Quantum Secret Sharing with Error Correction

Transcription

1 Commun. Theor. Phys. 58 (01) Vol. 58, No. 5, November 15, 01 Quntum Secret Shring with Error Correction Aziz Mouzli, 1, Ftih Merzk, nd Dmin Mrkhm 3 1 Electronic Deprtment, Ntionl Polytechnic School, Algiers, Algeri Electronic nd Computer Enginering Fculty, University of Science, nd Technology Houri Boumediéne, Algiers, Algeri 3 Lbortoire Tritement et Communiction de l Informtion CNRS Télécom PrisTech, Frnce (Received Jnury 19, 01; revised mnuscript received My, 01) Abstrct We investigte in this work quntum error correction on five-qubits grph stte used for secret shring through five noisy chnnels. We describe the procedure for the five, seven nd nine qubits codes. It is known tht the three codes lwys llow error recovery if only one mong the sent qubits is disturbed in the trnsmitting chnnel. However, if two qubits nd more re disturbed, then the correction will depend on the used code. We compre in this pper the three codes by computing the verge fidelity between the sent secret nd tht mesured by the receivers. We will tret the cse where, t most, two qubits re ffected in ech one of five depolrizing chnnels. PACS numbers: Ac, Hk, Lx Key words: quntum correction, grph stte, quntum secret shring, Feynmn progrm 1 Introduction The grph stte cn be very useful for severl quntum protocols s secret shring, mesurement-bsed computtion, error correction, teleporttion nd quntum communictions. Then, it would be in the future good wy to unify these topics in one formlism. For exmple, n output of quntum computtion considered s secret cn be included in grph stte, then protected by quntum error correcting code nd sent through noisy chnnel to severl receivers shring this secret. The quntum secret shring with grph stte is very well described in Ref. [1, prticulrly the five qubits grph stte. In this lst cse, only three receivers mong five will ccess the secret, the two others being considered s evesdroppers. In this work, we investigte the effects of the five, seven nd nine qubits codes used to protect five qubits grph stte contining secret nd sent by deler to five plyers. We will compre the fidelity to determine the best code for depolrizing chnnel where only one or two sent qubits re error ffected. Some of the results hve been obtined using simultor clled Feynmn Progrm, which is set of procedures supporting the definition nd mnipultion of n n-qubits system nd the unitry gtes cting on them. This progrm is described in detils in Refs. [ 5. Quntum Secret Shring Quntum secret shring (QSS) is quntum cryptogrphic protocol wherein deler shres privte or public quntum chnnels with ech plyer, while the plyers shre privte quntum or clssicl chnnels between ech other. The deler prepres n encoded version of the secret using qubits string, which he trnsmits to n plyers, only subset k of them cn collborte to reconstruct the secret. We cll (k, n) threshold secret shring protocol where ech plyer receives one equl shre of the encoded secret nd threshold of ny k plyers cn ccess the secret. This scheme is primitive protocol by which ny other secret shring is chieved. In this work, we treted the cse (k, n) = (3, 5) where the deler sends through five depolrizing chnnels, quntum secret encoded in five qubits grph stte. [1.1 Introduction to Grph Stte Grph sttes re n efficient tool for multiprtite quntum informtion processing tsk like secret shring. Also, they hve grphicl representtion which offers n intuitive picture of informtion flow. The secret to be shred is encoded onto clssicl lbels plced on vertices of the grph representing locl opertions. The entnglement of the grph stte llows these lbels to be shifted round, giving us the opportunity to see grphiclly which set of plyers cn ccess the secret. [1. Five-Qubit Grph