Quantum Network Code for Multiple-Unicast Network with Quantum Invertible Linear Operations

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1 Quantum Network Code for Multple-Uncast Network wth Quantum Invertble Lnear Operatons Seunghoan Song Graduate School of Mathematcs, Nagoya Unversty, Nagoya, Japan Masahto Hayash Graduate School of Mathematcs, Nagoya Unversty, Nagoya, Japan Centre for Quantum Technologes, Natonal Unversty of Sngapore, Sngapore, Sngapore Shenzhen Insttute for Quantum Scence and Engneerng, Southern Unversty of Scence and Technology, Shenzhen, Chna Abstract Ths paper consders the communcaton over a uantum multple-uncast network where r senderrecever pars communcate ndependent uantum states We concretely construct a uantum network code for the uantum multple-uncast network as a generalzaton of the code [Song and Hayash, arxv: , 208] for the uantum uncast network When the gven node operatons are restrcted to nvertble lnear operatons between bt bass states and the rates of transmssons and nterferences are restrcted, our code certanly transmts a uantum state for each sender-recever par by n-use of the network asymptotcally, whch guarantees no nformaton leakage to the other users Our code s mplemented only by the codng operaton n the senders and recevers and employs no classcal communcaton and no manpulaton of the node operatons Several networks that our code can be appled are also gven 202 ACM Subject Classfcaton Hardware Quantum communcaton and cryptography Keywords and phrases Quantum network code, Multple-uncast uantum network, Quantum nvertble lnear operaton Dgtal Object Identfer 04230/LIPIcsTQC2080 Acknowledgements SS s supported by Rotary Yoneyama Memoral Master Course Scholarshp (YM) Ths work was supported n part by a JSPS Grant-n-Ads for Scentfc Research (A) No7H0280 and for Scentfc Research (B) No6KT007, and Kayamor Foundaton of Informaton Scence Advancement Introducton When we transmt nformaton va network, t s useful to make codes by reflectng the network structure Such type of codng s called network codng and was ntated by Ahlswede et al [] Ths topc has been extensvely researched by many researchers Network codng employs computaton-and-forward n ntermedate nodes nstead of the nave routng method n tradtonal network communcaton For the uantum network, the paper [5] started the dscusson of the uantum network codng, and many papers [2, 9 2] have advanced the study of uantum network codng In the network codng, uncast network s the most basc network model that the entre network s used by a sender and a recever As one of the remarkable achevements of network Seunghoan Song and Masahto Hayash; lcensed under Creatve Commons Lcense CC-BY 3th Conference on the Theory of Quantum Computaton, Communcaton and Cryptography (TQC 208) Edtor: Stacey Jeffery; Artcle No 0; pp 0: 0:20 Lebnz Internatonal Proceedngs n Informatcs Schloss Dagstuhl Lebnz-Zentrum für Informatk, Dagstuhl Publshng, Germany

2 0:2 Quantum Network Code for Multple-Uncast Network wth QIL Operatons codng for the uncast network, on the classcal lnear network wth malcous adversares, the papers [6, 7] proposed codes that mplement the classcal communcaton by asymptotc n-use of the network In [6, 7], when the transmsson rate m n absence of attacks s at least the maxmum rate a of attack (e, a < m), the codes n [6, 7] mplement the rate m a communcaton asymptotcally As a uantum generalzaton of the codes n [6, 7], the paper [4] constructed a uantum network code that transmts a uantum state correctly and secretly by asymptotc n-use of the network Smlarly to [6, 7], when the transmsson rate m wthout attacks s at least twce of the maxmum number a of the attacked edges (e, 2a < m), the code n [4] mplements the rate m 2a uantum communcaton asymptotcally However, snce a network s used by several users n general, t s needed to treat the network model wth multple users nstead of the uncast network For ths purpose, the multple-uncast network has been researched, n whch dsjont r sender-recever pars (S, T ),, (S r, T r ) communcate over a network The paper [8] studed a uantum network code for the multple-uncast network The code n [8] transmts a state successfully for each use of the network However, [8] has a lmtaton that the code should manpulate the node operatons n the network and therefore the code depends on the network structure In addton, the code n [8] reures the free use of the classcal communcaton Ths paper proposes a uantum network code for the multple-uncast network whch s a generalzaton of the uncast uantum network code n [4] and overcomes the shortcomngs of the multple-uncast uantum network code n [8] In the same way as [4], the gven node operatons are nvertble lnear wth respect to the bt bass states, whch s called uantum nvertble lnear operatons descrbed n Secton 2, our code reures the asymptotc n-use of the network for the correct transmsson of the state, and the encodng and decodng operatons are performed on the nput and output uantum systems of the n-use of the network, respectvely On the other hand, dfferently from [8], our code can be mplemented wthout any manpulaton of the network operatons and any classcal communcaton Moreover, our code makes no nformaton leakage asymptotcally from a sender S to the recevers other than T because the correctness of the transmtted state guarantees no nformaton leakage [3] To dscuss the achevable rate by our code, we consder the stuaton that the nput states of all senders are the bt bass states Then, our network can be consdered as a classcal network, called bt classcal network, because a bt bass state s transformed to another bt bass state by our uantum node operatons In the bt classcal network, we assume that when the nputs of the senders other than S are to zero, the transmsson rate from S to T s m, whch s the same as the number of outgong edges of S and ncomng edges of T Also, when we defne the nterference rate by the rate of the transmtted nformaton to T from the senders other than S, we assume that the nterference rate to T s at most a n the bt classcal network In the same way, n case that the nput states of all senders are set to the phase bass states (defned n Secton 2), we call the network as phase classcal network In the phase classcal network, we also assume that the transmsson rate from S to T s m when the nputs of the senders other than S are zero Also, the nterference rate to T s at most a n the phase classcal network Under these constrants, f a + a < m, our code acheves the rate m a a uantum communcaton from S to T asymptotcally To help the understandng of the rates descrbed above, we explan the achevable transmsson rate from S to T n the network n Fg The bt and the phase classcal networks (Fg b and Fg c) are determned from the uantum network (Fg a) (see Secton 2) When X = X 2 = Y = Y 2 = 0, the transmsson rates from S to T are 2 for both networks, e, m = 2, whch s also the number of outgong edges of S and ncomng

