) photon pairs  possibility to measure coincidences

GAP Optique Geneva University  source of Aspect’s 1982 experiment

GAP Optique Geneva University 36 Photon pairs source (Geneva 1997)  Energy-time entanglement  diode laser  simple, compact, handy 40 x 45 x 15 cm 3  I pump = 8 mW  with waveguide in LiNbO 3 with quasi phase matching, I pump  8  W KNbO 3 F Laser L P  655nm output 1 output 2 lens filter crystal laser

GAP Optique Geneva University 37 Quantum non locality  the statistics of the correlations can‘t be described by local variables Quantum non locality b b j1j1j1j1 y1y1y1y1 y2y2y2y2 j2j2j2j2 y1y1y1y1 j1j1j1j1 y2y2y2y2 j2j2j2j2 _ analyzer single counts a-b

GAP Optique Geneva University 38 The qubit sphere and the time-bin qubit  qubit :  different properties : spin, polarization, time-bins  any qubit state can be created and measured in any basis variable coupler variable coupler les i   1 0   h AliceBob D 0 D 1 switch 1 0

GAP Optique Geneva University 39 The interferometers d FM C  single mode fibers  Michelson configuration  circulator C : second output port  Faraday mirrors FM: compensation of birefringence  temperature tuning enables phase change

GAP Optique Geneva University 40 entangled time-bin qubit  variable coupler non-linear crystal B s A s l B l A  depending on coupling ratio and phase , maximally and non-maximally entangled states can be created  extension to entanglement in higher dimensions is possible  robustness (bit-flip and phase errors) depends on separation of time-bins

GAP Optique Geneva University 41 test of Bell inequalities over 10 km d 1 KNbO km 9.3 km Genève Bellevue Bernex quantum channel classical channels F laser LP R++ R-+ R+- R-- d 2 APD & FS Z FM Z 4.5 km 7.3 km 10.9 km

GAP Optique Geneva University 42 results  15 Hz coincidences  S raw = 2.41 S net = 2.7  violation of Bell inequalities by 16 (25) standard- deviations  close to quantum- mechanical predictions  same result in the lab

GAP Optique Geneva University 43 le labo

GAP Optique Geneva University 44 Bell test over 50 km  With phase control we can choose four different settings  = 0 ° or 90 ° and  = -45 ° or 45 °  Violation of Bell inequalities: Violation of Bell inequalities by more than 15 

GAP Optique Geneva University 45 Qutrit Entanglement

GAP Optique Geneva University 46 Bell Violation I = / I(lhv) = 2 < I(2) = < I(3) = PRL 93, , 2004

GAP Optique Geneva University 47 Two-photon Fabry-Perot interferometer Aim : direct detection of high dimensional entanglement Coincidences D a - D b (red) and D a - D b ’ (blue) as function of time while varying the phase  a D. Stucki et al., quant-ph/ NLC : non linear crystal

GAP Optique Geneva University 48 Plasmon assisted entanglement transfer SS+LL SL LS difference of detection time e v e n t s TAC phase 15  BCB Si-waffer polarization direction 15  20nm BCB fiber 15 BCB Si-waffer polarization direction nm BCB fiber  a short lived phenomenon like a plasmon can be coherently excited at two times that differ by much more than its lifetime. At a macroscopic level this would lead to a “Schrödinger cat” in superposition of living at two epochs that differ by much more than a cat’s lifetime. 1 cm

GAP Optique Geneva University 49 Experimental QKD with entanglement AliceBob J. Franson, PRL 62, 2205, 1989 W. Tittel et al., PRL 81, , 1998 cw source NL crystal

GAP Optique Geneva University 50 QKD AliceBob G. Ribordy et al., Phys. Rev. A 63, , 2001 S. Fasel et al., European Physical Journal D, 30, , 2004 N. Gisin & N. Brunner, quant-ph// P.D. Townsend et al., Electr. Lett. 30, 809, 1994 R. Hughes et al., J. Modern Opt. 47, , 2000 A. Shields et al., Optics Express 13, 660, 2005

GAP Optique Geneva University 51 Quantum cryptography below lake Geneva Alice Bob F.M. PBS Applied Phys. Lett. 70, , Electron. Letters 33, , 1997; 34, , J. Modern optics 48, , 2001.

