Upper Bounds on the Quantum Capacity of Some Two-Qubit Channels Yingkai Ouyang, DepartmentofCombinatoricsandOptimization 2 1 UniversityofWaterloo,InstituteofQuantumComputing, 0 200UniversityAvenueWest, Waterloo,OntarioN2L3G1,Canada 2 ([email protected]) r p A April 25, 2012 4 2 Abstract ] h Evaluating the quantum capacity of the quantum channels is an important but dif- p ficult problem, even for channels of low input and output dimension. For example, - t depolarizingchannelsareimportantquantumchannels,butdonothavetightnumerical n bounds. We prove upper bounds on the quantum capacity of our two-qubit degrad- a able amplitude damping channels and its twirled counterparts, such as two-qubit Pauli u channels andfour dimension depolarizing channels. Exploiting thenotion of covariance q [ of quantum channels, we study the structure of our two-qubit degradable amplitude damping channels, and show that their quantum capacity is equivalent to the optimal 4 value of a concave program with linear constraints. v 7 3 3 1 Introduction 2 . 6 Thequantumcapacityofaquantumchannelisthemaximumrateatwhichquantuminforma- 0 tioncanbe transmitted reliablyacrossit, givenarbitrarilymany uses ofit [1]. The quantum 1 1 capacity of many classes of channels is undetermined. This makes evaluating bounds for : the quantum capacity of certain classes of quantum channels an interesting mathematical v problem. i X However, evaluating the quantum capacity of a quantum channel is in general an infi- r a nite dimension optimization problem, and hence difficult, even for quantum channels with low dimension input and output states. Depolarizing channels are particularly easy to p,d D describe,wherepanddarethenoisestrengthpanddimensionof respectively. Depolar- p,d D izingchannels mapaddimensioninputstateto aconvexcombinationofthe maximally p,d D mixed state of dimension d and the input state. In particular, d2 1 1 d2 1 d (ρ)=ρ 1 p − + p − Dp,d − d2 d d2 (cid:18) (cid:19) (cid:18) (cid:19) where1 istheidentitymatrixofdimensiond. Upperbounds[2,3,4,5,6]andlowerbounds d [7, 8, 9, 10] on Q( ) have been studied, but bounds on Q( ) are still not tight for p,2 p,2 D D p (0,1) despite more than a decade of research. Even less is known about Q( ) for ∈ 4 Dp,d d>2. 1 There are however quantum channels for which we know how to evaluate the quantum capacity–degradableandanti-degradablequantumchannels[11]. Looselyspeaking,degrad- able channels cansimulate the environmentwhile anti-degradablechannels canbe simulated by the environment. Degradable channels can also be used to obtain upper bounds on the quantum capacity of qubit depolarizing channels [5, 6]. In particular, Smith and Smolin [6] used the entire family of degradablequbit depolarizingchannels to obtainthe currently best upper bounds on qubit depolarizing channels. Qubit amplitude damping quantum channels are examples of qubit degradable