Stte The five qubits grph stte G given by Eq. (1) is schemtized in Fig. 1 where the vertices represent the qubits nd the edges the controlled-z gte. The grph stte Ψ G given by Eq. () nd contining the quntum secret Ψ s = α 0 β 1 = cos(θ/) 0 e iφ sin(θ/) 1, should be trnsmitted by deler to five plyers through five different chnnels. First, he constructs the stte G from n initil five qubits stte Ψ 0 = 00000, then pplies the Hdmrd gte H on ech qubit nd the controlled-z gte CZ on qubits [1,, [, 3, [3, 4, [4, 5, E-mil: zizmouzli@hotmil.com c 011 Chinese Physicl Society nd IOP Publishing Ltd

2 66 Communictions in Theoreticl Physics Vol. 58 [5, 1 to obtin: G = 1 i 4 CZ [i,i1 5. (1) The deler mkes n intriction between n dditionl qubit clled D nd ech of the five qubits, then dd to the obtined system the secret qubit S in the stte Ψ s = α 0 β 1. Then, he performs Bell mesurement on qubits D nd S nd obtins finlly: [1 [ Ψ G = α G β Z i G. () 1 i 5 the suitble recovering gte R g given in Tble 1 to ccess the secret stte. [1 We describe below this procedure. Eqution (4) cn be written: Ψ G = B Ψ 45 B Ψ b 45 where B Ψ c 45 B Ψ d 45, (5) Ψ 45 = 1 [α β B 01 45, Ψ b 45 = 1 [α β B 10 45, Ψ c 45 = 1 [α β B 00 45, Fig. 1 Five-qubits grph stte G. We obtin in the Dirc nottion: Ψ G = ( /8){(α β)[ (α β)[ ( α β)[ (α β)[ }. (3).3 Perfect Chnnel The grph stte Ψ G cn be decomposed in terms of Bell sttes B ij 13 nd B ij 45 : [1 Ψ G = ( 1 ){ B [α β B B [α β B B [α β B B [α β B )}, (4) where the Bell sttes re: B 00 = B 10 = , B 01 =, , B 11 =. (4b) The secret should be ccessible only for plyers 1,, nd 3, plyers 4 nd 5 being considered s evesdroppers. Plyers 1 nd 3 mesure their qubits in the Bell bsis nd trnsmit the result to plyer which pplies on its qubit Ψ d 45 = 1 [α β B (5b) Mesurement in the Bell bse { B ij 13 } gives only one term of the superposition in Eq. (5), then the globl density mtrix: ρ 1,...,5 = ( B ij B ij ) 13 ( Ψ x Ψ x ) 45 4 = (ρ ij) 13 (ρ x ) 45, (6) 4 with x =, b, c or d. The prtil trce over qubits (4, 5) gives the density mtrix of qubit : ρ = Ψ Ψ = P tr [(ρ x ) 45 4,5. (7) Then the secret stte: ρ = Rg ρ R g or Ψ = R g Ψ. (8) Tble 1 Secret recovering gte R g used by plyer versus the Bell stte B ij 13 mesured by plyers 1 nd 3. B ij 13 B 00 B 01 B 10 B 11 R g H ZH ZXH XH.4 Bit nd Phse Flip The sent qubits cn be ffected by error X, Z or Y represented respectively by the Puli mtrix ( ) ( ) X =, Z =, ( ) 0 1 Y = ixz = i 1 0 corresponding to rottion π round ox or oz or both in the block sphere. We hve pplied the procedure described in Subsec..3 nd hve obtined Tble giving the ffected secret stte Ψ E. Tble Tble Qubit stte Ψ E versus error on qubits 1,, nd 3. Error Ψ E X 1, X 3, Y Z 1, X, Z 3 Z, Y 1, Y 3 α 1 β 0 α 0 β 1 α 1 β 0

3 No. 5 Communictions in Theoreticl Physics Fidelity The fidelity is one of the mthemticl quntities which permits to know how close re two quntum sttes represented by the density mtrix σ nd ρ by mesuring distnce between them: [6 F(σ, ρ) = Tr( σρ σ). (9) In the cse of pure stte σ = Ψ Ψ nd n rbitrry stte ρ, the fidelity is the overlp between the two sttes: [6 F( Ψ, ρ) = Ψ ρ Ψ. (10) In this work we mesure the overlp between the correct secret stte σ s = Ψ s Ψ s nd the qubit stte ρ = Ψ Ψ mesured by plyer to ccess the secret. Then, the fidelity is function of the ngles (θ, φ) in the Block sphere nd the verge fidelity is: F = (1/4π) F(θ, φ)sin(θ)dθdφ, (11) with 0 θ π nd 0 φ π. We will describe below the procedure giving the fidelity. If ny Puli errors E =σ 0, σ x, σ y or σ z ffect the stte Ψ G in the trnsmission chnnel, we cn see from Eq. (4) tht Eq. (5) will keep the sme form nd become: Ψ E G = B Ψ E 45 B Ψ E b 45 B Ψ E c 45 B Ψ E d 45, (1) where Ψ E,b,c,d 45 re the new globl sttes of qubits system (, 4, 5) modified by the chnnel errors. We