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6. Chapter Quantum Communication 6.1 Quantum key distribution Brief history of cryptology Kryptos (hidden) + logos (word) ~400BC Spartans Scytale spiral strip of parchment tapered baton (transposition = rearrangement of plaintext by special permutation) Julius Caesar Caesar cipher (substitution = replacement of letters by other letters) AD BE COLDFROG two basic methods of encryption 1940's Germans ENIGMA ciphers broken by British team (Alan Turing) with the first digital computer (COLOSSUS) Crypt analysis was the midwife of computer science. plain text Ek (P) = C encryption algorithm key cryptogram Dk (C) = P decryption algorithm 1 The encryption and decryption algorithms are publicly known. The security of communication is ensured only by a secret key. Vernam's one time pad: plain text P key k c=P+k (mod 30) Claud Shannon: If the key is truly random, has the same length as the message and is never reused, the one-time pad cryptography is perfectly secure. What is wrong with classical cryptography? snag! How to distribute secret keys? Mathematical solution: Public key cryptography (Diffie-Hellman, Rivest-Shamir -Adleman (RSA)) public key private key Physical solution: Quantum cryptography Quantum threat: Shor's factoring and discrete log algorithms Grover's data search algorithm 2 6.1.1 Single photon QKD system (BB84 protocol) Single Photon Source Det 2 0 Alice or or 1 Quantum Channel PBS Det 1 Bob or General Quantum Measurement C. H. Bennett and G. Brassard (1984) Eve A. Intercept and resend attack Simple argument for security If Eve intercepts all single photons and measures their polarization states in a rectilinear basis and sends new single photons with identical polarization states, she must create 25% error, which can be easily detected. 1 1 1 = 2 4 2 probability that Alice and Bob choose a diagonal basis probability that Bob obtains a wrong result for a fake photon No measurement theory No cloning theory 3 B. General attacks and absolute security individual attack : individual unitary interaction + individual meas. collective attack : individual unitary interaction + collective meas. coherent attack : collective unitary interaction + collective meas. 1 2 3 n Alice's photons . . . . 1 2 3 Arbitrary Unitary Transform . . . . Quantum Bob's photons Eve's probe (m>n) . . . . Computer Eavesdropping m Quantum memory + POVM meas The absolute security against the most general coherent attack is proven for an ideal system. D. Mayers, J. ACM 48, 351 (2001) P. W. Shor and J. Preskill, Phys. Rev. Lett. 85, 441 (2000) C. Error correction (reconciliation) In a real communication system, errors are bound to occur. The error must be corrected through public discussion to achieve error free communication. Shannon's noisy channel coding theorem: n lim(r n) h( ) = [- log2 - (1- )log 2 (1 - )] length of the shifted key with an error rate of . minimum # of bits that must be exchanged publicly. 4 Parity check code: G. Brassard and L. Salvail: EUROCRYPT '93, Vol.765, ed. by T. Hellseth (Springer, Berlin, 1994) p.410 ......... block #1 block #2 block #3 The parity of each block is announced publicly between Alice & Bob. If their parities agree, they proceed to the next block. (The block size is chosen to have not more than two errors.) If their parities disagree, they recursively cut the block into two subgroups to find which sub-group has an error. Shuffle the positions of the bits and repeat the process. This scheme does not need to discard any bit from the shifted key but suffers from the additional leakage of information to Eve. D. Privacy amplification ( n - ) n binary matrix K Alice public channel Bob K= 0 1 1. 1 1 0. 1 0 1. n .... 0 .... 1 .... 0 n- random choice of 0 or 1 k final = K . k shifted (mod. 2) ( n - ) - bit error free string (n - bit) 5 is determined so that Eve has negligible information about the final key. IE ( final; KV )= H (k final )- H(k final KV ) k Eve's mutual information conditional entropy negligible The final key is secure. V: all the information available to Eve before privacy amplification; one from raw quantum transmission and the other from error correction. Experimental system Disturbance parameter nerror + D nD = innocent bit error rate n D =0 nrec nerror : # of error bits nD : # of dual fire events nrec : total # of received bits passive demodulation D : weighting parameter (=1/2 is OK for passive demodulation) 0 : error free = 1 2 : most random case 6 Collision probability Agreement between the error-free shifted key and Eve's key pc If If 1 2 + 2 - 2 2 1 pc = =0 , 2 (equivalent to random guess) 1 = , pc = 1 which means Eve can learn the entire which means Eve gets no information. 