# Loss-tolerant concatenated Bell-state measurement with encoded coherent-state qubits for long-range quantum communication

LLoss-tolerant concatenated Bell-state measurement with encodedcoherent-state qubits for long-range quantum communication

Seok-Hyung Lee, Seung-Woo Lee, and Hyunseok Jeong Department of Physics and Astronomy, Seoul National University, Seoul 08826, Republic of Korea Center for Quantum Information, Korean Institute of Science and Technology, Seoul 02792, Republic of Korea (Dated: February 26, 2021)The coherent-state qubit is a promising candidate for optical quantum information processingdue to its nearly deterministic nature of the Bell-state measurement (BSM). However, its non-orthogonality incurs diﬃculties such as failure of the BSM. One may use a large amplitude ( α ) forthe coherent state to minimize the failure probability, but the qubit then becomes more vulnerable todephasing by photon loss. We propose a hardware-eﬃcient concatenated BSM (CBSM) scheme withmodiﬁed parity encoding using coherent states with reasonably small amplitudes ( | α | (cid:47) α and suﬃciently low photon loss rates (e.g., (cid:47) I. INTRODUCTION

Optical systems are a competitive candidate forquantum information processing (QIP) due to theirlong coherence time and advantages in long-distancetransmission [1]. It is well known that they areparticularly promising for quantum communication.Single-photon states are usually considered for thecarriers of optical qubits such as vacuum-single-photon-qubit (single-rail encoding) [2] and polariza-tion qubit (dual-rail encoding) [3]. However, theseencoding schemes have a drawback that the Bell-state measurement (BSM) is non-deterministic withlinear optics [4, 5]. The BSM is essential for QIPtasks such as quantum teleportation [6, 7] and en-tanglement swapping [8, 9]. Quantum teleportationis widely employed not only for quantum commu-nication but also for all-optical quantum computa-tion with gate teleportation [3]. It is thus importantto overcome the problem of non-deterministic BSM.Several methods have been suggested using multi-ple photons for encoding [10, 11], ancillary states[12–14], coherent states [15–23], and hybrid states[24–26] to improve the success probability of BSM.Among them, in this paper, we focus on the schemeusing coherent-state qubits that enables one to per-form a nearly deterministic BSM with linear optics[15, 16, 27]. Early studies on coherent states as carriers ofquantum information focus on how to construct log-ical qubits and elementary logical gates [15, 17,18, 28–31]. In these works, the basis set is cho-sen either as {|± α (cid:105)} or as { N ± ( | α (cid:105) ± |− α (cid:105) ) } , where |± α (cid:105) are coherent states of amplitudes ± α and N ± are normalization factors. Various attempts to ob-tain fault-tolerance on QIP with coherent stateshave been made, starting from simple embedding onwell-known discrete-variable (DV) encoding schemes[19, 22], to exploiting the property of continuous-variable (CV) systems [23, 32–37], with some ex-perimental demonstration [38–41]. Recently, it wasclaimed that simple 1D repetition cat code enableshardware-eﬃcient topologically-protected quantumcomputation by exploiting the 2D phase space forlogical operations [42]. However, these studiesmainly deal with coherent-state qubits inside a cav-ity system, and they cannot be directly applied tofault-tolerant QIP in free-propagating optical ﬁelds.Our main goal is to investigate the possibility touse simple concatenated repetition codes, which canbe generated and manipulated with combinations ofwell-known elementary gates, for fault-tolerant QIPwith free-propagating coherent-state qubits.As mentioned before, the BSM of coherent-statequbits, where the basis is {|± α (cid:105)} , is nearly deter-ministic. However, due to the non-orthogonality ofthe basis set, a small but non-negligible probabil- a r X i v : . [ qu a n t - ph ] F e b ty of failure exists [15, 16]. One may use coherentstates with large values of α to solve this problem,but the qubit then becomes more vulnerable to de-phasing by photon loss [19]. It is impossible to ide-ally suppress both failures and dephasing simulta-neously with such an elementary coherent-state en-coding. In this paper, motivating by recent works on concatenated Bell-state measurement (CBSM) withmulti-photon polarization qubits [11, 43] and rep-etition cat code [42], we overcome these obstaclesby introducing the CBSM with modiﬁed parity en-coding employing coherent states. We propose anelaborately designed CBSM scheme with consider-ation of hardware-eﬃciency, and numerically showthat the scheme successfully suppresses both failuresand dephasing simultaneously with reasonably smallamplitudes (e.g., | α | <

2) of coherent states.One of the key applications with BSMs is long-distance quantum communication through quantumrepeaters [44]. In the initially proposed quantumrepeater schemes to generate Bell pairs between dis-tant parties [45–49], heralded entanglement genera-tion is required for suppressing transmission errors,which makes long-lived quantum memory essential[50]. Recently, quantum repeater schemes exploit-ing quantum error correction (QEC) have been sug-gested for suppressing errors, which do not requirelong-lived quantum memory, have been suggested[11, 43, 50–57], where a quantum repeater is built upwithout long-lived quantum memory by encoding in-formation with QEC codes, sending it by lossy chan-nel, and relaying the encoded information from eachstation to the next station with error corrections. Ineach repeater station, a fault-tolerant BSM can beused for QEC by teleporting the incoming lossy log-ical qubits with a generated logical Bell state [11].Later in this paper, we evaluate the performanceof the quantum repeater scheme using our CBSMscheme and show that it indeed enables quantum re-peater with high performance over distances longerthan 1000 km.The outline of the paper is as follows. In Sec. II,we review the BSM scheme of lossless coherent-statequbits, extend it to lossy cases, and evaluate its suc-cess, failure, and error rates against the coherent-state amplitude α and the photon survival rate. InSec. III, we present the modiﬁed parity encodingscheme employing coherent-state qubits, and showthe hierarchy relation between logical, block, andphysical level. In Sec. IV, we ﬁrst suggest an unopti-mized CBSM scheme which only uses simple major-ity votes and counting of measurement results, andanalyze the root of fault-tolerance of the scheme.After that, we propose an improved CBSM scheme BS PNPD

FIG. 1. The BSM scheme of coherent-state qubits [15].It uses one 50:50 beam splitter (BS) and two photon-number parity detectors (PNPDs). The result is de-termined by the measurement results of the PNPDs asEq. (2). which is elaborately designed considering hardwareeﬃciency. In Sec. V, we present the analytic ex-pressions of the probability distributions of CBSMresults, which are simple matrix forms enabling fastsampling of the results and can be generalized toany CBSM schemes. In Sec. VI, we show the re-sults of numerical calculations. We ﬁrst present aperformance analysis by the success, failure, and er-ror probabilities of CBSM. We then investigate theperformance of the quantum repeater scheme whichuses our CBSM scheme for error correction, as oneof the key applications of BSM. In Sec. VII, we de-scribe methods to prepare the logical qubits undermodiﬁed parity encoding and implement elementarylogical operations, which consist of several physical-level ingredients such as generation of superpositionsof coherent states (SCSs) and elementary gates un-der coherent-state basis. We also brieﬂy review re-cent progresses on realizations of these ingredients.We conclude with ﬁnal remarks in Sec. VIII

II. BELL-STATE MEASUREMENT OFLOSSY COHERENT-STATE QUBITS

We ﬁrst review the BSM scheme of losslesscoherent-state qubits encoded with basis | L (cid:105) := | α (cid:105) , | L (cid:105) := |− α (cid:105) . (1)The four Bell states of coherent-state qubits are | φ ± (cid:105) := N ± ( | α (cid:105) | α (cid:105) ± |− α (cid:105) |− α (cid:105) ) , | ψ ± (cid:105) := N ± ( | α (cid:105) |− α (cid:105) ± |− α (cid:105) | α (cid:105) ) , N ± := (cid:104) (cid:16) ± e − | α | (cid:17)(cid:105) − / are normaliza-tion factors. A BSM of lossless coherent-state qubitsis performed with a 50:50 beam splitter and two pho-ton number parity detectors (PNPDs) [15, 16], asseen in Fig. 1. The four Bell states can be deter-ministically identiﬁed from the results of the PNPDsunless both of the PNPDs do not detect any photonsas (even , → | φ + (cid:105) , (odd , → | φ − (cid:105) , (0 , even) → | ψ + (cid:105) , (0 , odd) → | ψ − (cid:105) . (2)In the case that both of the PNPDs do not detectphotons, which we call ‘failure,’ only the sign ( ± for | φ ± (cid:105) and | ψ ± (cid:105) ) of the Bell state can be determinedsince there exists ambiguity between | φ + (cid:105) and | ψ + (cid:105) .For realistic scenarios, we need to introduce pho-ton loss. We use the photon loss model by the Mas-ter equation under the Born-Markov approximationwith zero temperature [58]: ∂ρ∂τ = γ (cid:88) i (cid:18) ˆ a i ρ ˆ a † i −

