Practical long-distance quantum communication using concatenated entanglement swapping
aa r X i v : . [ qu a n t - ph ] M a y Practical long-distance quantum communicationusing concatenated entanglement swapping
Aeysha Khalique,
1, 2
Wolfgang Tittel, and Barry C. Sanders Institute for Quantum Science and Technology, University of Calgary, Alberta T2N 1N4, Canada Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, H-12 Islamabad, Pakistan Institute for Quantum Science and Technology and Department of Physics and Astronomy,University of Calgary, Alberta T2N 1N4, Canada (Dated: September 30, 2018)We construct a theory for long-distance quantum communication based on sharing entanglementthrough a linear chain of N elementary swapping segments of length L = Nl where l is the length ofeach elementary swap setup. Entanglement swapping is achieved by linear optics, photon countingand post-selection, and we include effects due to multi-photon sources, transmission loss and detectorinefficiencies and dark counts. Specifically we calculate the resultant four-mode state shared by thetwo parties at the two ends of the chain, and we derive the two-photon coincidence rate expected forthis state and thereby the visibility of this long-range entangled state. The expression is a nestedsum with each sum extending from zero to infinite photons, and we solve the case N = 2 exactly forthe ideal case (zero dark counts, unit-efficiency detectors and no transmission loss) and numericallyfor N = 2 in the non-ideal case with truncation at n max = 3 photons in each mode. For the generalcase, we show that the computational complexity for the numerical solution is n N max . PACS numbers:
I. INTRODUCTION
In practice long-distance quantum communicationbased on transmission of qubits suffers from a boundon transmission length because qubits can get lost alongthe way. Detector dark counts further complicate mat-ters by allowing detectors to register a spurious counteven if the original qubit is lost, and the combinationof loss and dark counts limits the distance to around200 km [1, 2]. Quantum repeaters provide a means toovercome this problem with a resource overhead that is atworst a polynomial function of the desired transmissiondistance L [3]. However, quantum repeaters are still in anearly stage or research [4–6]. On the other hand, entan-glement swapping [7] can be used to extend the distancefor quantum communication with current technology al-beit with a resource overhead that is exponential in L [8].This set-up is known as a ‘quantum relay’. Although anexponential overhead is daunting, it is still better than adistance bound that renders quantum communication atdistances greater than this bound impossible.Here we consider the practical quantum relay wherewe explicitly consider multi-photon events, transmissionloss and detector inefficiencies and dark counts. Our aimis to construct a formal theory for a linear chain of N elementary swapping segments subject to multi-photonevents and detector imperfections, with this theory deliv-ering an expression for the resultant entangled four-modestate at the ends of the chain post-selected on detec-tion records at entanglement swapping devices along thechain. This theory delivers not only the resultant statebut also the two-photon coincidence rate from which thetwo-photon visibility of the entangled state is calculated.Our results show that multiphoton effects are an impor-tant deleterious contributor to two-photon visibility for a long-distance array of concatenated entanglement swap-ping.Our theory requires that N = 2 ı for ı some positive in-teger due to symmetry in the calculations, and we solvethe expression exactly for N = 2 in the ideal case andnumerically for the non-ideal case and N = 2. Previ-ously only the N = 1 (equivalent to one swap) case hasbeen studied under practical conditions [9], and our workextends