2 string by a simple, intercept and resend attack compression factor final key n - = n[- log2 pc ]- - t - s : # of bits in the side information Eve obtains from error correction s,t : security parameters chosen by Alice and Bob Eve's mutual information In order to satisfy IE 1 , t must scale logarithmically with the size of the final key but a constant s is OK. 2- s IE 2 -t (n - ) + In2 E. Security of practical systems Poissonian photon source Lossy channel P (n) = n e n! n - n P ( m) = ( n ) m e - n m! 7 Eve's photon splitting attack photon splitter Alice QND measurement of photon number Bob ( n - 1 ) photons yes n>2? extract and keep one photon no block the signal and pretend a vacuum state If a channel loss is large, the photon arrival rate at Bob's station is much lower than the photon sending rate at Alice's station. Eve can replace a lossy fiber with a lossless fiber and block one photon pulse completely. If she finds the pulse with more than two photons, she will extract one photon and keep it until Alice and Bob disclose their modulation/demodulation bases. If the (n-1) transmitted photons make the average photon arrival rate which is determined by the channel loss, Eve can extract "complete" information about the key. The entire key is completely insecure. Fraction of detection events due to single photon states = n - nm : n worst case single photon probability n : total # of detection events nm : total # of multi-photon states that are injected into the quantum channel 8 Modified compression factor 2 1 = - log 2 + 2 - 2 - log2 pc - log 2 pc 2 Each multi-photon state reveals a bit of information to Eve. outer factor of Eve can create a larger error rate on the reminder of the key by the intercept and resend attack, while maintaining the same overall bit error rate. Detector dark count PBS DA Ps DB signal photon arrival rate per pulse = 1 error rate channel detector loss efficiency single photon source dark count makes an error with 50% dark count rate probability (two detectors independently 2dB 1 emit a dark 2 Ps + 2 dB count) total count rate dB : detector dark count per pulse _ Pe ~ If Pe > 1/4, Eve's intercept and resend attack for even a single photon source cannot be distinguished from the dark count induced innocent error. Even if a source emits exactly one photon per pulse, the secure communication becomes impossible once 2dB exceeds Ps = . The maximum channel loss < 2dB . 9 Eve's hybrid attack: intercept and resend attack for n = 1 and photon splitting attack for n 2. attenuated Poisson light P (n) = e- n ( n) n / n! average photon number sent by Alice (optimized) probability of finding one or more photoelectrons per pulse: Ps = n=1 _ _ _ - n ( e n) n / n! _ probability of finding a detector count: _ Pclick ~ Ps + 2dB probability for two or more photons per pulse sent by Alice: Pmulti = e- n ( n ) n / n! n=2 _ _ For all pulses with n 2, Eve performs a photon splitting attack. For part of pulses with n = 1, Eve performs an intercept & resend attack and block the remaining pulses. The secure communication becomes impossible if total dark count 1 _ 2dB > 2 one-half of the dark count makes an error 1 4 ( Pclick - Pmulti ) If Pmulti Pclick , she does not need to perform an intercept & resend attack. 10 probability that Eve performs an intercept & resend attack for part of the pulses with n = 1. _ B < 2d_ + n < 2 dB ( nopt = 2 dB) _ _ n 2 If n is too large, Eve's photon splitting attack is effective. _ If n is too small, Eve's intercept & resend attack is effective. F. BB84 with a sub-Poissonian photon source Second order correlation photon pulse contained in an interval [0, ] . g (0) = (2 ) 0 0 ^ ^ ^ a + (t )a+ (t )a(t )a(t ) dtdt ^ a + (t )a(t ) dt ^ ^ 0 2 ^ ^ [a(t ), a+ (t )]= (t - t ) ^ ^ n(n - 1) g ( 2 ) (0) = = n 2 ^ ^ n : average photon number per pulse i(i - 1)p(i) i= 2 n ^ 2 2p(i ) i= 2 i(i - 1) 2 ^ n 2 = 2 pm ^ 2 n probability sum for multi-photon states ^ 2 n g (2 ) (0) pm 2 > 1 (2 ) g (0) = 1 < 1 : super-Poisson source : Poisson source : sub-Poisson source 11 Communication rate pclick = psignal + pdark - psignal pdark psignal + pdark probability for detection event per pulse R = lim n- N N n = pclick N : # of bits in the shifted key n - : # of bits in the final key (after privacy amplification) s + t R = lim pclick - log 2 pc - - n n n s +t lim = 0 ( S ~ O (log n), t : constant ) n n n lim n = f ( ) h( ) Shannon limit degradation from Shannon limit ( 1) Usually, the error correcting code works below the Shannon limit. We need to disclose more bits than the coding theorem dictates. psignal , pdark , , n ^ psignal = p ( n ) 1 - (1 - ) n n =0 (1 - )n 1- n total optical loss In order for n-photons not to arrive at the detector, they must be all lost. The probability for this is (1 - )n . error rate due to imperfect optics = = psignal + pdark 2 pclick pclick - pm pclick : error rate : fraction of single photon states 12 ^ pclick = n + d 2 ^ n g( )(0 ) pm = 2 2 ^ : increases linearly with n ^ : increases quadratically with n 2 p(m 3) is negligibly small! ^ ^ If n is too high, R will drop due to an increase in pm . If n is too low, R will drop due to a decrease in pclick . ^ optimum n to maximize R -8 Examples: detector dark count 20 s-1 d = 4 10 measurement window 500 ps four detectors error rate due to imperfect optics = 0.01 source efficiency device= 1 (perfect) identical basis (50%) d, =1 (loss) 2 In order to avoid the multi-photon states, n must be decreased with increasing the channel loss. (loss) multiThere is no photon state. 