3 S Song and M Hayash 0:3 S S 2 X 2 b X b S S 2 X 2 X S S 2 Y 2 Y X b L(A ) X 2 b X A X 2 Y Â Y 2 X b X 2 + X b X 2 + X X Y 2 Y Y 2 T T 2 (a) Quantum network T T 2 (b) Bt classcal network T T 2 (c) Phase classcal network Fgure Toy example of a multple-uncast network In uantum network (a), b denote bt bass states and L(A ) s the network operaton (see Secton 2) The network (b) and (c) s the bt and phase classcal networks of the uantum network (a) edges of T Also, the nterference rates from S 2 to T are and 0 for the bt and the phase classcal networks, respectvely On ths network, f our code from S to T wth the rates (m, a, a ) = (2,, 0) s constructed, the condtons a, a 0 and a + a < m are satsfed, and therefore our code mplements the rate m a a = uantum transmsson from S to T asymptotcally In the practcal sense, our code can cope wth the node malfunctons n the followng case: on the multple-uncast network wth uantum nvertble lnear operatons, the network operatons are well-determned so that there s no nterference between all sender-recever pars, but three broken nodes apply uantum nvertble lnear operatons dfferent from the determned ones Moreover, let the transmsson rate m wthout nterferences from S to T be 00 and the number of outgong edges of the three broken nodes be 4 In ths case, 3 4 = 2 outgong edges of the three broken nodes transmt the unexpected nformaton whch mples the bt (phase) nterference rate s at most 2 Therefore, by our code wth m = 00 and a, a > 2, the sender S can transmt uantum states to the recever T correctly wth the rate 00 a a < 76 by asymptotcally many uses of the network The remanng of ths paper s organzed as follows Secton 2 ntroduces the formal descrpton of the uantum multple-uncast network wth uantum nvertble lnear operatons Secton 3 gves the man results of ths paper Based on the prelmnares n Secton 4, Secton 5 concretely constructs our code wth the free use of neglgble rate shared randomness The encoder and decoder of our code s gven n ths secton Secton 6 analyzes the correctness of the code n Secton 5 Then, Secton 7 constructs our code wthout the assumpton of shared randomness by attachng the secret and correctable communcaton protocol [5] to the code gven n Secton 5, whch proves the man result gven n Secton 3 Secton 8 gves several examples of the network that our code can be appled Secton 9 s the concluson of ths paper 2 Quantum Network wth Invertble Lnear Operatons Our code s desgned on the uantum network whch s a generalzaton of a classcal multple-uncast network In ths secton, we frst ntroduce the multple-uncast network wth classcal nvertble lnear operatons and generalze ths network as a network wth uantum nvertble lnear operatons The node operatons ntroduced n ths secton are dentcal to the operatons n [4, Secton II] T Q C 2 0 8

4 0:4 Quantum Network Code for Multple-Uncast Network wth QIL Operatons 2 Classcal Network wth Invertble Lnear Operatons Frst, we descrbe the multple-uncast network wth classcal nvertble lnear operatons The network topology s gven as a drected Graph G = (V, E) The r senders and r recevers are gven as r source nodes S,, S r and r termnal nodes T,, T r The sender S has m outgong edges and the recever T has m ncomng edges Defne m := m + + m r The ntermedate nodes are numbered from to c (= V 2r) accordngly to the order of the transmsson The ntermedate node numbered t has the same number k t of ncomng and outgong edges where k t m Next, we descrbe the transmsson and the operatons on ths network Each edge sends an element of the fnte feld F where s a power of a prme number p The t-th node operaton s descrbed as an nvertble lnear operaton A t from the nformaton on k t ncomng edges to that of k t outgong edges Snce node operatons are nvertble lnear, the entre network operaton s wrtten as K = A c A F m m For the network operaton K, we ntroduce the followng notaton: K, K,2 K,r K 2, K 2,2 K 2,r K :=, K r, K r,2 K r,r K,j F m mj Then, K,j s the network operaton from S to T j We assume rank K, = m whch means the nformaton from S to T s completely transmtted f there s no nterference When the network nputs by senders S,, S r are x F m,, x r F mr, the output y F m at the recever T ( =,, r) s wrtten as r y = K,j x j = K, x + K cz c, () j= K c :=[K, K, K,+ K,r ] F m (m m), z c :=[x T x T x T + x T r ] T F m m The second term K cz c of () s called the nterference to T, and rank K c s called the rate of the nterference to T Consder the n-use of the above network When the nputs by senders S,, S r are X F m n,, X r F mr n, the output Y F m n at the recever T ( =,, r) s r Y = K,j X j = K, X + K cz c, j= Z c :=[X T X T X T + X T r ] T F (m m) n 22 Quantum Network wth Invertble Lnear Operatons We generalze the multple-uncast network wth classcal nvertble lnear operatons to the network wth uantum nvertble lnear operatons In ths uantum network, the network topology s the same graph G = (V, E) Each edge transmts a uantum system H whch s -dmensonal Hlbert space spanned by the bt bass { x b } x F In n-use of the network, we treat the uantum system H m n spanned by the bt bass { X b } m X F n The sender S sends a uantum system H S := H m n and the recever T receves a uantum system H T := H m n To descrbe the uantum node operaton, we defne the followng uantum operatons

5 S Song and M Hayash 0:5 Defnton 2 (Quantum Invertble Lnear Operaton) For nvertble matrces A F m m and B F n n, two untares L(A) and R(B) are defned for any X F m n as L(A) X b := AX b, R(B) X b := XB b The operatons L(A) and R(B) are called uantum nvertble lnear operatons The t-th node operaton s gven as L(A t ) and t s called uantum nvertble lnear operaton The entre network operaton s wrtten as the untary L(K) = L(A c A ) = L(A c ) L(A ) When a state ρ on H S H Sr s transmtted by senders S,, S r, the network output σ T at H T s wrtten as σ T := Tr T,,T,T +,,T r L(K)ρL(K), where Tr T,,T,T +,,T r s the partal trace on the system H T H T H T+ H Tr When the nput state on the network s M b on H S H Sr, ths uantum network can be consdered as the classcal network n Subsecton 2 In the same way as the classcal network, we assume rank K, = m whch means S transmts any bt bass states completely to T f the nput states on source nodes S j (j ) are zero bt bass states Smlarly, rank K c s called the rate of the bt nterference to T We can dscuss the nterference smlarly on the phase bass { z p } z F defned n [3, Secton 82] by z p := x F ω tr xz x b, where ω := exp 2π p and tr y := Tr M y (y F ) wth the multplcaton map M y : x yx dentfyng the fnte feld F wth the vector space F t p For the analyss of the phase bass nterference, we gve Lemma 22 whch explans how node operatons L(A t ) are appled to the phase bass states Lemma 22 ( [4, Appendx A]) Let A F m m any M F m n, we have and B F n n L(A) M p = (A T ) M p, R(B) M p = M(B T ) p be nvertble matrces For For notatonal convenence, we denote  := (AT ) When the nput state s a phase bass state M p on H S H Sr, the network operaton L(K) s appled by L(K) M p = KM p In ths case, ths uantum network can also be consdered as a classcal network wth network operaton K = Âc  Then, K,j s defned from K n the same way as K,j K, K,2 K,r K 2, K2,2 K2,r K :=, K,j F m m j, K r, Kr,2 Kr,r K c :=[ K, K, K,+ K,r ] Smlarly to the condton rank K, = m, we also assume rank K, = m We also call rank K c the rate of phase nterference to T The transmsson rates from S to T are summarzed n Table T Q C 2 0 8