12 ˆ a † i ˆ a i ρ − ρ ˆ a † i ˆ a i (cid:19) , (3)where ρ ( τ ) is the density operator of system suﬀeringphoton loss as the function of time τ , γ is the decayconstant, and ˆ a i (ˆ a † i ) is the annihilation (creation)operator of the i th mode. It is known that this pho-ton loss model is equivalent with the beam splittermodel where each mode is independently mixed withthe vacuum state by a beam splitter with the trans-mittance t = e − γτ/ and the reﬂectance r = √ − t [59]: ˆ a ˆ b → ˆ a (cid:48) ˆ b (cid:48) = t − rr t ˆ a ˆ b . (4)Here, ˆ a (ˆ a (cid:48) ) is the annihilation operator of the input(output) mode, and ˆ b (ˆ b (cid:48) ) is that of the input (out-put) mode of the ancillary system which is initiallyin the vacuum state. The ﬁnal state after suﬀeringphoton loss is obtained by tracing out the ancillarysystem from the output state of the beam splitter.Considering the photon survival rate η = t , the ﬁ-nal state can be expressed in terms of η .Now, we consider the BSM on lossy coherent-statequbits. Precisely speaking, we deal with a situa-tion that the two coherent-state qubits suﬀer pho-ton losses before the BSM of Fig. 1 is performed.We ﬁrst rewrite each element of the BSM scheme inmathematical term: U BS is a unitary channel cor-responding to a 50:50 beam splitter, Λ η is a pho- ton loss channel with a survival rate η , and Π x for x ∈ { , , } is a projector deﬁned byΠ := | F (cid:105)(cid:104) F | , Π := (cid:88) n ≥ | n F (cid:105)(cid:104) n F | , Π := (cid:88) n ≥ | n F (cid:105)(cid:104) n F | , where | n F (cid:105) is the Fock state with a photon numberof n . A set of operators, M x,y := [ U BS ◦ ( Λ η ⊗ Λ η )] † (Π x ⊗ Π y )with x, y ∈ { , , } , then forms a positive-operatorvalued measure (POVM) corresponding to the BSMof lossy coherent-state qubits. Explicit forms ofthem are presented in Appendix A.Assuming the equal prior probability distributionof the four Bell states B = {| φ ± (cid:105) , | ψ ± (cid:105)} , we choosethe Bell state | B (cid:105) ∈ B which maximizes the poste-rior probability from the PNPD results ( x, y ): Pr ( B | x, y ) = Pr ( x, y | B ) Pr ( B ) (cid:80) | B (cid:48) (cid:105)∈B Pr ( x, y | B (cid:48) ) Pr ( B (cid:48) ) ∝ Pr ( x, y | B ) = (cid:104) B | M x,y | B (cid:105) . (5)In other words, we choose | B (cid:105) ∈ B satisfying | B (cid:105) = argmax | B (cid:48) (cid:105)∈B (cid:104) B (cid:48) | M x,y | B (cid:48) (cid:105) , (6)for the ﬁnal result of the BSM. A straightforwardanalysis with Eq. (6) and the POVM elements ofBSM presented in Appendix A shows the correspon- TABLE I. Correspondences between the pairs of thePNPD results and the resulting Bell states. The Bellstate | B (cid:105) ∈ {| φ ± (cid:105) , | ψ ± (cid:105)} is chosen to maximize the pos-terior probability Pr ( B | x, y ). Here, x and y indicate theresults of two PNPDs, where 0, 1, and 2 mean zero, odd,and even detection, respectively. The cases that both x and y are nonzero occur only when the loss rates ofthe two modes are diﬀerent. We also note that only thesign of the Bell state can be determined in the cases of x = y , which we call ’failure,’ since both | φ + (cid:105) and | ψ + (cid:105) maximize the posterior probability at the same time. x \ y φ + or ψ + ψ − ψ + φ − φ + or ψ + ψ − φ + φ − φ + or ψ + x and y can be nonzero at the sametime, while the probabilities of these cases vanishfor η = η .If the state before suﬀering the photon loss is oneof the four Bell states, there are ﬁve possible casesregarding the result of the measurement: success, X -error, Z -error, Y -error, and failure. If the resultingBell state is the same with the initial one, we call itsuccess. X -error corresponds to ‘letter ﬂip’, i.e., thechange of the letter ( φ or ψ ) in a Bell state such asfrom | φ + (cid:105) to | ψ + (cid:105) . Z -error corresponds to ‘sign ﬂip’,i.e., the change of the sign ( ± ) in a Bell state suchas from | φ + (cid:105) to | φ − (cid:105) . Y -error corresponds to simul-taneous symbol and sign ﬂips. The last case, failure,corresponds to the cases of x = y in Table I that theletter of the Bell state cannot be determined sinceboth | φ + (cid:105) and | ψ + (cid:105) maximize the posterior proba-bility at the same time. We would like to emphasizethat the sign still can be determined for this case.Now, we numerically analyze the success, failure,and error probabilities of BSM on coherent-statequbits. We consider a BSM on coherent-state qubitsperformed jointly on two systems which suﬀer in-ternal losses with the survival rates of η and thephotons of the second system travel the distance of L = 1 km before the measurement. The photon sur-vival rates of the two systems are then η := η and η := η e − L /L att , respectively, where L att = 22 kmis the attenuation length.Figure 2 shows the success, failure, and error prob-abilities of the BSM in this situation against the am-plitude α of the coherent state and the internal pho-ton survival rate η . It shows the well-known factthat the success probability is higher than that of aBSM on multi-photon polarization qubits with thesame photon number. Also, the failure and Z -errorprobabilities have a trade-oﬀ relation with changing α ; when α increases, failures get less probable while Z -errors get more probable. It is because coherentstates with large amplitudes have less overlaps withthe vacuum state and are more vulnerable to de-phasing by photon loss. Furthermore, we would liketo emphasize that the error is strongly biased, i.e.,the X - and Y -error probabilities are much smallerthan the failure and Z -error probabilities regardlessof α and η : p x , p y (cid:47) − . They even vanish if η = η , which is the consequence from the fact thatboth x and y in Table I can be nonzero simultane-ously only when the two photon survival rates arediﬀerent. This fact is important for constructing a Prob. (a) (b) p i p poli p fail p Z FIG. 2. The success ( p i ), failure ( p fail ), and Z -error probabilities ( p z ) of BSM on coherent-state qubitsagainst (a) α (ﬁxing η = 0 .

99) (b) η (ﬁxing α = 1).We set the photon survival rates of the two systemsas η := η and η := η e − L/L att , where L := 1 kmand L att := 22 km. It corresponds to the situation thatboth systems suﬀer internal losses with the photon sur-vival rates of η and the photons of the second systemtravel the distance of L := 1 km before the measure-ment. The blue solid line is the success probability p i ,the green dash-dotted line is the failure probability p fail ,and the red dotted line is the Z -error probability p z .Also, the gray dashed line is the success probability ofBSM on multi-photon polarization qubits for diﬀerentphoton numbers [10], which is plotted for comparison,where α is now the amplitude of the coherent state whichhas the same photon number with the qubit. The X -error ( p x ) and Y -error probabilities ( p y ) are not plottedsince they are much smaller than other probabilities re-gardless of α and η : p x , p y (cid:47) − hardware-eﬃcient CBSM scheme in Sec. IV C. III. MODIFIED PARITY ENCODINGSCHEME WITH COHERENT-STATEQUBITS

Now, we present the encoding scheme we use forour CBSM scheme. We modify the parity state en-coding or generalized Shor’s encoding [11, 60] for thecoherent-state qubit. The modiﬁed parity encodingis deﬁned as follows.

Deﬁnition 1.

The basis qubits {| L (cid:105) , | L (cid:105)} of( n, m, α ) modiﬁed parity encoding scheme where n and m are odd integers and α is a complex numberare deﬁned as: | L (cid:105) := (cid:104) N ( m ) (cid:110)(cid:12)(cid:12) (cid:101) + (cid:11) ⊗ m + (cid:12)(cid:12) (cid:101) − (cid:11) ⊗ m (cid:111)(cid:105) ⊗ n , | L (cid:105) := (cid:104) N ( m ) (cid:110)(cid:12)(cid:12) (cid:101) + (cid:11) ⊗ m − (cid:12)(cid:12) (cid:101) − (cid:11) ⊗ m (cid:111)(cid:105) ⊗ n , (cid:12)(cid:12) (cid:101) ± (cid:11) := | α (cid:105) ± |− α (cid:105) are unnormalizedSCSs (we use the tilde above the ket to de-note that it is unnormalized) and N ( m ) := (cid:104) m (cid:110)(cid:16) e − | α | (cid:17) m + (cid:16) − e − | α | (cid:17) m (cid:111)(cid:105) − / . Thisencoding scheme coincides the original coherent-state encoding in Eq. (1) when n = m = 1.The modiﬁed parity encoding has a hierarchystructure of Hilbert spaces: logical , block , and phys-ical level. The logical-level space is the total Hilbertspace spanned by {| L (cid:105) , | L (cid:105)} . It can be dividedinto n block-level spaces (referred as blocks), eachof which is spanned by (cid:8)(cid:12)(cid:12) ± ( m ) (cid:11)(cid:9) where (cid:12)(cid:12) ± ( m ) (cid:11) := N ( m ) (cid:110)(cid:0)(cid:12)(cid:12) ˜+ (cid:11)(cid:1) ⊗ m ± (cid:0)(cid:12)(cid:12) ˜ − (cid:11)(cid:1) ⊗ m (cid:111) . A block is again di-vided into m physical-level spaces (referred as PLSs),each of which is spanned by |± α (cid:105) .We also deﬁne four Bell states for each level as fol-lowing, where normalization constants are omitted: a. Logical level: | Φ ± (cid:105) := | L (cid:105) | L (cid:105) ± | L (cid:105) | L (cid:105)| Ψ ± (cid:105) := | L (cid:105) | L (cid:105) ± | L (cid:105) | L (cid:105) b. Block level: (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:69) := (cid:12)(cid:12)(cid:12) + ( m ) (cid:69) (cid:12)(cid:12)(cid:12) + ( m ) (cid:69) ± (cid:12)(cid:12)(cid:12) − ( m ) (cid:69) (cid:12)(cid:12)(cid:12) − ( m ) (cid:69)(cid:12)(cid:12)(cid:12) ψ ( m ) ± (cid:69) := (cid:12)(cid:12)(cid:12) + ( m ) (cid:69) (cid:12)(cid:12)(cid:12) − ( m ) (cid:69) ± (cid:12)(cid:12)(cid:12) − ( m ) (cid:69) (cid:12)(cid:12)(cid:12) + ( m ) (cid:69) c. Physical level: | φ ± (cid:105) := | α (cid:105) | α (cid:105) ± |− α (cid:105) |− α (cid:105)| ψ ± (cid:105) := | α (cid:105) |− α (cid:105) ± |− α (cid:105) | α (cid:105) Each logical-level Bell state can be decomposedinto block-level Bell states: (cid:12)(cid:12) Φ +( − ) (cid:11) = ˜ N ± ,n,m × (cid:88) k =even(odd) ≤ n P (cid:34)(cid:12)(cid:12)(cid:12)(cid:12) (cid:103) φ ( m ) − (cid:29) ⊗ k (cid:12)(cid:12)(cid:12)(cid:12) (cid:103) φ ( m )+ (cid:29) ⊗ n − k (cid:35) , (7a) (cid:12)(cid:12) Ψ +( − ) (cid:11) = ˜ N ± ,n,m × (cid:88) k =even(odd) ≤ n P (cid:34)(cid:12)(cid:12)(cid:12)(cid:12) (cid:93) ψ ( m ) − (cid:29) ⊗ k (cid:12)(cid:12)(cid:12)(cid:12) (cid:93) ψ ( m )+ (cid:29) ⊗ n − k (cid:35) , (7b) where˜ N ± ,n,m := 1 √ n − (cid:2) ± u ( α, m ) n (cid:3) − , (8) (cid:12)(cid:12)(cid:12)(cid:12) (cid:103) φ ( m ) ± (cid:29) := (cid:2) ± u ( α, m ) (cid:3) (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:69) , (9) (cid:12)(cid:12)(cid:12)(cid:12) (cid:93) ψ ( m ) ± (cid:29) := (cid:2) ± u ( α, m ) (cid:3) (cid:12)(cid:12)(cid:12) ψ ( m ) ± (cid:69) ,u ( α, m ) := (cid:16) e − | α | (cid:17) m − (cid:16) − e − | α | (cid:17) m (cid:0) e − | α | (cid:1) m + (cid:0) − e − | α | (cid:1) m , (10)and P [ · ] is the summation of all the possible per-mutations of the tensor product inside the squarebracket.Similarly, each block-level Bell state can be de-composed into physical-level Bell states: (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:69) = ˜ N ± , ,m √ (cid:88) l =even ≤ m P (cid:104) | ψ ± (cid:105) ⊗ l | φ ± (cid:105) ⊗ m − l (cid:105) , (11a) (cid:12)(cid:12)(cid:12) ψ ( m ) ± (cid:69) = ˜ N ± , ,m √ (cid:88) l =odd ≤ m P (cid:104) | ψ ± (cid:105) ⊗ l | φ ± (cid:105) ⊗ m − l (cid:105) . (11b)The core of CBSM is contained in Eqs. (7) and(11); they make it possible to perform a logical BSMby the combination of n block-level BSMs, each ofwhich is again performed by the combination of m physical-level BSMs.The equations also show that, in a lossless sys-tem, a CBSM does not incur any logical error, i.e.,the only possible cases are success and failure. Thisproperty is important since failures are detectablewhereas logical errors are not. Hence, the modi-ﬁed parity encoding in Deﬁnition 1 is the naturalextension of the original coherent-state encoding inEq. (1), in the sense that this desired property stillremains. If we use other states such as normalizedSCSs or coherent states in place of unnormalizedSCSs (cid:12)(cid:12) (cid:101) ± (cid:11) for the encoding, this property no longerexists. IV. CONCATENATED BELL-STATEMEASUREMENT WITH ENCODEDCOHERENT-STATE QUBITS

