Quantum Communication GAP Optique Geneva University Nicolas Gisin

Quantum Communication GAP Optique Geneva University § Nicolas Gisin Quantum cryptography: • • Q crypto: RMP 74, 145 -195, 2002 Q cloning: RMP 77, 1225 -1256, 2005 BB 84 and uncertainty relations Ekert and entanglement no cloning theorem BB 84 Ekert Implementations Eve: optimal individual attack Error correction, privacy amplification, advantage distillation § Quantum Teleportation § Optimal and generalized quantum measurements • • • principle, connection to optimal state estimation and cloning experiments quantum relays and quantum repeaters optimal quantum cloning POVMs (tetrahedron, unambiguous state discrimination) weak measurements 1

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

§ 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 GAP Optique Geneva University 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". 3

§ The "quantum communication channel" is not used to transmit a message (information), only a "key" is transmitted (no information). GAP Optique Geneva University § 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. 4

GAP Optique Geneva University Modern Cryptology Secrecy is based on: Complexity theory The key is public Information theory The key is secrete The public key contains the decoding key, but it is very difficult to find (one way functions) The key contains the decoding key: Only the two partners have a copy ! The security is not proven (no one knows whether one way functions exist) Example: 127 x 229 = 29083 The security is proven (Shannon theorem) Example: Message: 011001001 Key: 110100110 Coded message: 101101111 5

25% errors GAP Optique Geneva University BB 84 protocol: Eve 6

Security from Heisenberg uncertainty relations Alice Bob GAP Optique Geneva University 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 7

GAP Optique Geneva University 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 BB 84 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. ). 8

Eve: optimal individual attack GAP Optique Geneva University IAE 1 -IAB 9

Quantum Communication § Quantum Communication is the art of transferring a Q state from one place to another. Example: GAP Optique Geneva University • 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 10

a=x, z Ekert protocol (E 91) b=x, z source GAP Optique Geneva University a 0, 1 b 0, 1 Theorem: let If AB is pure, then 11

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

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

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

BB 84 E 91 a=x, z source GAP Optique Geneva University a 0, 1 Alice b=x, z source b 0, 1 Indistinguishable from a single photon source. The qubit is coded in the a-basis And holds the bit value given by Alice results. 15

16 GAP Optique Geneva University

Experimental Realization § Single photon source GAP Optique Geneva University • laser pulses strongly attenuated ( 0. 1 photon/pulse) • photon pair source (parametric downconversion) • true single-photon source § Polarization or phase control during the single photon propagation • parallel transport of the polarization state (Berry topological phase) no vibrations • fluctuations of the birefringence thermal and mechanical stability • depolarization mode dispersion smaller than the source coherence • Stability of the interferometers coding for the phase § Single photon detection • avalanche photodiode (Germanium or In. Ga. As) in Geiger mode dark counts • based on supraconductors requires cryostats 17

Telecommunication wavelengths § Attenuation ( transparency) GAP Optique Geneva University [ m] [d. B/km] T 10 km 0. 8 2 1% 1. 3 0. 35 44% 1. 55 0. 2 63% § Chromatic dispersion § Components available Two windows 18

Single Photon Generation (1) • Attenuated Laser Pulse Poissonian Distribution 100% Probability GAP Optique Geneva University Attenuating Medium 80% Mean = 1 Mean = 0. 1 60% 40% 20% 0% " 0 or 1 or 2 or. . . " rather than 1 0 1 2 3 4 5 Number of photons per pulse • Simple, handy, uses reliable technology today’s best solution 19

Avalanche photodiodes § Single-photon detection with avalanches in Geiger GAP Optique Geneva University mode macroscopic avalanche triggered by single-photon Silicon: 1000 nm Germanium: 1450 nm In. Ga. As/In. P: 1600 nm 20

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

GAP Optique Geneva University Efficiency and Dark Counts 22

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

24 GAP Optique Geneva University

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

Phase Coding Basis 1: A = 0; p Basis 2: A = p/2; 3 p/2 Bases GAP Optique Geneva University § Single-photon interference Basis: B = 0; p/2 Compatible: Alice A Di ( A- B = np) Bob Di A Incompatible: Alice and Bob ? ? ( A- B = p/2) 26