6 0:6 Quantum Network Code for Multple-Uncast Network wth QIL Operatons Table Defntons of Informaton Rates Rate m = rank K, = rank K, rank K c rank K c a a Meanng Bt (phase) transmsson rates from S to T wthout nterference Rate of nterference to T Rate of phase nterference to T Maxmum rate of bt nterference to T Maxmum rate of phase nterference to T 3 Man Results In ths secton, we propose two man theorems of ths paper The two theorems state the exstence of our code wth and wthout neglgble rate shared randomness, respectvely The codes stated n the theorems are concretely constructed n Secton 5 and 7, respectvely The theorems are stated wth respect to the completely mxed state ρ mx and the entanglement fdelty F 2 e (ρ, κ) := x κ ι R ( x x ) x for the uantum channel κ and a purfcaton x of the state ρ Theorem 3 Consder the transmsson from the sender S to the recever T over a uantum multple-uncast network wth uantum nvertble lnear operatons gven n Secton 2 Let m be the bt and phase transmsson rates from S to T wthout nterferences (m = rank K, = rank K, ), and a, a be the upper bounds of the bt and phase nterferences, respectvely (rank K c a, rank K c a ) When the condton a + a < m holds and the sender S and recever T can share the randomness whose rate s neglgble n comparson wth the block-length n, there exsts a uantum network code whose rate s m a a and the entanglement fdelty Fe 2 (ρ mx, κ ) satsfes n( Fe 2 (ρ mx, κ )) 0 where κ s the uantum code protocol from sender S to recever T Secton 5 constructs the code stated n Theorem 3 and Secton 6 shows that ths code has the performance n Theorem 3 Note that ths code does not depend on the detaled network structure, but depends only on the nformaton rates m, a and a As explaned n [4, Secton III], our code has no nformaton leakage from the condton n( F 2 e (ρ mx, κ )) 0 Although Theorem 3 assumed the free use of the neglgble rate shared randomness, t s possble to desgn the code of the same performance wthout neglgble rate shared randomness as follows The paper [5] gves the secret and correctable classcal network communcaton protocol for the classcal network wth malcous attacks, when the transmsson rate s more than the sum of the rate of attacks and the rate of nformaton leakage By applyng the protocol n [5] to our uantum network wth bt bass states, the neglgble rate shared randomness can be generated By ths method, we have the followng Theorem 32 and the detals are explaned n Secton 7 Theorem 32 Consder the transmsson from the sender S to the recever T over a uantum multple-uncast network wth uantum nvertble lnear operatons gven n Secton 2 Let m be the bt and phase transmsson rates from S to T wthout nterferences (m = rank K, = rank K, ), and a, a be the upper bounds of the bt and phase nterferences, respectvely (rank K c a, rank K c a ) When a + a < m, there exsts a uantum network code whose rate s m a a and the entanglement fdelty F e 2 (ρ mx, κ ) satsfes n( Fe 2 (ρ mx, κ )) 0 where κ s the uantum code protocol from sender S to recever T

7 S Song and M Hayash 0:7 (Prvate Randomness U,) (Shared Randomness SR ) ρ Encoder E SR,R S (ρ ) σ T S Quantum Network T (Multple-Uncast) T Decoder D SR (σ T ) S r T r Fgure 2 Overvew of code protocol from a sender S to a recever T States ρ and D SR (σ T ) are n code space H code 4 Prelmnares for Code Constructon Before code constructon, we prepare the extended uantum system, notatons, and CSS code used n our code 4 Extended Quantum System Although the unt uantum system for the network transmsson s H, our code s constructed based on the extended uantum system H descrbed below Frst, dependently on the block-length n, we choose a power := α to satsfy n (n ) m /( ) m max{a,a} 0 (eg = O(n +(max{a,a }+2)/(m max{a,a })) ) where n := n/α Let F be the α-dmensonal feld extenson of F Smlarly, let H := H α be the uantum system spanned by { x b } x F Then, the n-use of the network over H can be consdered as the n -use of the network over H The uantum nvertble lnear operatons (Defnton 2) can also be defned for nvertble matrces A F m m and B F n n as L (A) X b = AX b, R (B) X b = XB b, for any X F m n 42 Notatons for Quantum Systems and States n Our Code We ntroduce notatons used n our code By the n-use of the network, the sender S transmts the system H S = H m n and the recever T receves the system H T = H m n, whch are dentcal to H m n We partton the uantum system H m n as H A H B H C := H m m H m m H m (n 2m ) Furthermore, we partton the systems H A, H B, H C by H A = H A H A2 H A3 := H a m H (m a a ) m H a m, H B = H B H B2 H B3 := H a m H (m a a ) m H a m, H C = H C H C2 H C3 := H a (n 2m ) H (m a a ) (n 2m ) H a (n 2m ) For states φ H A, ψ H A2, and ϕ H A3, the tensor product state n H A s T Q C 2 0 8

8 0:8 Quantum Network Code for Multple-Uncast Network wth QIL Operatons denoted as φ ψ := φ ψ ϕ H A (2) ϕ The bt or phase bass state of (X, Y, Z) F a m X Y Z b X b := Y b, Z b X Y Z p F(m a a ) m Fa m s denoted as X p := Y p (3) Z p We also ntroduce notatons for the states n H B and H C n the same way as (2) and (3) In the followng, we denote the k l zero matrx as 0 k,l 43 CSS Code n Our Code In our code constructon, we use the CSS code defned n ths subsecton whch s smlarly defned from [4, Subsecton IV-B] Defne two classcal codes C, C 2 F m (n 2m ) whch satsfy C C2 as 0 a,n 2m C := X 2 F m (n 2m ) X 2 F (m a a ) (n 2m ), X 3 F a (n 2m ), X 3 X C 2 := X 2 F m (n 2m ) X F a (n 2m ), X 2 F (m a a ) (n 2m ) 0 a,n 2m For any [M ] C /C2 where M F (m a a ) (n 2m ), defne the uantum state [M ] b H C by [M ] b := C 2 Y C 2 0 a,n 2m M 0 a,n 2m + Y b 0 a,n 2m b = M b 0 a,n 2m p Wth the above defntons, the code space s gven as H code := H C2 = H (m a a ) (n 2m ) and a pure state φ H code s encoded as a superposton of the states [M ] b n ths CSS code by 0 a,n 2m b φ H C 0 a,n 2m p 5 Code Constructon wth Neglgble Rate Shared Randomness In ths secton, we construct our code that allows a sender S to transmt a state ρ on H code = H (m a a ) (n 2m ) correctly to a recever T by n-use of the network when the encoder and decoder share the neglgble rate random varable SR := (R, V ) The encoder and decoder are defned dependng on the prvate randomness U, owned by encoder and the randomness SR shared between the encoder and decoder These