Now, we suggest concatenated Bell-state measure-ment (CBSM) schemes with the modiﬁed parity5 SM Logical levelBSM BSM ⫶ BSM BSM Block levelBSM BSM ⫶ BSM BS PNPD

Physical levelBSM

12n 12m

FIG. 3. Schematic ﬁgure of CBSM schemes withcoherent-state qubits. The scheme is done in concate-nated manner: each logical-level BSM (BSM ) is doneby the combination of n block-level BSMs (BSM ). Eachblock-level BSM is again done by the combination of m physical-level BSMs (BSM ). encoding presented in the previous section. Theschematic ﬁgure of the CBSM schemes is shown inFig. 3. As mentioned in the previous section, eachlogical-level BSM is done by the composition of n block-level BSMs and each block-level BSM is doneby the composition of m physical-level BSMs. Weﬁrst consider an unoptimized scheme which consistsof simple counting of measurement results. We thenpresent a hardware-eﬃcient scheme which can sig-niﬁcantly reduce the expected cost of the CBSM de-ﬁned in terms of the expected number of physical-level BSMs used for a single CBSM. A. Unoptmized CBSM scheme

Here, we suggest a CBSM scheme which is un-optimized but much simpler than the hardware-eﬃcient scheme presented in the next subsection. Itis straightforward to justify the scheme with Eqs. (7)and (11). The interpretation of the measurement re-sults in the scheme is summarized in Table II.

1. Physical level:

BSM For a physical-level BSM (referred as BSM ), weuse the BSM scheme for single lossy coherent-statequbit presented in Fig. 1 and Table I. Remark thatthe sign of the Bell state is always determinable,while its letter is not determinable if the results ofthe two PNPDs are the same, i.e., x = y in Table I.

2. Block level:

BSM A block-level BSM (referred to BSM ) is done byperforming BSM on each PLS in the block. Thesign of the block-level Bell state is determined bythe majority vote of the signs of the BSM results.Its letter is determined by the parity of the numberof BSM results with ψ letter: φ ( ψ ) if the numberis even (odd).Since m is odd, the sign of the block-level Bellstate is always determinable. The letter is not deter-minable if at least one BSM fails, which we regardthat the BSM fails.

3. Logical level:

BSM A logical-level BSM (referred as BSM ) is doneby performing BSM on each block. The sign of thelogical-level Bell state is determined by the parity ofthe number of BSM results with minus sign: plus(minus) if the number is even (odd). Its letter isdetermined by the majority vote of the letters of theBSM results excluding the failed ones.Again, the sign of the logical-level Bell state isalways determinable. Its letter is not determinableif all the BSM s fail or the resulting block-level Bellstates have the same number of both letters. Weregard these cases as failure of BSM . B. Fault-tolerance of concatenated Bell-statemeasurement

Now, we investigate fault-tolerance of the unopti-mized CBSM scheme suggested in the previous sub-section. We argue that the physical-level and block-level repetitions contribute to suppressing logical er-rors and failures, respectively.First, Z ( X )-errors in the logical level are sup-pressed by the majority vote at the block (logical)level. Remark that the sign (letter) of a logical-levelBell state is determined only by the signs (letters) ofthe Bell states of the lower levels, as described in Ta-ble II. Z -errors (sign ﬂips) in the physical level canbe corrected by the majority vote in the block level,so do not cause a logical-level Z -error with a highprobability. Similarly, X -errors (letter ﬂips) in thephysical level can be corrected by the majority votein the logical level, so also do not cause a logical-level X -error with a high probability. Since Z -errors aremuch more common than X -errors in the physical6 ABLE II. Interpretation of the measurement results in the unoptimized CBSM scheme. It is also valid in thehardware-eﬃcient CBSM scheme, if we consider the results of BSM (BSM ) and BSM sign0 (BSM sign1 ) together whendetermining the sign of the block (logical) level Bell state. Level Sign ( ± ) Letter ( φ or ψ )Physical(BSM ) BSM scheme of original coherent-state qubitsBlock(BSM ) Majority vote of the signsof the BSM results Number of BSM results with ( − ) sign: φ if even, ψ if oddLogical(BSM ) Number of BSM results with ψ letter:(+) if even, ( − ) if odd Majority vote of the lettersof the BSM resultslevel ( p x /p z (cid:47) − ), we can infer that the physical-level repetition is crucial for fault-tolerance.However, we cannot assure that the repetitionsalways suppress logical errors. Although Z -errorscan be corrected by the physical-level repetition, theblock-level repetition has a rather negative eﬀect onit. Due to the error correction by the physical-levelrepetition, a block-level BSM result does not have a Z -error with a high probability. However, any singleremained Z -error among the block-level BSM resultscan cause a Z -error in the logical level. Therefore,a large value of the size of the block-level repetition( n ) leads to vulnerability of the CBSM to Z -errors.A similar logic applies to X -errors; the physical-levelrepetition has a negative eﬀect on it.Next, we consider failures in the logical level. Asexplained in the previous subsection, a BSM fails ifall the BSM s fail or the results of the BSM s havethe same number of both letters, and a BSM failsif any single BSM fails. The block-level repetitionthus suppresses failures of the CBSM, whereas thephysical-level repetition makes it vulnerable to fail-ures.In summary, ignoring X -errors which are muchmore uncommon than Z -errors and failures, thephysical(block)-level repetition contributes to mak-ing the CBSM tolerant to Z -errors (failures) but vul-nerable to failures ( Z -errors). Despite these negativeeﬀects, we numerically show in Sec. VI that a highsuccess probability are still achievable if the survivalrate of photons is high enough and the amplitude ofthe coherent state is large enough. C. Improved hardware-eﬃcient CBSM scheme

In this subsection, we suggest an improved CBSMscheme which is elaborately designed consideringhardware eﬃciency. We explicitly deﬁne the cost ofa single trial of CBSM in the last part of this section, but we ﬁrst regard it as the number of physical-levelBSMs used for it. Note that the cost is generallynot determined by the CBSM scheme alone; it canbe diﬀerent for each trial of CBSM.The unoptimized scheme in Sec. IV A always re-quires nm physical-level BSMs, and here we suggesta way to decrease the number. The core idea is thatit is redundant to perform ‘full’ BSMs for all thePLSs or blocks, where the term ‘full’ is used to em-phasize that the BSM captures both sign and letterinformation of the Bell state. For some PLSs orblocks, it is enough to get only the sign ( ± ) infor-mation of the Bell state or even do not measure itat all. Especially for the logical level, it is enough toperform full BSMs only for the ﬁrst few blocks dueto the biased noise. The hardware-eﬃcient CBSMscheme which is presented from now on is summa-rized in Fig. 4.

1. Physical level:

BSM and BSM sign0

BSM is completely same with the scheme givenin Sec. II. Using a 50:50 beam splitter and two PN-PDs (see Fig. 4(a)), one of the four Bell states canbe identiﬁed according to the results of the PNPDs,unless the two results are the same (failure). In thecase of failure, only sign information of the Bell statecan be captured.However, we need another ingredient in the phys-ical level for the hardware-eﬃcient CBSM scheme:partial physical-level BSM identifying only the sign( ± ) of the physical-level Bell state, which we denoteBSM sign0 (see Fig. 4(b)). For BSM sign0 , one needs tomeasure the parity of x + y in the Table I. Therefore,only one PNPD is needed for a BSM sign0 instead oftwo.7 S PNPD

BS PNPD s+1 n 𝑗 not-failed BSM ! BSM !, (e)

𝐁𝐒𝐌 (c) 𝐁𝐒𝐌 d+1 m BSM ’ Case 1. No failed

BSM ! d+1 f BSM ’ f+1 m Do nothing

Case 2. 𝑑 ≤ 𝑓 f+1 d

BSM ’ d+1 m BSM ’,

Do nothing

Case 3. 𝑑 > 𝑓 d+1 m BSM ’,

Do nothing (d)

𝐁𝐒𝐌 (a)

𝐁𝐒𝐌 (b) 𝐁𝐒𝐌

FIG. 4. Overview of the hardware-eﬃcient CBSM scheme. (a) For a physical-level BSM (BSM ), a 50:50 beamsplitter and two PNPDs are used. (b) For a physical-level partial BSM detecting only the sign (BSM sign0 ), a singlePNPD is necessary, instead of two. (c) For a block-level BSM (BSM ), one of BSM and BSM sign0 is performed oneach PLS one by one. We ﬁrst deﬁne positive integers d and f . d is the index of the ﬁrst PLS such that (cid:100) m/ (cid:101) of the physical-level BSM results until that PLS have the same sign. f is the index of the ﬁrst PLS such that thecorresponding physical-level BSM fails, which is deﬁned only if such a PLS exists. (Case 1) If there are no failedBSM s ( f is not deﬁned), BSM s are performed on the entire PLSs. (Case 2) If d ≤ f , BSM s are performed on theﬁrst f PLSs and the remained PLSs are left untouched. (Case 3) If d > f , one performs BSM s for the ﬁrst f PLSsand BSM sign0 s for the next d − f PLSs. The remained PLSs are left untouched. The reason to be able to do nothingfor the last several PLSs in

Case 2 and

Case 3 is that these two cases correspond to the failure of the BSM , somore physical-level BSMs are meaningless if the sign of the block-level Bell state is determined. (d) For a block-levelpartial BSM detecting only the sign (BSM sign1 ), BSM sign0 s are performed for the ﬁrst d PLSs and the remained PLSsare left untouched. (e) For a logical-level BSM (BSM ), BSM s are performed one by one until j not-failed BSM results are obtained, where j is a controllable positive integer referred as the letter solidity parameter . BSM sign1 s arethen performed for the left blocks.