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GAP Optique Geneva University 53 Limits of Q crypto distance  100 km Secret bit per pulse Q channel loss Detector noise - distance - bit rate

GAP Optique Geneva University 54 PNS Attack: the idea Alice Bob0 ph 1 phLosses Quantum memory Lossless channel (e.g. teleportation) QND measurement of photon number  PNS (photon-number splitting):  The photons that reach Bob are unperturbed  Constraint for Eve: do not introduce more losses than expected  PNS is important for long-distance QKD 1 ph0 ph2 ph 90,5%9%0.5% 0 ph 1 phEve!!!

GAP Optique Geneva University 55 Limits of Q crypto distance  100 km Secret bit per pulse Q channel loss Detector noise - distance - bit rate  50 km

GAP Optique Geneva University 56 1-photon Q crypto AliceBob 2-  single-photon source 31 km Results: (PRA 63,012309, 2001 and S. Fasel et al., quant-ph/0403xxx) CDC IF : P(1) = 0.5 … 0.7, P(2)  & g 2  0.1

GAP Optique Geneva University 57 Generalized measurements: POVM A set {P  } defines a POVM iff 1.P   0 2.   P  =1 The result  happens with probability Tr(  P  ) Example: unambiguous discrimination between 2 non-orthogonal Q states  POVM with 3 outcomes: 1. the state was definitively the first one 2. the state was definitively the second one 3. inconclusive result  minimal probability of an inconclusive result = (1-sin(  ))/2 where cos(  ) is the overlap PRA 54, 3783, 1996

GAP Optique Geneva University 58 A new protocol: SARG Joint patent UniGE + id Quantique pending PRL 92, , 2004; Phys. Rev. A 69, , 2004 The quantum protol is identical to the BB84 During the public discussion phase of the new protocol Alice doesn’t announce bases but sets of non-orthogonal states  even if Eve hold a copy, she can’t find out the bit with certainty  More robust against PNS attacks !

GAP Optique Geneva University 59 Distance [km] Secret key rate, log10 [bits/pulse] SARG BB84 Perfect detectors  =1, D=0 Typical detector  =0.1, D=10 -5 SARG vs BB84 PNS, optimal , detector efficiency , dark counts D 67km = Geneva-Lausanne  =  =  exp = 0.2

GAP Optique Geneva University 60 Protocols for high secret bit rate Alice Bob bit rate at emission goal: > 1 Gbit/s channel loss « no » loss in Bob’s optics detector + noise  secret bit rate goal: > 1 Mbit/s

GAP Optique Geneva University 61 protocols for high secret bit rate: an example (patent pending) Wish list: low loss at Bob’s side use one of the 2 bases more frequently make that basis simple telecom compatible resistant to PNS attacks does not work with single photons t B D B D M1 D M2 Laser IM bit 0 bit 1 decoy sequence quant-ph/ APL 87, , 2005

GAP Optique Geneva University 62 First results : Pulse rate 434 Mhz Link loss (25km) 5 dB QBER optical 1% QBER tot <4% quant-ph/ APL 87, , 2005

GAP Optique Geneva University 63 GHz Telecom QKD 1.27 GHz up-conversion detector: pump = 600 nm Rob Thew et al., 2005

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GAP Optique Geneva University 65 Bell measurement

GAP Optique Geneva University 66 Bell measurement D1D D2D p 1/16 1/81/21/8 D1D D2D p 1/16 1/4 1/8 D1D D2D p 1/ 8 D1D D2D p 1/ 4 1/8 P success = ½ 3 Bell states are detected!

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GAP Optique Geneva University 68 Q repeaters & relays * entanglement * Bell measurement.. QND measurement + Q memory * entanglement * Bell measurement. ?? REPEATER RELAY H. Briegel, W. Dür, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998) J. D. Franson et al, PRA 66,052307,2002; D. Collins et al., quant-ph/

GAP Optique Geneva University 69 3-photon: Q teleportation & Q relays  EPR  Bell 2 bits U 

GAP Optique Geneva University 70 The Geneva Teleportation experiment over 3x2 km Photon = particle (atom) of light Polarized photon (  structured photon) Unpolarized photon (  unstructured  dust)

GAP Optique Geneva University metres 2 km of optical fibre 2 km of optical fibre Two entangled photons

GAP Optique Geneva University metres 2 km of optical fibre

GAP Optique Geneva University metres Bell measurement (partial) the 2 photons interact 4 possible results: 0, 90, 180, 270 degrees