channels. Thesechannelsmodelspontaneousdecayintwo-levelquantumsystems,andhenceknowledge of their quantum capacity is a physically relevant problem [12]. An interesting fact is that thesechannelsarecovariant[13]withrespecttothediagonalPaulimatrices[14]. Thisnotion of covariance can help us simplify the evaluation of the quantum capacity of these channels. Inthispaper,westudyupperboundsonthequantumcapacityofsometwo-qubitquantum channels. Weintroducetwo-qubitamplitudedampingchannelswhichextendqubitamplitude damping channels, and obtain a simple procedure to evaluate their quantum capacity, by proving that they are covariant with respect to diagonal Pauli matrices. We also Pauli-twirl two-qubit amplitude damping channels using Smith and Smolin’s methodology [6] to obtain upper bounds on the quantum capacity of some two-qubit Pauli channels, which we state as Theorem 4. We then apply Theorem 4 to obtain upper bounds on the quantum capacity of four dimension depolarizing channels (see Corollary 2) and the quantum capacity of locally symmetric Pauli channels (see Corollary 3). The organization of this paper is as follows. In Section 2 we review the background material needed for this paper. In Section 3 we introduce a family of two-qubit amplitude damping channels and prove conditions for which they are degradable in Lemma 2. In Section 4, we discuss the relation of quantum channels that are covariant with respect to diagonal Pauli matrices and the quantum capacity of certain channels. In Theorem 1 we provethatthe quantumcapacityofadegradablequantumchannelcanbe simplifiedifitand its complementary channel are both covariant with respect to diagonal Pauli matrices. In Theorem2we provethatchannelswithKrausoperatorsthatallcanbe writtenasaproduct ofa diagonalmatrix and a Paulimatrix satisfy our requirementfor covariance. In Section5, we introduce results related to twirling of quantum channels. In particular, Lemma 6 gives anupperboundonthequantumcapacityofquantumchannelsthatareinvariantunderPauli twirling. In Section 6, we give upper bounds on the quantum capacity of some two-qubit channels. 2 Preliminaries We use the notation ‘:=’ to mean that the expression on its left hand side is defined as the expression on its right hand side. Define η(z) := zlog z where z [0,1] and η(0) := 0. Define the Pauli group on m qubits modulo phas−es, to2be := ∈1,X,Y,Z m. For all m ⊗ P , define wt(P) to be the weight of P, which is the nPumber {of qubits }on which the m op∈eraPtor P acts non-trivially on. For any V,W , define m ∈P +1 , WV=VW c[W,V]:= . (2.1) 1 , WV= VW (cid:26) − − Let D be the set of all diagonal matrices of size d. d ForacomplexseparableHilbertspace ,letB( ) be thesetofboundedlinearoperators K K mapping to . In this paper, we only deal with finite Hilbert spaces. Define the set of K K 2 quantum states on Hilbert space to be D( ) where D( ) is the set of all positive semi- definite and trace one operators inKB( ). LeKt D(Cd) BK(Cd) be the set of all dimension d quantum states. A quantum channelKΦ : B( ) ⊂B( ) is a completely positive and A B H → H trace-preserving(CPT) linear map, and can be written in its Kraus form [15] Φ(ρ)= AkρA†k, A†kAk =1dA k k X X where d =dim( ). We can also write down the action of a quantum channel Φ in terms A A of an isometry onHthe input state. Now define an isometry W :B( ) B( ) A E B H → H ⊗H W= k A . k | i⊗ k X Here k is an orthornormal set, and spans a Hilbert space that we define as the E {| i} H environment. Then observe that WρW† = |jihk|⊗AjρA†k. j,k X and that Tr (WρW )=Φ(ρ) E † H Then we can define the complementary channel ΦC :B( ) B( ) [16] as A E H → H ΦC(ρ)=Tr B(WρW†). H Sincewearefreetochoosetheorthornormalbasisoftheenvironment ,ΦC isonlydefined E up to a unitary. We use the abovedefinition as our canonicalone. LetHΦC(ρ)= µRµρRµ†. Onecancheck[17]thatthej-throwofR istheµ-throwofA . Forcompleteness,weinclude µ j P the proof here. ΦC(ρ)=Tr (WρW ) B † H =TrHB |jihk|⊗AjρA†k j,k X   = |jihk|Tr AjρA†k Xj,k (cid:16) (cid:17) = |ji hµ| AjρA†k |µihk| Xj,k Xµ (cid:16) (cid:17) =  |jihµ|Ajρ A†k|µihk|! µ j k X X X = RµρRµ†  (2.2) µ X where R = j µA . µ j| ih | j P 2.1 Quantum Capacity For a quantum channel Φ:B( ) B( ), define A B H → H I (Φ,ρ):=S(Φ(ρ)) S(ΦC(ρ)) coh − 3 and I (Φ):= max I (Φ,ρ) coh coh ρ D( A) ∈ H where S(ρ):=Tr(ρlog ρ) 2 isthevonNeumannentropyofagivendensitymatrixρ. I isafunctioncalledthecoherent coh information first introduced by [18]. It was shown by [19, 11, 20] that the quantum capacity of Φ is 1 Q(Φ)= lim I (Φ n), (2.3) coh ⊗ n n →∞ and the expression on the right hand side of (2.3) exists1. 2.2 Pauli Twirling Twirlingisaspecialwaytoobtainconvexcombinationsofasetofchannels. Givenachannel Φ : B(Cd) B(Cd) and a set of unitary operators B(Cd), the twirl of Φ with respect → U ⊂ to is an operator Φ that maps ρ to the state U U 1 Φ (ρ):= U†Φ(UρU†)U. (2.4) U |U|U X∈U When d = 2m, we may twirl channel Φ with respect to to get a channel Φˆ = Φ . We call Φˆ the Pauli twirl of Φ, and Φˆ maps an m qubit statePρmto Pm 1 Φˆ(ρ):=Φ (ρ)= PΦ(PρP)P. (2.5) Pm 22m PX∈Pm Let the set of Kraus operators of Φ be . The authors in [22] proved that the set of the Kraus operators of Φˆ is A Tr(PA)/2m 2P: P . (2.6) m  | | ∈P  sAX∈A    2.3 Degradable Channels A channel : B(Cd) B(Cd′) is degradable [16] if there exists a quantum operation Ψ N → such that Ψ = C, ◦N N that is Ψ( (ρ))= C(ρ), ρ D(Cd) N N ∀ ∈ and is antidegradable if its complementary channel C is degradable. A channel is a N T degradable extension [6] of channel if is degradable and there exists a quantum N T operation Ψ such that Ψ = . From [6] and [5] we know that if is a degradable ◦T N T extensionofsomechannel,thenQ( )=I ( ).Degradableextensionsallowustoconstruct coh T T upper bounds of quantum channels in the following way. 1seeAppendixAof[21] 4 Lemma 1 (Smith, Smolin [6]) Let ,..., :B( ) B( ) be degradable channels. 