note tht Ψ E,b,c,d 45 E Ψ,b,c,d 45 s chnnel error cn ffect qubits (1, 3) s well s qubits (, 4, 5). In fct, if n error ffects qubits (1, 3) or (4, 5), then their globl stte will simply chnge to nother Bell stte. Similrly, if qubit is ffected, then its stte will only switch to one of the forms ppering in Eq. (4). After mesuring on the Bell sttes of qubits (1, 3) only one term will remin in Eq. (1): ( 1 Ψ E G = B ij 13 Ψ ) E x 45. (13) The corresponding ffected density mtrix is: ( 1 ) ρ E 1,...,5 = ( B ij B ij ) 13 ( Ψ E x 4 ΨE x ) 45 ( 1 ) = ρ E 1,...,5 = (ρ ij ) 13 (ρ E x 4 ) 45. (14) The prtil trce over qubits (4, 5) gives the mesured density mtrix of qubit : ρ E = Ψ E Ψ E = P tr [(ρ E x ) 45 4,5. (15) Then the ffected secret stte mesured by plyer two: ρ E = R G ρ E R G = Ψ E ΨE. (16) We multiply by the secret stte Ψ s = α 0 β 1 to obtin the fidelity: F(θ, φ) = Ψ s ρ E Ψ s. (17) Tble 3 gives the fidelity F(θ, φ) clculted by Feynmn progrm for ll the errors on qubits i = 1, or 3. Figure shows the fidelity F(θ, φ 0 = 0 or π/) function of the ngle θ for error occurring with probbility P = 1 on one, two nd three noisy chnnels. We note for exmple tht if Ψ s = ( /)( 0 1 ) the fidelity is the best (F = 1) for error X 1 nd the worst (F = 0) for error Z 1. Also, we deduce from Eq. (4) tht ny errors on qubits 4 nd 5 do not ffect the secret stte giving then fidelity equl to one. Fig. Fidelity F(θ,φ) with 0 < θ < π, φ = π/ for errors ε nd φ = 0 for errors ε c. We note tht for errors ε d we hve F(θ, φ) = 1 for 0 < (θ, φ) < π. Tble 3 Fidelity nd mesured stte Ψ versus errors on qubits i = 1,,3. The error groups re depicted on Tble 4. Error Ψ F(θ, φ) F ε α 1 β 0 sin(θ) sinφ 1/3 ε b α 0 β 1 cos (θ) 1/3 ε c α 1 β 0 sin(θ) cos φ 1/3 ε d α 0 β Depolrizing Chnnel The depolrizing chnnel is prticulr model for the noise on quntum systems. In this process, the globl density mtrix ρ is replced by mixed one ρ(p) function of the probbility P tht Puli error E ij = (σ 1j = σ xj, σ j = σ yj or σ 3j = σ zj ) ffects ny qubit j in the n- qubits system. For one-qubit system the mtrix density is given by Eq. (18) [ nd for the n-qubits system it cn be generlized by Eq. (19): ρ 1 (P)=(1 P)ρ P [XρX Y ρy ZρZ, (18) 3 ρ n (P)=(1 P) n ρ P k [ 1 jl n P n 3 n 1 i 3 [ 1 jl n 1 i 3 (Π 1 l k σ ij l ) (1 P)n k 3k ρ(π 1 l k σ ijl ) (Π 1 l n σij l ) ρ(π 1 l n σ ijl ). (19)

4 664 Communictions in Theoreticl Physics Vol. 58 Consider now the cse where the five qubits re sent by the deler through five depolrizing chnnels. Suppose the probbility P tht ny single error occurs on ny qubit is the sme in the five chnnels. Then we cn use Eq. (19) s if the deler sends the five qubits through only one depolrized chnnel. We describe below the procedure to obtin the verge fidelity F (P), considering ll the possible errors in the five trnsmitting noisy chnnels. We begin by writing the ffected density mtrix ρ E 1,...,5(P) received by the five plyers: ρ E 1,...,5(P) = (1 P) 5 ρ 1,...,5 P (1 P)4 3 [ρ E1 1,...,5 ρe 1,...,5 ρe3 1,...,5 ρe4 1,...,5 ρe5 1,...,5 P 9 (1 P)3 [ρ E1E 1,...,5 ρe1e3 1,...,5 ρe1e4 1,...,5 ρ E1E5 1,...,5 ρee3 1,...,5 ρee4 1,...,5 ρee5 1,...,5 ρ E3E4 1,...,5 ρe3e5 1,...,5 ρe4e5 1,...,5 P 3 (1 P) 7 [ρ E1EE3 1,...,5 ρ E1EE4 1,...,5 ρ E1EE5 1,...,5 ρ E1E3E4 1,...,5 ρ E1E3E5 1,...,5 ρ E1E4E5 1,...,5 ρ EE3E4 1,...,5 ρ EE3E5 1,...,5 ρ EE4E5 1,...,5 ρ E3E4E5 1,...,5 P 4 (1 P) 81 [ρ E1EE3E4 1,...,5 ρ E1EE3E5 1,...,5 ρ E1EE4E5 ρ E1E3E4E5 1,...,5 ρ EE3E4E5 1,...,5 P 5 1,...,5. (0) 43 ρe1ee3e4e5 With ρ Ei 1,...,5 the density mtrix ffected by errors on qubit i : ρ Ei 1,...,5 = X iρ 1,...,5 X i Y i ρ 1,...,5 Y i Z i ρ 1,...,5 Z i. (1) The density mtrix ρ EiEj 1,...,5, ρeieje k 1,...,5, ρ EiEjE ke l 1,...,5, nd ρ EiEjE ke l E m 1,...,5 re summtion of respectively 9, 7, 81, nd 43 terms nd represent