2. Block level:

BSM and BSM sign1

For a block-level BSM, we perform one of BSM or BSM sign0 on each PLS, one by one in order. Theprocess is not parallel, since the determination be-tween BSM and BSM sign0 is aﬀected by the previousmeasurement results. We ﬁrst deﬁne a positive in-teger d ≤ m by the index of the ﬁrst PLS such that (cid:100) m/ (cid:101) of the physical-level BSM results until thatPLS have the same sign. In other words, the resultof the majority vote of the signs is already deter-mined until d th physical-level BSM, and thus thesign information is no longer necessary. Also, we de- ﬁne a positive integer f ≤ m by the index of the ﬁrstPLS such that the corresponding BSM fails, whichis deﬁned only if such a PLS exists.Three cases are possible on BSM : no failedphysical-level BSMs ( f is not deﬁned), d ≤ f , and d > f (see Fig. 4(c)). (Case 1) If there are nofailed physical-level BSMs, it is same with the un-optimized scheme; BSM s are performed for all thePLSs. (Case 2) If d ≤ f , BSM s are performedfor the ﬁrst f PLSs. The remained m − f PLSs areleft untouched. (Case 3) If d > f , BSM s are per-formed for the ﬁrst f PLSs, and then BSM sign0 s areperformed for the next d − f PLSs. The remained8 − d PLSs are left untouched.For all the three cases, the sign of the block-levelBell state is determined by the signs of the ﬁrst d BSM (or BSM sign0 ) results. However, the letter isdetermined only for the ﬁrst case by the parity ofthe number of BSM results with letter ψ . For thesecond and third case, there exists a failed BSM ,so the number of results with letter ψ is ambiguous.Hence, the BSM fails in these two cases. This isthe reason to be able to do nothing on the last sev-eral PLSs after the sign of the block-level BSM isdetermined.Like the physical level, we also consider partialblock-level BSM which determines only the sign ofthe block-level Bell state (BSM sign1 ) (see Fig. 4(d)).For BSM sign1 , BSM sign0 s are performed for the ﬁrst d PLSs, and the remained PLSs are left untouched.The sign of the block-level Bell state is determinedby the majority vote of the results of the ﬁrst d BSM sign0 results.

3. Logical level:

BSM For a logical-level BSM (BSM ) (see Fig. 4(e)),BSM s are performed one by one until we get j not-failed results. j is a controllable positive integer re-ferred as the letter solidity parameter which meansthat high values of j lead to high probabilities to getcorrect letter information. After that, BSM sign1 s areperformed for the remained blocks.The sign of the resulting logical-level Bell state isdetermined by the parity of the number of BSM or BSM sign1 results with minus sign. The letter isdetermined by the majority vote of the letters amongthe ﬁrst j not-failed BSM results.Note the diﬀerence between BSM and BSM : ForBSM , the majority vote is taken for the ﬁrst j not-failed BSM s with a ﬁxed j , while for BSM , themajority vote is taken when the result of the major-ity vote on the total PLSs is deﬁnitely determined.This asymmetry comes from the fact that the noiseis strongly biased; X -errors are much less likely tooccur compared to Z -errors in BSM as shown inFig. 2. Therefore, when taking the majority vote ofthe letters of the BSM results, it is enough to useonly a few BSM results to correct X -errors. Onthe other hands, the majority vote of the signs ofthe physical-level BSM results should be taken for alarge number of PLSs.

4. Calculation of the cost

At the beginning of this subsection, we regard thecost of a single CBSM by the number of physical-level BSMs used for the measurement. However,considering that PNPDs are the most diﬃcult el-ements when implementing the BSM scheme and aBSM sign0 uses one of them while a BSM uses two, itis reasonable to assign each BSM sign0 half the cost ofone BSM . Deﬁnition 2.

The cost function C of a single trialof CBSM is deﬁned by C := N BSM + 12 N BSM sign0 , (12)where N BSM and N BSM sign0 are the number ofBSM s and BSM sign0 s used for the CBSM, re-spectively. Also, we deﬁne the expected cost C exp ( n, m, α, j ; η ) by the expectation value of thecost C for the CBSM scheme speciﬁed by the pa-rameters ( n, m, α, j ) and the photon survival rate η , with the assumption that the initial state beforesuﬀering photon loss is one of the four logical Bellstates with equal probabilities.We use the expected cost C exp as a measureof hardware-eﬃciency of a CBSM scheme. It isstraightforward to see that the CBSM scheme in theprevious subsection has a less expected cost than theunoptimized one in Sec. IV A. Not only that, it isdesigned to minimize the expected cost. For BSM ,the numbers of BSM and BSM sign0 are minimizedwhile keeping the result to be the same with that ofthe corresponding BSM in the unoptimized scheme.For BSM , the expected cost is determined by thecontrollable letter solidity parameter j . D. Parallelization of concatenated Bell-statemeasurement

The two CBSM schemes in Sec. IV A and IV Care processed in a completely or partially distributedmanner, which makes eﬃcient information process-ing possible by parallelization. The unoptimizedscheme is done in a completely distributed manner,i.e., a BSM is split by 2 nm BSM s, each of whichis performed independently. The BSM results arecollected classically to deduct the logical-level BSMresult.The hardware-eﬃcient scheme also can be done ina partially distributed manner allowing partial par-allelization, with requirements of classical communi-9ation channels between diﬀerent PLSs and blocks.In a BSM , BSM s can be done parallelly for the ﬁrst j blocks, then one by one until obtaining j not-failedBSM results, where j is the letter solidity param-eter. BSM sign1 s for the remained blocks also can bedone parallelly. In a BSM , BSM s should be doneone by one until a BSM fails, so BSM s in all thethree cases cannot be done parallelly. Case 3 can bepartially parallelized only if f < m/

2: BSM sign0 s canbe done parallelly for ( f +1)th to (cid:100) m/ (cid:101) th PLS since d is always larger than m/

2. In BSM sign1 , BSM sign0 can be done parallelly for the ﬁrst (cid:100) m/ (cid:101) PLSs, thenone by one for the remained PLSs.Therefore, the hardware eﬃciency is the result ofthe sacriﬁce of parallelization. We can still widenthe range of parallelization by adjusting the schemeappropriately at the expense of reducing hardwareeﬃciency. For example, in a BSM , BSM s can bedone for the ﬁrst j blocks, not for the ﬁrst not-failed j blocks. Moreover, in a BSM and BSM sign1 , insteadof determining the type of BSM (BSM or BSM sign0 )separately for each PLS, we can divide the PLSsinto several groups and perform BSMs with the sametype parallelly on PLSs in each group. However, weuse the original hardware-eﬃcient CBSM scheme forthe numerical simulation in Sec. VI to ﬁgure out thebest possible performance. V. PROBABILITY DISTRIBUTIONS OFCONCATENATED BELL-STATEMEASUREMENT RESULTS

In this section, we present the analytic expres-sions of the probability distributions of CBSM re-sults conditioning to the initial Bell state before suf-fering photon loss. We only consider the unopti-mized CBSM scheme, since the measurement resultsof the hardware-eﬃcient CBSM scheme is the di-rect consequence of those of the unoptimized scheme.Here, we show only the ﬁnal results. A brief outlinefor inducing the results is presented in Appendix C.The results of this section have two importantmeanings. First, the probability distributions arewritten in simple matrix-form expressions, whichmakes it possible to sample arbitrary CBSM resultsat a high rate, since a matrix calculation can be donemuch faster on a computer compared to calculatingthe same thing by simple loops. Second, the resultscan be easily generalized to any CBSM schemes withother encoding methods such as multi-photon polar-ization encoding [11].

A. Probability distributions of block-levelresults

We ﬁrst ﬁnd the probability distributions of block-level BSM results, conditioning to the initial block-level Bell state. A single BSM result can be ex-pressed by two vectors x , y ∈ { , , , } m , where the i th elements of them are the two PNPD results of the i th PLS. What we want is the conditional probabil-ity Pr ( x , y | B ) for | B (cid:105) ∈ B := (cid:110)(cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:69) , (cid:12)(cid:12)(cid:12) ψ ( m ) ± (cid:69)(cid:111) .First, we deﬁne 4 × ˜M ± x,y for x, y ∈{ , , , } as: ˜M ± x,y := M ± M ± M ± M ± M ± M ± M ± M ± M ± M ± M ± M ± M ± M ± M ± M ± , where M ± := (cid:104) φ ± | ˆ M x,y | φ ± (cid:105) ,M ± := (cid:104) φ ± | ˆ M x,y | ψ ± (cid:105) ,M ± := (cid:104) ψ ± | ˆ M x,y | ψ ± (cid:105) are the matrix elements of POVM elements of BSM and can be calculated from Eqs. (A2) in AppendixA. The conditional probability Pr ( x , y | B ), wherethe k th element of x ( y ) is x k ( y k ), is then: Pr (cid:16) x , y (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:17) = 12 ˜ N ± (1 , m ) v ± m ( x , y ) , (13a) Pr (cid:16) x , y (cid:12)(cid:12)(cid:12) ψ ( m ) ± (cid:17) = 12 ˜ N ± (1 , m ) v ± m ( x , y ) , (13b)where ˜ N ± (1 , m ) is deﬁned in Eq. (8) and v ± mµ ( x , y )is the µ th element of a four-dimensional vector v ± m ( x , y ) = ˜M ± x m ,y m · · · ˜M ± x ,y (1 , , , T .A brief outline for inducing these results is pre-sented in Appendix B 1. B. Probability distributions of logical-levelresults