1 k A B N N H → H k Then for all non-negative λ ,...,λ such that λ = 1, there exists a quantum channel 1 k i=1 i :B( ) B( ), a degradable extension of λ , such that T HA → HB ⊗HC P i iNi k kP Q λ Q( ) λ Q( ). (2.7) i i i i N !≤ T ≤ N i=1 i=1 X X 2.4 No-Cloning Bound Cerf’sno-cloningbounds[2]canbe usedtoobtainupper boundsonthe quantumcapacityof depolarizingchannels. Inparticular,byCerf’sresult, isadegradableandanti-degradable p,d D channel when d d2 1 d2 1 d 1 p= − = − = − . (2.8) 2d+2 d2 2d(d+1) 2d 3 Two-qubit Amplitude Damping Channels Define the linear map Φa :B(C4) B(C4) → 3 Φa(ρ):= KiρK†i (3.1) i=0 X where 3 K0 = a0,i i i | ih | i=0 X K1 =a1,1 0 1 +a1,2 2 3 | ih | | ih | K2 =a2,1 0 2 +a2,2 1 3 | ih | | ih | K3 =a3,1 0 3 (3.2) | ih | where a=(a ,a ,a ,a ,a ,a ,a ,a ,a ) 0. When 0,0 0,1 0,2 0,3 1,1 1,2 2,1 2,2 3,1 ≥ a =1, a2 +a2 =1, a2 +a2 =1, a2 +a2 +a2 +a2 =1, (3.3) 0,0 0,1 1,1 0,2 2,1 0,3 1,2 2,2 3,1 then Φa is also a quantum channel. For x,y,z 0,1 2y z 0, define Φ :B(C4) B(C4) where x,y,z ≥ − − ≥ → Φ :=Φ (3.4) x,y,z (1,s1,s1,s2,√x,√y,√x,√y,√z) to be a quantum channel with Kraus operators A0 = 0 0 +s1(1 1 + 2 2)+s2 3 3 | ih | | ih | | ih | | ih | A1 =√x0 1 +√y 2 3 | ih | | ih | A2 =√x0 2 +√y 1 3 | ih | | ih | A3 =√z 0 3 (3.5) | ih | where s = √1 x,s = √1 2y z. For x,y,z 0,1 2y z 0, we call Φ a 1 2 x,y,z − − − ≥ − − ≥ two-qubit amplitude damping channel. Observe Φγ,γ(1 γ),γ2 = Φγ Φγ where Φγ is the − ⊗ 5 qubit amplitude damping channel with Kraus operators √γ 0 1 and 0 0 +√1 γ 1 1 | ih | | ih | − | ih | for γ [0,1]. ∈ 2 Let K be the family of quantum channels with Kraus operators of the form (3.2). Then we prove the following. Theorem 1 K if and only if C K. K∈ K ∈ Proof: Using the recipe of [17], if the Kraus operators of have the form of (3.2), then the K channel C has the Kraus operators K 3 K′0 =a0,0 0 0 + ai,1 i i | ih | | ih | i=1 X K =a 0 1 +a 2 3 ′1 0,1 2,2 | ih | | ih | K =a 0 2 +a 1 3 ′2 0,2 1,2 | ih | | ih | K =a 0 3 (3.6) ′3 0,3 | ih | which is also of the form (3.2). This proves the forward direction. Since ( C)C = , the K K reverse implication also holds. ⊓⊔ ThelinearmapΦ canalsobeadegradablequantumchannelforsuitablevaluesofx,y,z. x,y,z Lemma 2 Φ is adegradable channelwithdegrading map when x,y,z 0and2y+z< x,y,z G ≥ 1,x< 1. 2 Proof: Note that Φ is a quantum channel for x,y,z 0 and 2y+z < 1 which Kraus x,y,z ≥ operators given by (3.5). Also note that its complementary channel ΦC = Φ x,y,z 1 x,y,1 2y z − − − has the Kraus operators R0 = 0 0 +√x1 1 +√x2 2 +√z 3 3 | ih | | ih | | ih | | ih | R1 =√1 x0 1 +√y 2 3 − | ih | | ih | R2 =√1 x0 2 +√y 1 3 − | ih | | ih | R3 = 1 2y z 0 3. − − | ih | p Consider another amplitude damping channel =Φ with Kraus operators g,h,k G G0 = 0 0 + 1 g(1 1 + 2 2)+√1 2h k 3 3 | ih | − | ih | | ih | − − | ih | G1 =√g 0 1p+√h2 3 | ih | | ih | G2 =√g 0 2 +√h1 3 | ih | | ih | G3 =√k 0 3 | ih | where 1 2x gy g = − , h= 1 x (1 2y z) − − − z k =1 2h . (3.7) − − 