the density mtrix ffected by error on two, three, four nd five qubits. After mesuring on the Bell sttes of qubits (1, 3) we obtin: ρ E 45(P) = (1 P) 5 ρ 45 P (1 P)4 3 [ρ E1 45 ρe 45 ρe3 45 ρe4 45 ρe5 45 P 9 (1 P)3 [ρ E1E 45 ρ E1E3 45 ρ E1E4 45 ρ E1E5 45 ρ EE3 45 ρ EE4 45 ρ EE5 45 ρ E3E4 45 ρ E3E5 45 ρ E4E5 45 P 3 7 (1 P) [ρ E1EE3 45 ρ E1EE4 45 ρ E1EE5 45 ρ E1E3E4 45 ρ E1E3E5 45 ρ E1E4E5 45 ρ EE3E4 45 ρ EE3E5 45 ρ EE4E5 45 ρ E3E4E5 45 P 4 (1 P)[ρE1EE3E4 45 ρ E1EE3E ρ E1EE4E5 ρ E1E3E4E5 45 ρ EE3E4E5 45 P 5 43 ρe1ee3e4e5 45. () With ρ 45 = (ρ ) 45, (ρ b ) 45, (ρ c ) 45 or (ρ d ) 45 nd ρ E 45 = (ρ E ) 45, (ρ E b ) 45, (ρ E c ) 45 or (ρ E d ) 45. After trcing over qubits (4, 5) nd multiplying by the recovering gte R g we obtin: ρ E (P) = (1 P) 5 ρ P (1 P)4 3 [ρ E1 ρ E ρ E3 ρ E4 ρ E5 P 9 (1 P)3 [ρ E1E ρ E1E3 ρ E1E4 ρ E1E5 ρ EE3 ρ EE4 ρ EE5 ρ E3E4 ρ E3E5 ρ E4E5 P 3 (1 P) 7 [ρ E1EE3 ρ E1EE4 ρ E1EE5 ρ E1E3E4 ρ E1E3E5 ρ E1E4E5 ρ EE3E4 ρ EE3E5 ρ EE4E5 ρ E3E4E5 P 4 (1 P)[ρE1EE3E4 81 ρ E1EE3E5 ρ E1EE4E5 ρ E1E3E4E5 ρ EE3E4E5 P 5 43 ρe1ee3e4e5. (3) With ρ = Ψ s Ψ s the correct secret nd ρ E = (ρe ), (ρ E b ), (ρ E c ) or (ρ E d ) the secret stte disturbed by error E = E i, E i E j, E i E j E k, E i E j E k E l, or E i E j E k E l E m. We multiply by the secret stte nd integrte over (θ, φ) to obtin the verge fidelity: Ψ s ρ E Ψ s = (1 P) 5 P (1 P)4 3 With [F E1 P F E F E3 9 (1 P)3 [F E1E F E1E5 F E3E4 [F E1EE3 F E1E3E4 F EE3E4 F EE3 F E3E5 F E4 F E1E3 F EE4 F E1EE4 F E1E3E5 F EE3E5 F E5 F E1E4 F EE5 F E4E5 P 3 (1 P) 7 F E1EE5 F E1E4E5 F EE4E5 F E3E4E5 P 4 E1EE3E4 (1 P)[F 81 F E1EE3E5 F E1EE4E5 F E1E3E4E5 F EE3E4E5 P 5 43 F E1EE3E4E5. (4) Ψ s ρ Ψ s = 1, nd F E = Ψ s ρ E Ψ s. (5) We cn write Eq. (4) s Ψ s ρ E Ψ s = (1 P) 5 P 3 (1 P)4 [A P 9 (1 P)3 [B P 3 7 (1 P) [C P 4 81 (1 P)[DP5 43 [E.(6) We deduce from Tbles 3 nd 4 the vlues of F E contined in Tble 5:

5 No. 5 Communictions in Theoreticl Physics 665 Tble 4 Error groups with sme verge fidelity. ε X 1, X 3, Y, X 1 X Z 3, Z 1 X X 3, Y 1 X, Y 1 Z 3, Z 1 Y 3, X Y 3, Y 1 Y Y 3, Y 1 Z X 3, X 1 Y X 3, X 1 Z Y 3, Z 1 Y Z 3 ε b Z 1, X, Z 3, X 1 Z, Z X 3, X 1 X X 3, Z 1 X Z 3, Y 1 Y, Y 1 X 3, Y Y 3, X 1 Y 3, Y 1 X Y 3, Y 1 Z Z 3, X 1 Y Z 3, Z 1 Y X 3, Z 1 Z Y 3 ε c Z, Y 1, Y 3, X 1 X, X X 3, Z Z 3, X 1 Z 3, Z 1 Z, Z 1 X 3, Z 1 Z Z 3, X 1 Z X 3, Y Z 3, Z 1 Y, Y 1 Y X 3, Y 1 X Z 3, Y 1 Z Y 3, X 1 Y Y 3, Z 1 X Y 3 ε d X 1 X 3, Z 1 Z 3, Z 1 X, X Z 3, X 1 Z Z 3, Z 1 Z X 3, Y 1 Y 3, Y 1 Z, Y X 3, X 1 Y, Z Y 3, Y 1 Y Z 3, Y 1 X X 3, Z 1 Y Y 3, X 1 X Y 3, Tble 5 Vlues of F E for ll the possible errors. F E Vlues F E 1, F E, F E 3 (3 1/3)= 1 F E 4, F E 5 (3 1)= 3 F E 1E, F E 1E 3, F E E 3 (6 1/3)(3 1)= 5 F E 1E 4, F E 1E 5, F E E 4, (9 1/3)= 3 F E E 5, F E 3E 4, F E 3E 5 F E 4E 5 (9 1)= 9 F E 1E E 3 (1 1/3)(6 1)= 13 F E 1E E 4, F E 1E 3 E 4, F E E 3 E 4, [(6 3) 1/3[(3 3) 1 = 15 F E 1E E 5, F E 1E 3 E 5, F E E 3 E 5 F E 1E 4 E 5, F E E 4 E 5, F E 3E 4 E 5 (3 9) 1/3= 9 F E 1E E 3 E 4, F E 1E E 3 E 5 [(1 3) 1/3[(6 3) 1 = 39 F E 1E E 4 E 5, F E 1E 3 E 4 E 5, F E E 3 E 4 E 5 (6 9) 1/3(3 9) 1 = 45 F E 1E E 3 E 4 E 5 (1 9) 1/3(6 9) 1 = 117 The clculus gives: A = 9, B = 4, C = 130, D = 13, E = 117. (7) The verge fidelity is then: F (P) = (1 P) 5 3P(1 P) P (1 P) 3 Finlly: P 3 (1 P) 71 7 P 4 (1 P) 13 7 P 5. (8) F (P) = 1 P 8 3 P 3 7 P 3. (9) 3 The Five-Qubit Code We describe below the chnnel error correction by the five qubits code. This code is described in Refs. [6 7 nd uses five qubits to protect one of them in superposed stte from ny error X, Y or Z. We note tht when using ny of the three codes, the fidelity is lwys equl to one if only one mong the five sent qubits is disturbed in the trnsmiting chnnel. However, if two sent qubits nd more re disturbed during the trnsmission, then the fidelity will vry upon the used code. 3.1 