arXiv:0712.2471v1 [quant-ph] 14 Dec 2007 aaiyo unu hne sntknown. classi not private is the channel quantum [4], a formula of single-letter kno capacity is a channel by broadcast given classical and a of capacity private the capacity information quantum the namely information, information mutual the of generalization channel of number large asymptotically an uses. use to general, in distributi equa input is all channel to over noisy information mutual a a maximum of the capacity through to 194 the his transmitted that with showing be field [1] the paper can created Shannon information Indeed, channel. which noisy at rate the h uulifraini end suul codn to according usual, as defined, is information mutual The h e aeo B4wt n-a otpoesn n quantu th and post-processing demonstrates one-way rate latter with error the phas BB84 bit of of independent rate study key with Our the interest channel noise. particular amplitude the of and and channels channel app We two formul depolarizing channel. to quantum up single-letter general technique powerful a this of a generating capacities for the by technique on bounds simple described a provides are This capacities one-way aiyi ie y[6] by given is pacity S where Here U with where ( N h unu aaiyi ie y[] 6,[7]: [6], [5], by given is capacity quantum The quantum apparent the that known been long has it However, finding is theory information of problem central the Perhaps Abstract igeuse single a ρ C B nioercetninof extension isometric an ω p ) (1) − | ABE φ ( S AB N esuyetnin faqatmcanlwhose channel quantum a of extensions study We — ( ρ I ) osntyedasnl-etrfruafrthe for formula single-letter a yield not does , i AB diieetnin faqatmchannel quantum a of extensions Additive = c ≡ q ( Q ftecanl vntog h noigneeds, encoding the though even channel, the of N antexceed cannot saprfiainof purification a is ) P { C lim = p φ , with p x x,t ( , n | = ) N B JWto eerhCenter Research Watson TJ IBM ϕ →∞ sup p .I I. x lim = ) ( i} S t I | ,X ( [email protected] x n 1 c NTRODUCTION 11KthwnRoad Kitchawan 1101 ρ okonN 10598 NY Yorktown ) n → max )= p I →∞ ( T reeSmith Graeme ⊗N H x N ( − ) (1 φ I | Q n 1 t n ( / ( (i.e., Tr( ih T I C | 2 t φ 2,[] iial,though Similarly, [3]. 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( = ) = ) B ) T ω N B JWto eerhCenter Research Watson TJ IBM B T = ) = Q q C fteei eodchannel second a is there if (1) DTV AND DDITIVE by p [email protected] Q = S Q ( ecall We 11KthwnRoad Kitchawan 1101 ( T ( okonN 10598 NY Yorktown T ( H (1) N S ) ω onA Smolin A. John ) ewl o useti prahhere. approach this pursue not will we , B ( (1 nodrt n pe onson bounds upper find to order in = ) ( T can T ) ω / hni scerwihsaew are we state which clear is it When . ) ) 2 ω B . Q T T 1 − vlaecpct o oechannels— some for capacity evaluate (1) Tr = + an 2 q D ( S N (1 i.e. EGRADABLE diieextension additive ( B T ) − ≡ ) ω ω q BT B )) max C φ − p (1) − ,adue h notation the used and ), R S ( H I T ( c uhthat such (2 = ) BT ( E N q XTENSIONS (1 φ , ) ω C C − ) BT faquantum a of p p , ( ( q N T hr we where , N )) ) ) However, . . = lohave also R◦T tion (5) r- d e l Definition 2 A channel , with isometric extension U : A Proof: First, let be the partial trace on the flagging BE is called degradableNif there is a degrading map such→ system so that = R . To see that is degradable, note D N R ◦ T T that = , where (ρ) = TrB UρU †. is called the that the complementary channel