Now, we consider the probability distributions oflogical-level results conditioning to the initial logical-level Bell state, which is the goal of this section. Asingle CBSM result can be expressed by two matrices X , Y ∈ { , , , } n × m , where the ( i, k ) elements ofthem are the two PNPD results of the k th PLS of the10 th block. What we want is the conditional proba-bility Pr ( X , Y | B ) for | B (cid:105) ∈ B := {| Φ ± (cid:105) , | Ψ ± (cid:105)} .We ﬁrst deﬁne 2 × ˜L φ x , y and ˜L ψ x , y where x , y ∈ { , , , } m in the similar way with the block-level case: ˜L φ ( ψ ) x , y := L φ ( ψ )+ L φ ( ψ ) − L φ ( ψ ) − L φ ( ψ )+ , where L φ ( ψ ) ± := (cid:2) ± u ( α, m ) (cid:3) × (cid:68) φ ( m ) ± (cid:16) ψ ( m ) ± (cid:17)(cid:12)(cid:12)(cid:12) m (cid:79) k =1 ˆ M x k ,y k (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:16) ψ ( m ) ± (cid:17)(cid:69) , (14) u ( α, m ) is deﬁned in Eq. (10), and x k ( y k ) is the k thelement of x ( y ). We note that the RHS of Eq. (14)can be calculated from Eqs. (13). The conditionalprobability Pr ( X , Y | B ), where the i th row vectorof X ( Y ) is x i ( y i ), is then Pr ( X , Y | Φ + (Ψ + )) = ˜ N + ( n, m ) w φ ( ψ ) n ( X , Y ) , Pr ( X , Y | Φ − (Ψ − )) = ˜ N − ( n, m ) w φ ( ψ ) n ( X , Y ) , where ˜ N ± ( n, m ) is deﬁned in Eq. (8) and w φ ( ψ ) nµ ( X , Y ) is the µ th element of thetwo-dimensional vector w φ ( ψ ) n ( X , Y ) := ˜L φ ( ψ ) x n , y n · · · ˜L φ ( ψ ) x , y (1 , T . A brief outline for in-ducing these results is presented in AppendixB 2.In conclusion, one can calculate the probabilitydistributions of CBSM results by systematical ma-trix operations as described in this and the previoussubsection. The probability distributions then canbe used to sample the CBSM results for numericalcalculations. VI. NUMERICAL CALCULATIONS

In this section, we show the results of the numeri-cal calculations. We use the Monte-Carlo method forthe simulation: sampling the measurement resultsrandomly and counting the number of successes, er-rors, and failures. We sample the result of eachphysical-level BSM one by one in order, which is ex-ponentially faster than sampling the entire measure-ment results at once. The detailed method for sam-pling the CBSM results using the results of Sec. Vis presented in Appendix C. Remark that there are four free parameters re-lated to the hardware-eﬃcient CBSM scheme: n , m , α , and j . n and m determine the block-leveland physical-level repetition size of the scheme, re-spectively. α is the amplitude of the coherent stateconstituting the logical basis. j is the letter solidityparameter which is the number of not-failed blocksused for the majority vote of letters in BSM . A. Performance analysis

Now, we analyze the performance of the hardware-eﬃcient CBSM scheme suggested in Sec. IV C bycalculating numerically the success, error, and fail-ure probabilities of the scheme with various settingsof the parameters ( n, m, α, j ). For the simulation,we assume that both systems have the same photonsurvival rates η . We use the Monte-Carlo method asmentioned before. For each trial, we ﬁrst choose oneof the four Bell states as the initial state with equalprobabilities, sample the physical-level BSM resultswith respect to the selected initial state, and deter-mine the logical Bell state by the hardware-eﬃcientCBSM scheme. Repeating this trials many times, wedetermine the success ( p i ), Z -error ( p z ), and failureprobabilities ( p fail ) of the CBSM scheme. We alsocalculate the expected cost C exp deﬁned in Deﬁni-tion 2 .Figure 5 illustrates the success probability p i ofCBSM with coherent-state qubits and polarizationqubits [11] against the photon survival rate η fordiﬀerent ranges of the expected cost C exp , where p i is maximized for each η and C exp . Figure 5(a)shows that the repetition indeed enhances the per-formance if η (cid:39) . η is close to unity. For example, if η = 0 . p i = 0 .

80 without repetition, but it reaches 0.90 withjust a little repetition ( C exp ≤ < C exp ≤

35. In other words, it is the clearevidence that high success rates close to unity areachievable by CBSM if the photon survival rate issuﬃciently high. Meanwhile, comparing Fig. 5(a)and (b), we can see that the CBSM with coherent-state qubits outperforms that with multi-photon po-larization qubits when the repetition size is relativelysmall ( C exp ≤ η = 0 .

99, theCBSM with coherent-state qubits achieves p i = 0 . C exp ≤

5, while that with multi-photon polariza-tion qubits reaches only p i = 0 . p i ), Z -error( p z ), and failure probabilities ( p fail ) against n and m , for two diﬀerent values of α : α = 1 . .6 0.7 0.8 0.9 1.00.40.60.81.0 p i (a) Coherent-state qubit (b) Polarization qubit No repetition1 < C exp C exp C exp FIG. 5. Success probabilities p i of CBSM with (a)coherent-state qubits and (b) multi-photon polarizationqubits against the photon survival rate η for diﬀerentranges of the expected cost C exp . For coherent-statequbits, the amplitude α is ﬁxed to α = 1 . j is chosen to maximize p i for each η and range of C exp . For polarization qubits, we followthe CBSM scheme proposed in [11]. In this case, we de-ﬁne C exp := nm , which is the number of physical-levelBSMs used for one CBSM. (a) shows that the repetitionindeed contributes to enhance the success probability.Comparing (a) and (b), we can see that the CBSM withcoherent-state qubits outperforms that with polarizationqubits when the repetition size is relatively small. From this ﬁgure, we can check the dependence of p z and p fail on the repetition; it clearly shows thatthe physical-level repetition ( m >

1) suppresses Z -errors and the block-level repetition ( n >

1) sup-presses failures, as argued in Sec. IV B. Moreover,the negative eﬀects discussed in Sec. IV B that thephysical(block)-level repetition makes the CBSMvulnerable to failures ( Z -errors) are also shown inthe ﬁgure, and in spite of them, the success proba-bility close to unity still can be achieved.Lastly, the success probability p i against α and theexpected cost C exp for four diﬀerent survival rates( η = 1, 0.99, 0.95, and 0.9) is plotted in Fig. 7. Theﬁgure shows that the success probability over 0 . η ≥ .

95 and appropriate val-ues of α , if suﬃciently large costs of the CBSM isavailable. In lossless case ( η = 1), the success prob-ability reaches very close to unity for any α (cid:39) . α andcost are required for reaching high success probabil-ities. In detail, to reach p i > .

98, we need α (cid:39) . η = 0 .

99 and α (cid:39) . η = 0 .

95. Nonethe- n m (a) = 1.2 n (b) = 1.6 p i n m (c) = 1.2 n (d) = 1.6 p z n m (e) = 1.2 n (f) = 1.6 p fail FIG. 6. Success ( p i ), Z -error ( p z ), and failure probabil-ities p fail of CBSM against the repetition sizes n and m for coherent-state amplitudes α = 1 . α = 1 . η is ﬁxed to 0.99, and j is se-lected to maximize p i for each ( n, m ) point. It clearlyshows that physical-level repetition suppresses Z -errorand block-level repetition suppress failure. less, the ﬁgure also indicates that a higher value of α does not always guarantee a higher success ratedue to dephasing by photon loss, which is especiallyevident in (c) η = 0 . B. Quantum repeater with concatenatedBell-state measurement

In this subsection, we investigate the performanceof the quantum repeater scheme which uses the sug-gested CBSM scheme for quantum error correction,as one of the key applications of BSM.12 C exp (a) = 1 (b) = 0.99 (c) = 0.95 (d) = 0.9 p i FIG. 7. Success probabilities p i against the coherent-state amplitude α and the expected cost C exp for (a) η = 1,(b) η = 0 .

99, (c) η = 0 .

95, and (d) η = 0 .

9, where η is the photon survival rate of both parties. p i is selected bymax { p i ( n, m, α , j ) | C exp ( n, m, α , j ; η ) ∈ [ C − , C + 2) } for each point ( α , C ). The ﬁgure indicates that a largevalue of α does not always guarantee a high success probability, which is especially evident in (c). |𝜓 ! ⟩ RepeaterStation 𝐿 ! (a) (b) Incoming |Φ ! ⟩ CBSM

OutgoingRepeaterStation RepeaterStation RepeaterStation

Repeater Station

Classicalcommunication

FIG. 8. (a) Schematic of quantum information transmission through the quantum repeater scheme. Quantuminformation encoded in modiﬁed parity encoding is transmitted to the other end. It passes through multiple repeaterstations where the interval is L . (b) Schematic of processes inside a repeater station. A Bell state (cid:12)(cid:12) Φ + (cid:11) is preparedinside the station and a CBSM is performed between the incoming qubit and one side of the Bell state. The quantuminformation inside the incoming qubit is then teleported to the other side of the Bell state, which is then transmittedto the next repeater station. The measurement result of the CBSM is sent classically to the ﬁnal end for recoveringthe original quantum information. Due to fault-tolerance of the CBSM scheme, each repeater station can correctpossible logical errors from photon loss which the incoming qubit suﬀers.

1. Network design

We follow the network design in Ref. [11], whichsuggests an all-optical quantum network with quan-tum repeater exploiting the CBSM scheme withmulti-photon polarization qubits. As shown inFig. 8(a), we consider one-way quantum communica-tion with which a qubit encoded by the modiﬁed par-ity encoding is transmitted to the other end. Whiletraveling between two ends with the total distanceof L , the qubit passes through multiple repeater sta-tions separated by intervals of L . Figure 8(b) illus-trates the processes inside each repeater station. Ineach of them, a Bell state | Φ + (cid:105) is prepared and aCBSM is performed jointly on the incoming qubit and one side of the Bell state. The quantum infor-mation in the incoming qubit is then teleported tothe other side of the Bell state, which is transmittedto the next station. The measurement result of theCBSM in each station is sent classically to the ﬁnalend for recovering the original quantum information.Because of fault-tolerance of the CBSM scheme, eachrepeater station can correct possible logical errorsoriginated from photon loss, which makes a long-range transmission of quantum information possible.We assume two sources of photon loss: internalloss in each repeater station and loss during trans-mission between stations with survival rates of η and η L := e − L /L att , respectively, where L att =22 km is the attenuation length. Therefore, the sur-13 n m (a) Optimizing n (b) = 1.4 n (c) = 1.9 Q tot /10 n m (d) Optimizing , L = 0.7 km n (e) = 1.4, L = 0.7 km n (f) = 1.9, L = 0.7 km Rt FIG. 9. (top) Eﬀective total cost Q tot and (bottom) expected key length Rt of the quantum repeater againstrepetition sizes n and m , for three diﬀerent settings of the coherent-state amplitude α : optimizing α , ﬁxing α = 1 . α = 1 .