that work but in a nontrivial way. In particu-lar our calculations only work for N = 2 , , , . . . , withthis restriction to make the calculations easier to per-form, whereas the previous result effectively considersonly ı = 0. Entanglement swapping concatenation hasbeen studied before but without multiphoton effects [10].We determine the computational complexity for thenumerical simulation as a function of N and the trunca-tion n max of photon number in each mode. Specificallythe algorithm that we developed is inefficient as its run-time scales as n N max . By truncating at n max = 3 we areable to solve for N = 2 using a message-passing interfaceparallel program on a supercomputer. For a fixed set ofparameters and with truncation to a maximum of threephotons for each of the 16 modes, the Fortran programrunning at 2.66 GHz on an Intel Xeon E5430 quad-coreprocessor with 8 GB of memory required approximatelysix hours to compute two-photon coincidence probabilityon a single core. Hence code parallelization was neces-sary to deliver probabilities for wide range of inputs inreasonable time, by making use of multiple cores. Thegiven inputs included parametric down conversion pumprate χ , dark-count rate ℘ , detector efficiency η , trans-mission loss and polarization rotator angles ˜ α for Aliceon the left end of the chain of entanglement swappingsand ˜ δ for Bob on the right end.Our paper proceeds as follows. In Sec. II we provide FIG. 1: N = 1 entanglement swapping. Entangled states areprepared in each pair of modes by two parametric down con-verters (PDCs). Bell measurement on spatial modes b and c are used to conditionally prepare an entangled state betweenmodes a and d . Bell measurement comprises of combiningmodes b and c on beam splitter B followed by polarization sep-aration at one polarizing beam splitter (PBS) for each mode.The resultant photon counts at each of the four detectors (the‘detector four tuple’) is { q, r, s, t } . a background on practical entanglement swapping. Thisbackground concerns only the N = 1 case and intro-duces the concepts required for subsequent analysis. The N = 2 case is developed in Sec. III including comput-ing the conditioned entangled state shared between Aliceon the left and Bob on the right and the resultant two-photon coincidence probability and visibility. In Sec. IVwe derive the general state and visibility formulæ for ar-bitrary number N concatenations of entanglement swap-ping. We discuss the complexity for numerically solvingthe general case in this section as well. In Sec. V wesummarize the result and present our conclusions. II. BACKGROUND: PRACTICALENTANGLEMENT SWAPPING
In this section we reprise the case of a single entan-glement swapping ( N = 1) under practical conditions [9]as these results inform us in how to solve the case of N = 2 ı , for ı any positive integer, in subsequent sec-tions. The setup is shown in Fig. 1. For simplicity weassume that qubits are encoded into polarization states.However our formalism is general and can also be appliedto other realization of qubits e.g., time bin qubits. Twoparametric downconversion (PDC) sources produce two-mode entangled states. Ideally each of these two-modeentangled states corresponds to an entangled pair of pho-tons, but the realistic case involves the vacuum state | vac i and higher-order Fock states. The ‘right’ mode from the‘left’ PDC and the ‘left’ mode of the ‘right’ PDC aresubjected to joint Bell-state measurements. The Bell-state measurement conditions the resultant state sharedby modes a and d , which should then be in an entangledstate despite the two modes a and d not being entangled initially, hence the term “entanglement swapping”.Assuming that PDC generates a pure state, the quan-tum state prepared by the two PDC sources is given as | χ i = exp[ iχ (ˆ a † H ˆ b † H + ˆ a H ˆ b H ) ⊗ exp[ iχ (ˆ a † V ˆ b † V + ˆ a V ˆ b V ) ⊗ exp[ iχ (ˆ c † H ˆ d † H + ˆ c H ˆ d H ) ⊗ exp[ iχ (ˆ c † V ˆ d † V + ˆ c V ˆ d V ) | vac i , (1)and the mixed-state case is readily generalized by mak-ing an incoherent mixture of the pure states (1). Giventhat four imperfect detectors (a detector fourtuple) (i.e.,detectors having non-unit efficiency η and non-zero darkcount rates ℘ ) in the Bell measurement yield readout { q, r, s, t } , as explained in Fig. 1, the posteriori condi-tional probability for any readout ( ijkl ) that four idealdetectors would have yielded, is P qrstijkl := p ( ijkl | qrst ) ≡ p ( qrst | ijkl ) p ( ijkl ) ∞ P i ′ ,j ′ ,k ′ ,l ′ =0 p ( qrst | i ′ j ′ k ′ l ′ ) p ( i ′ j ′ k ′ l ′ ) . (2)Note that as in [9], we include all transmission loss intothe detector efficiency. As the detectors are independent, P ( qrst | ijkl ) = p ( q | i ) p ( r | j ) p ( s | k ) p ( t | l ) . (3)For photon number discriminating detector with effi-ciency η and dark count probability ℘ , p ( q | i ) = (1 − η )(1 − ℘ )1 − η (1 − ℘ ) (cid:18) η − η (cid:19) q (1 − η ) i G ( i, q ; η, ℘ )(4)for i ≥ q and p ( q | i ) = (1 − η )(1 − ℘ )1 − η (1 − ℘ ) (cid:20) − ηη b ( η, ℘ ) (cid:21) q − i η i G ( i, q ; η, ℘ )(5)for q > i where b ( η, ℘ ) := (cid:20) − ηη℘ (cid:21) − . (6)Also G ( κ, λ ; η, ℘ ) = ∞ X n =0 (cid:18) κλ (cid:19)(cid:18) κ − λ + nκ − λ (cid:19) [ b ( η, ℘ )] n × (cid:20) F (cid:18) − n, − λ ; κ − λ + 1; η − η (cid:19)(cid:21) (7)for κ ≥ λ and G ( κ, λ ; η, ℘ ) := 0 for κ < λ . In the aboveequations, F is the hypergeometric function.For threshold detector there are two possibilities, clickor no-click, which result in p (no click | i ) = p ( q = 0 | i )=(1 − ℘ )[1 − η (1 − ℘ )] i (8) p (click | i ) =1 − p (no click | i )=1 − (1 − ℘ )[1 − η (1 − ℘ )] i (9)The state of the remaining modes a and d after record-ing photons counts { q, r, s, t } is the mixed state ρ qrst = X i,j,k,l P qrstijkl | Φ ijkl ih Φ ijkl | , (10)with | Φ ijkl i the state corresponding to the count { i, j, k, l } for perfect detectors { b H , b V , c V , c H } . Thisideal state is | Φ ijkl i = 1 p i + j + k + l i ! j ! k ! l ! i X µ =0 j X ν =0 k X κ =0 l X λ =0 ( − µ + ν × (cid:18) iµ (cid:19)(cid:18) jν (cid:19)(cid:18) kκ (cid:19)(cid:18) lλ (cid:19) ˆ a † µ + λ H × ˆ a † ν + κ V ˆ d † i + l − µ − λ H ˆ d † j + k − ν − κ V | vac i . (11)The visibility of the modes a and d is calculated by pass-ing these two modes through polarizer rotators and mea-suring the coincidence count at the detectors after pass-ing through polarizer beam splitters. This procedure re-flects the standard experimental approach. III. CONCATENATING TWOENTANGLEMENT SWAPPINGS
In this section we develop the theory of concatenatedentanglement swapping under practical conditions. Theconcept of concatenated entanglement swapping is thattwo or more entanglement swapping processes are com-bined into a single entanglement-swapping procedure. Inthis section we only deal with the case of N = 2 concate-nated elementary entanglement swappings, which we areable to solve approximately in the numerical case. Be-yond N = 2 is the subject of Sec. IV where we providethe formalism but do not solve numerically. A. Conditionally prepared state
In Fig. 2 we depict the case of N = 2 concatenatedentanglement swapping. For the two elementary swapsshown in Fig. 2, we take the states of swap 1 and 2 as in Eq. (11). The modes a and d are combined on the 3 rd Bell-state measurement setup comprising a beam splitter U B , polarizing beam splitters and a fourtuple of detec-tors. The operators ˆ a , H and ˆ d , H transform under theaction of the beam splitter as U † B ˆ a † , H U B = 1 √ (cid:16) ˆ a † , H − ˆ d † , H (cid:17) (12)and U † B ˆ d † , H U B = 1 √ (cid:16) ˆ a † , H + ˆ d † , H (cid:17) , (13) FIG. 2: Concatenation of N = 2 elementary segments com-prising of entanglement swapping setup: The cap on the innerarms represents a Bell-state measurement. Quantum commu-nication distance is doubled by concatenating two elementaryswaps by means of a third swap. Modes a and d from the1st and 2nd swap are subjected to a Bell-state measurementto swap the entanglement to the right and left most arms. and ˆ a † , V and ˆ d † , V transform analogously.Taking the readout at the detectors for the two hori-zontal modes a , H and d , H to be | i , l i , we obtain h i , l | U B ˆ a † µ + λ , H ˆ d † i + j − µ − λ , H | vac i = µ + λ X γ =0 (cid:18) µ + λ γ (cid:19)(cid:18) i + l − µ − λ i − γ (cid:19) × ( − µ + λ − γ s i ! l !