2 dB 2dB 13 Source efficiency ndevice () = 1 () = 10-1 () = 10-2 () = 10-3 Artificial attenuation at the source is required to suppress the multi-photon state. This is a remarkable result, because device < 1 can achieve the same Why? cutoff loss as device = 1. G. Closed form approximations for cutoff loss Allowable error rate (or channel loss) = 25% error due to intercept & resend attack fraction of single photon states in the shifted keys 1 4 Eve can intercept and resend all single photon states, while a photon splitting attack on the multiple photon states. = = psignal + pdark 2 pclick pclick - pmult pclick cutoff loss (An entire message is insecure) optimize n ^ ^ (2 ) 1 d n g (0) : cutoff loss = + ^ 2 1- 4 n 14 Ideal single photon source: device =1 and g (2 )(0) = 0 ^ n = 1 : optimum average photon number ideal = d : maximum transmission loss 1 - 4 Non-ideal photon source: device =1 but g (2 )(0) > 0 ^ n = 2d : optimum average photon number g(2 ) (0) g (2 )(0) is invariant against a linear loss. non-ideal 2dg(2 )(0) : minimum transmission coefficient = 1 - 4 2d ^ If device exceeds n = g(2) (0) , which is usually much smaller than one, one can always achieve the above limit. Minimum transmission and optimum photon number exact closed form 15 6.1.2 Entangled photon-pair QKD system (Ekert91 protocol) Det 2 Quantum Channel Quantum Channel Det 2 PBS Det 1 Alice Det 1 PBS EPR-Bell source Bob Arbitrary Quantum State Preparation A. K. Ekert (1991) Eve A source emits pairs of photons in a polarization singlet state: ab = 1 V 2 ( a H b - H a V b ) Alice and Bob perform measurements on polarizations along one of three directions given by unit vectors ai and bj (i, j=1, 2, 3) which lie in the x-y plane, perpendicular to the trajectory of the photons. 3a a 2 3b 1a 2b 1b Alice and Bob choose the orientation of the analyzers randomly and independently for each pair of incoming photons. Each measurement can yield two results, +1 (transmission) and 1 (reflection), with respect to the polarization beam splitter (PBS). 16 After the transmission has taken place, Alice and Bob can announce in public the orientations of the analyzers and divide the results into two separate groups: a first group for which they used the same orientation and a second group for which they used the different orientation. How to create secret keys ? Correlation between Alice and Bob: According to quantum mechanics, E ( ai , b j ) = p++ + p-- - p+- - p-+ E ( ai , b j ) = - cos 2 ia - b = - ai i b j j ( ) For the first group, i.e. two pairs of same analyzer orientation (a2, b1) and (a3, b2), perfect anti-correlation of the results is expected: E ( a2 , b1 ) = E ( a3 , b2 ) = -1 Generation of secret keys How to detect eavesdropping ? Alice and Bob reveal publicly the results they obtained from the second group. Bell's inequality of Clauser, Horne, Shimony and Holt type: S = E ( a1 , b1 ) - E ( a1 , b3 ) + E ( a3 , b1 ) + E ( a3 , b3 ) According to quantum mechanics, S = -2 2 This result is obtained only when there is no eavesdropping (for a pure singlet state). 17 Eavesdropper's intercept and resend attack: S = ( na , nb ) dna dnb ( a1 i na )( b1 i nb ) - ( a1 i na )( b3 i nb ) + ( a3 i na )( b1 i nb ) + ( a3 i na )( b3 i nb ) na , nb : unit vectors oriented along the directions of Eve's measurement axes ( na , nb ) : Eve's strategy for the probability of intercepting and measuring along a given direction. If only one photon (a) is measured by Eve, one may put nb = - na . S = ( na , nb ) dna dnb 2 na i nb - 2S 2 (for any strategy ( na , nb ) ) Contradiction to quantum mechanics Eve's replacing a singlet state with a fake photon-pair: ^ = p ( a , b ) a a b b d a d b 2 - 2 18 S = 2 p ( a , b ) d a d b cos 2 1a - a cos 2 1b - b - 2 { ( ) ( ) ( + cos 2 ( + cos 2 ( - 2 - cos 2 1a - a cos 2 3b - b a 3 a 3 - a - a ) ( ) cos 2 ( ) cos 2 ( b 1 b 3 - b - b ) ) ) } = 2 p ( a , b ) d a d b 2 cos 2 ( a - b ) - 2S 2 (for any p ( a , b ) ) contradiction to quantum mechanics 19 6.1.3 Entangled photon-pair QKD system (BBM92 protocol) Entangled Photon Source Det 2 Quantum Channel Quantum Channel Det 2 PBS or Det 1 Alice Det 1 PBS or EPR-Bell source Bob Arbitrary Quantum State Preparation Eve polarization singlet state: 12 = 1 2 C. H. Bennett, G. Brassard and N. D. Mermin (1992) [ 1 2- 1 2 ] perfect correlation in V-H basis logical "0" logical "1" = 1 [ 1 2 - 1 2 2 ] perfect correlation in 45 basis logical "0" logical "1" Alice & Bob independently and randomly choose the demodulation basis. If their bases match, they keep the results as shifted keys. A key does not exist in the transmission channel. Rather it is created by a projective measurement process. Similar to BB84 protocol but free from Eve's photon splitting attack. 