of is complementaryD ◦ N N channel ofN . N T N = i i (12) b b b T Ni ⊗ | ih | We call an additive extension of a that i X is degradable a degradable extension. Degradable extensions and that letting b c have the additional property that their coherent information is = i i , (13) an upper bound for the private classical capacity as well as D Di ⊗ | ih | i the . X Our main tool will be the following simple theorem, which where = , we have = . DiTi Ti D ◦ T T bounds the capacity of a quantum channel in terms of the Finally, letting φ be the optimal input state for , we find capacity of its additive extensions. b b T Q(1)( ) = S p (φ) i i Theorem 3 The quantum capacity of a channel satisfies T iNi ⊗ | ih | N i ! X Q( ) Q(1)( ), (6) N ≤ T S p (φ) i i − iNi ⊗ | ih | for all additive extensions, , of . Furthermore, if is i ! T N T X degradable, the private classical capacity of satisfies = p S ( b(φ)) S (φ) N i Ni − Ni (1) i Cp( ) Q ( ). (7) X    (1) N ≤ T piQ ( i), b (1) ≤ N Proof: Q ( ) = Q( ) and Q( ) Q( ) follows i X immediately from theT fact thatT can beN obtained≤ T from by so that by Theorem 2 the result follows. apply . If is degradable, itN was shown in [11] that T R T C ( )= Q(1)( ), (8) B. No-cloning bounds p T T We next show that no-cloning bounds [15], [16] are a special so that we hvae C ( ) C ( )= Q(1)( ). p N ≤ p T T case. III. KNOWN UPPER BOUNDS Suppose is antidegradable, meaning there is a channel such thatN = . In this case, we can define a zero- In this section we show that the two strongest techniques capacityD degradableD◦ N extensionN of as follows. Let have for upper bounding the capacities of a quantum channel are N N isometric extensionb U : A BE, d = max(dB, dE), and F1 encompassed by our approach. The first technique, established → and F2 be d-dimensional spaces with B F1 and E F2. in [10], [13] and best for low noise levels, is to decompose ⊂ ⊂ Then define isometry V : A F1F2C1C2 as the channel into a convex combination of degradable channels. → The second, first studied in [14], [15], [16], is a no-cloning 1 1 V φ = U φ 01 C C + (SWAPF F U φ ) 10 C C type argument that can sometimes be used to show that a very | i √2 | i| i 1 2 √2 1 2 | i | i 1 2 noisy channel has zero capacity. This gives a degradable extension of , (ρ) = N T Tr V ρV †, which can be degraded to . A. Convex combinations of degradable channels F2C2 N Lemma 4 Suppose we have C. Convexity of bounds = p , (9) We now show that if we have upper bounds for the capacity N iNi i of two channels, both obtained from a degradable extension, X the convex combination of the bounds is an upper bound for where i is degradable with degrading map i. Then N D the capacity of the corresponding convex combination of the = pi i i i (10) channels. More concretely, suppose and are degradable T N ⊗ | ih | T0 T1 i extensions of and , respectively. Then, X N0 N1 is a degradable extension of , and N = p 0 0 0 + (1 p) 1 1 1 (14) T T ⊗ | ih | − T ⊗ | ih | (1) Q( ) piQ ( i). (11) is a degradable extension of = p + (1 p) , and N ≤ N 0 1 i N N − N X satisfies We will call a flagged degradable extension on since i (1) c c T N Q ( ) = pI ( 0, φ)+(1 p)I ( 1, φ) (15) keeps track of which i actually occurred in the decomposi- T T − T N (1) (1) tion of . pQ ( 0)+(1 p)Q ( 1). (16) N ≤ T − T IV. BOUNDS ON SPECIFIC CHANNELS A. Depolarizing Channel In this section we will evaluate explicit upper bounds on the The depolarizing