GAP Optique Geneva University metres Bell measurement (partial) the 2 photons interact 4 possible results: 0, 90, 180, 270 degrees Perfect Correlation The correlation is independent of the quantum state which may be unknown or even entangled with a fourth photon

GAP Optique Geneva University 75 Quantum teleportation     z    x    y  Bell 2 bits U 

GAP Optique Geneva University 76 What is teleported ?  According to Aristotle, objects are constituted by matter and form, ie by elementary particles and quantum states.  Matter and energy can not be teleported from one place to another: they can not be transferred from one place to another without passing through intermediate locations.  However, quantum states, the ultimate structure of objects, can be teleported. Accordingly, objects can be transferred from one place to another without ever existing anywhere in between! But only the structure is teleported, the matter stays at the source and has to be already present at the final location.

GAP Optique Geneva University 77 Implications of entanglement  The world can’t be understood in terms of “little billiard balls”.  The world is nonlocal (but the nonlocality can’t be used to signal faster than light).  Quantum physics offers new ways of processing information.

GAP Optique Geneva University 78 2 km of optical fiber Alice Alice:creation of qubits to be teleported Alice 55 m Bob Bob:analysis of the teleported qubit, 55 m from Charlie Bob Charlie Charlie:the Bell measurement Charlie fs 710 nm Experimental setup creation of entangled qubits coincidence electronics & LBO RG WDM RG WDM Ge InGaAs LBO BS InGaAs f s l a s e r sync out 1. 3 m  1. 3 m  1. 5 m  1. 5 m  2km

GAP Optique Geneva University 79results Equatorial states Raw visibility : V raw = 55 ± 5 % = 77.5 ± 2.5 % = 78 ± 3% = 77 ± 3% North & south poles mean fidelity: F poles =77.5 ± 3 % 77.5 ±2.5 % Mean Fidelity » 67 % (no entanglement)

GAP Optique Geneva University 80 Size of the classical communication One proton in one cm 3 at a temperature of 300 K:  bits protons in one cm 3 at a temperature of 300 K  x 155  bits To be compared to today’s optical fiber communication in labs: 1 Tbyte x 1024 WDW channels x 1000 fibers  bits/sec.   1 hour !! bits

GAP Optique Geneva University 81 1: EPR 2: Distribute 3: Create Qubit 4: Prepare BSM 5: BSM 6: Send result 7: Store photon 8: Wait for BSM 9: Analysis

GAP Optique Geneva University 82 1: EPR 2: Distribute 3: Create Qubit 4: Prepare BSM 5: BSM 6: Send result 7: Store photon 8: Wait for BSM 9: Analysis Laser fs LBO & PC n n+1 n n+1

GAP Optique Geneva University 83 Entanglement swapping Bell state measurement Entangled photons that never interacted EPR source 3-Bell-state analyzer N.Brunner et al., quant-ph/ independent sources

GAP Optique Geneva University 84 Superposition basis: results V = (80 ± 4) % F  90 % 78 hours of measurement ! Deriedmatten, Marcikic et al., PRA 71, 05302, 2005

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GAP Optique Geneva University 86 Coin tossing at a distance correlated Non correlated Each side the results are random the statistics of the correlations can‘t be described by local variables Quantum non locality

GAP Optique Geneva University 87 LMR Siparam  Prob( resultats =) GGG100 % GGR1/3 GRG1/3 RGG1/3 GRR1/3 RGR1/3 RRG1/3 RRR100 % Arbitr. mixture  1/3 Bell Inequality 1/4 3/4 Bell’s inequality: (D. Mermin, Am. J. Phys. 49, , 1981)

GAP Optique Geneva University 88 Bell inequality Locality  In particular: a.b+a.b’+a’.b-a’.b’=a.(b+b’)+a’.(b-b’)  2 E(a,b)= a( ).b( )  ( ) d S=E(a,b)+E(a,b’)+E(a’,b)-E(a’,b’)  2Bell inequality

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GAP Optique Geneva University 90 Generalized measurements: POVM A set {P  } defines a POVM iff 1.P   0 2.   P  =1 The result  happens with probability Tr(  P  ) Example: where the m  are the 4 vectors of the thetrahedron

GAP Optique Geneva University 91 input PBS 50% 33.3% /2 D1D1 D2D2 D3D3 D4D4

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GAP Optique Geneva University 93 Non-locality according to Newton  Newton was very conscious of an unpleasant characteristics of his theory of universal gravitation :  A stone moved on the moon would immediately affect the gravitational field on earth.  Newton didn’t like this non-local aspect of his theory at all, but, due to a lack of alternatives, physics had to live with it until 1915.