20 A. Eve's attack Eve can intercept and block the EPR-Bell pair from the source, and prepare a new photon-pair for which she has at least "partial information" (three partially entangled photons). This necessarily introduce a bit error between Alice & Bob which is absent in a perfect singlet state without eavesdropping. However, in a practical system, a bit error is caused by innocent imperfection of the system components. This bit error cannot be distinguished from the error caused by the eavesdropping. B. False coincidence parametric down-converter Bob Alice entangled pair #1 entangled pair #2 t Poissonian photon-pair source t no danger for photon splitting attack but introduces innocent bit errors (no common information stored) independent 21 average # of photon-pairs per pulse true coincidence rate: P true = 2 2 n _ (survival of the photon-pair) four detectors in each station false coincidence rate: P false = 2 2n 2 + 8 n d + 16 d survival of one photon and one dark count _ _ 2 survival of un-paired two photons simultaneous two dark counts (4d 4d) 2 = 1 2 P false P true+ P false p(n)n(n - 1) = n n=2 - n = n 2 (Poisson) This innocent error is considered solely as the error introduced by eavesdropping. (It is assumed that Eve can replace a practical system with an ideal system with no transmission loss, perfect detector efficiency and no dark count.) Eve can perform an intercept and resend attack by taking advantage of innocent bit errors. C. BBM92 with ideal entangled photon-pair source This source emits exactly one entangled photon-pair per pulse. Coincidence probability pcoin = ptrue + p false true coincidence from a photon-pair ( negligible dual fire events) false coincidence x Alice source L-x Bob 22 -( x 10 ) , loss coefficient (dB km) ) Tx = 10 ptrue = Tx TL- x ( : constant (independent of x) = TL p false = 4Tx d + 4TL - x d + 16d one photon and one dark count 2 two dark counts minimum when x = L 2 The source should be located at a mid-point. p false = 8TL 2 d + 16d ( min) 2 Error rate and collision probability = pc p false 2 + ptrue pcoin 1 + 2 - 2 2 2 represents an amount of information leaked to Eve in a raw quantum transmission. Final key after privacy amplification n - = n - log 2 pc - -t-s n ( 1 2 entropy function = represents an amount of information leaked to Eve in error correction Npcoin : length of the error corrected key 2 ( N : total # of signal pulses) : probability for the same analyzer orientation) Alice & Bob choose the same basis with a 50% probability. Communication rate n - pcoin R = lim = {- log2 pc + f ( )[ log2 + (1- ) log2 (1 - )]} N N 2 23 D. BBM92 with a parametric down-converter non-collinear Type II phase matching: I= i (2 )Ve ( i - a - b ) ^ (a b^ + + x y ^y ^ + a+ bx+ + h.c. ) pump wave amplitude signal idler x,y: polarization of the photon = exp = e 1 i ( 2 T 0 I( t )dt 0 tanh a x+b y+ + a + b + ^ ^ ^y ^x ) 0 cosh 1 = tanhn n 2 cosh n= 0 [ ax n by +n ay n bx ] = f ( (2 ),V,T ) If is sufficiently small, is a linear superposition of a vacuum state and EPR-Bell state. Linear loss in a transmission line ^ ^ ^ a TL a + 1- TL c 2 2 ^ ^ ^ b TL b + 1- TL d 2 2 vacuum state received state density matrix 24 ^ ^ ^0 ^ ^ ^ ^ ^ AB = A + B0a 0b + C(ua b + 0a ub ) + ^a ^b ^ + Du u + (1- A - B - 2C - D)D ^ 0 : vacuum state 1 One of the photon-pair is lost, 2 so the remaining ^ D : more than one photon in either a or b mode photon is left in (dual fire) random mixture. ^ ^ u = I : unpolarized photon state 2TL tanh 1 2 A= 4 2 4 cosh 2 1- tanh 1 - TL 2 2 2 ( ) 2 true coincidence ( ) 2T ( - T ) 1 tanh 1 C= cosh 1- tanh ( - T ) 1 4T ( - T ) tanh 1 1 D= cos 1 - tanh ( - T ) 1 L 2 L 2 4 2 L 2 2 2 L 2 4 L 2 4 2 4 2 L 2 B= 1 1 2 2 cosh 4 2 1- tanh 1 - TL 2 no detection event at both detectors 2 3 only one detector receives a photon two detectors receive uncorrelated photons ptrue = A p false = 16d 2 B + 8dC + D 25 Numerical example: 0.8 m Si-APD system d = 5 10 -8 (optimized for each channel loss) = 0.01 26 BB84 QKD experiment with a single photon source Alice He Cryo. dot Pinhole Lens Lens Counter Spec. slit /2 plate Det 0 Sm fiber EOM Lens PBS grating Data Gen. Flip Mirror Laser Pulse TIA Amp. channel 50-50 /4 plate PBS BSP Det 4 Det 1 Det 2 Det 3 Bob TIA 0.5 0.4 0.3 0.2 0.1 0 Alice R Alice L Alice H Alice - Bob -H V Bob -R Bob -L Bob -V Communication rate 70KHz Error rate 3% E. Waks et al., Nature 420, 762 (2002) 27 Laser (sim) Turnstile (sim) Device Turnstile Device (Measured) Laser (Measured) E. Waks et al., Nature 420, 762 (2002) 28 Energy-time entanglement + + = 1 i ( + ) s 1 s 2 +e l 1l 2 (+ +) [ 2 ] (+ ) ( ) ( +) H. Zbinden et al., Experimental Quantum Computation and Information, Proceedings of the International School of Physics "Enrico Fermi" (IOS Press, Amsterdam, 2003) pp.217-232. 