(2) i + l × δ i + l ,µ + λ + i + l − µ − λ . (14)The transformation for vertical modes is similar. UsingEq. (14) and taking the readout at the 3 rd fourtuple ofdetectors as { i , j , k , l } , the unnormalized state of theremaining modes d and a is (cid:12)(cid:12) Φ ′ ijkl (cid:11) = h i j k l | U B | Φ i j k l Φ i j k l i = Y p =1 √ i p + j p + k p + l p p i p ! j p ! k p ! l p ! (tanh χ ) i p + j p + k p + l p cosh χ i p X µ p =0 j p X ν p =0 k p X κ p =0 l p X λ p =0 ( − µ p + ν p (cid:18) i p µ p (cid:19)(cid:18) j p ν p (cid:19)(cid:18) k p κ p (cid:19)(cid:18) l p λ p (cid:19) × Ω( µ , λ , i , l )Ω( ν , κ , j , k ) √ i ! j ! k ! l !( √ i + j + k + l δ i + l ,µ + λ + i + l − µ − λ δ j + k ,ν + κ + j + k − ν − κ × ˆ d † i + l − µ − λ , H ˆ d † j + k − ν − κ , V ˆ a † µ + λ , H ˆ a † ν + κ , V | vac i , (15)for i = ( i i i ), j = ( j j j ), k = ( k k k )and l = ( l l l ) andΩ( µ , λ , i , l ) = µ + λ X γ =0 (cid:18) µ + λ γ (cid:19) × (cid:18) i + l − µ − λ i − γ (cid:19) ( − µ + λ − γ . (16)Eq. (16) reduces toΩ( µ , λ , i , l ) = ( − µ + λ (cid:18) i + l − µ − λ i (cid:19) × F ( − µ − λ , − i ; l − µ − λ + 1; −
1) (17)for l − µ − λ ≥ µ , λ , i , l ) = ( − l (cid:18) µ + λ µ + λ − l (cid:19) × F ( − i − l + µ + λ ; l ; µ + λ − l + 1; − l − µ − λ <
0, and analogously for Ω( ν , κ , j , k ).The quantum state after actual readout { q, r, s, t } ,with q = ( q q q ) and similar for { r, s, t } , at the threefourtuples of detectors is ρ = Y u =1 ∞ X i u ,j u ,k u ,l u =0 P qrstijkl | Φ ′ i h Φ ′ | (19)with P qrstijkl = p ( qrst | ijkl ) Q u =1 ∞ P i u ,j u ,k u ,l u =0 p ( qrst | ijkl ) h Φ | Φ i . (20)Entanglement verification can be done by measuring thecoincidence rate, for various combinations of local projec-tion measurements, which in turn is done by introducing variable polarization rotators in the spatial paths of themodes d and a and then detecting them after passingthrough polarizing beam splitters as done in [9]. Thepolarization rotators are given by unitary operationsˆ U a (˜ α ) = exp (cid:20) i ˜ α (cid:16) ˆ a † ,V ˆ a ,H + ˆ a ,V ˆ a † ,H (cid:17)(cid:21) (21)and ˆ U d (˜ δ ) = exp (cid:20) i ˜ δ (cid:16) ˆ d † ,V ˆ d ,H + ˆ d ,V ˆ d † ,H (cid:17)(cid:21) . (22)Given the imperfect Bell-state measurement events { q, r, s, t } on the three detector fourtuples, theconditional probability that ideal measurements ofmodes a ,H , a ,V , d ,V and d ,H , would have yielded theresult { i ′ , j ′ , k ′ , l ′ } is p ( i ′ j ′ k ′ l ′ | qrst ) =Tr { ( | i ′ j ′ k ′ l ′ ih i ′ j ′ k ′ l ′ | ) ˆ U a (˜ α ) ⊗ ˆ U d (˜ δ ) ρ ˆ U † d (˜ δ ) ˆ U † a (˜ α ) } = Y u =1 ∞ X i u ,j u ,k u ,l u =0 W ijkl i ′ j ′ k ′ l ′ × P qrstijkl , (23)with P qrstijkl given by Eq. (20) and W ijkl i ′ j ′ k ′ l ′ := |h i ′ j ′ k ′ l ′ | ˆ U a N (˜ α ) ⊗ ˆ U d (˜ δ ) | Φ i| = (cid:12)(cid:12)(cid:12) A ijkl i ′ j ′ k ′ l ′ (cid:12)(cid:12)(cid:12) (24)is the transition probability. Here A ijkl i ′ j ′ k ′ l ′ = Y p =1 p i p + j p + k p + l p i p ! j p ! k p ! l p ! (tanh χ ) i p + j p + k p + l p cosh N χ i p X µ p =0 j p X ν p =0 k p X κ p =0 l p X λ p =0 ( − µ p + ν p (cid:18) i p µ p (cid:19)(cid:18) j p ν p (cid:19)(cid:18) k p κ p (cid:19)(cid:18) l p λ p (cid:19) × Ω( µ , λ , i , l )Ω( ν κ , j , k ) √ i ! j ! k ! l !( √ i + j + k + l δ i + l ,µ + λ + i + l − µ − λ δ j + k ,ν + κ + j + k − ν − κ × ( ν + κ )!