9. We ﬁx the total distance L = 1000 km and the internal photon survival rate in each station η = 0 .

99. For calculating Rt , we also ﬁx the station interval L = 0 . n, m ) point, other parameterssuch as the letter solidity parameter j and the station interval L (only for Q tot ) are selected to minimize Q tot ormaximize Rt . The ’X’ marks in (a) and (d) indicate the optimal point where Q tot is minimized. The parameters atthis point are ( n, m, α, j ) = (3 , , . ,

1) and L = 0 . Rt = 0 . ± .

02 and Q tot = (1 . ± . × atthis point, where the range is the 95% conﬁdence interval. vival rates of two systems on which CBSM is jointlyperformed is η := η e − L /L att and η := η .

2. Quantiﬁcation of the performance

One way to quantify the performance of a quan-tum repeater scheme is the asymptotic key gener-ation rate R of quantum key distribution (QKD),which is the expected length of a fully secure keythat can be produced per unit time [43, 61]. Moreprecisely, it is the product of the raw-key rate, whichis the length of a raw key that can be produced perunit time, and the secret fraction, which is the frac-tion of the length of a fully secure key to the length of a raw key in the asymptotic case of N → ∞ where N is the number of signals [61]. We use Rt as themeasure of performance where t is the time takenin one repeater station, which we call the expectedkey length . The expected key length is given by [43]: Rt = max [ P s { − h ( Q ) } , , (15)where P s is the probability not to fail during theentire transmission, Q is the average quantum biterror rate (QBER), and h ( Q ) := − Q log ( Q ) − (1 − Q ) log (1 − Q ) is the binary entropy function. Theprobability P s is given by: P s = (1 − p fail ) L/L , .2 1.6 2.0 2.4024 Q tot /10 (a) L (km) Q tot /10 (b) Rt Rt Q tot Rt FIG. 10. Optimal eﬀective total cost Q tot and thecorresponding expected key length Rt against (a) thecoherent-state amplitude α and (b) the repeater stationinterval L . For each point, other parameters ( n , m , j , L for (a) / n , m , j , α for (b)) are selected to minimize Q tot , and the value of Rt corresponds to that optimalset of parameters. Overall, Q tot is minimum at α = 1 . L = 0 . where p fail is the failure probability of a CBSM ina single repeater station. The average QBER Q isdeﬁned by Q = ( Q X + Q X ) /

2, where Q X and Q Z are given by: Q X/Z = 12 (cid:34) − (cid:18) p i ∓ p x ± p z − p y p i + p x + p z + p y (cid:19) L/L (cid:35) , where p i , p x , p y , and p z are the success, X -error, Y -error, and Z -error probabilities of a CBSM in asingle repeater station, respectively.We also deﬁne the eﬀective total cost Q tot of thequantum repeater by: Q tot := C exp Rt × LL , (16)where C exp is the expected cost of CBSM in a singlerepeater station deﬁned in Deﬁnition 2 . Q tot quan-tiﬁes the expected total cost of CBSM to generate asecret key with unit length. In the numerical calcula-tions, we try to ﬁnd the set of parameters ( n, m, α, j )and station interval L which minimizes Q tot .

3. Results

We ﬁnd the optimal parameter sets which mini-mize the eﬀective total cost Q tot for the total dis-tance L = 1000 km and L = 10000 km. The param-eter sets and the corresponding eﬀective total costs Q tot and expected key lengths Rt are: L = 1000 km:( n, m, α, j ) = (3 , , . , , L = 0 . → Q tot = (1 . ± . × ,Rt = 0 . ± . L = 10000 km:( n, m, α, j ) = (5 , , . , , L = 0 . → Q tot = (2 . ± . × ,Rt = 0 . ± . Q tot and Rt of the quantum re-peater against the repetition sizes n and m when L = 1000 km, for diﬀerent settings of the coherent-state amplitude α . Here, α , L , and the letter so-lidity parameter j are selected to minimize Q tot ormaximize Rt if they are not ﬁxed explicitly. Fig-ure 9(c) indicates that Rt arbitrarily close to unitycan be obtained for suﬃciently large values of n and m . Particularly, m should be suﬃciently large toﬁx Z -errors. However, since X -errors are very rarecompared to failures and Z -errors, n does not needto be very large, although it should be larger than 1to suppress failures.Comparing the second and third columns of Fig. 9,CBSM with a small value of α requires a relativelylarge value of n to reach low Q tot and high Rt . Thisis the consequence of the fact that BSM of coherent-state qubits with a small value of α has a higherfailure probability than that with a large value of α , and the eﬀect of failures can be mitigated by in-creasing n as discussed in Sec. IV B. Meanwhile, theminimal attainable Q tot is smaller for α = 1 . α = 1 .

4. The dependence of the performance ofthe repeater network to α is more clearly shown inFig. 10(a). Here, Q tot is minimal at α = 1 .

9; thisindicates that the parity code with α > . Q tot and the corresponding Rt to the station interval L in Fig. 10(b). It shows that Q tot is minimal when L is around 0.6–1.0 km.Our repeater scheme shows the similar scale ofperformance with CBSM based on multi-photon po-larization qubits, where Q tot = 6 . × and thecorresponding key generation rate is 0.70 with thesame condition of the total distance and photon loss15 ψ (cid:105) | ψ (cid:105) L | α (cid:105) + |− α (cid:105)| α (cid:105) + |− α (cid:105)| α (cid:105)| α (cid:105) + |− α (cid:105)| α (cid:105) + |− α (cid:105)| α (cid:105)| α (cid:105) + |− α (cid:105)| α (cid:105) + |− α (cid:105) FIG. 11. Encoding circuit of the modiﬁed parity codedeﬁned in

Deﬁnition 1 , for n = m = 3 case. Here, | ψ (cid:105) is the desired qubit encoded in coherent-state basis( | (cid:105) = | α (cid:105) , | (cid:105) = |− α (cid:105) ). and all the controlled-not gatesare also under coherent-state basis. | α (cid:105) + |− α (cid:105) is theSCS, where the normalization constant is omitted. rate [11], although the precise comparison is im-possible due to the diﬀerence of the physical-levelBSM schemes. Although we cannot say our repeaterscheme is better than that in Ref. [11], it is still a re-markable result considering that the scheme in Ref.[11] outperforms recent advanced matter-based andall-optical based schemes [11]. VII. IMPLEMENTATION OF THEMODIFIED PARITY ENCODING

In this section, we discuss implementations of themodiﬁed parity encoding and its elementary opera-tions. Here, a logical gate or measurement meansa gate or measurement in modiﬁed parity encodingbasis {| L (cid:105) , | L (cid:105)} , whereas a physical gate or mea-surement means a gate or measurement in coherent-state basis {|± α (cid:105)} . A. Logical-level implementations

Here, we investigate the ways to encode a logicalqubit and to implement logic gates and measure- For CBSM with multi-photon polarization qubits, we use C exp = nm , the number of physical-level BSMs for oneCBSM, in the deﬁnition of Q tot [Eq. (16)]. ments, in terms of physical operations. Encoding.

The encoding circuit of a logical qubitis illustrated in Fig. 11 for n = m = 3 case. The de-sired qubit encoded in coherent-state basis is pre-pared at the ﬁrst PLS of the ﬁrst block. First,controlled-not (CNOT) gates are operated betweenthe ﬁrst PLS of the ﬁrst block (control) and thePLSs of the other blocks (target). After that, foreach block, CNOT gates are operated between theﬁrst PLS (target) and the other ones (control). Theencoding circuit for arbitrary values of n and m gen-erally requires n − | α (cid:105) , n ( m −

1) copies of the SCS N + ( | α (cid:105) + |− α (cid:105) ), and nm − X L and Z L gate. A logical X gate ( X L ) can bedecomposed into n physical X gates, while a logical Z gate ( Z L ) can be decomposed into m physical Z gates: X L = n (cid:89) i =1 X ik for any k ≤ n, (17a) Z L = m (cid:89) k =1 Z ik for any i ≤ m, (17b)where X ik ( Z ik ) is a physical X ( Z ) gate on the k thPLS of the i th block. X L and Z L gates are used inthe quantum repeater scheme discussed in Sec. VI Bto recover the original quantum information fromthe transmitted state and the classical informationon the CBSM results at the end of the network.We note that they are not necessary for the CBSMscheme itself. X L and Z L measurement. A X L ( Z L ) measure-ment is done by the combination of n ( m ) physical X ( Z ) measurements as seen in Eqs. (17). However,this procedure is not fault-tolerant, since a singlephysical-level Z ( X )-error before the measurement ora single physical measurement error causes an er-ror in the measurement. In order to obtain fault-tolerance, one needs to perform multiple measure-ments for diﬀerent k ( i )’s in Eqs. (17). B. Physical-level implementations

Now, we review the recent progress on implemen-tations of physical-level ingredients required for ourscheme including the logical operations discussed inthe previous subsection. We need to deal with SCSs,CNOT gates, X ( Z ) gates, X ( Z ) measurements, andPNPDs.16 uperpositions of coherent states. SCSs (oftencalled Schr¨odinger’s cat states) in free-propagatingoptical ﬁelds are required for encoding logical qubits.It was known that SCSs may be produced using astrong nonlinearity [62] or a precise photon-resolvingdetector [63, 64] although it was experimentallyhighly demanding. Later, the possibilities of gen-erating SCSs using realistic detectors [65, 66] or aweak nonlinearity [67–69] have been explored. Free-propagating SCSs with amplitudes of | α | (cid:47) CNOT gates.

Physical CNOT gates are also re-quired for encoding logical qubits. Gate telepor-tation has been mainly studied for implementingCNOT gates [17, 18], which requires particular two-mode or three-mode entangled states as resourcesand BSMs under coherent-state basis. An alterna-tive way proposed by Marek and Fiur´aˇsek employssingle-photon subtractions as the driving force, butit has a limitation of being non-deterministic [75]. X and Z gates. Physical X ( Z ) gates are re-quired for logical X ( Z ) gates. Implementation of an X gate is straightforward; ˆ X = exp (cid:0) iπ ˆ a † ˆ a (cid:1) , whereˆ a is the annihilation operator, is just swifting theelectromagnetic wave’s phase by π [18]. Implemen-tation of a Z gate is more complicated due to its non-unitarity. An approximate Z gate can be achievedvia nonlinear medium [17], gate teleportation withresources of SCSs [18, 21], or single-photon subtrac-tion [75]. The single-photon subtraction method wasexperimentally demonstrated in [76]. X and Z measurements. Physical X ( Z ) mea-surements are required for logical X ( Z ) measure-ments. An X measurement can be approxi-mately implemented via heterodyne measurement[77], while a perfect measurement is impossible dueto the non-orthogonality between | α (cid:105) and |− α (cid:105) . A Z measurement is the same as measuring the parity ofthe photon number, which is what exactly a PNPDdoes. Photon-number parity detectors.