29 BBM92 QKD experiment W. Tittel et al., PRL 84, 4737 (2000) Alice & Bob public communication side peaks s p s A A s B : "0" l pl e i( + ) s l B : "1" central peak p l A p l s B A + e i l s B optimize , , + + : "0" : "1" (4 dB) (10 dB) H. Zbinden et al., Experimental Quantum Computation and Information, Proceedings of the International School of Physics "Enrico Fermi" (IOS Press, Amsterdam, 2003) pp.217-232. 30 6.2 Quantum teleportation How to transfer a qubit of information to a (unknown) distant point? Alice unknown state 3 Bell state analysis 1 2 two bits of classical information ^ U 3 Bob 1 EPR-Bell pair 23 A. Principle of operation - spin singlet state: 23 = 1 2 ( 2 - 3 1 2 3 ) unkown input state: 1 = a 1 + b - initial state: 1 23 ( |a|2+|b| 2 =1) no classical correlation, no quantum entanglement between the particle 1 and 2 & 3. No measurement on either member of the EPR pair, or both together, can yield any information about 1 . Instead, Alice performs a measurement on the joint system of particles 1 & 2 in the EPR-Bell basis. - 12 = + 12 - 12 + 12 1 - 1 2 1 2 2 1 = + 1 2 1 2 2 1 = - 1 2 1 2 2 1 = + 1 2 1 2 2 31 - 1 23 = a 1 + b ( = 1 - 12 2 ) 12 ( - ) ( -a - b ) + ( -a + b ) 1 2 3 2 3 3 3 + 12 3 3 + 12 a 3 + b - ( 3 3 )+ (a + 12 3 -b 3 )] = 1 - 12 (- 2 )+ + 12 -1 0 0 1 3 0 1 - + 12 1 0 0 - 1 + + 12 3 3 1 0 irrelevant phase If the Bell state analysis reports - 12 no need to manipulate the particle 3 180 rotation about z-axis 1 0 0 -1 - - 3 + 12 3 - 12 180 rotation about x-axis 0 1 1 0 3 + 12 180 rotation about y-axis 0 -i i 0 -i 3 The particle 3 is now prepared in an unknown state except an irrelevant phase factor. Since the input unknown state 1 is destroyed and the obtained two classical bits of information do not contain any information about 1 , the quantum teleportation does not conflict with "no cloning theorem" and "no measurement theorem." In quantum teleportation with EPR-Bell pairs shared beforehand by both parties, Alice does not need to know the exact location of Bob since she 32 can broadcast the Bell measurement results. B. Bell-state analyser in linear optics Collision of the two identical particles 1 & 2 at 50-50% beam splitter PBS PBS d a c b 50-50% beam splitter Bosonic particle 1 symmetrization postulate - - 12 12 = 1 2 - 1 2 a 1 b 2 - b 1 a 2 2 spin-singlet state orbital-singlet state anti-symmetric spin wavefunction anti-symmetric orbital wavefunction overall wavefunction is symmetric beam splitter = Hadamard transformation H= 1 1 1 2 1 -1 a b operates only on orbital state 1 (c + d 2 1 Hb = (c - d 2 Ha = ) ) 33 - H 12 = 1 c 2 1 d 2 -d 1 - c 2 = 12 - - 12 12 is an eigenstate of the beam splitter Hamiltonian. + 12 - 12 + 12 + 12 + - 12 = 12 + 12 1 [a 1 b 2 + b 1 a 2 2 ] spin-triplet states orbital-triplet state symmetric spin wavefunction symmetric orbital wavefunction overall wavefunction is symmetric 1 c 1 c 2 - d 1 d 2 2 - If one photon exits at each port c & d, the input state is 12 . + H 12 = If two photons exit together either at port c or d, and also if these two + photons have the orthogonal polarizations, the input state is 12 . If two photons exit together either at port c or d, and also if these two - photons have the same polarization, the input state is either 12 + or 12 . (inconclusive result) - + In order to distinguish 12 and 12 , a nonlinear gate is needed. In general Bell state analysis consisting of C-NOT gate and Hadamard gate, the two particles do not need to be identical quantum particles. 34 C. Full Bell state analyzer 12 1 in H C-NOT gate Hadamard gate out 12 out 2 12 12 12 12 1 [0 2 1 = [0 in 2 1 = [0 in 2 1 = [0 in 2 = in 1 0 2 + 1 1 1 2 ] 12 0 2 - 1 1 1 2 ] 12 12+ 110 12- 110 2 = 010 = 110 2 1 out 2 1 ] ] 12 out = 0 112 = 1112 1 2 12 out One C-NOT gate followed by a Hadamard gate realizes a full Bell state analyzer. However, optical nonlinearity is too weak to construct a C-NOT gate. How to trap a photonic qubit in more nonlinear, more stable physical qubit? ( Quantum memory, Photon trapping) Quantum Memory 35 D. Entanglement swapping Since teleportation is a linear operation applied to the quantum state , it will work not only with pure states but also with mixed states and entangled states. For example, let Alice's original particle 2 be part of an EPR singlet with another particle 1. Alice Bell state analyzer 1 EPR-Bell source 2 3 classical communication Bob ^ U 4 EPR-Bell source 1234 = 1 1 - - 1 2 3 4 1 2 3 4 2 2 1 + + - - + + - - = 23 14 - 23 14 - 23 14 + 23 14 2 - + ^ U = 14 14 - + ^ U = 14 14 - - ^ U = 14 14 The Bell state analysis performed on the particles 2 & 3 projects out one of the four EPR-Bell states in the particles 1 & 4, even though they are uncorrelated before the measurement. After the unitary operation for the particle 4, the particles 1 and 4 would be left in a singlet state. This process is cascaded to arbitrary number of Bell state analyzers. 