Difficulties with Phase Coding § Stability of a 20 km long interferometer? Coincidences GAP Optique Geneva University Time Window long -long short -short Time (ns) 0 short - long + long - short -3 -2 -1 0 1 2 3 Problems: • stabilization of the path difference active feedback control • stability of the interfering polarization states 27

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

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

GAP Optique Geneva University QC over 67 km, QBER 5% RMP 74, 145 -195, 2002, Quant-ph/0101098 + aerial cable (in Ste Croix, Jura) ! D. Stucki et al. , New Journal of Physics 4, 41. 1 -41. 8, 2002. Quant-ph/0203118 30

§ Company established in 2001 • Spin-off from the University of Geneva GAP Optique Geneva University § Products • Quantum Cryptography (optical fiber system) • Quantum Random Number Generator • Single-photon detector module (1. 3 m and 1. 55 m) § Contact information email: info@idquantique. com web: http: //www. idquantique. com 31

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

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

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

GAP Optique Geneva University 2 - source of Aspect’s 1982 experiment 35

Photon pairs source (Geneva 1997) r se a L m 5 n GAP Optique Geneva University 65 output 1 F L P KNb. O 3 § output 2 crystal Energy-time entanglement lens § § diode laser simple, compact, handy 40 x 45 x 15 cm 3 filter laser § § Ipump = 8 m. W with waveguide in Li. Nb. O 3 with quasi phase matching, Ipump 8 W 36

GAP Optique Geneva University _ y j single counts Quantum non locality b y j analyzer a-b § the statistics of the correlations can‘t be described by local variables Quantum non locality 37

The qubit sphere and the time-bin qubit q qubit : q different properties : spin, polarization, timebins GAP Optique Geneva University q any qubit state can be created and measured in any basis Alice y = s + e i l j 1 h 1 0 variable coupler Bob D 0 0 D 1 switch variable coupler 38

The interferometers FM C GAP Optique Geneva University 1 d 2 3 FM § § § single mode fibers Michelson configuration circulator C : second output port Faraday mirrors FM: compensation of birefringence temperature tuning enables phase change 39

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

test of Bell inequalities over 10 km Bellevue km 5. 4 FM d 1 GAP Optique Geneva University tum an u q km 8. 1 Genève las er P F L KNb. O 3 R++ R-+ R+R-- an APD 1 - tum classical channels APD 2 km ch APD 2+ an 7. 3 k FM APD 1 + & 9. 3 qu Z 10. 9 km l ne an ch ne Z l FS d 2 FS m Bernex 41

results § § 1. 0 correlation coefficient GAP Optique Geneva University 0. 5 0. 0 -0. 5 V = (85. 3 ± 0. 9)% raw 15 Hz coincidences Sraw = 2. 41 Snet = 2. 7 § violation of Bell inequalities by 16 (25) standarddeviations § close to quantummechanical predictions § same result in the lab V = (95. 5 ± 1) % net. 0 1000 4000 7000 10000 13000 time [sec] 42

GAP Optique Geneva University le labo 43

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

GAP Optique Geneva University Qutrit Entanglement 45

Bell Violation PRL 93, 010503, 2004 GAP Optique Geneva University I(lhv) = 2 < I(2) = 2. 829 < I(3) = 2. 872 I = 2. 784 +/- 0. 023 46

Two-photon Fabry-Perot interferometer GAP Optique Geneva University Aim : direct detection of high dimensional entanglement NLC : non linear crystal Coincidences Da. Db (red) and Da. Db’ (blue) as function of time while varying the phase a D. Stucki et al. , quant-ph/0502169 47

Plasmon assisted entanglement transfer polarization direction 20 nm GAP Optique Geneva University phase BCB 15 Si-waffer SS+LL 1 cm TAC events fiber LS SL difference of detection time 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. 48

Experimental QKD with entanglement cw source GAP Optique Geneva University Alice Bob NL crystal J. Franson, PRL 62, 2205, 1989 W. Tittel et al. , PRL 81, 3563 -3566, 1998 49

QKD GAP Optique Geneva University Alice Bob G. Ribordy et al. , Phys. Rev. A 63, 012309, 2001 S. Fasel et al. , European Physical Journal D, 30, 143 -148, 2004 P. D. Townsend et al. , Electr. Lett. 30, 809, 1994 R. Hughes et al. , J. Modern Opt. 47, 533 -547 , 2000 A. Shields et al. , Optics Express 13, 660, 2005 N. Gisin & N. Brunner, quant-ph//0312011 50