channel of error probability p is given by private classical and quantum capacities of the depolarizing p p p channel and Pauli channels with independent amplitude and p(ρ)=(1 p)ρ + XρX + Y ρY + ZρZ . phase noise (which we also call “the BB84 channel”, because N − 3 3 3 of its relevance for BB84). In each case, we will use a flagged It is particularly nice to study since it has the property that degradable extension of the channel of interest, based on for any unitary U the convex decomposition into degradable channels used in [10][11]. The advantage we obtain over this previous work p(UρU †)= U p(ρ)U † . (22) N N is, essentially, due to the fact that our upper bound involves a maximization of the average coherent informations of the The following theorem, together with the subsequent corol- elements of our decomposition, all with respect to the same lary, provides the strongest upper bounds to date on the state. In [10][11], the corresponding bound is the average of capacity of the depolarizing channel. We provide a proof of the individual maxima, allowing different reference states for the theorem below, after establishing two essential lemmas. each channel in the decomposition, which generally leads to a weaker bound. Theorem 6 The capacity of the depolarizing channel with Throughout this section, we will use the following special error probability p satisfies property of coherent information for degradable channels, which was first proved in [17]. It will assist in the evaluation Q( p) co [∆(p), 1 4p] , (23) N ≤ − of coherent informations for specific degradable extensions below. where 1 1 Lemma 5 Let be degradable. Then ∆(p) = min H [1+ sin u sin v] H [1+ cos u cos v] , 2 − 2 N     pIc( , φ )+(1 p)Ic( , φ ) Ic( , pφ + (1 p)φ ). N 0 − N 1 ≤ N 0 − 1 with the minimization over (u, v) such that In other words, Ic( , φ) is concave as a function of φ. cos2(u/2)cos2(v/2) = 1 p, and co[f (p),f (p) ...f (p)] N − 1 2 n Proof: Writing out the entropies involved explicitly, what denotes the maximal convex function that is less than or we would like to prove is that equal to all fi(p) ,i =1 ...n. pS ( (φ )) + (1 p)S ( (φ )) N 0 − N 0 Corollary 7 S p (φ0)+(1 p) (φ1) − N − N 1 γ(p) γ(p)   Q( p) co 1 H(p),H( − ) H( ), 1 4p , ≤ N ≤ − 2 − 2 −   pS (φ0) + (1 p)S (φ0) N − N where γ(p)=4√1 p(1 √1 p).     − − − S pb (φ0)+(1 p) (φb1) , (17) − N − N Proof: Corollary 7 follows from the theorem by noting which, letting U be the isometric extension of  and b b N that the first two terms inside the square brackets are special ρVBE = p 0 0 V Uφ0U † + (1 p) 1 1 V Uφ1U †, (18) cases of ∆ for values of (u, v) corresponding to amplitude | ih | ⊗ − | ih | ⊗ damping and to dephasing channels, and using the fact that is equivalent to the true minimum is always bounded by particular cases. H(V B)ρV B H(V E)ρVE . (19) To establish Theorem 6 we will use the following flagged | ≤ | Noting that H(U B) is nondecreasing under operations on B degradable extension of the depolarizing channel: (which is a simple| consequence of the strong subadditivity dep 1 of quantum entropy), and there is a that maps to (ρ)= c† (u,v)(cρc†)c c c , (24) B E T(u,v) N ⊗ | ih | D c completes the proof. |C| X∈C We will also have use for , the most general degrad- N(u,v) where is the set of unitaries which map I,X,Y,Z able qubit channel (up to unitary operations on the input and I,X,Y,ZC under conjugation (the Clifford group).