Bit nd Phse Flip The deler protects ech of the three qubits (1,, 3) with four ncills s showed in Fig. 3 nd sends them through three noisy chnnels, which introduce bit or phse flip or both with probbility P = 1. Fig. 3 Trnsmission of five protected qubits through five chnnels. We note tht in generl the deler will protect the five qubits s he does not know which plyers will ccess the secret. With The grph stte (3) cn be written s follow: Ψ G = ( /8){[α 0 β 1 i Ψ jklm [α 0 β 1 i Ψ b jklm [β 0 α 1 i Ψ c jklm [β 0 α 1 i Ψ d jklm. (30) [i = 1, (j, k, l, m) =, 3, 4, 5, [i =, (j, k, l, m) = 1, 3, 4, 5 or [i = 3, (j, k, l, m) = 1,, 4, 5. (31)

6 666 Communictions in Theoreticl Physics Vol. 58 The sttes Ψ jklm, Ψ b jklm, Ψ c jklm, Ψ d jklm hve different expressions depending on the vlue of i. We suppose tht the deler knows tht the secret will be ccessed by plyers (1,, 3). Then he will not protect the qubits sent to plyers (4, 5) s the chnnel errors on them do not ffect the ccessed secret. To protect qubit i the deler dds four ncills ech one in the stte 0. After pplying the coding circuit we obtin the coded grph stte Ψ G1 sent by the deler, where 0 l nd 1 l re the logicl qubits: [6 Ψ G1 = ( /8){[α 0 l β 1 l i Ψ jklm [α 0 l β 1 l i Ψ b jklm [β 0 l α 1 l i Ψ c jklm [β 0 l α 1 l i Ψ d jklm. (3) After syndrome mesurement, correction nd decoding, the plyers suppress the ncills. If the qubit i is ffected by error E i = X i, Y i or Z i then the grph stte becomes: Ψ Ei G = E i Ψ G = ( /8){E i [α 0 β 1 i Ψ jklm E i [α 0 β 1 i Ψ b jklm E i [β 0 α 1 i Ψ c jklm E i [β 0 α 1 i Ψ d jklm. (33) Tble 6 gives the syndromes S for the five qubits code of single nd double errors occurring in the logicl qubit i. The double errors re corrected s the single error hving sme syndrome. The third column gives the error E i ffecting the to be protected physicl qubit i fter decoding obtined by Feynmn Progrm. Tble 6 The error E i ffecting the physicl qubit i versus errors on the logicl qubit i. The ncills re designed by j ndthe to be protected qubit by i with i = 1,, 3 nd j = 1,,3, 4. Error S E i X i,(z Z 3 ),(X 3 Z 4, Z 1 X ) 0101 I i, (X i ), (Z i ) X 1, (Z 3 Z 4 ),(Z i X 4, Z X 3 ) 0010 I i, (X i ), (Z i ) X,(Z i Z 4 ),(X i Z 1, Z 3 X 4 ) 1001 I i, (X i ), (Z i ) X 3,(Z i Z 1 ), (X i Z 4, X 1 Z ) 0100 I i, (X i ), (Z i ) X 4,(Z 1 Z ), (X Z 3, Z i X 1 ) 1010 I i, (X i ), (Z i ) Z i,(x 1 X 4 ), (X Z 4, Z 1 X 3 ) 1000 I i, (X i ), (Z i ) Z 1,(X i X ), (Z i X 3, Z X 4 ) 1100 I i, (X i ), (Z i ) Z, (X 1 X 3 ), (X i Z 3, Z 1 X 4 ) 0110 I i, (X i ), (Z i ) Z 3,(X X 4 ),(X i Z, X 1 Z 4 ) 0011 I i, (X i ), (Z i ) Z 4, (X i X 3 ), (X 1 Z 3, Z i X ) 0001 I i, (X i ), (Z i ) Y i,(x X 3, Z 1 Z 4 ) 1101 I i, (Y i ) Y 1,(X 3 X 4, Z i Z ) 1110 I i, (Y i ) Y,(X i X 4, Z 1 Z 3 ) 1111 I i, (Y i ) Y 3,(X i X 1, Z Z 4 ) 0111 I i, (Y i ) Y 4,(X 1 X, Z i Z 3 ) 1011 I i, (Y i ) 3. Depolrizing Chnnel Consider three double errors E k E l, E m E n nd E o E p occurring respectively in chnnels 1,, nd 3 on ny qubits (k, l, m, n, o, p) nd corrected s the three single error with similr syndrome. The qubits (4, 5) re not protected nd cn be ffected by ny error X, Y or Z. If the probbility tht chnnel error occurs on one qubit is equl to P, then the density mtrix received by the five plyers is: ρ E (13)(45) = (1 P)8 ρ (13)(45) P 3 (1 P)7[ ρ E x (13)(45) P 3 (1 P)6[ E ρ xe y (13)(45) P 4 P 5 P (1 P)5[ E ρ xe ye z [ E ρ xe ye ze ue v (13)(45) P 6 (13)(45) 3 4 (1 P)4[ E ρ xe ye ze u (13)(45) 3 6 (1 P)[ E ρ xe ye ze ue ve w (13)(45) P 8 (1 P)3 35 P 7 [ [ 3 7 (1 P) E ρ xe ye ze ue ve we s E (13)(45) ρ k E l E me ne oe pe 4E (13)(45). (34) The nottion (13) represents the logicl qubits 1,, nd 3, ech one protected by four ncills nd: ρ E x (13)(45) =ρe k (13)(45) ρe l (13)(45) ρem (13)(45) ρen (13)(45) ρeo (13)(45) ρep (13)(45) ρe4 (13)(45) ρe5 (13)(45), (35) ρ E k (13)(45) = ρx k (13)(45) ρy k (13)(45) ρz k (13)(45) ; ρx k (13)(45) = X k ρ (13)(45)X k. The summtions ρ Ex (13)(45), ρ ExEy (13)(45), ρ ExEyEz (13)(45), ρ ExEyEzEu (13)(45), ρ ExEyEzEuEv (13)(45), ρ ExEyEzEuEvEw (13)(45), nd E ρ xe ye ze ue ve we s (13)(45) contin respectively 8X3, 8X3, 56X3 3, 70X3 4, 56X3 5, 8X 3 6, nd 8X3 7 terms. The expression ρ E ke l E me ne oe pe 4E 5 (13)(45) is the summtion of 3 8 terms. After decoding nd suppressing the ncills we obtin by using Tbles 6 the ffected grph stte: [ ρ E (1,...,5) = (1 P) 8 18 P 3 (1 P)7 108 P 9 (1 P)6 16 P 3 7 (1 P)5 ρ (1,...,5) [3 P 9 (1 P)6 36 P 3 7 (1 P)5 108 P 4 81 (1 P)4 [ρ E1 (1,...,5) ρe (1,...,5) ρe3 (1,...,5) [9 P 4 81 (1 P)4 54 P 5 43 (1 P)3[ ρ E1E (1,...,5) ρe1e3 (1,...,5) ρee3 (1,...,5) [ 1 7 P 6 (1 P) [ρ E1EE3 (1,...,5) (35b)

7 No. 5 Communictions in Theoreticl Physics (1 P)7 18 P 9 (1 P)6 108 P 3 7 (1 P)5 16 P 4 81 (1 P)4 [ρ E4 (1,...,5) ρe5 (1,...,5) 9 (1 P)6 18 P 3 7 (1 P)5 108 P 4 81 (1 P)4 16 P 5 [ 43 (1 P)3 [ρ E4E5 (1,...,5) 36 P 4 81 (1 P)4 108 P 5 43 (1 P)3 [ρ E1E4 (1,...,5) ρe1e5 (1,...,5) ρee4 (1,...,5) ρee5 (1,...,5) ρe3e4 (1,...,5) ρe3e5 (1,...,5) 3 P 3 (1 P)5 7 [3 P 4 81 (1 P)4 36 P 5 43 (1 P)3 108 P 6 [ 3 6 (1 P) [ρ E1E4E5 (1,...,5) ρee4e5 (1,...,5) ρe3e4e5 (1,...,5) 9 P 5 (1 P) P (1 P) [ρ E1EE4 (1,...,5) ρe1ee5 (1,...,5) ρe1e3e4 (1,...,5) ρe1e3e5 (1,...,5) ρee3e4 (1,...,5) ρee3e5 (1,...,5) [9 P (1 P) 54 P (1 P) [ρ E1EE4E5 (1,...,5) ρ E1E3E4E5 (1,...,5) ρ EE3E4E5 (1,...,5) [7 P (1 P) [ρ E1EE3E4 (1,...,5) ρ E1EE3E5 (1,...,5) P [7ρE1EE3E4E5 (1,...,5). (36) After mesuring on the Bell sttes of qubits (1, 3), trcing over qubits (4, 5), multiplying by the recovering gte nd the secret stte nd integrting we obtin the verge fidelity: F (P) = Ψ s ρ E Ψ s = [(1 P) 8 18 P 3 (1 P)7 108 P (1 P)5 [3 P 9 (1 P)6 36 P 3 [9 P 4 81 (1 P)4 54 P 5 3 (1 P)7 18 P 7 (1 P)5 108 P 4 43 (1 P)3 [F E1E 9 (1 P)6 16 P (1 P)4 [F E1 F E1E3 F E F E3 [ 1 F EE3 7 P 6 (1 P) [F E1EE3 F E5 9 (1 P)6 108 P 3 7 (1 P)5 16 P 4 81 (1 P)4 [F E4 9 (1 P)6 18 P 3 7 (1 P)5 108 P 4 81 (1 P)4 16 P 5 43 (1 P)3 [F E4E5 [3 P 3 7 (1 P)5 36 P 4 81 (1 P)4 108 P 5 43 (1 P)3 [F E1E4 F E1E5 F EE4 [3 P 4 81 (1 P)4 36 P 5 43 (1 P)3 108 P (1 P) [F E1E4E5 F EE4E5 [9 P 5 43 (1 P)3 54 P (1 P) [F E1EE4 F E1EE5 F E1E3E4 F E1E3E5 [9 P (1 P) 54 P (1 P) [F E1EE4E5 F E1E3E4E5 F EE3E4E5 [7 P (1 P) [F E1EE3E4 F E1EE3E5 7 P 8 F EE5 F E3E4E5 F E3E4 F EE3E4 F EE3E5 F E3E5 E1EE3E4E5 [F 38. (37) We deduce from Tble 5: F (P) = [(1 P) 8 18 P 3 (1 P)7 108 P 9 (1 P)6 16 P 3 (1 P)5 7 [3 P 9 (1 P)6 36 P 3 7 (1 P)5 108 P 4 81 (1 P)4 [3 [9 P 4 81 (1 P)4 54 P 5 43 (1 P)3 [15 [ 1 7 P 6 (1 P) [13 3 (1 P)7 18 P 9 (1 P)6 108 P 3 7 (1 P)5 16 P 4 81 (1 P)4 [6 9 (1 P)6 18 P 3 7 (1 P)5 108 P 4 81 (1 P)4 16 P 5 43 (1 P)3 [9 [3 P 3 7 (1 P)5 36 P 4 81 (1 P)4 108 P 5 43 (1 P)3 [18 [3 P 4 (1 P) P 5 43 (1 P)3 108 P (1 P) [7 [9 P 5 43 (1 P)3 54 P (1 P) [90 [9 P (1 P) 54 P (1 P) [135 [7 P (1 P) [78 7 P 8 [117. (38) 38

8 668 Communictions in Theoreticl Physics Vol. 58 Then: F (P) = (1 P) 8 8P(1 P) 7 6P (1 P) 6 44P 3 (1 P) P 4 (1 P) P 5 (1 P) P 6 (1 P) P 7 (1 P) 13 7 P 8. (39) 4 The Seven-Qubit Code This code is described in Ref. [8 nd uses five qubits to protect one of them ginst n X, Y or Z error. Tbles 7 9 depict the error E i on the protected qubit i (fter correction) for different chnnel errors E. Tble 7 Syndromes nd the error E i ffecting the protected qubits i fter correction by n opertors X k or Y k (k = 1,..., 7). E S E i E S E i X 1,(X X 3, X 4 X 5, X 6 X 7 ) I i,(x i ) Y I i X, (X 1 X 3, X 4 X 6, X 5 X 7 ) I i,(x i ) Y I i X 3, (X 1 X, X 5 X 6, X 4 X 7 ) I i,(x i ) Y I i X 4,(X 1 X 5, X X 6, X 3 X 7 ) I i, (X i ) Y I i X 5,(X 1 X 4, X X 7, X 3 X6) I i,(x i ) Y I i X 6,(X 1 X 7, X X 4, X 3 X 5 ) I i,(x