36 2 bits Bell state analyser 1 2 bits Bell state analyser 2 2 bits Bell state analyser 3 1 EPR-Bell source 2 3 2 bits EPR-Bell source 4 5 EPR-Bell source 6 ...... Bell state analyser N ^ U EPR-Bell source 2N +2 N 2 bits of classical information projects ...... 2N +1 - + - 1,2 N + 2 , 1,2 N + 2 , 1,2 N + 2 or + 1,2 N + 2 In the end, the particle 1 and 2N +2 are prepared in a EPR singlet state. ^ unitary operation U on particle 2N +2 37 E. BBM92 QKD with entanglement swapping EPR-Bell state analyzer p true 1 2 = 2 fiber loss from a source to a Bell analyzer (two photons simultaneously arrive at Bell state analyzer with probability of 2) only 50% success probability of linear optics Bell analyzer p false = 6 d + 12d 2 one of two photons arrive and one dark count in other detectors (6 = 2 3) which photon? which remaining detectors? two dark counts (12 = 4 3) first dark count second dark count ptrue g = true p + p false ^ I + (1- g) 4 two photons 1 and 4 are left in random polarization 38 ^ ^ 14 = g + or - depending on the measurement result Since the photons 2 and 3 are lost into reservoirs with a probability (1-g), the remaining photons 1 and 4 are left in the maximally mixed states. N entanglement swaps each photon arrival probability = 10 L - 10 (2 N +2 ) : fiber loss (dB/km) : detector efficiency L : total length between Alice and Bob ^ ^ 1,2 N + 2 = g + (1 - g N N ) ^ I 4 Probability for all N Bell state analyzers to report a measurement result: pBell = (p true +p false N ) ptrue = pBell g N 2 The arrival probability of photon 1 and photon 2N + 2 p false = p Bell g N (8 d + 16d 2 ) + (1- gN ) 2 error in the Alice/Bob detector error in the swap process [ ] (passive demodulation with four detectors) p false 2 + ptrue error rate = pcoin communication rate pcoin = ptrue + p false R= pcoin - { log 2 pc + f ( )[ log 2 + (1- )log2 (1 - )]} 2 39 Bell state analyzer efficiency ( = 1/2 ) detector dark count (4 d) photon arrival rate dark count E. Waks et al., PRA 65, 052310 (2002) Entanglement swapping with linear optics suffers from enormous reduction of efficiency due to the loss of a transmission line. Quantum repeater 40 F. Quantum teleportation of N-state particle F1. Maximally entangled states Consider systems A and B with dim HA = dim HB = d. The state 1 d = jA jB d j ^ ^ and the states produced by local unitary operations, U AU B d , are called maximally entangled states. All maximally entangled states have the same marginal density operator (maximally mixed state): ^ A = 1 j d j A A ^ j , B = 1 j d j B B j All maximally entangled states are inter-convertible by unitary operations on system A (B) alone. F2. Complementary bases Consider a system with d-dimensional space H . Take a standard basis (Z-basis) { j} d -1 j =0 d -1 j =0 d -1 j =0 . Define unitary operators ^ X = j +1 j ^ Z = exp ( 2 i / d ) j j where d 0 . j 41 ^ ^ ^ X d = Zd = I ^ ^ ^ ^ Z m X l = exp ( 2 i / d ) lm X l Z m Define 0 1 d j j ^ k Zk 0 ^ Since X k = exp - ( 2 i / d ) k k , k and called a complementary basis to { j { } } d -1 k =0 is an orthonormal basis d -1 j =0 . ^ In quantum optics, Z -basis corresponds to photon number eigenstate ^ and X -basis corresponds to Pegg-Barnett phase eigenstate. F3. Bell basis Consider systems A and B with dim HA = dim HB = d. 1 0,0 AB j A j B :maximally entangled state d j l ,m Since AB ^ l ^m X A Z B 0,0 = exp ( 2 i / d ) l l ,m ^ ^- Z A Z B1 l ,m ^ ^ X A X B l ,m AB AB AB = exp - ( 2 i / d ) m l ,m AB { l , m AB } forms an orthonormal basis (Bell basis) on l ,m ^ l ^m = X A Z A 0,0 ^m ^ - = Z B X B l 0,0 HA HB . This basis states can be converted to one another by local unitary operations to system A (B) alone: AB AB AB 42 F4. Entanglement swapping dim HA = dim HB = dim HA = dim HC = d 1 : two Bell pairs 0,0 AB 0,0 AC = j A j B k A k C d j ,k Alice holds A and A , while Bob holds B. Systems B and C have no correlations initially. Alice measures system A A on the Bell basis l ,m . { AA } AA 0,0 0,0 AB 0,0 AC d j ,k 1 = 0,0 BC d = 1 d jk j B k C If the measurement result is 0,0 AA , systems B and C are projected onto a maximally entangled state 0,0 . BC AA l ,m 0,0 AB 0,0 AC AB AB ^ l ^- = AA 0,0 X A Z A m 0,0 ^- ^ - = AA 0,0 Z B m X B l 0,0 = 1 ^ - m ^ -l Z B X B 0,0 d BC 0,0 0,0 AC AC If the measurement result is l ,m , systems B and C are ^- ^ - projected onto a maximally entangled state Z B m X B l 0,0 . AA BC Thus, if Alice sends outcome (l, m) faithfully to Bob through a ^ ^ classical channel, Bob can undo the unitary operation Z - m X - l to obtain 0,0 B B BC . 