9 S Song and M Hayash 0:9 random varables are unformly chosen from the values or matrces satsfyng the followng respectve condtons: the varable R := (R,, R,2 ) F (m a) m F(m a ) m satsfes rank R, = m a and rank R,2 = m a, the random varable V := (V,,, V,4m ) conssts of 4m values V,,, V,4m F 4m and the random varable U, F m m satsfes rank U, = m Next, we construct the encoder E SR,U, U,, the encoder E SR,U, to H S and decoder D SR Dependng on SR and of the sender S s defned as an sometry channel from H code = H m n Dependng on SR, the decoder D SR of the recever T s defned as a TP-CP map from H T = H m n to H code Note that the randomness SR s shared between the encoder and the decoder Because SR conssts of αm (2m a a + 4) elements of F, the sze of the shared randomness SR s sublnear wth respect to n (e, neglgble) 5 Encoder E SR,U, of the sender S The encoder E SR,U, conssts of three steps In the followng, we descrbe the encodng of the state φ n H code Step E The sometry map U R,0 encodes the state φ wth the CSS code defned n Subsecton 43 and the uantum systems H A and H B as φ := U R,0 φ = 0 a,m R, R 0 a,m b,2 φ H A H B H C = H S 0 a,m p b 0 a,m Step E2 By uantum nvertble lnear operaton L (U, ), the encoder maps φ to φ 2 := L (U, ) φ Step E3 From random varable V = (V,,, V,4m ), defne matrces Q,;j,k := (V,k ) j, Q,2;j,k := (V,m+k) j for j n 2m, k m, and Q,3;j,k := (V,2m+k) j, Q,4;j,k := (V,3m+k) j for j, k m Wth these matrces, defne the matrx U V,2 n Fn as U V,2 := p I m 0 m,m 0 m,n 2m I m 0 m,m 0 m,n 2m Q T,3 Q,4 I m 0 m,n 2m 0 m,m I m Q T,2 0 n 2m,m 0 n 2m,m I n 2m 0 n 2m,m 0 n 2m,m I n 2m I m 0 m,m 0 m,n 2m 0 m,m I m 0 m,n 2m, Q, 0 n 2m,m I n 2m where I d s the dentty matrx of sze d By uantum nvertble lnear operaton R (U V,2 ), the encoder maps φ 2 to R (U V,2 ) φ 2 By above three steps, the encoder E SR,U, s descrbed as an sometry map E SR,U, : φ R (U V,2 )L (U, )U R,0 φ H S T Q C 2 0 8

10 0:0 Quantum Network Code for Multple-Uncast Network wth QIL Operatons 52 Decoder D SR of the recever T Decoder D SR ψ H T conssts of two steps In the followng, we descrbe the decodng of the state Step D (U V,2 ) = Snce (U V,2 ) can be constructed from shared randomness V by I m 0 m,m 0 m,n 2m I m 0 m,m 0 m,n 2m 0 m,m I m 0 m,n 2m 0 m,m I m Q T,2 Q, 0 n 2m,m I n 2m 0 n 2m,m 0 n 2m,m I n 2m I m 0 m,m 0 m,n 2m Q T,3 Q,4 I m 0 m,n 2m, 0 n 2m,m 0 n 2m,m I n 2m the decoder apples the reverse operaton R (U V,2 ) = R ((U V,2 ) ) of Step E3 as ψ := R (U V,2 ) ψ Step D2 Perform the bt and phase bass measurements on H A and H B, respectvely, and let O,, O,2 F m m be the respectve measurement outcomes By Gaussan elmnaton, fnd nvertble matrces D R,,O,,, D R,2,O,2,2 F m m satsfyng P W, D R,,O,, O, = 0 a,m R,, P W,2 D R,2,O,2,2 O,2 = R,2 0 a,m (4) where P W s the projecton from F m to the subspace W, the subspace W, conssts of the vectors whose -st,, a -th elements are zero and the subspace W,2 conssts of the vectors whose (m a + )-st,, m -th elements are zero The case of non-exstence of D R,,O,, nor D R,2,O,2,2 means decodng falure, whch mples that the decoder performs no more operatons Also, when D R,,O,, and D R,2,O,2,2 are not determned unuely, the decoder chooses D R,,O,, and D R,2,O,2,2 determnstcally dependng on O,, R, and O,2, R,2, respectvely Based on D R,,O,, and D R,2,O,2,2 found by (4), the decoder apples L (D R,,O,, ) and L ( D R,2,O,2,2 ) consecutvely to ψ, and the resultant state on H code s the output of Step D2 Then, Step D2 s wrtten as the followng TP-CP map D R D R ( ψ ψ ) := Tr C,C3 U O,,O,2 F m m R,O,,O,2 D where the matrces U R,O,,O,2 D and σ O,,O,2 are defned as U R,O,,O,2 D σ O,,O,2 :=L ( D R,2,O,2,2 )L (D R,,O,, ), : σ O,,O,2 (U R,O,,O,2 D ), := Tr A,B ψ ψ ( O, bb O, O,2 pp O,2 I C ), wth the dentty operator I C on H C