PNPDs are re-quired for physical-level BSMs and Z measurements.There exist two ways to realize a PNPD: detect-ing the parity of the photon number directly (di-rect measurement), or detecting it indirectly by mea-suring the photon number (indirect measurement).Regarding the direct measurement, parity measure-ments in cavities have been demonstrated and real-ized via Rydberg atom interacting with photons [78],Ramsey interferometry [39, 79], or strong nonlinear Hamiltonian of a Josephson circuit [36]. However,parity measurements of propagating waves have notbeen covered much yet except a few studies suchas parity measurement via strong nonlinear opticalswitching devices [80, 81] or a cavity QED system re-alized in superconducting circuits [82]. Indirect mea-surement, or photon-number-resolving (PNR) detec-tion, is a more actively studied topic due to its wideavailability [83]. PNR schemes can be classiﬁed intotwo categories: inherent PNR detectors and multi-plexed single-photon detectors. Transition edge sen-sors (TESs) are promising candidates for inherentPNR detectors [84–88], which can distinguish up to12 photons with an estimated detection eﬃciency of0.98 [89]. While inherent PNR detectors generallydemand tricky conditions [90], multiplexed single-photon detectors exploit several inexpensive single-photon detectors [91–96]. However, it is currentlydiﬃcult to achieve a suﬃciently high eﬃciency withmultiplexed single-photon detectors, e.g., one cannotresolve more than three photons with better-than-guessing quality using ideal click detectors with aneight-segment detector [83]. VIII. CONCLUSION

Bell-state measurement (BSM) is an essential el-ement for optical quantum information processing,particularly for long-range communication througha quantum repeater. The original coherent-statequbit with basis {|± α (cid:105)} enables one to performnearly deterministic BSM, but it is vulnerable todephasing by photon loss especially for large val-ues of amplitude α of coherent states required toreduce non-orthogonality. Fault-tolerant operationswith encoded coherent-state qubits have been stud-ied mainly with cavity systems, but this cannot bedirectly applied to free-propagating ﬁelds.In this paper, we have explored the possibilityto use such encoded coherent-state qubits for long-range quantum communication by designing an ap-propriate encoding scheme and fault-tolerant BSMscheme. We have presented the modiﬁed parity en-coding which is a natural extension of the origi-nal coherent-state encoding, and also suggested ahardware-eﬃcient concatenated Bell-state measure-ment (CBSM) scheme in a completely or partiallydistributed manner. We have argued and numer-ically veriﬁed that the CBSM scheme successfullysuppresses both failures and dephasing simultane-ously. We have also shown that SCSs with rea-sonable values of the amplitude such as α (cid:47) α (cid:47) ACKNOWLEDGMENTS

This work was supported by NationalResearch Foundation of Korea grantsfunded by the Korea government (NRF- 2020R1A2C1008609, NRF-2019M3E4A1080074and NRF-2020K2A9A1A06102946) via the Institutefor Applied Physics at Seoul National University.S.W.L. acknowledges support from the National Re-search Foundation of Korea (2020M3E4A1079939)and the KIST institutional program (2E31021).

Appendix A: Positive-operator valued measureelements of Bell-state measurement oncoherent-state qubits in lossy environment

Here, we explicitly present the positive-operatorvalued measure (POVM) elements of BSM under thebasis of {|± α (cid:105)} in lossy environment, which is dealtin Sec. II. The set of operators { M x,y | x, y ∈ { , , }} where M x,y := [ U BS ◦ ( Λ η ⊗ Λ η )] † (Π x ⊗ Π y )forms a POVM corresponding to the BSM ofcoherent-state qubits, where U BS is a unitary chan-nel corresponding to a 50:50 beam splitter, Λ η is aphoton loss channel with a survival rate of η , andΠ x is a projector deﬁned byΠ := | F (cid:105)(cid:104) F | , Π := (cid:88) n ≥ | n F (cid:105)(cid:104) n F | , Π := (cid:88) n ≥ | n F (cid:105)(cid:104) n F | , where | n F (cid:105) is the Fock state with a photon numberof n . The photon loss channel Λ η transforms | α (cid:105)(cid:104) α | and | α (cid:105)(cid:104)− α | as follows:Λ η ( | α (cid:105)(cid:104) α | ) = |√ ηα (cid:105)(cid:104)√ ηα | , Λ η ( | α (cid:105)(cid:104)− α | ) = e − − η ) | α | |√ ηα (cid:105)(cid:104)−√ ηα | . (A1)With these relations, we ﬁnd the analytic expres-sions of the matrix elements of each POVM element18 x,y as: (cid:104) φ ± | M x,y | φ ± (cid:105) = c ± (cid:104) ± ( − x + y e − − η − η ) | α | (cid:105) × f x ( η + ) f y ( η − ) , (A2a) (cid:104) ψ ± | M x,y | ψ ± (cid:105) = c ± (cid:104) ± ( − x + y e − − η − η ) | α | (cid:105) × f x ( η − ) f y ( η + ) , (A2b) (cid:104) φ ± | M x,y | ψ ± (cid:105) = c ± (cid:104) ± ( − x + y e − − η ) | α | + e − − η ) | α | (cid:105) × f x (cid:0) √ η + η − (cid:1) f y (cid:0) √ η + η − (cid:1) , (A2c) (cid:104) φ + | M x,y | ψ − (cid:105) = (cid:104) ψ + | M x,y | ψ − (cid:105) = (cid:104) φ ± | M x,y | ψ ∓ (cid:105) = 0 , (A2d)where c ± := e − ( η + η ) | α | ± e − | α | , η ± := (cid:0) √ η ± √ η (cid:1) ,f i ( η ) := i = 0 , sinh (cid:0) η | α | (cid:1) if i = 1 , cosh (cid:0) η | α | (cid:1) − i = 2 . Appendix B: Derivation of the probabilitydistributions of concatenated Bell-statemeasurement results

In this appendix, we show a brief outline to inducethe analytic expressions of the probability distribu-tions of CBSM results conditioning to the initial Bellstates before the measurement. We only consider theunoptimized CBSM scheme, since the measurementresults of the hardware-eﬃcient CBSM scheme arethe direct consequences of those of the unoptimizedscheme.

1. Derivation of the probability distributionsof block-level results

We ﬁrst ﬁnd the probability distributions of block-level BSM results, conditioning to the initial block-level Bell state. A single BSM result can be ex-pressed by two vectors x , y ∈ { , , , } m , wherethe i th elements of them are the two PNPD resultsof the i th PLS. We want to ﬁnd Pr ( x , y | B ) for | B (cid:105) ∈ B := (cid:110)(cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:69) , (cid:12)(cid:12)(cid:12) ψ ( m ) ± (cid:69)(cid:111) From Eqs. (5) and (11a), the conditional proba-bility for the initial state of | B (cid:105) = (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:69) is express as: Pr (cid:16) x , y (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:17) = (cid:68) φ ( m ) ± (cid:12)(cid:12)(cid:12) m (cid:79) i =1 M x i ,y i (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:69) = 12 ˜ N ± (1 , m ) (cid:88) l,l (cid:48) =even ≤ m g ± m,l,l (cid:48) ( x , y ) , (B1)where ˜ N ± (1 , m ) is deﬁned in Eq. (8). The function g ± m,l,l (cid:48) ( x , y ) is deﬁned as: g ± m,l,l (cid:48) ( x , y ) := (cid:88) (cid:78) mi =1 | P i (cid:105)∈ Perm [ | ψ ± (cid:105) ⊗ l | φ ± (cid:105) ⊗ m − l ] (cid:78) mi =1 | P (cid:48) i (cid:105) ∈ Perm (cid:104) | ψ ± (cid:105) ⊗ l (cid:48) | φ ± (cid:105) ⊗ m − l (cid:48) (cid:105) (cid:34) m (cid:89) i =1 (cid:104) P i | M x i ,y i | P (cid:48) i (cid:105) (cid:35) , where Perm[ · ] is the set of all the permutations oftensor products inside the square bracket. The func-tion g ± m,l,l (cid:48) has a recurrence relation: (omit x and y for simplicity) g ± m,l,l (cid:48) = g ± m − ,l,l (cid:48) M ( m ) ± + (cid:104) g ± m − ,l,l (cid:48) − + g ± m − ,l − ,l (cid:48) (cid:105) M ( m ) ± + g ± m − ,l − ,l (cid:48) − M ( m ) ± (B2)where M ( k ) ± := (cid:104) φ ± | ˆ M x k ,y k | φ ± (cid:105) , (B3a) M ( k ) ± := (cid:104) φ ± | ˆ M x k ,y k | ψ ± (cid:105) , (B3b) M ( k ) ± := (cid:104) ψ ± | ˆ M x k ,y k | ψ ± (cid:105) , (B3c)which can be calculated from Eqs. (A2). Now, wedeﬁne a vector v ± m ( x , y ): (Note that g ± m,l,l (cid:48) is a func-tion of x and y .) v ± m := (cid:88) l,l (cid:48) :even ≤ m g ± m,l,l (cid:48) , (cid:88) l :even ≤ ml (cid:48) :odd ≤ m g ± m,l,l (cid:48) , (cid:88) l :odd ≤ ml (cid:48) :even ≤ m g ± m,l,l (cid:48) , (cid:88) l,l (cid:48) :odd ≤ m g ± m,l,l (cid:48) T (B4)19rom Eq. (B2), we get a recurrence relation of v ± m : v ± m = M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± M ( m ) ± v ± m − := ˜M ± x m ,y m v ± m − . (B5)Considering the initial condition at m = 1, v ± m ( x , y )is written as: v ± m ( x , y ) = ˜M ± x m ,y m · · · ˜M ± x ,y (1 , , , T . (B6)Finally, Pr (cid:16) x , y (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:17) is written in terms of thevector v ± m using Eqs. (B1) and (B4): Pr (cid:16) x , y (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:17) = 12 ˜ N ± (1 , m ) v ± m ( x , y ) ., (B7a)where v ± mi is the i th element of v ± m . In the similarway, Pr (cid:16) x , y (cid:12)(cid:12)(cid:12) ψ ( m ) ± (cid:17) is written as: Pr (cid:16) x , y (cid:12)(cid:12)(cid:12) ψ ( m ) ± (cid:17) = 12 ˜ N ± (1 , m ) v ± m ( x , y ) . (B7b)In conclusion, the conditional probability distri-bution of CBSM results conditioning to the inputblock-level Bell state is obtained from Eqs. (B7) withEqs. (A2), (B3), (B5), and (B6), all of which arewritten in simple matrix forms.