GAP Optique Geneva University 94 Let’s read Newton’s words: That Gravity should be innate, inherent and essential to Matter, so that one Body may act upon another at a Distance thro’ a Vacuum, without the mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity, that I believe no Man who has in philosophical Matters a competent Faculty of thinking, can ever fall into it. Gravity must be caused by an Agent acting constantly according to certain Laws, but whether this Agent be material or immaterial, I have left to the Consideration of my Readers. Isaac Newton Papers & Letters on Natural Philosophy and related documents Edited by Bernard Cohen, assisted by Robert E. Schofield Harvard University Press, Cambridge, Massachusetts, 1958

GAP Optique Geneva University 95  Today, thanks to Einstein, gravitation is no longer considered as a kind of action at a distance. A moon-quake triggers a bunch of gravitons that propagate through space and « informs » Earth. The propagation is very fast, but at finite speed, the speed of light, i.e. about 1 second from the moon to our Earth. Einstein, the greatest mechanical engineer

GAP Optique Geneva University 96  In 1905, Einstein also gave a description of Brownian motion: the statistics of collisions between invisible atoms and molecules support the atomic hypothesis:  Still in 1905, Einstein gave a mechanical explanation of the photo-electric effect: Einstein, the greatest mechanical engineer

GAP Optique Geneva University 97 Classical physics: Nature is made out of many little “billiard balls” that mechanically bang into each other Quantum physics: Named by historical accident quantum mechanics, the new physics is precisely characterized by the fact that it does not provide a mechanical description of Nature

GAP Optique Geneva University 98  Einstein was very conscious of an "unpleasant" characteristic of quantum physics :  Spatially separated systems behave as a single entity: they are not logically separated.  Acting “ here ” has an apparent, immediate, effect “ there ”.  Einstein-Podolski-Rosen argued that this being obviously impossible, quantum physics is incomplete.  Most physicists didn’t like this non-local aspect of quantum theory, but again, due to a lack of alternatives, … it remained in the curiosity-lab. Non-locality according to Einstein

GAP Optique Geneva University 99 Non-locality for non-physicists

GAP Optique Geneva University 100 Joint conditional probability Events at 2 separated locations. Not under the professor’s control Settings (experimental conditions). Under the professor’s control Bob b y Alice a x Quantum exams

GAP Optique Geneva University 101 Quantum exam #1  Suppose Alice is asked to output the question received by Bob, and vice-versa.  Can they succeed?  Clearly, not! Why? Because it would imply signaling (arbitrarily fast communication) and every physicists knows – since Einstein – that this is impossible. And even long before Einstein, Newton and others had the strong intuition that signaling is impossible.  The relativistic no-signaling condition implies that some conditional probabilities (i.e. some exams) are impossible !

GAP Optique Geneva University 102 Quantum exam #2  Suppose that Alice and Bob are asked to always output the same answer, whenever they receive the same question.  Can they succeed?  Clearly yes! It suffice that Alice and Bob prepare a common strategy before being spatially separated; i.e. they should prepare one precise answer for each question.  Is there an alternative strategy? No, as all students preparing exams know.  Some conditional probabilities can be explained in the frame of classical physics only with common causes.

GAP Optique Geneva University 103 Quantum exam #3: binary case  But now, assume that A&B should always output the same value, except when both receive the input 1  Formally a+b=x y modulo 2  Can they succeed? Note that the exam doesn’t require signaling.  If A’s output is predetermined by some strategy, then this would allow signaling.  Consequently, A’s output has to be random. Similarly, B’s output has to be random.  A and B’s randomness should be the same whenever x.y=0, but should be opposite whenever x=y=1.  This is impossible, although there is no signaling.  How close to a+b=x y can they come? Can they achieve a probability larger than 50%?