11 S Song and M Hayash 0: By above two steps, the decoder D SR D SR s descrbed as ( ( ψ ψ ) := D R R (U V,2 ) ψ ψ R (U V,2 ) ) Snce the sze of the shared randomness SR s sublnear wth respect to n, our code s mplemented wth neglgble rate shared randomness 6 Correctness of Our Code In ths secton, we confrm that our code correctly transmts the state from the sender S to the recever T As s mentoned n Secton 3, we show the condton n( F 2 e (ρ mx, κ )) 0 whch mples the correctness of our code Frst, we descrbe the uantum code protocol κ from S to T, whch s an ntegraton of the encodng, transmsson, and decodng The encodng and decodng n κ s gven by the probablstc mxture of the code n Secton 5 dependng on the unformly chosen random varables SR and U, Then, the code protocol κ s wrtten as, for the state ρ on H code, κ (ρ ) := ( ( N DSR Tr L(K) T,,T,T +,,T r SR,U, E SR,U, (ρ ) ρ c )L(K) ), where ρ c s the state n H S H S H S+ H Sr of senders other than S, and N := 4m + {U, F m m rank U, = m } + {R, F (m a) m rank R, = m a } + {R,2 F (m a ) m rank R, = m a } As explaned n [4, Secton IV], Fe 2 (ρ mx, κ ) s upper bounded by the sum of the bt error probablty and the phase error probablty The bt error probablty s the probablty that a bt bass state X b H code s sent but the bt bass measurement outcome on the decoder output s not X In the smlar way, the phase error probablty s defned for the phase bass We wll show n Subsectons 62 and 63 that( the bt and phase error probabltes ( { are upper bounded by O max, Therefore, we have ( n( Fe 2 (ρ mx, κ )) no max Snce s taken n Secton 4 to satsfy (n ) m ( ) m a }) and O { max (n, ) m ( ) m a } ), respectvely {, (n ) m ( ) m max{a,a } }) (5) n (n ) m ( ) m max{a,a } 0, the RHS of (5) converges to 0 and therefore n( F 2 e (ρ mx, κ )) 0 Ths completes the proof of Theorem 3 6 Notaton and Lemmas for Bt and Phase Error Probabltes In ths subsecton, we prepare a notaton and lemmas for provng the upper bounds of the bt and phase error probabltes The upper bounds of these probabltes are shown separately n Subsectons 62 and 63 We ntroduce the notaton X := (X A, X B, X C ) F k m Fk m 2m ) Fk (n for X F k n wth arbtrary postve nteger k Also, we prepare the followng lemmas Lemma 6 For ntegers d 0 d + d 2, let W F d0 be a d -dmensonal subspace and W 2 F d0 be a d 2-dmensonal subspace Assume the followng three condtons (Γ) W W 2 = {0 d0,} T Q C 2 0 8

12 0:2 Quantum Network Code for Multple-Uncast Network wth QIL Operatons (Γ2) Let m d + d 2 The vectors x,, x m W and y,, y m W 2 satsfy span((x, y ),, (x m, y m )) = W W 2 (Γ3) Let W F d0 be a d -dmensonal subspace and r,, r m W There exsts an nvertble lnear map A : W W whch maps [x,, x m ] = A[r,, r m ] Then, the followng two statements hold ( ) There exsts nvertble lnear map D : F d0 Fd0 P W D[(x, y ),, (x m, y m )] = A [x,, x m ] = [r,, r m ] (6) that ( 2) For the above lnear map D, t holds for any x W and y W 2 that P W D(x, y) = A x (7) Proof Frst, we show the tem ( ) Let W 3 be a subspace of F d0 that satsfes W W 2 W 3 = F d0 If D s defned as D W = A and D W2 W 3 (W 2 W 3 ) = W, we obtan (6), e, ( ) from P W D((x, y )) = P W (D W (x ) + D W2 W 3 (y )) = A x = r Next, we show that the tem ( 2) Snce arbtrary (x, y) W W 2 s spanned by (x, y ),, (x m, y m ), E (6) mples (7), whch yelds ( 2) Lemma 62 ( [4, Lemma 7]) For ntegers d a d b + d c, fx a d b -dmensonal subspace W F da, and randomly choose a d c-dmensonal subspace R F da wth the unform dstrbuton Then, we have Pr[W R = {0 da,}] = O( d b +d c d a ) Lemma 63 For d d, ] ( Pr [rank[t,, t d ] = d t,, t d F d O ) Proof From d d, we have ] Pr [rank[t,, t d ] = d t,, t d F d Pr [rank[t,, t d ] = d t,, t d F d ] (8) On the other hand, the RHS of (8) s euvalent to the probablty to choose d ndependent vectors n F d : ] Pr [rank[t,, t d ] = d t,, t d F d = d d By combnng the above neualty and eualty, we have the lemma d d d ( ) = O d d Lemma 64 ( [4, Lemmas 72 and 74]) For the random matrx U V,2 defned n Step E3, we have max 0 n, x F n max 0 n, x F n Pr[x T ((U V,2 ) ) A =0,m ] Pr[x T ((Û V,2 ) ) B =0,m ] ( n 2m ) m, ( n 2m ) m

13 S Song and M Hayash 0:3 62 Bt Error Probablty In ths subsecton, we show that when arbtrary bt bass state M b H code s the nput state of ( the { sender S, the}) orgnal message M s correctly recovered wth probablty at least O max, (n ) m ( ) m a Step : We derve a necessary condton for correct decodng of the orgnal message M n bt bass When arbtrary bt bass state M b H code s the nput state of the sender S, the encoded state s wrtten as 0 a,m 0 a,n 2m E SR,R ( M b ) = U, Ē R M U V,2,, Ē 2 Ē F m m (n 2m ),Ē2 Fa where we gnore the normalzng factors and phase factors Note that bt state measurement on network output system H T = H m n commutes wth the decodng operaton D SR on H T Therefore, n the analyss of the bt error probablty, we take the method to perform bt state measurement to H T frst, and then apply the decodng operaton correspondng to D SR, nstead of decodng frst and performng bt state measurement By performng the bt bass measurement to the network output σ T = κ ( M bb M ), we have the followng measurement outcome Y : 0 a,m 0 a,n 2m Y = K, U, Ē M R, Ē 2 where Ē F m m, Ē 2 F a (n 2m ) U V,2 + K cz, and Z F(m m) n By Step D, Y s decoded to 0 a,m 0 a,n 2m Ȳ = Y (U V,2 ) = K, U, Ē M + K cz(u V,2 R ), Ē 2 The measurement outcome O, n Step D2 s O, = Ȳ A = K, U, 0 a,m R, + (K cz(u V,2 ) ) A b Snce the decoder knows O, and R,, the matrx D R,,O,, s found by Gaussan elmnaton to the left euaton of (4) whch s wrtten as P W, D R,,O,, O, =P W, D R,,O,, K,U, 0a,m + (K cz(u V,2 R ) ) A = 0a,m (9), Therefore, f the matrx D R,,O,, derved n (9) satsfes the followng euaton P W, D R,,O,, Ȳ C =P W, D R,,O,, K,U, 0 a,n 2m M +(K cz(u V,2 ) ) C = 0 a,n 2m M, (0) Ē 2 Ē 2 the orgnal message M s correctly recovered R, T Q C 2 0 8