Quantum cryptography below lake Geneva GAP Optique Geneva University Alice Bob PBS F. M. Applied Phys. Lett. 70, 793 -795, 1997. Electron. Letters 33, 586 -588, 1997; 34, 2116 -2117, 1998. J. Modern optics 48, 2009 -2021, 2001. 51

52 GAP Optique Geneva University

Secret bit per pulse - distance - bit rate Q cha nn el l oss 10 -6 GAP Optique Geneva University 10 -2 Limits of Q crypto Detector noise 100 km distance 53

PNS Attack: the idea GAP Optique Geneva University 90, 5% 9% 0. 5% Alice 0 ph 1 ph 2 ph QND measurement of photon number 0 ph 1 ph Eve!!! Losses Bob Lossless channel (e. g. teleportation) Quantum memory è 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 54

10 -6 Secret bit per pulse GAP Optique Geneva University 10 -2 Limits of Q crypto Qc - distance - bit rate han nel los s Detector noise 50 km 100 km distance 55

1 -photon Q crypto Alice Bob GAP Optique Geneva University 2 - IF 31 km CDC single-photon source : P(1) = 0. 5 … 0. 7, P(2) 0. 015 & g 2 0. 1 Results: (PRA 63, 012309, 2001 and S. Fasel et al. , quant-ph/0403 xxx) 56

Generalized measurements: POVM GAP Optique Geneva University 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 57

A new protocol: SARG GAP Optique Geneva University The quantum protol is identical to the BB 84 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 ! Joint patent Uni. GE + id Quantique pending PRL 92, 057901, 2004; Phys. Rev. A 69, 012309, 2004 58

SARG vs BB 84 Secret key rate, log 10 [bits/pulse] GAP Optique Geneva University PNS, optimal , detector efficiency , dark counts D Perfect detectors =1, D=0 Typical detector =0. 1, D=10 -5 = 0. 335 mexp = 0. 2 = 0. 014 SARG BB 84 Distance [km] 67 km = Geneva-Lausanne 59

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

protocols for high secret bit rate: an example (patent pending) GAP Optique Geneva University quant-ph/0411022 Wish list: • low loss at Bob’s side APL 87, 194105, 2005 • 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 Laser IM bit 0 bit 1 decoy sequence DB DM 1 DM 2 61

Pulse rate 434 Mhz Link loss (25 km) 5 d. B QBERoptical 1% QBERtot <4% GAP Optique Geneva University First results : quant-ph/0411022 62 APL 87, 194105, 2005

GAP Optique Geneva University GHz Telecom QKD 1. 27 GHz L Raw Ra te (a ve) QBER 25 k m 2. 62 M H z 1. 2 % L 50 k m Raw Ra te (a ve) QBER 530 k. H z 7. 3 % up-conversion detector: R(secre t[es tim ) 1550 ated] + 980 pump = 600 nm 500 k. H z R( secre t[es tim ated] ) 75 k. H z Rob Thew et al. , 2005 63

64 GAP Optique Geneva University

GAP Optique Geneva University Bell measurement 65

GAP Optique Geneva University Bell measurement D 1 00 D 2 00 p 1/16 D 1 00 D 2 p 22 1/16 1 2 D 1 22 0 1 2 D 2 1/16 1/8 1/2 1/8 p 22 00 1/16 0 11 22 1/16 1/4 01 12 01 1/8 0 2 D 1 11 0 2 D 2 1/4 1/8 p 1/8 11 1 0 2 1 12 0 1 1 2 1/8 1/8 1/8 02 11 1/4 1/8 2 0 02 0 2 1/8 1/8 Psuccess = ½ 3 Bell states are detected! 66

67 GAP Optique Geneva University

Q repeaters & relays * . . * entanglement J. D. Franson et al, PRA 66, 052307, 2002; D. Collins et al. , quant-ph/0311101 Bell measurement REPEATER GAP Optique Geneva University RELAY Bell measurement * . ? ? * entanglement QND measurement + Q memory H. Briegel, W. Dür, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998) 68

