Quantum Communication. Serge Massar Université Libre de Bruxelles

Transcription

1 Quantum Communication Serge Massar Université Libre de Bruxelles

2 Plan Why Quantum Communication? Prepare and Measure schemes QKD Using Entanglement Teleportation Communication Complexity And now what? Talk: Theoretical Concepts & Illustrative Experiments

3 Quantum Communication Why? How?

4 Quantum Communication Why? Quantum Crypto Q. Key Distribution Other protocols Coin Tossing, etc Communication Complexity Foundations of Physics How?

5 Quantum Communication Why? Quantum Crypto Q. Key Distribution Other protocols Coin Tossing, etc Communication Complexity Foundations of Physics How? Photons ω E( x, t) = Aucos t k. x ϕ u Polarization ω, t Frequency/Energy k, x Momentum/Position A, ϕ Amplitude/Phase Wavelength Visible: Free Space Near IR: fiber optics λ 1.5μm

6 Protocols in which a single qubit is Prepared Sent Measured ALICE 0 U ψ BOB Quantum Key Distribution

7 Quantum Key Distribution Alice Eve Bob Alice and Bob want to share a secret key rrr... r { 0,1} N N rrr r

8 Quantum Key Distribution Alice Eve Bob Alice and Bob want to share a secret key rrr... r { 0,1} N N rrr r Eve should not learn the key If Eve tries to learn the key, she is detected

9 Quantum Key Distribution Alice Eve Bob Alice and Bob want to share a secret key rrr... r { 0,1} N N rrr Eve should not learn the key If Eve tries to learn the key, she is detected r Use quantum communication & uncertainty principle / no cloning theorem

10 QKD If Alice prepares: two orthogonal states If Bob measures: in basis i 0 e ϕ i Send to Bob ± 1 0 ± e ϕ 1 Then Bob learns which state was prepared by Alice

11 !!!But Eve can also learn the state by Measuring in same basis!!! QKD If Alice prepares: two orthogonal states If Bob measures: in basis i 0 e ϕ i Send to Bob ± 1 0 ± e ϕ 1 Then Bob learns which state was prepared by Alice

12 QKD: Trick Alice randomly prepares Bob randomly measures in bit=0,1 0 ± 1 bit=0,1 0 ± i 1 Send to Bob 0 ± 1 0 basis 0 ± i 1 90 basis Now Eve is stymied. In which basis to measure?!!if she learns information, she disturbs the state!!

13 QKD: Trick Alice randomly prepares Bob randomly measures in bit=0,1 0 ± 1 bit=0,1 0 ± i 1 Send to Bob 0 ± 1 0 basis 0 ± i 1 90 basis Now Eve is stymied. In which basis to measure?!!if she learns information, she disturbs the state!! Alice and Bob can obtain a secret key by revealing publicly at a later stage the basis used. If the basis are the same, the prepared and measured state constitute the secret key.

14 QKD: Example Emission φ Bit Sent Mst Basis Mst Result Key X X 1 X X

15 QKD needs single photon states In practice: Attenuated coherent states «The poor man s single photon source» α 2 α n 2 α α 2 = e n α n n! 2

16 First QKD Experiment Propagation distance: 30cm Key rate 1 bit/s Journal of Cryptology 5, 3-28 (1992)

17 Quantum Cryptography Today Key distribution over 50km of optical fiber Secret key rate: 1Mbit/s Continuous operation for 36hours Technique used: time bins A. R. Dixon et al., Applied Physics Letters, 96, (2010)

18 Experiments with entangled photons ALICE BOB 1 φ + = d k k A k B

19 Experiments with entangled photons ALICE BOB Why? 1 φ + = k d Q. Comm. Over longer distances: Slightly further than prepare and measure schemes First step towards quantum repeaters Fundamental test of Nature: Quantum Non Locality Device Independent Quantum Cryptography k A k B

20 Non Locality: Aspect type experiment X Input =Measurement Settings Y Alice Bob D φ A φ B D D D a Output=Measurement Outcomes b P(ab XY) = P(A outcome & B outcome A mst setting & B mst setting)