14 0:4 Quantum Network Code for Multple-Uncast Network wth QIL Operatons Step 2: In the next step, we show that the condtons (Γ), (Γ2) and (Γ3) of Lemma 6 n the followng case mply E (0); W := col K,U, 0a,m, W2 := col ( K cz(u V R ) ),2, W := W,, m := m,, [x,, x m] := K,U, 0a,m, [y,, y m] := (K cz(u V,2 R ) ) A,, [r,, r m] := 0a,m R,, A := (K,U,) W, (d 0, d, d 2) := (m, m a, rank K cz), where col(t ) of the matrx T s the column space of T and W, s defned n Step D2 of Subsecton 52 Applyng Lemma 6, we show that E (0) holds f the condtons (Γ), (Γ2) and (Γ3) are satsfed Assume that (Γ), (Γ2) and (Γ3) are satsfed Then, the condton ( ) holds whch mples the exstence of D R,,O,, n (9) Moreover, ( 2) mples that for any r W, y W 2 and x = K, U, r W, t holds P W D R,,O,, (x + y) = A x = ( (K, U, ) W ) (K, U, r) = r, and ths yelds (0) Step 3: In the thrd ( step, { we show that }) the relatons (Γ), (Γ2) and (Γ3) hold at least wth (n probablty O max, ) m, whch proves the desred statement by combnng ( ) m a the concluson of Steps and 2 Step 3-: In ths substep, we show that the probablty satsfyng (Γ), (Γ2) and (Γ3) s obtaned by Pr[(Γ) (Γ2) (Γ3)] = Pr[(Γ)] Pr[(Γ2 )] Pr[(Γ2) (Γ2 ) (Γ)], () where the condton (Γ2 ) s gven as (Γ2 ) rank K cz((u V,2 ) ) A = rank K cz E () s derved by the followng reductons: Pr[(Γ) (Γ2) (Γ3)] (a) = Pr[(Γ) (Γ2)] (b) = Pr[(Γ)] Pr[(Γ2) (Γ)] (c) = Pr[(Γ)] Pr[(Γ2) (Γ2 ) (Γ)] (d) = Pr[(Γ)] Pr[(Γ2 ) (Γ)] Pr[(Γ2) (Γ2 ) (Γ)] (e) = Pr[(Γ)] Pr[(Γ2 )] Pr[(Γ2) (Γ2 ) (Γ)] The eualty (a) follows from the fact that (Γ3) s always satsfed for A defned n Step 2, and (b) and (d) are trval (c) s obtaned because (Γ2 ) s a necessary condton for (Γ2) Snce span(y,, y m ) = W 2 s a necessary condton for (Γ2) n Lemma 6, the condton (Γ2 ) s also necessary for (Γ2) from rank K cz((u V,2 ) ) A =rank(k cz(u V,2 ) ) A =dmspan(y,, y m ) =dm W 2 =rank K cz(u V,2 ) =rank K cz The eualty (e) follows from the fact that (Γ) and (Γ2 ) are ndependent, whch wll be shown by Pr[(Γ) (Γ2 )] = Pr[(Γ)] n Step 3-2

15 S Song and M Hayash 0:5 Step 3-2: In ths step, we prove Pr[(Γ)] O(/ ) and Pr[(Γ) (Γ2 )] = Pr[(Γ)] Fx R, and U V,2 Then, W s randomly chosen d -dmensonal subspace under unform dstrbuton and W 2 s fxed d 2 -dmensonal subspace Therefore, Lemma 62 can be appled wth (d a, d b, d c, W) := (d 0, d 2, d, W 2 ) and Pr[(Γ)] = O( d 2+d d 0 ) O(/ ) On the other hand, snce Pr[(Γ)] does not depend on U V,2 but Pr[(Γ2)] depends only on U V,2, we have Pr[(Γ) (Γ2 )] = Pr[(Γ)] Step 3-3: In ths step, we show Pr[(Γ2 )] nm m a The condton (Γ2 ) holds f and only f x T K cz((u V,2 ) ) A 0,m for any vector x F m satsfyng xt K cz 0,n (consderng K c, Z and ((U V,2 ) ) A as lnear maps on row vector spaces, ths s euvalent that ((U V,2 ) ) A has trval kernel {0,n } for the mage of K cz) Therefore, by applyng Lemma 64 for all dstnct x T K cz whch s not zero vector, we have ( n 2m Pr[(Γ2 )] rank K c Z ) m a ( n 2m ) m nm m a Step 3-4: Now we evaluate the probablty Pr[(Γ2) (Γ2 ) (Γ)] O(/ ) Fx the random varable U V,2 so that (Γ2 ) holds n the followng Defne matrces T x = [x (),, x (d+d 2)], T y = [y (),, y (d+d 2)] and T = T x + T y F d0 (d+d2) where : {,, d + d 2 } {,, m} s an njectve ndex functon such that y (),, y (d2) are lnearly ndependent e, rank T y = d 2 Then, we have Pr [ (Γ2) (Γ2 ) (Γ) ] Pr[span ( (x (),y () ),, (x (d +d 2 ), y (d +d 2 )) ) =W W 2 (Γ2 ) (Γ)] (a) = Pr [ rank T = d +d 2 (Γ2 ) (Γ) ] = Pr [ ker T = {0 d +d 2,} (Γ2 ) (Γ) ] (b) = Pr [ ker T x ker T y = {0 d +d 2,} (Γ2 ) (Γ) ], where (a) follows from span ( (x (), y () ),, (x (d+d 2), y (d+d 2)) ) W W 2, and (b) follows from the condton (Γ) Snce rank T x d follows from ts defnton and the dmenson of ker T y s d, the condton rank T x = d s a necessary condton for ker T x ker T y = {0 d+d 2,} Therefore, we have Pr[ker T x ker T y = {0 d+d 2,} (Γ2 ) (Γ)] = Pr[ker T x ker T y rank T x = d (Γ2 ) (Γ)] Pr[rank T x = d (Γ2 ) (Γ)] (2) By applyng Lemma 62 for (d a, d b, d c, W) := (d +d 2, d =dm ker T y, d 2 =dm ker T x, ker T y ), the frst multplcand of (2) euals to O(/ ) From Pr[rank T x = d (Γ2 ) (Γ)] Pr [ ] rank[t,, t d+d 2 ] = d t,, t d+d 2 F d and Lemma 63, the second multplcand of (2) s greater than or eual to O(/ ) Therefore, Pr[(Γ2) (Γ2 ) (Γ)] O(/ ) In summary, we obtan Pr[(Γ) (Γ2) (Γ3)] = Pr[(Γ)] Pr[(Γ2 )] Pr[(Γ2) (Γ2 ) (Γ)] ( ( )) ( ) ( ( )) ( O nm { m a O = O max, (n ) }) m ( m a ) 63 Phase Error Probablty In ths subsecton, we show that the orgnal ( message M n the phase bass s correctly { } ) (n recovered wth probablty at least O max ) m, ( ) m a T Q C 2 0 8