2. Derivation of the probability distributionsof logical-level results

Now, we consider the probability distributions oflogical-level results conditioning to the initial logical-level Bell state, which is the goal of this appendix. Asingle CBSM result can be expressed by two matri-ces X , Y ∈ { , , , } n × m , where the ( i, k ) elementsof them are the two PNPD results of the k th PLSof the i th block. What we want to ﬁnd is the prob-ability distribution Pr ( X , Y | B ) for | B (cid:105) ∈ B := {| Φ ± (cid:105) , | Ψ ± (cid:105)} .Because of the similarity of Eqs. (7) and (11),we can follow the almost same logical structurewith the previous subsection when ﬁnding the ex-pressions of the probability distributions. However,there exist three main diﬀerences between the blockand logical level. First, the roles of the lettersand signs are inverted between the two sets of the equations. Second, there are unnormalized statesin the summations of Eqs. (7), unlike Eqs. (11).Lastly, (cid:68) φ ( m )+ (cid:16) ψ ( m )+ (cid:17)(cid:12)(cid:12)(cid:12)(cid:78) mk =1 ˆ M x k ,y k (cid:12)(cid:12)(cid:12) φ ( m ) − (cid:16) ψ ( m ) − (cid:17)(cid:69) vanish unlike the corresponding one in block level,i.e., (cid:104) φ ± | ˆ M x,y | ψ ± (cid:105) in Eq. B3b.Considering the diﬀerences, we deﬁne 2 × ˜L φ x , y and ˜L ψ x , y where x , y ∈ { , , , } m , insteadof 4 × ˜L φ ( ψ ) x , y := L φ ( ψ )+ L φ ( ψ ) − L φ ( ψ ) − L φ ( ψ )+ , where L φ ( ψ ) ± := (cid:2) ± u ( α, m ) (cid:3) × (cid:68) φ ( m ) ± (cid:16) ψ ( m ) ± (cid:17)(cid:12)(cid:12)(cid:12) m (cid:79) k =1 ˆ M x k ,y k (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:16) ψ ( m ) ± (cid:17)(cid:69) , (B8) u ( α, m ) is deﬁned in Eq. (10), and x k ( y k ) is the k thelement of x ( y ). We do not need 4 × (cid:78) mk =1 ˆ M x k ,y k be-tween two Bell states of diﬀerent signs vanish. Wealso note that the RHS of Eq. (B8) can be calcu-lated from Eqs. (B7). The conditional probability Pr ( X , Y | B ), where the i th row vector of X ( Y ) is x i ( y i ), is then Pr ( X , Y | Φ + (Ψ + )) = ˜ N + ( n, m ) w φ ( ψ ) n ( X , Y ) , (B9a) Pr ( X , Y | Φ − (Ψ − )) = ˜ N − ( n, m ) w φ ( ψ ) n ( X , Y ) , (B9b)where ˜ N ± ( n, m ) is deﬁned in Eq. (8) and w φ ( ψ ) nµ ( X , Y ) is the µ th element of the two-dimensional vector w φ ( ψ ) n ( X , Y ) deﬁned by: w φ ( ψ ) n ( X , Y ) := ˜L φ ( ψ ) x n , y n · · · ˜L φ ( ψ ) x , y (1 , T . (B10) Appendix C: Method for sampling concatenatedBell-state measurement results

In this appendix, we explain the method to sam-ple CBSM results. Since we have the analytic ex-pressions of the probability distributions of measure-ment results [Eqs. (B9)], it is possible to sample ar-bitrary CBSM results, each of which is composed of2 nm PNPD results. However, since the number of20BSM results increases exponentially on n and m , itis computationally expensive to use this method. In-stead of that, denoting ( p, q ) the q th PLS of the p thblock, we sample the results for each PLS in order:(1 , → (1 , → (1 , → · · · → (1 , m ) → (2 , →· · · → (2 , m ) → · · · → ( n, m ). Therefore, we needthe conditional probability of getting each ( p, q ) re-sult conditioning to all the results before ( p, q ).The conditional probability we want is Pr ( x pq , y pq | x , y , · · · , x p (cid:48) q (cid:48) , y p (cid:48) q (cid:48) ; B ) ∝ Pr ( x , y , · · · , x pq , y pq | B )= Pr ( x , y , · · · , x p − , y p − ,x p , y p , · · · , x pq , y pq | B ) , (C1)where ( p (cid:48) , q (cid:48) ) = (cid:40) ( p, q −

1) if q > , ( p − , m ) otherwise ,x ik and y ik are the measurement results of the two( i, k ) PNPDs, | B (cid:105) ∈ B , and x i ( y i ) is a vectorwhose k th element is x ik ( y ik ). Note that the pro-portionality is valid only when x , y , ..., x p (cid:48) q (cid:48) , y p (cid:48) q (cid:48) are ﬁxed. From now on, we use the proportion-ality notation while assuming this condition. UsingEq. (7a) and the fact from Eq. (A2d) that the crossterms of (cid:78) ms =1 M x rs ,y rs between Bell states with dif-ferent signs vanish, it is deduced that the RHS ofEq. (C1) with | B (cid:105) = | Φ ± (cid:105) is Pr ( x , y , · · · , x p − , y p − ,x p , y p , · · · , x pq , y pq (cid:12)(cid:12) Φ +( − ) (cid:1) ∝ (cid:88) k :even(odd) ≤ n (cid:88) (cid:78) nr =1 (cid:12)(cid:12)(cid:12) P ( m ) r (cid:69) ∈ Perm [ | Φ − (cid:105) ⊗ k | Φ + (cid:105) ⊗ n − k ] (cid:104) C n − k )+ C j − h ( p,q ) P , ··· ,P p (cid:105) , (C2)where C ± := (cid:2) ± u ( α, m ) (cid:3) / , the normalizationconstant in Eq. (9). Also, h ( p,q ) P , ··· ,P p := (cid:34) p − (cid:89) r =1 (cid:68) P ( m ) r (cid:12)(cid:12)(cid:12) m (cid:79) s =1 M rs (cid:12)(cid:12)(cid:12) P ( m ) r (cid:69)(cid:35) × (cid:68) P ( m ) p (cid:12)(cid:12)(cid:12)(cid:32) q (cid:79) s =1 M ps (cid:33) ⊗ I ⊗ m − q (cid:12)(cid:12)(cid:12) P ( m ) p (cid:69) , (C3)where M rs := M x rs ,y rs and I is the identity operatorin a single PLS. Again, using Eq. (11a), the last part of the RHS of the above deﬁnition is: (cid:68) φ ( m ) ± (cid:12)(cid:12)(cid:12)(cid:32) q (cid:79) s =1 M ps (cid:33) ⊗ I ⊗ m − q (cid:12)(cid:12)(cid:12) φ ( m ) ± (cid:69) ∝ (cid:88) l,l (cid:48) :even ≤ m (cid:88) (cid:78) ms =1 | P s (cid:105)∈ Perm [ | ψ ± (cid:105) ⊗ l | φ ± (cid:105) ⊗ m − l ] (cid:78) ms =1 | P (cid:48) s (cid:105) ∈ Perm (cid:104) | ψ ± (cid:105) ⊗ l (cid:48) | φ ± (cid:105) ⊗ m − l (cid:48) (cid:105) (cid:34) q (cid:89) s =1 (cid:104) P s | M ps | P (cid:48) s (cid:105) m (cid:89) s = q +1 (cid:104) P s | P (cid:48) s (cid:105) (cid:35) := ξ φ ± ,p,q . After transforming the RHS of the above equationappropriately with using the fact that (cid:104) φ − | ψ − (cid:105) van-ishes while (cid:104) φ + | ψ + (cid:105) does not, we obtain: ξ φ + ,p,q = R + q (cid:0) v + q + v + q (cid:1) + R − q (cid:0) v + q + v + q (cid:1) if q < m,v + m if q = m, (C4a) ξ φ − ,p,q = (cid:40) v − q + v − q if q < m,v − m if q = m, (C4b)where R ± q := (1 + (cid:104) φ + | ψ + (cid:105) ) m − q ± (1 − (cid:104) φ + | ψ + (cid:105) ) m − q and v ± qi is the i th element of vector v ± q ( x , y , · · · , x p , y p ) calculated from Eq. (B6).Substituting these on Eq. (C3) and transforming itappropriately, Eqs. (C1) and (C2) become: Pr ( x pq , y pq | x , y , · · · , x p (cid:48) q (cid:48) , y p (cid:48) q (cid:48) ; Φ ± ) ∝ Pr ( x , y , · · · , x p − , y p − ,x p , y p , · · · , x pq , y pq | Φ ± ) ∝ C ξ φ + ,p,q (cid:16) D ± p w φp − , + D ∓ p w φp − , (cid:17) + C − ξ φ − ,p,q (cid:16) D ± p w φp − , + D ∓ p w φp − , (cid:17) if p < n,C ± ξ φ ± ,p,q w φn − , + C ∓ ξ φ ∓ ,p,q w φn − , if p = n, (C5)where D ± p := (cid:0) C + C − (cid:1) n − p ± (cid:0) C − C − (cid:1) n − p and w φp − ,i is the i th element of vector w φp − ( x , y , · · · , x p − , y p − ) deﬁned in Eqs. (B10).The probability distribution for the initial state of | B (cid:105) = | Ψ ± (cid:105) is obtained in very similar way with the21bove arguments. The result is as follows: Pr ( x pq , y pq | x , y , · · · , x p (cid:48) q (cid:48) , y p (cid:48) q (cid:48) ; Ψ ± ) ∝ C ξ ψ + ,p,q (cid:16) D ± p w ψp − , + D ∓ p w ψp − , (cid:17) + C − ξ ψ − ,p,q (cid:16) D ± p w ψp − , + D ∓ p w ψp − , (cid:17) if p < n,C ± ξ ψ ± ,p,q w ψn − , + C ∓ ξ ψ ∓ ,p,q w ψn − , if p = n, (C6)where w ψp − ,i is the i th element of vector w ψp − ( x , y , · · · , x p − , y p − ) deﬁned in Eq. (B10), and ξ ψ + ,p,q = R + q (cid:0) v + q + v + q (cid:1) + R − q (cid:0) v + q + v + q (cid:1) if q < m,v + m if q = m, (C7a) ξ ψ − ,p,q = (cid:40) v − q + v − q if q < m,v − m if q = m. (C7b)In summary, the probability distributions of ( p, q )results conditioning to the previous measurement re-sults (1 , , · · · , ( p (cid:48) , q (cid:48) ) and the initial logical-levelBell state can be obtained from Eqs. (C5) and (C6)together with Eqs. (C4) and (C7). We use theseprobability distributions to sample each physicallevel one by one in order. 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