( j + k − ν − κ )! r j ′ ! k ′ ! i ′ ! l ′ ! Min[ j ′ ,ν + κ ] X n a =0 Min[ k ′ ,j + k − ν − κ ] X n d =0 ( i tan ˜ α ν + κ + j ′ − n a (cos ˜ α i ′ + j ′ − n a × ( i tan ˜ δ k ′ + j + k − ν − κ − n d (cos ˜ δ l ′ + k ′ − n d ( i ′ + j ′ − n a )!( l ′ + k ′ − nd )! n a ! n d !( j ′ − n a )!( k ′ − n d )!( ν + κ − n a )!( j + k − ν − κ − n d )! × δ i ′ + j ′ ,µ + ν + κ + λ δ k ′ + l ′ ,i + j + k + l − µ − ν − κ − λ . (25)The conditional probability to observe the event { q ′ , r ′ , s ′ , t ′ } on modes a ,H , a ,V , d ,V and d ,H withnonideal imperfect detectors, given imperfect Bell-statemeasurement events { q, r, s, t } at the three detectorfourtuple is Q := p ( q ′ r ′ s ′ t ′ | qrst )= ∞ X i ′ ,j ′ ,k ′ ,l ′ =0 p ( q ′ r ′ s ′ t ′ | i ′ j ′ k ′ l ′ ) p ( i ′ j ′ k ′ l ′ | qrst ) . (26)This equation is used to calculate the two-photon visibil-ity. B. Reduction of states under ideal detectors
In this subsection we consider the case of ideal detec-tors to show a reduction of the expressions in the pre-vious subsection to well known Bell state results. Ourmodel incorporates transmission loss into the detector-efficiency parameter. Hence unit-efficiency detectectionimplies zero transmission loss. For a single swap with im-perfect threshold detectors, a non-ideal projection ontothe Bell state | ψ − i bc is achieved whenever Bell-state mea-surement events { q, r, s, t } = { , , , } or { , , , } areobtained.Let us consider the outcome { , , , } . Thus, for idealdetectors with unit efficiency and zero dark counts, P qrstijkl = δ qi δ rj δ sk δ tl , (27)and the state in Eq. (19) reduces to a single component.For a single swap, Eq. (11) yields | Φ i = 1 √ (cid:18) | i − | i√ | i − | i√ (cid:19) . (28)Hence, there is another term superposed to a perfect four-mode singlet | ψ − i := 1 √ | i − | i ) . (29) The perfect Bell state | ψ − i results from each source pro-ducing exactly one pair and the other two terms resultwhen one source produces two pairs and the other sourceproduces vaccuum. The probability for each of thesethree alternatives is proportional to χ , which explainswhy the resultant state of remaining modes a and d inEq. (28) does not depend on χ .For two concatenated elementary swaps, with the con-dition that all three detector fourtuples yield { , , , } ,the renormalized state | Φ ′ ijkl i from Eq. (15) yields (cid:12)(cid:12) Φ ′ ijkl (cid:11) = 1 √ (cid:18) | i − | i√ | i − | i√ (cid:19) , (30)which is the same state as Eq. (28). Under these con-ditions, the conditional probabilities, Q and Q ,of recording the events { , , , } and { , , , } , respec-tively, on modes a , H , a , V , d , V and d , H are calculatedfrom Eq. (26) as Q = Q = A cos ˜ α − ˜ δ ! (31)and the corresponding probabilities, Q and Q forthe events { , , , } and { , , , } are Q = Q = A sin ˜ α − ˜ δ ! . (32)Here, A is the normalization factor. The correlation func-tion is P (˜ α, ˜ δ ) := Q + Q − Q − Q Q + Q + Q + Q = cos(˜ α − ˜ δ ) , (33)with ˜ α and ˜ δ characterizing the polarization rotators.We numerically simulate the two swaps case for idealdetectors with fixed ˜ α and varying ˜ δ with truncationat n max = 1. P (˜ α, ˜ δ ) thus obtained is consistent withEq. (33) as shown in Fig. 3. -Π - Π Π Π- - ∆Ž P H Α Ž , ∆ Ž L FIG. 3: Numerically evaluated correlation function P (˜ α, ˜ δ )with ˜ α = 0 (solid curve) and ˜ α = π/ α − ˜ δ . C. Visibility for two concatenated elementaryswaps
For imperfect detectors and losses, visibility decreasesas the number of entanglement-swapping concatenationsincreases. We calculate the visibility for the N = 2 casefrom our model derived in Subsec. III A. For Bell-statemeasurement events { , , , } or { , , , } on all threedetector fourtuples, which yields the singlet state (29),the two-fold coincidences Q + Q and Q + Q are calculated numerically for dark count probability ℘ = 1 × − , efficiency η = 0 .