GAP Optique Geneva University 104 Prob(a+b=x y)=? P(a+b=x y|x=0,y=0) + P(a+b=x y|x=0,y=1) + P(a+b=x y|x=1,y=0) + P(a+b=x y|x=1,y=1) optimal for Alice and Bob sharing quantum entanglement   2+  2  3.41 Quantum correlations (entanglement) allows one to perform some tasks, including some useful tasks, that are classically impossible ! optimal for classical Alice and Bob CHSH-Bell inequality:  3

GAP Optique Geneva University 105 Entanglement is everywhere! old wisedom: entanglement is like a dream, as soon as one tries to tell it to a friend, it evaporates! Entanglement is fragile ! recent experiments: Entanglement is not that fragile ! Entanglement is everywhere, but hard to detect. Can entanglement be derived from a more primitive concept? Can Q physics be studied from the outside ? This new wisedom raises new questions:

GAP Optique Geneva University 106 Q concepts without Hilbert space  Can entanglement, non-locality, no-cloning, uncertainty relations, cryptography, etc be derived from one primitive concept ?  Can all these be studied « from the outside », i.e. without all the Hilbert space artillery? Theoretical Physics

GAP Optique Geneva University 107 binary local correlations polytope of local correlations p(a,b|x,y) all facets correspond to the CHSH-Bell  :  P  3 QM Bob b y Alice a x p(a,b|x,y)

GAP Optique Geneva University 108 CHSH-Bell inequality No better inequality is known to detect non-locality of Werner 2 qubit states !!! P(a+b=x y|x=0,y=0) + P(a+b=x y|x=0,y=1) + P(a+b=x y|x=1,y=0) + P(a+b=x y|x=1,y=1)  3 use non-signaling to remove the output 1: P(0,1|x,y)=P(a=0|x)-P(0,0|x,y) P(1,1|x,y)=1-P(a=0|x)-P(b=0|y)+P(0,0|x,y)  P(00|00)+P(00|01)+P(00|10)-P(00|11)  P(a=0|0)+P(b=0|0) y x 00

GAP Optique Geneva University 109 detection loophole P(00|00)+P(00|01)+P(00|10)-P(00|11)  P(a=0|0)+P(b=0|0) detection efficiency    2 P(00|00)+  2 P(00|01)+  2 P(00|10)-  2 P(00|11)   P(a=0|0)+  P(b=0|0) a violation requires:  threshold for max entangled qubit pair  82%  threshold decreases for partially entangled qubit pairs towards 2/3 ! (P. H. Eberhard, Phys. Rev. A 47, R747,1993)  find better inequalities

GAP Optique Geneva University 110 The new inequality for qubits with 3 settings This is the only new inequality for 3 inputs and binary outputs y x  0 I3322 = Collins & Gisin, J.Phys.A 37, 1775, 2004

GAP Optique Geneva University trace(B  )  For each , let CHSH be the critical weight such that  (  )= CHSH P cos(  )|00>+sin(  )|11> + (1- CHSH ) P |01> is at the limit of violating the CHSH inequality

GAP Optique Geneva University 112 Non-locality without signaling polytope of local correlations p(a,b|x,y) facet corresponding to the CHSH-Bell  :  P  3 QM facet corresponding to the no-signaling  : a+b=xy J.Barrett et al, quant-ph/ set of correlations p(a,b|x,y) s.t. 1. p(a|x,y)=  b p(a,b|x,y) = p(a|x) 2. p(b|x,y)=  a p(a,b|x,y) = p(b|y) binary case: unique extremal point! one above each CHSH- Bell inequality Bob b y Alice a x p(a,b|x,y)

GAP Optique Geneva University 113 A unit of non-locality, or non-locality without the Hilbert space artillery Non Local Machine AliceBob x a b y a + b= x.y A single bit of communication suffice to simulate the NL Machine (assuming shared randomness). But the NL Machine does not allow any communication. Hence, the NL Machine is a strickly weaker ressource than communication.

GAP Optique Geneva University 114 no-cloning theorem without quantum Non local Machine x y z b c a If then b+c=x(y+z), and Alice can signal to B-C a+b=x.y a+c=x.z L.Masanes, A.Acin, NG quant-ph/ Non-signaling no-cloning theorem

GAP Optique Geneva University 115 From Bell inequality to cryptography polytope of local correlations facet corresponding to the CHSH-Bell  :  P  3 facet corresponding to the no-signaling  : a+b=xy QM isotropic correlations 1 0  2-1 CHSH Q-crypto protocol Alice Bob sifting: 1-way all bits are kept noisy even without Eve 1-way distillation intrinsic info > 0 2-way ??? A.Acin, L.Masanes, NG quant-ph/  secure QKD against individual attacks by any post-quantum non-signaling Eve !