16 0:6 Quantum Network Code for Multple-Uncast Network wth QIL Operatons Step : We derve a necessary condton for correct decodng of the orgnal message M n phase bass For the analyss of the phase error probablty, we apply the same dscusson as the bt error probablty When a phase bass state M p H code s the nput state of sender S, the encoded state s wrtten as E SR,R ( M p ) = Ē Fm m,ē 2 Fa (n 2m ) Û, Ē where we gnore normalzng factors and phase factors R,2 Ē 2 M 0 a,m 0 a,n 2m Û V,2 Snce phase bass measurement and decodng operaton D SR commutes, we frst apply phase bass measurement, and then decode the measurement outcome for the analyss of the phase error probablty Then, the phase bass measurement outcome Y on the network output of T s wrtten as Y = K, Û, Ē R,2 Ē 2 M 0 a,m 0 a,n 2m where Ē F m m, Ē 2 F a (n 2m ) Ȳ = Y (Û V,2 ) = K, Û, Ē Û V,2 + K cz, and Z By Step D, Y F(m m) n s decoded to R,2 Ē 2 M 0 a,m 0 a,n 2m + K cz(û V,2 ) p, By Step D2, the measurement outcome O,2 s gven as O,2 = Ȳ B = K, Û, R,2 + 0 a,m ( K cz(û V,2 ) ) B, and D R,2,O,2,2 s found by Gaussan elmnaton to the rght euaton of (4) whch s wrtten as P W,2 D R,2,O,2,2 O,2 =P W,2 D R,2,O,2 K, Û,,2 R,2 +( K cz(û V,2 ) ) B = R,2 (3) 0 a,m 0 a,m Thus, the correct estmate of M s obtaned when the followng relaton holds for D R,2,O,2,2 derved n (3): P W,2 D R,2,O,2,2 Ȳ C =P W,2 D R,2,O,2 K, Û,,2 Ē 2 M +( K cz(û V,2 ) ) C = Ē 2 M (4) 0 a,n 2m 0 a,n 2m Step 2: ( In the next step, we show that the euaton (4) holds wth probablty at least { } ) (n O max ) m, whch shows the desred statement by combnng Step, ( ) m a In the same way as Subsecton 62, the condtons (Γ), (Γ2) and (Γ3) of Lemma 6 n

17 S Song and M Hayash 0:7 the followng case mply E (4); W := col K, Û, R,2, W2 := col( K cz(û V,2 ), ) W := W,2, m := m, 0 a,m [x,, x m] := K, Û, [r,, r m] := R,2 0 a,m R,2 0 a,m, [y,, y m] := ( K cz(û V,2 ) ) B,, A := ( K, Û,) W, (d 0, d, d 2) := (m, m a, rank K cz), where W,2 s defned n Step D2 of Subsecton 52 Also, ( n the same way, the condtons { } ) (n (Γ), (Γ2) and (Γ3) holds wth probablty at least O max ) m, ( ) m a 7 Code Constructon Wthout Free Classcal Communcaton We show that our code n Theorem 3 can be mplemented wthout the assumpton of neglgble rate shared randomness The paper [5] shows the followng Proposton 7 by constructng a secret and correctable classcal communcaton protocol for the classcal uncast lnear network Due to the relaton between the phase error and the nformaton leakage n the bt bass [4, Lemma 59], we fnd that the dmenson of leaked nformaton s a n the nformaton transmsson from the sender S to the recever T We apply Proposton 7 to the sender-recever par (S, T ) wth c := a and c 2 := a Therefore, the protocol of Proposton 7 can be mplemented n our multple-uncast network by preparng the nput state of S n the bt bass By attachng Proposton 7 to our code n the above method, we can mplement our code satsfyng Theorem 32 Proposton 7 ( [5, Theorem ]) Let be the sze of the fnte feld whch s the nformaton unt of the network edges We assume the neualty c + c 2 < c 0 for the classcal network where c 0 s the transmsson rate from the sender S to the recever T, c s the rate of nose njecton, and c 2 s the rate of nformaton leakage Defne 2 := c0 Then, there exsts a k-bt transmsson protocol of block-length n := c 0 (c 0 c 2 + )k over F 2 such that P err kc 0 / 2 and I(M; E) = 0, where P err s the error probablty and I(M; E) s the mutual nformaton between the message M F k 2 and the leaked nformaton E The proof of Theorem 32 takes a smlar method to the proof of [4, Theorem 32] Proof of Theorem 32 To construct the code satsfyng the condtons of Theorem 32, we generate the shared randomness SR by Proposton 7 and then apply the code n Secton 5 To apply Proposton 7 n our uantum network, we prepare the nput state as a bt bass state Gven a block-length n, we take = β such that β = 2 log 2 log 2 n m log 2 e, 2 /(log n) 2 = m /(log n)2, and = α such that α = (m+2) log 2 n log 2 e, /n m+2 Frst, by the protocol of Proposton 7 wth (c 0, c, c 2 ) := (m, a, a ), the sender S and the recever T share the randomness SR Snce SR conssts of m (2m a a + 4) elements of F, the number of bts to be shared s k = m (2m a a + 4) log 2 = m (2m a a + 4) (m + 2) log 2 n log 2 m (m + 2)(2m a a + 4) log 2 n log 2 T Q C 2 0 8