3 -photon: Q teleportation & Q relays 2 bits Bell GAP Optique Geneva University Ä y 2 km U EPR Classical channel Charlie Alice BSM 2 km EPR source 2 km Bob y 69

The Geneva Teleportation experiment over 3 x 2 km GAP Optique Geneva University Photon = particle (atom) of light Polarized photon ( structured photon) Unpolarized photon ( unstructured dust) 70

GAP Optique Geneva University 55 metres 2 km of optical fibre Two entangled photons 71

GAP Optique Geneva University 55 metres 2 km of optical fibre 72

GAP Optique Geneva University 55 metres Bell measurement (partial) the 2 photons interact 4 possible results: 0, 90, 180, 270 degrees 73

GAP Optique Geneva University 55 metres Bell measurement (partial) the 2 photons interact 4 possible results: 0, 90, 180, 270 degrees n o i t la e f r e P e rr o ct C The correlation is independent of the quantum state which may be unknown or even entangled with a fourth photon 74

Quantum teleportation 2 bits GAP Optique Geneva University Bell U z x y 75

What is teleported ? § According to Aristotle, objects are constituted by matter and form, ie by elementary particles and quantum states. GAP Optique Geneva University § 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. 76

Implications of entanglement § The world can’t be understood in terms of GAP Optique Geneva University “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. 77

Bob Experimental setup & In. Ga. As Charlie Ge 55 m In. Ga. As Alice: creation of qubits to Alice be teleported BS m 1. 5 WDM RG m m 1. 3 m 2 k Alice 1. 3 WDM RG m 1. 5 LBO creation of entangled qubits Charlie: the Bell measurement Charlie Bob: analysis of the Bob teleported qubit, 55 m from Charlie 2 km of optical fiber fs laser GAP Optique Geneva University m 2 k 2 km fs laser @ 710 nm coincidence electronics sync out 78

results Equatorial states GAP Optique Geneva University Mean Fidelity = 78 ± 3% = 77 ± 3% 77. 5 ± 2. 5 % mean fidelity: Fpoles=77. 5 ± 3 % North & south poles » 67 % (no entanglement) Raw visibility : Vraw= 55 ± 5 % = 77. 5 ± 2. 5 % 79

Size of the classical communication GAP Optique Geneva University One proton in one cm 3 at a temperature of 300 K: bits 1020 protons in one cm 3 at a temperature of 300 K 1020 x 155 1022 bits To be compared to today’s optical fiber communication in labs: 1 Tbyte x 1024 WDW channels x 1000 fibers 1019 bits/sec. 1 hour !! 80

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

GAP Optique Geneva University & LBO n n+1 PC LBO n n+1 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 82

Entanglement swapping GAP Optique Geneva University Entangled photons that never interacted 3 -Bell-state analyzer N. Brunner et al. , quant-ph/0510034 Bell state measurement 2 independent sources EPR source 83

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

85 GAP Optique Geneva University

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

GAP Optique Geneva University Bell’s inequality: (D. Mermin, Am. J. Phys. 49, 940 -943, 1981) Bob Left Alice same different Middle same different Right same different Left Middle 1/4 1/4 Right 100 % 0 % 1/4 LMR GGG GGR GRG RGG GRR RGR RRG RRR Arbitr. mixture 3/4 100 % 0 % 3/4 1/4 3/4 Si param ¹ Prob(resultats =) 100 % 1/3 1/3 1/3 100 % 1/3 3/4 100 % 0 % Bell Inequality 87

GAP Optique Geneva University 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’) 2 Bell inequality 88

89 GAP Optique Geneva University

GAP Optique Geneva University 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 90

GAP Optique Geneva University 50% input D 1 PBS D 2 /2 33. 3% D 3 D 4 91

92 GAP Optique Geneva University

Non-locality according to Newton GAP Optique Geneva University § 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. 93

GAP Optique Geneva University 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 94

Einstein, the greatest mechanical engineer GAP Optique Geneva University § Today, thanks to Einstein, gravitation is no longer considered as a kind of action at a distance. A moonquake 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. 95

Einstein, the greatest mechanical engineer § In 1905, Einstein also gave a description of Brownian GAP Optique Geneva University 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: 96

Classical physics: GAP Optique Geneva University 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 97

Non-locality according to Einstein GAP Optique Geneva University § 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. 98