04 and source brightness χ = 0 .
24. The detector efficiency includes the channelloss given as η = η × − αl/ with α the loss coefficient, l the distance that light travels and η the intrinsic detec-tor efficiency. As an example, for light with a wavelengthof 1550 nm propagating through a telecom optical fibre,the loss coefficient is approximately α = 0 . − and forstandard InGaAs avalanche photodiodes η = 0 .
15. Thus η = 0 .
04 corresponds to a distance of 30 km in one arm.This corresponds to a total distance of about 240 km be-tween left-most and right-most arm of the N = 2 setup.For superconducting detectors featuring η = 0 .
93 [11]these distances change to 70 km and 560 km respectively.The maximum number of photons in each mode are trun-cated at n max = 3. The results of the numerical simula-tion are shown in Fig. 4 for fixed angle ˜ α = π/
2. As ex-pected, the two curves for Q + Q and Q + Q are complementary. Visibility is given as V = max − minmax + min (34)with max denoting the maximum value of conditionaltwo-fold coincidence and min the minimum value. Forour numerical simulations, the visibility is about 32%.For the same detector parameters, loss and source bright-ness, the visibility for a single swap is about 70%. Wenow have expressions for the N = 2 concatenated entan-glement swapping case, shown that they reduce to known results for perfect detectors and numerically evaluatedvisibilities. The visibility in the N = 2 case is compared -Π - Π Π Π ∆Ž FIG. 4: Four-fold coincidence probabilities Q + Q (dashed curve) and Q + Q (solid curve) plotted asa function of polarization rotation angle ˜ δ for fixed rotationangle ˜ α = π/
2. Detector parameters are dark count rate ℘ = 1 × − and efficiency η = 0 .
04, and the pump parameteris χ = 0 .
24. The truncation is done at n max = 3 . Χ V i s i b ilit y FIG. 5: Comparison of the variation of visibility with sourcebrightness χ for single swap, N=1(dotted curve) and for twoelementary swaps, N=2 (solid curve). Both rotation anglesare fixed at ˜ α = ˜ δ = π/ N = 2 case, n max = 3 while for the N = 1 case, n max = 4. to that in the N = 1 case in Fig. 5. The visibility dropsmore rapidly for increasing χ for the N = 2 case.In the next section we develop the full formalism forthe case of arbitrary N . IV. CONCATENATION OF N SWAPPINGS
We now extend the treatment of two concatenationsto arbitrary distance and arbitrary number of swappings N = 2 ı for ı any positive integer. The 1 st and the 2 nd swaps are combined on the ( N + 1) st beam splitter, andthe ( N − st and N st swaps combine on the ( N + N ) st beam splitter etc. Thus 2 N − N − N = 1 swapping with a de-tector 2 N − ı rows and each m th row contains N/ m swaps. After these swappings the unnormalized state ofremaining modes d and a N is FIG. 6: Concatenation of N = 4 elementary swaps: Filled-in caps are each a detector fourtuple and each element of thebottom row is an entanglement swapping resulting from aBell measurement. The middle row depicts two N = 2 casescorresponding to concatenating N = 1 cases from the bottomrow. The filled-in cap in the middle row corresponds to adetector 12-tuple. The top row corresponds to N = 4 whichcombines the two N = 2 from the middle row. The filled-incap in the top row corresponds to a detector 28-tuple. | Φ i = N Y p =1 √ i p + j p + k p + l p p i p ! j p ! k p ! l p ! (tanh χ ) i p + j p + k p + l p cosh N χ i p X µ p =0 j p X ν p =0 k p X κ p =0 l p X λ p =0 ( − µ p + ν p (cid:18) i p µ p (cid:19)(cid:18) j p ν p (cid:19)(cid:18) k p κ p (cid:19)(cid:18) l p λ p (cid:19) × ı Y m =1 N/ m Y n =1 Ω( µ β mn , λ β mn , i α mn , l α mn )Ω( ν β mn , κ β mn , j α mn , k α mn ) p i α mn ! j α mn ! k α mn ! l α mn !