43 F5. Quantum teleportation Just apply C AC 0,0 A in the above argument. N -state particles in a completely entangled state 23 = 0,0 23 1 = N j =0 N -1 j 2 j 3 j = 0,1,N - 1 : orthonormal basis joint measurement 3 Alice 1 unknown state ^ U 2 entangled particle source 3 1 = cj j 1 j=0 N -1 Joint measurement for particle 1 and 2: A measurement result is 1 N -1 i 2 jn N eigenstate nm = n ,m = j 1 ( j + m ) mod N e 12 N j =0 1 ^ ^ n ,m 0,0 23 1 = Z1- m X 1- n 1 12 d classical communication from Alice to Bob (nm) 2 log2N (bits) 2 44 Unitary operation ^ ^ ^ U nm = X Z = ei 2 kn N k n 1 m 1 k =0 N -1 ( k + m ) mod N 3 = ck k k=0 N -1 3 : complete teleportation G. 4 way coding Protocol (1) Alice and Bob each obtain one particle of an entangled pair, say, in 1 + 12 = [0 1 1 2 + 1 1 0 2 ] 2 (2) Bob performs one of the four unitary transformations on his particle 2. + i) Identity operation 12 ^ ii) State exchange x 1 0 1 + = 12 = 2 [ 0 1 0 2 + 1 1 1 2 ] 1 0 1 0 -1 iii) State dependent phase shift - ^ z = 12 = 2 [ 0 1 1 2 - 1 1 0 0 -1 2 ] iv) ii)+iii) - ^ ^ z x 12 = 1 [0 1 0 2 2 - 1 1 1 2] 4 distinguishable messages = 2 bits of information 45 (3) Bob sends his particle 2 to Alice, who reads the encoded information by the Bell state analyzer. Impact Transmission of one qubit along a quantum channel realizes the transfer of two classical bits of information, if Alice and Bob share an entangled state initially. Fig. 3.7. in The Physics of Quantum Information, D. Bouwmeester, A. Ekert, and A. Zeilinger (Eds.), p.63 (Springer, New york, 2001) Table 3.1. in The Physics of Quantum Information, D. Bouwmeester, A. Ekert, and A. Zeilinger (Eds.), p.63 (Springer, New york, 2001) 46 Coincidence rates CHV ( ) and CHV' ( ) depending on the path length difference , for transmission of the state - . The constructive interference for the rate CHV' enables one to read the information associated with that state. Coincidence rates CHV ( ) and CHV' ( ) as functions of the path length difference when the state + is transmitted. For perfect tuning (=0) constructive interference occurs for CHV, allowing identification of the state sent. Coincidence rates CHH ( ), CHV ( ), and CHV' ( ) as functions of the path length detuning . The maximum in the rate CHH signifies the transmission of a third state - encoded in a two-state particle. CHH is smaller by a factor of 4 compared to the rates of Figures 3.8 and 3.9 due to a further reduced registration probability of - , see text. Fig. 3. 8. 3. 10. in The Physics of Quantum Information, D. Bouwmeester, A. Ekert, and A. Zeilinger (Eds.), p.63 (Springer, New york, 2001) 47 6.3 Quantum Repeater How to create an EPR-Bell pair at two distant points? Quantum repeater based on entanglement purification and swapping What is a key device for constructing a quantum repeater? Quantum memory connecting nuclear and photonic qubits A. Entanglement purification -- Principle -- How to purify a single particle state? ^ A = f 0 A A 0 + (1 - f ) 1 A A 1 ? ^ C = 0 C C 0 C - NOT 00 0 + (1 - f ) 1 0 C C 0 ^ Output: AC = f 0 A A C C A A 11C C 1 ^ A = 0 Projection by A A 0 Purified sub-ensemble 0 C C 0 There is a catch: By assuming we have an ancilla qubit c in a pure state 0 C already, the whole purification idea seems to be pointless. 48 What can we do if we do not have a purified ancilla qubit? ^ A = f 0 A A 0 + (1 - f ) 1 A A 1 ? 0 ^ C = f 0 C C 0 + (1 - f ) 1 C A A C C 0 C 1 2 2 ^ Output: AC = f 0 ( 0 + (1 - f ) 1 A A 1 0 ) C C 0 1 + f (1 - f ) 0 ^ A = f 0 Projection by ( A A 0 +1 A A 1 1C A A ) C A A 0 + (1 - f ) 1 f2 2 1 0 C C 0 f= f 2 + (1 - f ) 1 > f if f > 2 If we iterate this procedure, as indicated by the staircase, we are able to distill particles arbitrarily close to the pure state 0 , as A long as the initial ensemble is sufficiently large. 49 Alice and Bob share an ensemble of two-particle (mixed) entangled ^ states AB . How to purify mixed entangled states by local unitary operations, measurements and classical communication? ^ AB = f + A AB + + (1 - f ) + S S AB + B ^ AB = + AB + + + 1 00 AB + 11 AB AB 2 1 01 AB + 10 AB = AB 2 = ( ) ) ( f = AB 1 ^ AB is inseparable. < f <1 2 + + ^ AB + : entanglement fidelity with respect to AB AB Bi-lateral C-NOT operation: + AB + + AB AB AB + + ( 0 AB + A AB + AB AB + AB AB + B + Projection by 0 or 1 A 1 B result): + + purified! + + ^ If AB = f AB + (1 - f ) AB (mixed state) and we follow the same procedure (bi-lateral C-NOT operation + + ), a new ensemble is projective measurement + 50 AB described by ^ AB = f + AB + + (1 - f ) + AB + f= f 2 2 f 2 + (1 - f ) 1 > f if f > 2 or 1 1 In order to compare the outcomes of their measurements and to decide which pairs to keep 0 A 1 B or 1 A 0 B , Alice and Bob have to communicate discard and exchange "classical information". By iterating this procedure as indicated by the staircase before, Alice and Bob can distill an ensemble of pairs with entanglement fidelity f arbitrarily close to unity. If an initial state is a general mixed state ( (0 A 0 B ) A B ) or to ^ AB = f1 + - + f3 + + + f 4 - AB - , AB AB AB where f1 + f 2 + f 3 + f 4 = 1 and f > 1 , then the same method also 1 2 works. + + f2 - The scheme works as long as a state contains a sufficiently large fidelity for any maximally entangled state. 51 B. Quantum repeaters To build up EPR correlations in two quantum memories at two locations, single qubits (photons) need to be transmitted over a lossy channel. These photons cannot be amplified without destroying the quantum correlations. All we can do is to detect whether a photon has been absorbed in a transmission line and, if that is a case, repeat the transmission. A lossy channel is modeled as 0 1 A A 0 0 B B 0 0 A A 0 B A 1 B T1 + 0 0 B Ta T1 = e - = e - l / 2l0 ^ Ta = j ( ) b + j j c l0 = 2 2 j ( ) = 1 - e -2 j The probability for a successful transmission of a qubit from A to B is - l / l0 p (l ) = e The average number of required repetitions is n (l ) = 1 = el / l0 p (l ) (exponential resource) A A el / l0 repetitions N segments B B el / Nl0 repetitions/section 52 The average number of required repetitions is l n ( l ) = Nel / Nl0 nmin ( l ) = e1 (polynomial resource!) l0 N min = l / l0 There is a catch in this scheme: 1) the local operations at every checkpoint will introduce some noise. the fidelity of transmission across every segment is already limited to some maximum value Fmax 2) Both effects accumulate exponentially with the number of check points. Solution: Nested purification protocol Assume N=Ln (for some integer n) (i) first level Connect the pairs (initial fidelity F1) at all connection points except at CL, C2L, .......CN-L. (L=3 in above example) N L pairs of length L and fidelity FL<F1 53 Use M copies of these pairs, purify them to restore initial fidelity F1 (M=4 in above example) The total number of elementary pairs used up to this point is L M (=12 in above example) (ii) second level Connect L of these larger pairs at every connection point CN - L . CkL (k=1,2,.....) except at CL , C2 L , 2 2 2 N L 2 pairs of length L2 and fidelity FL Use M copies of these pairs, purify them to restore initial fidelity F1 The total number of elementary pairs up to this point is L2 M2 (=144 ). Now, the whole array of 32 42 pairs has been replaced by a single pair of fidelity F1 . continue until the n-th level A final pair between A and B of length N and fidelity F1 Total # of elementary pairs = ( LM )n = N 1+ log L M (polynomial resource!) 54 The purification is performed with an ensemble of identical EPR pairs stored in many, many quantum memories Can one do the purification with the aid of a single auxiliary pair at each level? Yes, one can. In this case, the vertical dimension (M copies) is translated into a temporal axis (number of repetitions). Parallel resources Mn and time T needed for creating a distant EPR pair via optical fibers (see text). Continental scale means 27 = 128 segments, intercontinental scale means 210 = 1024 segments. Error parameters are = p1 = p2 = 0.995. For (C), the resources grow only logarithmically, i.e. Mn = n + 1. 55 Number of Qubits 106 105 104 10 3 Quantum Computation 102 Quantum Simulation Quantum Metrology Quantum Repeater Quantum Cryptography Quantum Authentication 10 1 Difficulty 56 Quantum Processor #1 3 10 qubits 3 Quantum Processor #2 Quantum Teleportation Quantum 10 qubits Teleportation Quantum Teleportation 10 qubits 3 Quantum Processor #3 Photonic qubit network for creating initial EPR-Bell states (Duan - Lukin - Cirac - Zoller scheme, new Lukin scheme) Nuclear qubit memory for storing and purifying EPR-Bell states T2 25s at 300K ( 29 Si/ 28 Si) Electron spin for interfacing photonic qubits and nuclear qubits 57 T2 ~ 6 s at 300K (N-V center), T2 ~ 60ms at 4K ( 31 P:28 Si) > Nuclear spin Electron spin Excitonic dipole Photon 31P: Si nuclear spin 31P 19F: ZnSe 13C: NV center GaAs QD Ga and As nuclear spins nuclear spin 19F electron spin electron spin electron spin Hyperfine f interaction HF PL linewidth fPL PL lifetime rad PL efficiency Indistinguishable single photons Isotope engineering ~ 60MHz <10MHz ~10MHz <1GHz ~130MHz <150MHz 100KHz (collectively enhanced) <1GHz ~ 3msec <1nsec ~1 ~ 20 nsec <1nsec ~1 <10-4 (Anger recombination) <1 (Sydney, Maryland) (Wurzburg) (Stuttgart, Paris) (Washington, Munich, Zurich) 58

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