{ } → output) [18]. has Kraus operators { } N(u,v) 1 cos( 2 (v u)) 0 Lemma 8 A+ = − 1 (20) 0 cos( 2 (v + u))  1  (1) dep 1 1 0 sin( 2 (v + u)) Q ( (u,v))=H [1+ sin u sin v] H [1+ cos u cos v] A = 1 . (21) T 2 − 2 − sin( (v u)) 0      2 −  In [13], (u,v) was shown to be degradable when sin v Proof: The main step is to show that the coherent dep cos u . N | | ≤ information of is maximized by the maximally mixed | | T(u,v) state. To see this, first note that for any φ, we have 1 dep (ρ)= c† (cρc†)c (36) (X I) (u,v)(XφX)(X I) (25) N N ⊗ T ⊗ |C| c 1 X∈C = Xc† (cXφXc†)cX c c is a depolarizinge channel with the same entanglement fi- N(u,v) ⊗ | ih | c delity as . In other words, |C| X∈C 1 N = Xc† (cXφXc†)cX V cX cX V † N(u,v) ⊗ | ih | c + + + + + + + + |C| X∈C φ (I )( φ φ ) φ = φ (I )( φ φ ) φ , dep h | ⊗ N | ih | | i h | ⊗ N | ih | | i = (I V ) (φ)(I V †), (37) ⊗ T(u,v) ⊗ where φ+ = ( 1 )( 00 + 11 ). As a result,e letting = where we have chosen unitary V such that V cX = c . Since | i √2 | i | i Np(u,v) | i | i (u,v) define p(u, v), and using the fact that 1 p(u, v) = dep 1 N + + + + − (φ)= c† (cφc†)c c c , (26) φ (I p(u,v))( φ φ ) φ , we have T(u,v) N(u,v) ⊗ | ih | h | ⊗ N | ih | | i c e + + + + |C| X∈C 1 p(u, v) = φ (I )( φ φ ) φ b b − h | ⊗ N(u,v) | ih | | i by an identical argument we also get that + + + + = φ I A+ φ φ I A+† φ dep dep h | ⊗ | ih | ⊗ | i (XφX) = (X V ) (φ)(X V †). (27) + + + + T(u,v) ⊗ T(u,v) ⊗ + φ I A φ φ I A† φ . h | ⊗ −| ih | ⊗ −| i Thus, we have b b Using the fact that S dep (φ) = S dep (XφX) (28) (u,v) (u,v) + + + + 1 1 T T φ I A φ φ I A† φ = Tr A Tr A†  dep   dep  h | ⊗ | ih | ⊗ | i 2 2 S (u,v)(φ) = S (u,v)(XφX) , (29)     T T and A is traceless then gives c dep c dep   − so that I ( (u,vb), φ)= I ( (u,v),XφXb ), and similarly for Y 2 T T c dep 1 and Z. Using the concavity of I ( , φ) in φ, this gives us 1 p(u, v) = Tr A+ (38) T(u,v) − 2   Ic( dep , φ) (30) 1 2 T(u,v) = (cos((v u)/2) + cos((v + u)/2)) 1 c dep 2 − = I ( , P φP †) (31)   4 T(u,v) = cos2(v/2)cos2(u/2). P X∈P c dep 1 I , P φP † (32) ≤ T(u,v) 4 Proof: (of Theorem 6) Let 1 p(u, v) = P ! 2 2 dep − X∈P cos (u/2)cos (v/2), and be the channel described in 1 T(u,v) = Ic dep , I , (33) Eq. (24). Then, by Lemma 4, dep (ρ) is degradable, and T(u,v) 2 T(u,v)   by Lemma 9 tracing over the flag system degrades dep to where we have let = I,X,Y,Z denote the Pauli ma- T(u,v) . Thus, dep is a degradable extension of . trices. This shows thatP the{ maximum} coherent information is p(u,v) (u,v) p(u,v) NAs a result, by TheoremT 3 we have N achieved for the reference state I/2, where its value is (1) dep 1 Q( p(u,v)) Q ( (u,v)), (39) Ic dep , I = S( (I/2)) S( (I/2)) N ≤ T T(u,v) 2 N(u,v) − N(u,v)   whereas by Lemma 8 1 = H [1 + sin u sin v]b 1 1 2 Q(1)( dep )=H [1+ sin u sin v] H [1+ cos u cos v] .   T(u,v) 2 − 2 1     H [1 + cos u cos v] . (34) Furthermore, it was shown in [15] that becomes anti- − 2 Np   degradable when p =1/4, so that Q( 1/4)=0. Since both of these bounds are the resultN of arguments via a The following lemma shows that dep can be degraded to (u,v) degradable extension, their convex hull is also an upper bound a depolarizing channel, and computesT the error probability of for the capacity of p, which completes the proof. that channel as a function of u and v. N B. The BB84 Channel Lemma 9 Bennett-Brassard [12] is the most Tr dep (ρ)= (ρ), (35) 2 T(u,v) Np(u,v) widely studied and practically applied form of quantum cryp- where p(u, v)=1 cos2(u/2)cos2(v/2). tography. A simple bound on the achievable key rate is there- − fore quite useful. Here we evaluate the coherent information Proof: First, note that for any channel , it was shown sharable though a degradable extension of the BB84 channel, in [14] that N which also bounds the secret key rate of this protocol. 