18 0:8 Quantum Network Code for Multple-Uncast Network wth QIL Operatons ( log2 n The error probablty s P err (m / m ) m (m +2)(2m a a +4) log 2 n =O (log 2 n) ) 2 0, and the block-length over F s 2 n =m (m a +)kβ m (m a +) m (m +2)(2m a a log2 log +4) log 2 n 2 n, m log 2 whch mples n /n 0 Therefore, the sharng protocol s mplemented wth neglgble rate uses of the network Next, we apply the code n Secton 5 wth the extended feld of sze and n 2 := n n uses of the network The relaton n 2 /n = (n n )/n holds and therefore the feld sze satsfes n 2 (n 2) m /( ) m max{a,a} 0 where n 2 := n 2 /α Thus, ths code mplements the code n Theorem 32 8 Examples of Network In ths secton, we gve several network examples that our code can be appled Frst, as the most trval case, f rank K, = m and any dstnct sender-recever pars do not nterfere wth each other, e, K,j ( j) are zero matrces, the network operaton K s a block matrx Ths s the case where each par ndependently communcates In ths case, our code s mplemented wth the rate m 8 Smple Network n Fg In the network n Fg, the network and node operatons are descrbed as K = , K = , A = [ ] 0 When we consder the transmsson from S to T, the rates of bt and phase nterferences are [ ] [ ] 0 0 rank K c = rank =, rank 0 K 0 0 c = rank = In ths network, by constructng our code wth (m, a, a ) := (2,, 0), our codng protocol transmts the state of rate m a a = asymptotcally from S to T 82 Network wth Bt Interference from One Sender As a generalzaton of the network n Fg, consder the case where the network conssts of two sender-recever pars, and there s no bt nterference from the sender S to recever T 2 The network operaton of ths network can be descrbed by L(K) wth [ ] [ K, K K =,2 (K, K =,) T 0 m,m 2 0 m2,m K 2,2 (K2,2) T K,2(K T,) T (K T 2,2) In ths network, there s no phase nterference from the sender S 2 to recever T, and the other two rates rank K,2 and rank(k T 2,2) K T,2(K T,) concde from rank K,2 = rank K T,2 = rank(k T 2,2) K T,2(K T,) Therefore, by mplementng our code wth a, a ( =, 2) satsfyng rank K,2 a, a 2 < m and a = a 2 := 0, each sender-recever par can transmt the states ]

19 S Song and M Hayash 0:9 Moreover, we generalze the above network for arbtrary r sender-recever pars where the nterferences are generated only from one sender S In ths network, the network operaton s gven by the untary operator L(K) wth K defned as follows: K, K,2 K,3 K,r 0 m2,m K = K 2,2 0 m2,m 3 0 m2,m r, 0 mr,m 0 mr,m 2 0 mr,m 3 K r,r (K,) T 0 m,m 2 0 m,m 3 0 m,m r (K2,2) T K,2(K T,) T (K2,2) T 0 m2,m 3 0 m,m r K =, (Kr,r) T K,r(K T,) T 0 mr,m 2 0 mr,m 3 (Kr,r) T where the ranks of m m matrces K, are m, resepctvely In ths network, f a, a ( =,, r) are set to a rank[k,2 K,3 K,r ], a rank K, ( = 2,, r), and a = a 2 = a 3 = = a r 0 and the condton a + a < m holds, the sender S can send to the recever T the rate m a a state asymptotcally by our code 83 Network wth Two Way Bt Interferences In ths subsecton, we consder the code mplementaton over the network descrbed as follows: The sze s 3, there exsts two pars (S, T ) and (S 2, T 2 ) n the network, S, S 2, T, T 2 are connected to three edges, and the network operaton s gven by L(K) of K = [ ] K, K, =, K = K 2, K 2, Then, dfferently from the prevous examples, there are bt nterferences both from S to T 2 and from S 2 to T because K,2 and K 2, are not zero matrx In the above network, we construct our code for S to T wth (m, a, a ) := (3,, ) Then, our code mplements the rate m a a = 3 = uantum communcaton asymptotcally from the relatons rank K =rank K =m =3, rank K c =rank [ ] [ ] =, rank K c =rank = Concluson In ths paper, we have proposed a uantum network code for the multple-uncast network wth uantum nvertble lnear operatons As the constrants of nformaton rates, we assumed that the bt and phase transmsson rates from S to T wthout nterference are m (m = rank K, = rank K, ), the upper bounds of the bt and phase nterferences are a, a, respectvely (rank K c a, rank K c a ), and a + a < m holds Under these constrants, our code acheves the rate m a a uantum communcaton by asymptotc n-use of the network The neglgble rate shared randomness plays a crucal role n our code, and t s realzed by attachng the protocol n [5] T Q C 2 0 8

20 0:20 Quantum Network Code for Multple-Uncast Network wth QIL Operatons Our code can be appled for the multple-uncast network wth the malcous adversary When the eavesdropper attacks at most a edges connected wth the sender S and the recever T, f a + a + 2a < m holds, our code mplements the rate m a a 2a uantum communcatons asymptotcally Ths fact can be shown by ntegratng the methods n ths paper and [4] References Rudolf Ahlswede, Nng Ca, S-YR L, and Raymond W Yeung Network nformaton flow IEEE Transactons on nformaton theory, 46(4):204 26, Masahto Hayash Pror entanglement between senders enables perfect uantum network codng wth modfcaton physcal revew A, 76(4):04030, Masahto Hayash Group Representaton for Quantum Theory Sprnger, Masahto Hayash Group Theoretc Approach to Quantum Informaton Sprnger, Masahto Hayash, Kazuo Iwama, Harumch Nshmura, Rudy Raymond, and Shgeru Yamashta Quantum network codng In Annual Symposum on Theoretcal Aspects of Computer Scence, pages Sprnger, Masahto Hayash, Masak Owar, Go Kato, and Nng Ca Secrecy and robustness for actve attack n secure network codng In Proceedngs on 207 IEEE Internatonal Symposum on Informaton Theory (ISIT), pages 72 76, S Jagg, M Langberg, S Katt, T Ho, D Katab, M Medard, and M Effros Reslent network codng n the presence of byzantne adversares IEEE Transactons on Informaton Theory, 54(6): , June Go Kato, Masak Owar, and Masahto Hayash Sngle-shot secure uantum network codng for general multple uncast network wth free publc communcaton In Internatonal Conference on Informaton Theoretc Securty, pages Sprnger, Hrotada Kobayash, Franços Le Gall, Harumch Nshmura, and Martn Rötteler General scheme for perfect uantum network codng wth free classcal communcaton In Internatonal Collouum on Automata, Languages, and Programmng, pages Sprnger, Hrotada Kobayash, Franços Le Gall, Harumch Nshmura, and Martn Rötteler Perfect uantum network communcaton protocol based on classcal network codng In Proceedngs of 200 IEEE Internatonal Symposum on Informaton Theory (ISIT), pages , 200 Hrotada Kobayash, Franços Le Gall, Harumch Nshmura, and Martn Rötteler Constructng uantum network codng schemes from classcal nonlnear protocols In Proceedngs of 20 IEEE Internatonal Symposum on Informaton Theory (ISIT), pages 09 3, 20 2 Debbe Leung, Jonathan Oppenhem, and Andreas Wnter Quantum network communcaton the butterfly and beyond IEEE Transactons on Informaton Theory, 56(7): , Benjamn Schumacher Sendng entanglement through nosy uantum channels Physcal Revew A, 54(4):264, Seunghoan Song and Masahto Hayash Secure uantum network code wthout classcal communcaton arxv: , Hongy Yao, Danlo Slva, Sdharth Jagg, and Mchael Langberg Network codes reslent to jammng and eavesdroppng IEEE/ACM Transactons on Networkng, 22(6): , 204