21 Implications of Non Locality Local Hidden Variable Model Pab ( xy) = dλp( λ) Pa ( xλ) Pb ( yλ) If a lhv description is possible, P(ab xy) satisfies all Bell inequalities local deterministic description of measurements is possible If lhv description is impossible: (Quantum) Non Locality measurements results are random, must be secret detected by Bell inequality violation

22 Experiments with entangled particles Equivalence with remote state preparation Equivalence with prepare and measure

23 Equivalences between schemes u i v j i j U V φ + = j (Random) remote state preparation v j V u i i U T (Random) Prepare and measure scheme

24 Entangled Photon source Frequency Doubling ω 2ω Parametric Down Conversion ω 1 2ω ω 2 ω 1 + ω 2 = 2ω : Energy Conservation (approximate) Momentum Conservation (Phase Matching Condition)

25 Frequency Entanglement bpf D Laser PPLN ~Ω bβ ψ = dωf( ω) ω + ω ω ω 0 0 aα bpf D Normalised Coincidence rate Relative phase 2π

26 Nature Physics 3, (2007) 35 coincidences / s

27 Quantum Teleportation Bell Measurement ± φ ψ ± = 0 0 ± 1 1 = 0 1 ± 0 1 r ψ Ur ψ φ + =

28 Entanglement Swapping φ + = A D A D Bell Measurement ± φ ψ ± = 0 0 ± 1 1 B C B C = 0 1 ± 0 1 B C B C r Ur φ + = φ + = A B A B C D C D

29 Bell State Measurement with Photons a 1 a 2 D ( ab ab ab ab ) ψ = α + β + γ + δ Two photons Two modes in beam a Two modes in beam b b 1 b 2 D Coincident detection in both detectors implies that initial state was 1 1 ψ = ab 1 2 ab With probability 1/4 one measures a Bell sta

30 Experimental Quantum Teleportation. Telecomunication Wavelengths distance 55m, passing through a spool of 2km optical fiber Nature 421, (2003)

31 Quantum Communication with atoms and photons. Entanglement of two Yb+ ions Situated in 2 separate vacuum chambers separated by 1m 1 event every 10 minutes Advantages: Information can be stored Interfacable with quantum computer Detection loophole closed.

32 Quantum Communication with atoms and photons. Entanglement of two Yb+ ions Situated in 2 separate vacuum chambers separated by 1m 1 event every 10 minutes Advantages: Information can be stored Interfacable with quantum computer Detection loophole closed. Nature 449, (2007)

33 Quantum Communication Complexity input x input y ALICE PRIOR RESOURCE COMMUNICATION BOB ouput f(x,y) TASK:Minimum Communication to provide the correct output

34 Quantum Communication Complexity input x PRIOR RESOURCE Shared Randomness Entanglement input y ALICE ouput f(x,y) COMMUNICATION Cbits Qbits BOB TASK:Minimum Communication to provide the correct output

35 Example: Equality Input x ALICE { } 0,1 n Output x = y or x y PRIOR RESOURCE Shared Randomness Entanglement COMMUNICATION Cbits Qbits Input y { } BOB 0,1 n

36 Example: Equality Input x { } 0,1 n PRIOR RESOURCE Shared Randomness Entanglement Input y { } 0,1 n ALICE BOB Output x = y or x y COMMUNICATION Cbits Qbits No Error: n cbits of communication required Small Error probability & shared randomness Log(n) cbits of communication required Deutsch-Jozsa setting: either x=y or x differs from y in exactly n/2 positions O(0.007n) cbits required Log(n) qubits required Log(n) ebits + Log(n) cbits

37 Example: Sum mod 2π PROMISE θ = 0 or π (mod 2 π) i i θ 1 θ 2 θ 3 θ n A1 A2 A3 An Question: is θ = 0 or π (mod 2 π) i i PRIOR RESOURCES COMMUNICATION Bounded Error: requires O(n Log(n)) cbits n qubits 1GHZ state + n cbits

38 Experimental Realisation of Sum mod 2π PRA 72, R 2005

39 Faster Conclusion The future of Quantum Better detectors Further Via satellite (?) Repeaters Communication Interfacing with stationary qubits Quantum memories for light Error Correction