1.0 Proof: This follows, via Theorem 3, from the fact BB84 BB84 that q is a degradable extension of q together 0.8 withT Lemma 11, which evaluates this extension’sN coherent information. 0.6 Lemma 11 0.4 1 Q(1)( BB84)= H 2q(1 q) H(2q(1 q)) (45) Tq 2 − − − − 0.2   Proof: We would like to evaluate

0.05 0.10 0.15 0.20 0.25 c BB84 max I ( q , φ). (46) φ T Fig. 1. Bounds on the quantum capacity of the depolarizing channel with error probability p. The horizontal axis is the error probability, and the vertical For any φ, we have axis is the rate. The lowest line is the achivable rate using hashing [14]. The top line is the minimum of − H p [19] and − p [14]. The middle 1 ( ) 1 4 c BB84 c BB84 line is our new bound from Corollary 7. While the bound from Theorem 6 I ( q , φ)= I ( q , Y φY ) (47) is tighter everywhere, it, the bound plotted, and the bound from [10] all look T T essentially identical on this scale. so that, using the concavity of Ic in φ for degradable channels, we have 1 1 Define the following degradable channel Ic BB84, φ Ic BB84, φ + Y φY (48) Tq ≤ Tq 2 2   BB84 1 ad 1 ad  = (ρ) 0 0 + Y (Y ρY )Y 1 1 , As a result, we may take the optimal φ to be of the form Tq 2Nγ(q) ⊗ | ih | 2 Nγ(q) ⊗ | ih | ad 1 α with γ(q) =4q(1 q), and γ the amplitude damping φα = I + Y. (49) channel with Kraus operators− N 2 2 Now, 1 0 A0 = (40) c BB84 0 √1 γ I ( q , φα)= S 4q(1 q)(φα) S 1 4q(1 q)(φα) ,  −  T N − − N − − 0 √γ A = . (41) which can be written more explicitly as  1 0 0   1 ad H 1 γ2 + α2(1 γ) (50) The channel γ is a special case of (u,v) with u = v = 2 − − cos 1(√1 γN), which is degradable asN long as γ 1/2 [20].   − 1  p  By tracing− out the flag system, we find that ≤ H 1 (1 γ)2 + α2γ , − 2 − −   BB84 2  p  Tr2 q (ρ) = (1 q) ρ + q(1 q)XρX c BB84 c BB84 T − − from which we see that I ( q , φα) = I ( q , φ α). 2 +q ZρZ + q(1 q)Y ρY. Using the concavity of Ic again,T we see T − − With suitable unitary rotations on the input and output, this 1 I ( BB84) = Ic( BB84, I) (51) can be transformed to Tq Tq 2 c BB84 I ( q , φα), (52) BB84(ρ) (1 q)2ρ + q(1 q)XρX (42) ≥ T Nq ≡ − − +q(1 q)ZρZ + q2Y ρY. (43) and evaluating Eq. (50) for α =0 gives the result. − The channel BB84 is such that its private classical capacity V. ADDITIVE EXTENSIONS AND SYMMETRIC ASSISTANCE Nq is equal to the maximal achievable key rate in BB84 with There is an entertaining connection to the capacity of a one-way postprocessing and quantum bit error rate q. Since quantum channel with symmetric assistance [10]. We first BB84 is a degradable extension of an equivalent channel, its briefly summarize the main finding of [10], using slightly Tq coherent information provides an upper bound on the key rate more streamlined notation. Letting ′ = span (i, j) i

1 (1) n n Q( ) = lim Q ( ⊗ ⊗ ) (55) n N ⊗ A →∞ n N ⊗ A 1 (1) n (1) (n 1) = lim Q ( ⊗ )+ Q ( ⊗ − ) (56) n n N ⊗ A A ⊗ A →∞  1  = lim (nQ(1)( )) = Q(1)( ) . (57) n →∞ n N ⊗ A N ⊗ A Therefore is an additive extension of for any . JAS thanksN ⊗ ARO A contract DAAD19-01-C-0056.N N

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