GAP Optique Geneva University 116 From Bell inequality to cryptography polytope of local correlations facet corresponding to the CHSH-Bell  :  P  3 facet corresponding to the no-signaling  : a+b=xy QM isotropic correlations 1 0  way distillation intrinsic info > 0 2-way ??? A.Acin, L.Masanes, NG quant-ph/  secure QKD against individual attacks by any post-quantum non-signaling Eve ! Uncertainty relations, i.e. information / disturbance trade-off: I(E:B|x=0) = fct(QBER x=1 ) I(E:B|x=1 ) = fct(QBER x=0 ) V. Scarani

GAP Optique Geneva University 117 Simulating entanglement with a few bits of communication (+ shared randomnes) Alice  Bob  & define measurement bases. The output  &  should reproduce the Q statistics: Case of singlet: 8 bits, Brassard,Cleve,Tapp, PRL 83, bits, Steiner, Phys.Lett. A270, , Gisins Phys.Lett. A260, 323, bit! Toner & Bacon, PRL 91, , bit: impossible (Bell inequality) … but …

GAP Optique Geneva University 118 Simulating singlets with the NL Machine Alice Bob Non local Machine b Given &, the statistics of  &  is that of the singlet state: a

GAP Optique Geneva University hint for the proof:

GAP Optique Geneva University 120 x=0 x=1

GAP Optique Geneva University 121 (0,0) (0,1) (1,1) (1,0) (1,1) (0,1) (0,0) (x,y)

GAP Optique Geneva University 122 (0,0) (0,1) (1,1) (1,0) (1,1) (0,1) (0,0)  = a +1  = a  =b+1  =b « cqfd » (x,y)

GAP Optique Geneva University 123 Partially entangled states seem more nonlocal than the max entangled ones ! Simulating partial entanglement Bell inequalities are more violated by partially entangled states than by max entangled ones (for dim > 2 & all known cases). When testing Bell inequality, the use of a partially entangled state provides more information per experimental run than the use of max entangled states. (T. Acin, R. Gilles & N. Gisin, PRL 95, , 2005 ) Partially entangled states are more robust against the detection loophole (P. H. Eberhard, Phys. Rev. A 47, R747,1993)

GAP Optique Geneva University 124 How to prove that some correlation can’t be simulated with a single use of the nonlocal machine ?  same idea as Bell inequality, i.e. 1. List all possible strategies 2. Notice that they constitute a convex set 3. Notice that this convex set has a finite number of extremal points (vertices), i.e. it’s a polytope 4. Find the polytope’s facets 5. Express the facets as inequalities

GAP Optique Geneva University 125 A i A r B j B r x a y b a+b=xy For given A i and, there are 6 extremal local strategies: 1. r A =0 3. X =0 and r A =a 5. X =0 and r A =a+1 2. r A =1 4. X =1 and r A =a 6. X =1 and r A =a+1 For 2 settings per side, there are 6 4 strategies defining 264 different vertices. The polytope is the same as the “no-signaling polytope” studied by J. Barrett et al in quant-ph/ Consequently, no quantum state can violate such a 2-settings inequality

GAP Optique Geneva University 126 The 1 nl-bit inequality For 3 settings per side: there are 6 6 strategies defining 3880 different vertices. There is a unique new inequality: y x y x P( r A =0|x) P( r A = r B =0|x,y) P( r B =0|y)  0 Recall: for standard Bell inequalities (i.e with no nonlocal machines) and 3 settings per side, there is also a unique new inequality: y x  0 I3322 = Collins & Gisin, J.Phys.A 37, 1775, 2004

GAP Optique Geneva University 127 CHSH D D NLM I3322 D D NLM Geometric intuition

GAP Optique Geneva University 128 Very partially entangled states do violate the 1-nl bit-Bell inequality :  Very partially entangled states can’t be simulated with only 1 nl-bit  Partially entangled states are more nonlocal than the singlet ! partial ent. max ent.