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

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

Quantum exam #1 GAP Optique Geneva University § 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 ! 101

Quantum exam #2 GAP Optique Geneva University § 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. 102

Quantum exam #3: binary case § But now, assume that A&B should always output the same GAP Optique Geneva University § § § 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%? 103

GAP Optique Geneva University Prob(a+b=x • y)=? optimal for classical Alice and Bob CHSH-Bell inequality: 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 2+ 2 3. 41 optimal for Alice and Bob sharing quantum entanglement Quantum correlations (entanglement) allows one to perform some tasks, including some useful tasks, that are classically impossible ! 104

GAP Optique Geneva University 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. This new wisedom raises new questions: Can entanglement be derived from a more primitive concept? Can Q physics be studied from the outside ? 105

Theoretical Physics Q concepts without Hilbert space GAP Optique Geneva University § 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? 106

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

CHSH-Bell inequality GAP Optique Geneva University 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) x -1 y 0 -1 +1 +1 0 +1 -1 0 No better inequality is known to detect non-locality of Werner 2 qubit states !!! 108

detection loophole GAP Optique Geneva University 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, R 747, 1993) find better inequalities 109

The new inequality for qubits with 3 settings GAP Optique Geneva University This is the only new inequality for 3 inputs and binary outputs. I 3322 = x -1 y 0 0 -2 +1 +1 +1 -1 0 0 Collins & Gisin, J. Phys. A 37, 1775, 2004 110

For each , let CHSH be the critical weight such that ( )= CHSH Pcos( )|00>+sin( )|11> + (1 - CHSH) P|01> is at the limit of violating the CHSH inequality 1. 03 1. 01 1. 00 trace(B ) GAP Optique Geneva University 1. 02 0. 99 0. 98 0. 97 0. 96 0. 95 0. 0 0. 2 0. 4 0. 6 0. 8 1. 0 1. 2 1. 4 1. 6 q 111

Non-locality without signaling GAP Optique Geneva University J. Barrett x et al, quant-ph/0404097 y set of correlations p(a, b|x, y) s. t. 1. p(a|x, y)= bp(a, b|x, y) = p(a|x) 2. p(b|x, y)= ap(a, b|x, y) = p(b|y) Alice Bob facet corresponding to the no-signaling : binary case: unique extremal point! a+b=xy one above each CHSHa p(a, b|x, y) b Bell inequality QM facet corresponding to the CHSH-Bell : P 3 polytope of local correlations p(a, b|x, y) 112

A unit of non-locality, or non-locality without the Hilbert space artillery GAP Optique Geneva University x Alice Bob y Non Local Machine a a + b= x. y b 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. 113

no-cloning theorem without quantum L. Masanes, A. Acin, NG quant-ph/0508016 x ac l M ca lo y a lo ca z l M ac h ine No n GAP Optique Geneva University hi ne No n b c a+b=x. y If then b+c=x(y+z), and Alice can signal to B-C a+c=x. z Non-signaling no-cloning theorem 114

From Bell inequality to cryptography CHSH Q-crypto protocol Alice Bob sifting: • 1 -way • all bits are kept • noisy even without Eve A. Acin, L. Masanes, NG quant-ph/0510094 GAP Optique Geneva University facet corresponding to the no-signaling : a+b=xy intrinsic info > 0 1 -way distillation 2 -way ? ? ? 1 secure QKD against individual 2 -1 QM attacks by any post-quantum non-signaling Eve ! 0 isotropic correlations polytope of local correlations facet corresponding to the CHSH-Bell : P 3 115

From Bell inequality to cryptography A. Acin, L. Masanes, NG quant-ph/0510094 Uncertainty relations, i. e. information / disturbance trade-off: I(E: B|x=0) = fct(QBERx=1) I(E: B|x=1 ) = fct(QBERx=0) GAP Optique Geneva University facet corresponding to the no-signaling : V. Scarani a+b=xy intrinsic info > 0 1 -way distillation 2 -way ? ? ? 1 secure QKD against individual 2 -1 QM attacks by any post-quantum non-signaling Eve ! 0 isotropic correlations polytope of local correlations facet corresponding to the CHSH-Bell : P 3 116