i ) Y I i X 7,(X 1 X 6, X X 5, X 3 X 4 ) I i,(x i ) Y I i Tble 8 Syndromes nd the error ffecting the protected qubits fter correction by n opertor Z k (k = 1,..., 7). Errors S E i Z1, (Z6Z7, ZZ 3, Z4Z 5) I i,(z i ) Z,(Z1Z 3, Z4Z 6, Z5Z 7) I i,(z i ) Z 3,(Z4Z 7, Z1Z, Z5Z 6) I i,(z i ) Z 4,(Z3Z 7, Z1Z 5, ZZ 6) I i,(z i ) Z 5, (ZZ7, Z1Z 4, Z3Z 6) I i,(z i ) Z 6,(Z1Z 7, ZZ 4, Z3Z 5) I i,(z i ) Z 7,(Z1Z 6, ZZ 5, Z3Z 4) I i,(z i ) Tble 9 Syndromes of double chnnels errors Z k X l not ffecting the protected qubits fter correction. Errors S Errors S Errors S X 1 Z Z 1 X Z 1 X X 1 Z Z X X1Z X 1 Z Z 3 X X1Z X 1 Z Z 1 X X Z X Z Z X X Z X 3 Z Z 1 X X Z X 3 Z Z X X Z X 3 Z Z 3 X X Z X 4 Z Z 1 X Z X X 5 Z Z X X 4 Z X 6 Z Z 3 X X 4 Z X 6 Z Z 4 X Z 4 X X 5 Z Z 3 X X 3 Z The Nine-Qubit Code This code clled the Shor code nd explined in Ref. [6 uses nine qubits to protect one of them in superposed stte from ny error X, Y or Z. The simultion with Feynmn Progrm gives in Tbles the error E i ffecting the protected qubits (fter correction) for different single nd double chnnels errors E on logicl qubit i.

9 No. 5 Communictions in Theoreticl Physics 669 Tble 10 Syndromes S nd the E i error ffecting the protected qubits fter correction by n opertors X k (k = 1,..., 9). E S E i Y 1, X 1 Z, I i Y, X Z 1, I i Y 3, X 3 Z 1, I i Y 4, X 4 Z 5, I i Y 5, X 5 Z 4, I i Y 6, X 6 Z 4, I i Y 7, X 7 Z 8, I i Y 8, X 8 Z 7, I i Y 9, X 9 Z 7, I i Tble 11 Syndromes S nd the E i ffecting the protected qubits fter correction by n opertors Y k (k = 1,..., 9). E S E i Y 1, X 1 Z, I i Y, X Z 1, I i Y 3, X 3 Z 1, I i Y 4, X 4 Z 5, I i Y 5, X 5 Z 4, I i Y 6, X 6 Z 4, I i Y 7, X 7 Z 8, I i Y 8, X 8 Z 7, I i Y 9, X 9 Z 7, I i Tble 1 Syndromes S nd the E i ffecting the protected qubits fter correction by n opertor Z k (k = 1,..., 9). Errors S E i (Z 1, Z, Z 3 ), (Z 4 Z7,8,9, Z 5Z7,8,9, Z6Z7,8,9) (I i ),(X i ) (Z 4, Z 5, Z 6 ), (Z1Z7,8,9, ZZ7,8,9, Z3Z7,8,9) (I i ),(X i ) (Z 7, Z 8, Z 9 ), (Z 1Z 4,5,6, ZZ 4,5,6, Z3Z 4,5,6 ) (I i ),(X i ) (Z1Z,3, ZZ3, Z 4Z5,6, Z5Z6, Z7Z8,9, Z8Z9) (I i ) Tble 13 Syndromes S of double chnnels errors X k X l not ffecting the protected qubits fter correction. E S E S E S X1X XX X4X X1X XX X4X X1X XX X4X X1X X3X X5X X1X X3X X5X X1X X3X X5X XX X3X X 6 X XX X3X X6X XX X 3 X X6X

10 670 Communictions in Theoreticl Physics Vol. 58 Tble 14 Syndromes of double chnnels errors X k Z l not ffecting the protected qubits fter correction. E S E S X 1 Z 4,5, X 4 Z 1,, X 1 Z 7,8, X 5 Z 1,, X Z 7,8, X 6 Z 1,, X Z 4,5, X 7 Z 1,, X 3 Z 4,5, X 8 Z 1,, X 3 Z 7,8, X 9 Z 1,, X 4 Z 7,8, X 7 Z 4,5, X 5 Z 7,8, X 8 Z 4,5, X 6 Z 7,8, X 9 Z 4,5, Comprison Among the Three Codes The procedure giving the verge fidelity described in Sec. 4 is the sme for the seven nd nine qubits codes. The difference comes from the number n of double chnnel errors hving n exclusive syndrome llowing their recovery, then letting the protected qubit error free. We suppose tht triple chnnel errors nd more re very unlikely. We deduce from Tbles 6 14 nd for ech code the next frctions of recoverble double chnnel errors: n 5 N 5 = 0, n 7 = 39 N 7 81, n 9 = 108 N (40) With N 5 =40, N 7 = 81, nd N 9 = 144 the totl number of double errors for ech code. We cn deduce the verge fidelity by chnging the vlue F = 1/3 in Tble 3 by n verge vlue f n nd obtin for ech code: f 5 = 1 3, f 7= (n 7 1 (N 7 n 7 ) 1/3 = , f 9 = (n 9 1 (N 9 n 9 ) 1/3 = (41) We substitute in Tble 5 the vlue 1/3 by the verge vlue f n (except in the lst rrow s for P = 1 the errors re unrecoverble whtever is the code). Eqution (39) becomes: F (P) = Ψ s ρ E Ψ s = [(1 P) 8 18 P 3 (1 P)7 108 P 9 (1 P)6 16 P 3 (1 P)5 7 [3 P 9 (1 P)6 36 P 3 7 (1 P)5 108 P 4 81 (1 P)4 [9f n [9 P 4 81 (1 P)4 54 P 5 43 (1 P)3 [3(6f n 3) [ 1 7 P 6 (1 P) [1f n 6 3 (1 P)7 18 P 9 (1 P)6 