GAP Optique Geneva University 129

GAP Optique Geneva University 130

GAP Optique Geneva University 131 The I3322-Bell inequality is not monogamous A B C There exists a 3-qubit state  ABC, such that A-B violates the I3322-Bell inequality and A-C violates it also. (see D. Collins et al., J.Phys. A 37, , 2004)  ABC

GAP Optique Geneva University 132

GAP Optique Geneva University 133 Quantum Cryptography guaranties confidentiality  Bell’s inequalities are violated  Quantum correlation can’t be explained by local variables

GAP Optique Geneva University 134 Alice Eavedropping (cloning) machine U Bob Eve Bell states Error operator:

GAP Optique Geneva University 135 Where: N. Cerf et al., PRL 84,4497,2000 & 88,127902,2002

GAP Optique Geneva University 136 Case d=2 (qubits): Classical random variables: AliceBobEve X=0,1Y=0,1Z=[Z 1,Z 2 ] Z 1 =X+Y Z 2 =X with prob. Conditional mutual information:

GAP Optique Geneva University 137 Optimal individual attack on BB84 Page 182 à 185 de Rev.Mod.Phys. 74, 145, 2002

GAP Optique Geneva University 138 I AE  1-I AB I AE Bell inequ. violatedBell inequ. not violated Eve: optimal individual attack

GAP Optique Geneva University 139 Advantage distillation AliceBob X 0 =1Y 0 X 1 =1Y 1 X 2 =0Y 2 X 3 =1Y 3…. X j Y j Alice announces {0,1,3}, Bob accepts iff Y 0 = Y 1 = Y 3 Eve can’t do better than a majority vote! Alice and Bob take advantage of their public authenticated channel Theorem: if the intrinsic information vanishes, then advantage distillation does not produce a secert key. Theorem: In arbitrary dimensions d and either the case of 2 bases or of d+1 bases: Advantage distillation produces a secret key iff Alice and Bob are not separated. N. Gisin & S. Wolf, PRL 83, , 1999

GAP Optique Geneva University 140 I AE 2-wayAlice quantum. Inf. Proc.and sufficeBob separated Bell inequality: can be never violated or classical 1-way class. Inf. Proc. suffice D0D0

GAP Optique Geneva University 141 Quantum Cryptography  AB P(A,B,E) measurement  AB Entanglement distillation shared secret bit measurement Secret key distillation Entanglement Q nonlocality I(A:B), I(A:E) I(A:B|E) I(A:B  E) intrinsic info. Where is Eve ?In the Q scenario one assumes that Eve holds the entire universe except the Q systems under Alice and Bob’s direct control. Ie Eve holds the purification of .

GAP Optique Geneva University 142 Intrinsic information Eve Alice Bob ¼ 0 1 ¼ 1 0 ¼ 1 1 ¼ I(A:B|E) = 1 0 E  E e 1

GAP Optique Geneva University 143 Intrinsic information Eve Alice Bob 0 1 e 0 0 ¼ ¼ 0 1 ¼ ¼ 1 0 ¼ ¼ 1 1 ¼ ¼ I(A:B|E) = 1 0 E  E e 1 I(A:B|E) = 0 Intrinsic information: I(A:B  E) = Min I(A:B|E) E  E

GAP Optique Geneva University 144 Intrinsic info  entanglement Theorem: Let P(A,B,E) be a probability distribution shared between Alice, Bob and Eve after measuring a quantum state  ABE. I(A:B  E) > 0 iff  AB is entangled N. Gisin and S. Wolf, PRL 83, , S. Wolf and N. Gisin, Proceedings of Crypto 2000, pp Theorem: If moreover Alice and Bob hold qubits, then  AB is entangled iff P(A,B,E) is such that Alice and Bob can distil a secret key A. Acin, L. Masanes and N. Gisin, PRL 91, , 2004.

GAP Optique Geneva University 145 Quantum Cryptography  AB P(A,B,E) measurement  AB Entanglement distillation shared secret bit measurement Secret key distillation Entanglement Q nonlocality I(A:B), I(A:E) I(A:B|E) I(A:B  E) intrinsic info. In the binary case, the diagram commutes. A counter example in dimension 3 is known. The existence of bound information is conjectured.

GAP Optique Geneva University 146 What is secure ? Quantum cryptography is technically ready to provide absolute secure key distribution between two end-points: Secure QKD channel Bob Alice Where are Alice’s and Bob’s boundaries ?? At the quantum/classical split: and old question in a modern setting!

GAP Optique Geneva University 147 How to improve Q crypto ? Q repeater

GAP Optique Geneva University 148 port 1 horizontal pol. vertical pol. port 2-45° pol. +45° pol. port 2 horizontal pol. vertical pol. Faraday effect port 3

GAP Optique Geneva University 149 Rayleighback-scaterrings  delay line Perfect interference (V  99%) without any adjustments, since: both pulses travel the same path in inverse order both pulses have exactly the same polarisation thanks to FM Alice Bob Drawback 1: Rayleigh backscattering Drawback 2: Trojan horse attacks