( √ i αmn + j αmn + k αmn + l αmn × δ i αmn + l αmn ,µ βmn + λ βmn + i βmn +1 + l βmn +1 − µ βmn +1 − λ βmn +1 δ j αmn + k αmn ,ν βmn + κ βmn + j βmn +1 + k βmn +1 − ν βmn +1 − κ βmn +1 × ˆ d † i + l − µ − λ , H ˆ d † j + k − ν − κ , V ˆ a † µ N + λ N N, H ˆ a † ν N + κ N N, V | vac i . (35)Here, α mn = N (cid:18) − ( ) m − (cid:19) + n β mn = 2 m − (2 n − l α − µ β mn − λ β mn ≥ µ β mn , λ β mn , i α mn , l α mn )=( − µ βmn + λ βmn (cid:18) i α mn + l α mn − µ β mn − λ β mn i α mn (cid:19) × F ( − µ β mn − λ β mn , − i α mn ; l α mn − µ β mn − λ β mn + 1; − l α mn − µ β mn − λ β mn < µ β mn , λ β mn , i α mn , l α mn ) =( − l αmn (cid:18) µ β mn + λ β mn µ β mn + λ β mn − l α mn (cid:19) × F ( − i α mn − l α mn + µ β mn + λ β mn ; l α mn ; µ β mn + λ β mn − l α mn + 1; − . (38)The form of Ω( ν β , κ β , j α , k α ) is analogous. The expres-sion for A ijkl i ′ j ′ k ′ l ′ for this general case is calculated as A ijkl i ′ j ′ k ′ l ′ = N Y p =1 p i p + j p + k p + l p i p ! j p ! k p ! l p ! (tanh χ ) i p + j p + k p + l p cosh N χ i p X µ p =0 j p X ν p =0 k p X κ p =0 l p X λ p =0 ( − µ p + ν p (cid:18) i p µ p (cid:19)(cid:18) j p ν p (cid:19)(cid:18) k p κ p (cid:19)(cid:18) l p λ p (cid:19) × ı Y m =1 N/ m Y n =1 Ω( µ β mn , λ β mn , i α mn , l α mn )Ω( ν β mn κ β mn , j α mn , k α mn ) p i α mn ! j α mn ! k α mn ! l α mn !( √ i αmn + j αmn + k αmn + l αmn × δ i αmn + l αmn ,µ βmn + λ βmn + i βmn +1 + l βmn +1 − µ βmn +1 − λ βmn +1 δ j αmn + k αmn ,ν βmn + κ βmn + j βmn +1 + k βmn +1 − ν βmn +1 − κ βmn +1 × ( ν N + κ N )!( j + k − ν − κ )! r j ′ ! k ′ ! i ′ ! l ′ ! Min[ j ′ ,ν N + κ N ] X n a =0 Min[ k ′ ,j + k − ν − κ ] X n d =0 (cid:18) i tan ˜ α (cid:19) ν N + κ N + j ′ − n a × (cid:18) cos ˜ α (cid:19) i ′ + j ′ − n a i tan ˜ δ ! k ′ + j + k − ν − κ − n d cos ˜ δ ! l ′ + k ′ − n d × ( i ′ + j ′ − n a )!( l ′ + k ′ − nd )! n a ! n d !( j ′ − n a )!( k ′ − n d )!( ν N + κ N − n a )!( j + k − ν − κ − n d )! × δ i ′ + j ′ ,µ N + ν N + κ N + λ N δ k ′ + l ′ ,i + j + k + l − µ − ν − κ − λ . (39)The two-fold coincidences and hence visibility can be cal-culated using Eq. (26). The complexity of the algorithmto calculate the same scales as n N max , where N is the num-ber of swaps and n max is the truncation of multiphotonincidences. V. CONCLUSIONS
We have developed a theoretical framework for con-catenated entanglement swapping for imperfect detec-tors, sources and loss. Our theory will be valuable inmodelling a new generation of long-distance quantumcommunication experiments based on the quantum re-lay as well as quantum repeaters, which both rely onconcatenated swapping. Our theory assumes that thenumber of entanglement swapping operations is of theform N = 2 ı for ı a positive integer, hence does notreduce to the previous practical entanglement swappinganalysis of N = 1 [9] for which ı = 0. We develop the N = 2 case extensively and show that it reduces to theknown perfect-detector case, and we solve numericallyfor a truncation of n max = 3 photons per mode. Thetruncation at n max = 3 is reliable for small values of χ , including χ = 0 .