108 P 3 7 (1 P)5 16 P 4 81 (1 P)4 [6 9 (1 P)6 18 P 3 7 (1 P)5 108 P 4 81 (1 P)4 16 P 5 43 (1 P)3 [9 [3 P 3 (1 P) P 4 81 (1 P)4 108 P 5 43 (1 P)3 [54f n [3 P 4 81 (1 P)4 36 P 5 43 (1 P)3 108 P (1 P) [81f n [9 P 5 43 (1 P)3 54 P (1 P) [54(f n 1) [9 P (1 P) 54 P (1 P) [81(f n 1 [7 P (1 P) [18(7f n ) 7 P 8 [117. (4) 38 We obtin: F (P) = (1 P) 8 8P(1 P) 7 (3f n 5)P (1 P) 6 7 Summry (18f n 38)P 3 (1 P) 5 (41f n 9)P 4 (1 P) 4 (44f n 1)P 5 (1 P) 3 ( 05fn 47 ) P 6 (1 P) 9 ( 50fn ) P 7 (1 P) P 8. (43) Tble 15 summrizes ll the results nd Fig. 4 compres the verge fidelity without nd with correction by the three codes. Figure 1 shows logiclly tht the verge fidelity is decresing with P without nd with correction by ny code. The vlues of fidelity re lwys better nd the decrese is slower when using codes. The best verge fidelity is given by the nine qubits code, followed by the seven qubits then the five qubits code. The reson is tht for the five qubits code ll the double errors (in logicl qubit) let the protected qubits ffected, while some of them could be covered when using the two other codes. Fig. 4 (Color online) Fidelity without nd with correction by the five, seven nd nine qubits codes.

11 No. 5 Communictions in Theoreticl Physics 671 We consider in this work tht triple errors (in logicl qubits) nd more re very unlikely, so tht syndrome mesurement llows (in seven nd nine qubits code) recovering errors. We note tht if P = 1, then the verge fidelity F(P = 1) =13/7= is the sme regrdless the used code. Tble 15 Fidelity without nd with correction by the five, seven nd nine qubits codes. The symbol C 0 corresponds to no correction. Code F (P) C 0 1 P 8 3 P 3 7 P3 (1 P) 8 8P(1 P) 7 (3f n 5)P (1 P) 6 (18f n 38)P 3 (1 P) 5 (41f n 9)P 4 (1 P) 4 C n ( ) ( ) (44f n 1)P 5 (1 P) 3 05fn 47 P 6 (1 P) 7 50fn P (1 P) P8 (1 P) 8 8P(1 P) 7 6P (1 P) 6 44P 3 (1 P) P4 (1 P) 4 C P5 (1 P) P6 (1 P) P7 (1 P) 13 7 P8 (1 P) 8 8P(1 P) P (1 P) P3 (1 P) P4 (1 P) 4 C P5 (1 P) P6 (1 P) P7 (1 P) 13 7 P8 (1 P) 8 8P(1 P) 7 55 P (1 P) 6 53P 3 (1 P) P4 (1 P) 4 C P5 (1 P) P6 (1 P) P7 (1 P) 13 7 P8 8 Conclusion This work ws devoted to error correction for quntum secret shring. The results show tht the nine qubits code gives the best fidelity, followed by the seven, then the five qubits code, regrdless the depolrizing chnnel error probbility P. This conclusion seems to confirm the simultion work done in Ref. [9 where errors were introduced by the correction process itself. We conclude tht higher is the ncills number better is the fidelity. The reson is tht the nine qubits code offers higher frction of double chnnel errors letting unffected the received useful qubit. In fct, s only single nd double errors hve been considered, the nine qubits code gives specific syndrome for higher number of double errors, then llowing their recovery which led to fidelity equl to one. We hve supposed tht triple errors nd more re very unlikely nd then with negligible effect on the obtined results. Acknowledgments We would like to thnk very much Mrs. S. Fritzsch nd T. Rdtke for providing us with the version 4 (008) of Feynmn Progrm. References [1 Dmin Mrkhm nd Brry C. Snders, Phys. Rev. A 78 (008) [ T. Rdtke nd S. Fritzsche, Compt. Phys. Commun. 173 (005) 91. [3 T. Rdtke nd S. Fritzsche, Compt. Phys. Commun. 175 (006) 145. [4 T. Rdtke nd S. Fritzsche, Compt. Phys. Commun. 176 (007) 617. [5 T. Rdtke nd S Fritzsche, Compt. Phys. Commun. 179 (008) 647. [6 M.A. Nielsen nd I.L. Chung, Quntum Computtion nd Informtion, Cmbridge University Press, Cmbridge (000). [7 Rymond Lflmme, et l., Phys. Rev. Lett. 77 (1996) 198. [8 A.M. Stene, Proc. R. Soc. Lond. A 45 (1996) 551, qunt-ph/ [9 Jumpei Niw, Keiji Mtsumoto, nd Hiroshi Imi, Simulting the Effects of Quntum Error-correction Schemes, rxiv:qunt-ph/011071v1, 13 Nov. (00).