Simulating entanglement with a few bits of communication (+ shared randomnes) GAP Optique Geneva University & define measurement bases. The output & should reproduce the Q statistics: Alice Bob Case of singlet: 8 bits, Brassard, Cleve, Tapp, PRL 83, 1874 1999 2 bits, Steiner, Phys. Lett. A 270, 239 2000, Gisins Phys. Lett. A 260, 323, 1999 1 bit! Toner & Bacon, PRL 91, 187904, 2003 0 bit: impossible (Bell inequality) … but … 117

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

GAP Optique Geneva University hint for the proof: 1 2 119

GAP Optique Geneva University x=0 x=1 x=0 120

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

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

Simulating partial entanglement GAP Optique Geneva University Partially entangled states seem more nonlocal than the max entangled ones ! Partially entangled states are more robust against the detection loophole (P. H. Eberhard, Phys. Rev. A 47, R 747, 1993) 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, 210402, 2005 ) 123

How to prove that some correlation can’t be simulated with a single use of the nonlocal machine ? GAP Optique Geneva University 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 124

Ai Bj x y a+b=xy GAP Optique Geneva University a r. A b r. B For given Ai 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 64 strategies defining 264 different vertices. The polytope is the same as the “no-signaling polytope” studied by J. Barrett et al in quant-ph/0404097 Consequently, no quantum state can violate such a 2 -settings inequality 125

The 1 nl-bit inequality For 3 settings per side: x -2 y 0 0 -2 +1 +1 +1 -1 0 x y P(r. B =0|y) GAP Optique Geneva University • there are 66 strategies defining 3880 different vertices. • There is a unique new inequality: 0 P(r. A =0|x) P(r. A = r. B =0|x, y) Recall: for standard Bell inequalities (i. e with no nonlocal machines) and 3 settings per side, there is also a unique x -1 new inequality: 0 0 y I 3322 = Collins & Gisin, J. Phys. A 37, 1775, 2004 -2 +1 +1 +1 -1 0 0 126

GAP Optique Geneva University Geometric intuition NLM D CHSH D NLM D I 3322 127

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

129 GAP Optique Geneva University

130 GAP Optique Geneva University

The I 3322 -Bell inequality is not monogamous There exists a 3 -qubit state ABC, such that A-B violates the I 3322 -Bell inequality and A-C violates it also. GAP Optique Geneva University A ABC B C (see D. Collins et al. , J. Phys. A 37, 1775 -1787, 2004) 131

132 GAP Optique Geneva University

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

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

GAP Optique Geneva University Where: N. Cerf et al. , PRL 84, 4497, 2000 & 88, 127902, 2002135

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

Optimal individual attack on BB 84 GAP Optique Geneva University Page 182 à 185 de Rev. Mod. Phys. 74, 145, 2002 137

Eve: optimal individual attack GAP Optique Geneva University IAE 1 -IAB IAE Bell inequ. violated Bell inequ. not violated 138

GAP Optique Geneva University Advantage distillation Alice X 0=1 X 1=1 X 2=0 X 3=1 Bob Y 0 Y 1 Y 2 Y 3 …. Xj Yj 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, 4200 -4203, 1999 139

GAP Optique Geneva University IAE 2 -way quantum. Inf. Proc. suffice 1 -way class. Inf. Proc. suffice Bell inequality: can be never violated Alice and Bob separated or classical D 0 140

Quantum Cryptography GAP Optique Geneva University Entanglement Q nonlocality AB measurement Entanglement distillation AB Where is Eve ? P(A, B, E) I(A: B), I(A: E) I(A: B|E) I(A: B E) intrinsic info. Secret key distillation measurement shared secret bit 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 . 141

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

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

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. GAP Optique Geneva University I(A: B E) > 0 iff AB is entangled N. Gisin and S. Wolf, PRL 83, 4200 -4203, 1999. S. Wolf and N. Gisin, Proceedings of Crypto 2000, pp 482 -500 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, 167901, 2004. 144

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

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

How to improve Q crypto ? GAP Optique Geneva University Effect on distance Effect on bit rate Feasibility Detectors 1 - u source Q channel Protocols Q relays Q repeater 147

horizontal vertical pol. -45° pol. PBS@45° GAP Optique Geneva University port 2 Faraday effect PBS@0° port 1 vertical horizontal pol. +45° pol. port 3 148

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