1 2y z − − 6 When x,y,z 0 and 2y+z <1,x< 1, is a valid quantum operation. ≥ 2 G Wewanttofindtheconditionswhere Φ =ΦC whichmeansthat isadegrading G◦ x,y,z x,y,z G map that takes the output state of Φ to the output state of ΦC . By the Kraus x,y,z x,y,z representation, (Φx,y,z(ρ))= GkAℓρAℓ†Gk†. G k,ℓ∈{X0,1,2,3} Hence inthis representationΦC = Φ is a quantum channelwith the sixteen Kraus x,y,z G◦ x,y,z operators G A k ℓ for k,ℓ 0,1,2,3 . Now we evaluate GkAℓ. ∈{ } 1 2x G1A3 =G1A1 =0,G1A2 = − y 0 3 1 x | ih | r − 1 2x G2A3 =G2A2 =0,G2A1 = − y 0 3 1 x | ih | r − G3A3 =G3A2 =G3A1 =0. Also we have 1 2x G1A0 =√1 2x0 1 + − y 2 3 − | ih | 1 x | ih | r − 1 2x G2A0 =√1 2x0 2 + − y 1 3 − | ih | 1 x | ih | r − 1 x 2y(2 3x) G3A0 = − − − 0 3. 1 x | ih | r − Moreover xy G0A1 =√x0 1 + 2 3 | ih | 1 x| ih | r − xy G0A2 =√x0 2 + 1 3 | ih | 1 x| ih | r − G0A3 =√z 0 3. | ih | Observe then that G0A1 = x G1A0 and G0A2 = x G2A0. Thus applying the Kraus 1 2x 1 2x operators GiA0 and G0Ai isqeq−uivalent to applying qthe−Kraus operator Ri for i 1,2 . Similarly, applying the Kraus operators G1A2,G2A1 and G3A0 is equivalent to app∈lyin{g th}e Kraus operator R3. Moreover, since 1 g = x and (1 2h k)(1 2y z)= z, we have that G0A0 = R0. Therefore we have sh−own th1a−txΦx,y,z is−degr−adable−with−degrading map G when x,y,z 0 and 2y+z <1,x< 1. ≥ 2 ⊓⊔ 4 Covariant Channels 4.1 Quantum Capacity of Special Degradable Channels In this paper we work with degradable channels Φ that map m qubits to m qubits. Since the coherent information of degradable channels is additive [16], the evaluation of Q(Φ) is a 7 finite optimization problem. In particular Q(Φ)= max I (Φ,ρ). ρ D(C2m) coh ∈ The main point of this section is the following theorem, which extends Wolf and P´erez- Garc´ıa’smethod [14], and generalizesan observationmade by Giovannetti and Fazio [12] for the qubit amplitude damping channel. Theorem 1 Assume that Φ is a degradable channel. Also assume that Φ and ΦC satisfy the conditions of Lemma 3. Let D2m D2m be the set of diagonal density matrices of size 2m. ⊂ Then e Q(Φ)= max I (Φ,ρ). coh e ρ D2m ∈ Propertiesofquantumchannelscovariantwithrespecttoalocallycompactgroupwerestud- ied by Holevo in [13]. We also study properties of special covariant quantum channels. In particular,we generalize the covarianceproperty of qubit channels in equation 7 of [14], and givesufficientconditionsforaquantumchanneltobe covariantunderconjugationbycertain diagonal matrices. Lemma 3 (Covariance) Let Φ : B(C2m) B(C2m) be a linear map such that for all W , → m ∈P Φ(W)= aW,VVW (4.1) V 1,Z ⊗m ∈{X} for some aW,V C. Then for all Λ 1,Z ⊗m and for all ρ D(C2m), we have ∈ ∈{ } ∈ Φ(ΛρΛ)=ΛΦ(ρ)Λ. Proof: Φ(ΛWΛ)=( 1)c[W,V]Φ(( 1)c[W,V]ΛWΛ) − − =( 1)c[W,V] aW,VV(( 1)c[W,V]ΛWΛ) − − V 1,Z ⊗m ∈{X} = aW,VV(ΛWΛ) V 1,Z ⊗m ∈{X} =Λ aW,VVWΛ, ∵ΛV=VΛ V 1,Z ⊗m ∈{X} =Λ aW,VVW Λ   V 1,Z ⊗m ∈{X} =ΛΦ(W)Λ  (4.2) ThefirstequalityfollowsfromthelinearityofΦ,thesecondequalityfollowsbecause( 