24, but fails to deliver correct results forlarger values of χ .Although the general case is exponentially expensive tosolve in terms of N , which is proportional to the length ofthe channel, we are optimistic that further simplificationcan be found. The expressions are complicated nestedproducts and sums of many terms, but we have sampledthose terms and find that most are negligibly small. If astrategy can be found to eliminate all small terms, thenthe computations could be performed for N higher than2 as we report here. Despite the current limitation ofworking with N = 2 and not beyond, the N = 2 caseis directly relevant to soon-to-be-realized concatenatedentanglement-swapping experiments. Acknowledgments
We acknowledge valuable discussions with MichaelLamoreux and Artur Scherer and financial support fromAITF and NSERC. BCS appreciates Senior Fellow sup-port from CIFAR. This research has been enabled by theuse of computing resources provided by WestGrid andCompute/Calcul Canada. [1] D. Stucki, N. Walenta, F. Vannel, R.T. Thew, N. Gisin,H. Zbinden, S. Gray, C.R. Towery, and S. Ten. High rate,long distance quantum key distribution over 250 km ofultra low loss fibres.
New J. Phys. , 11:075003–075011,July 2009. [2] T.-Y. Chen, J. Wang, Y. Liu, W.-Q. Cai, X. Wan, L.-K. Chen, J.-H. Wang, S.-B. Liu, H. Liang, L. Yang, C.-Z. Peng, Z.-B. Chen, and J.-W Pan. 200km decoy-statequantum key distribution with photon polarization.
Opt.Express , 18(8):8587–8594, 2010. [3] L.-M. Duan, M.D. Lukin, J.I. Cirac, and P. Zoller. Long-distance quantum communication with atomic ensemblesand linear optics.
Nature , 414:413–418, 2001.[4] N. Sangouard, C. Simon, H. de Riedmatten, andN. Gisin. Quantum repeaters based on atomic ensem-bles and linear optics.
Rev. Mod. Phys. , 83:33–80, Mar2011.[5] J. Min´aˇr, H. de Riedmatten, and N. Sangouard. Quan-tum repeaters based on heralded qubit amplifiers.
Phys.Rev. A , 85:032313–032319, Mar 2012.[6] S. Abruzzo, S. Bratzik, N.K. Bernardes, H. Kamper-mann, P. van Loock, and D. Bruss. Quantum repeatersand quantum key distribution: analysis of secret keyrates. arxiv:1208.2201 , 2012.[7] J.-W. Pan, D. Bouwmeester, H. Weinfurter, andA. Zeilinger. Experimental entanglement swapping: En-tangling photons that never interacted.
Phys. Rev. Lett. , 80:3891–3894, May 1998.[8] N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden. Quan-tum cryptography.
Rev. Mod. Phys. , 74:145–195, Mar2002.[9] A. Scherer, R.B. Howard, B.C. Sanders, and W. Tittel.Quantum states prepared by realistic entanglement swap-ping.
Phys. Rev. A , 80:062310–062329, Dec 2009.[10] D. Collins, N. Gisin, and H. de Riedmatten. Quantumrelays for long distance quantum cryptography.
J. Mod.Optic. , 52:735–753, March 2005.[11] F. Marsili, V.B. Verma, J.A. Stern, S. Harrington, A.E.Lita, T. Gerrits, I. Vayshenker, B. Baek, M.D. Shaw,R.P. Mirin, and S.W. Nam. Detecting single infraredphotons with 93% system efficiency.