1)c[W,V]ΛWΛ= W, and the third equality follows from rearrangingthe scalar terms. − Now every m-qubit density matrix ρ can be expanded in the Pauli basis, so we can write ρ= cPP PX∈Pm 8 for cP C. Therefore ∈ Φ(ΛρΛ)=Φ Λ cPPΛ , by definition of ρ PX∈Pm ! = cPΦ(ΛPΛ) , by linearity of Φ PX∈Pm = cPΛΦ(P)Λ , by (4.2) PX∈Pm =ΛΦ cPP Λ , by linearity of Φ PX∈Pm ! =ΛΦ(ρ)Λ , by definition of ρ . (4.3) ⊓⊔ Itmaynotbe immediately obviousthatquantumchannelsofthe form(4.1)exist,andhence we provide several examples of such quantum channels as well as sufficient conditions for their existence in Section 4.2. To prove Theorem 1 we need another lemma. Lemma 4 (Invariance) Suppose that Φ and ΦC are quantum channels that satisfy the re- quirements of Lemma 3. Then for all Λ 1,Z m and for all ρ D(C2m), we have ⊗ ∈{ } ∈ I (Φ,ΛρΛ)=I (Φ,ρ). coh coh Proof: I (Φ,ΛρΛ) coh =S(Φ(ΛρΛ)) S(ΦC(ΛρΛ)) − =S(ΛΦ(ρ)Λ) S(ΛΦC(ρ)Λ) , by Lemma 3 − =S(Φ(ρ)) S(ΦC(ρ)) , by unitary invariance of S − =I (Φ,ρ). (4.4) coh ⊓⊔ Generalizing a step by [14], let : B(C2) B(C2) be the completely dephasing qubit channel with Kraus operators 1N1 and 1 Z.→For all ρ D(C2m), by decomposing ρ in the √2 √2 ∈ Pauli basis, one can verify that m(ρ)=ρ ⊗ ′ N where the diagonal entries of ρ are equal to the diagonal entries of ρ and the off-diagonal ′ entries of ρ are zero. Note that I (Φ,ρ) is a concave function of ρ for Φ degradable [23]. ′ coh Now we prove Theorem 1. 9 Proof of Theorem 1: I (Φ, m(ρ)) coh ⊗ N 1 =I Φ, AρA coh 2m † A 1,Z ⊗m ∈{X} 1   ≥2m Icoh(Φ,AρA†) ,by concavity [23] A 1,Z ⊗m ∈{X} 1 = I (Φ,ρ) ,by Lemma 4 2m coh A 1,Z ⊗m ∈{X} =I (Φ,ρ) coh Therefore, optimizing the coherent information of Φ over diagonal states suffices to obtain Q(Φ). ⊓⊔ 4.2 Examples of Covariant Channels Therearemanywell-knownexamplesofquantumchannelsoftheform(4.1). Themainresult of this subsection is Theorem 2, which gives a sufficient condition for a quantum channel to satisfy (4.1) and hence satisfy the covariance property of Lemma 3. Theorem 2 Suppose that a quantum channel Φ : B(C2m) B(C2m) has set of Kraus → operators K such that K=VKPK, K K (4.5) ∈ for some VK D2m and PK m. Then Φ satisfies the condition of (4.1). ∈ ∈P We postpone the proof of Theorem 2 to Section 4.3. Corollary 1 The linear map Φa defined by (3.2) that is a quantum channel satisfies the condition of (4.1). Proof: By Theorem 2, it suffices to show that every Kraus operator of Φ can be written in the form Ki =ViPi (4.6) for some Vi D4 and Pi 2 for all i 0,1,2,3 . We define the vectors 0 , 1 , 2 , 3 to ∈ ∈ P ∈ { } | i | i | i | i be the two qubit states 0,0 , 0,1 , 1,0 , 1,1 respectively. Let us have | i | i | i | i 3 V0 = a0,i i i, P0 =1 1 | ih | ⊗ i=0 X V1 =a1,1 0 0 a1,2 2 2, P1 =Z X | ih |− | ih | ⊗ V2 =a2,1 0 0 a2,2 1 1, P2 =X Z | ih |− | ih | ⊗ V3 = 0 0, P3 =X X. (4.7) | ih | ⊗ Using equations (A.14), (A.15), (A.12), (A.13), (A.11) in the Appendix, we can verify that (4.6) holds. ⊓⊔ 10