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DAMTP-2002-5 Countering quantum noise with supplementary classical information Jonathan Barrett Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA (January 15, 2002) Weconsider situations inwhich i) Alicewishes tosend quantuminformation toBob viaanoisy quantum channel,ii) Alice hasa classical description of thestates she wishes to send and iii) Alice can make use of a finite amount of noiseless classical information. After setting up the problem in general, we focus attention on one specific scenario in which Alice sends a known qubit down a depolarizing channel along with a noiseless cbit. We describe a protocol which we conjecture is 2 optimalandcalculatetheaveragefidelityobtained. Asurprisingamountofstructureisrevealedeven 0 for this simple case which suggests that relationships between quantum and classical information 0 could in general be very intricate. 2 n PACS number(s): 03.67.-a a J I. INTRODUCTION [6,7]. But note that (i) in general it is not possible to 7 distill a perfect singlet from a finite number of mixed 1 states, so the resulting states are still noisy [8], (ii) some Inthe study ofquantuminformationtheoryit is often 1 assumedthatclassicalinformationiseffectivelynoiseless, states only admit distillation if collective operations are allowed,thatisoperationsonmorethanonepairatonce, v free and unlimited. In this context many problems be- 3 come trivial. For example, consider a situation in which (for example this is true of Werner states [8,9]) and this 7 Alicewantsto‘teleport’[1]aquantumstate,whoseiden- maybe impracticalina givensituation andmostimpor- 0 tantly (iii) distillation itself involves the sending of cbits tity is known to her, to Bob (this has come to be known 1 and if this is expensive, distillation may not be the best as ‘remote state preparation’[2–4]). If classicalinforma- 0 option. tion is considered to be free, then no teleportation-type 2 Itmayrarelybethecasethatthesendingof(relatively 0 procedure is actually needed. Alice can simply call Bob noiseless)cbitsisexpensivecomparedwiththesendingof / on the telephone and tell him what the state is. If they h (potentiallynoisy)qubits. Evenifso,byassumingalways don’tcarehowlongthecalllaststhenBobcanconstruct p that classical information is effectively free and thereby a state arbitrarily close to Alice’s original. - t Remote state preparation becomes non-trivial if we not bothering to count it, we may miss out on interest- n ing theoretical relations between quantum and classical a wish to restrict the amount of classical information that information. u AlicecansendtoBob. Ofcourse,ifAliceandBobsharea q perfectsinglet(oroneebit),thentheycanachieveperfect : teleportation with the transmission of only two classical v i bits (cbits). But in [2], it is shown that if Alice and Bob II. THE PROBLEM X don’t mind using up a large amount of entanglement, r then in the asymptotic limit as the number of states be- With the above in mind, we consider the following a ing teleported tends to infinity, they can get away with problem. Alice and Bob are separated by a noisy quan- sending only one cbit per qubit and still retain arbitrar- tum channel. Alice sends into the channel some quan- ily good fidelity. ‘Large amount’ of entanglement means tumstate,drawnfromanensemble p , ψ (wherethe j j { | i} thattheamountneededincreasesexponentiallywiththe state ψ is drawn with probability p and Alice and j j | i number of qubits being sent. It is also shown that this Bob both know the ensemble). Alice is given a classi- exponential increase becomes a mere multiplying factor cal description of which state went into the channel and if we allow cbits to be sent from Bob to Alice. Further, canalsosendn noiselessclassicalbits whichencode part an upper bound is plotted for how many cbits must be of this information. Bob then performs some operation sent if we wish to use less than one ebit per qubit trans- on the state which he receives in an effort to undo the mitted. In [5], an optimal procedure is given for this effects of the noise. If Bob’s eventual state, given that less-than-one-ebit case. ψ was sent, is σ , then the average fidelity is given by j j Otherissueswillariseifweconsiderallquantumchan- |F¯ =i p ψ σ ψ . We wish to describe a scheme for j jh j| j| ji nels to be noisy and thus prevent the sharing of perfect AlicePand Bob which will optimize this quantity. singlets. Wemightconsider,inthecontextofsomegiven We note that one might consider an alternative sce- situation, how the transmission of extra cbits can offset nario in which Alice is trying to prepare states remotely this problem. Of course, if imperfect singlets are shared and can generate any state she wants to be sent into the then one option is always to try to distill better singlets channel. ThedifferenceliesinthefactthatAlicemay,in 1 this alternative case, generate and send a state which is the state she is sending lies in. Bob has a choice of 2n differentfromtheonewhichshewantsBobultimatelyto possible quantum operations to perform. Which one he end up with. Here we do not investigate this possibility performs is determined by the values of the n classical but concentrate only on the scenario in which the state bits. The problem is to find the scheme which leads to entering the channel is always identical with the state the maximum value for F¯. Alice wishes Bob to prepare. This would be the case if We can split this problem into two. The first part is Alice has no controloverwhat goes into the channel but to determine, for a general ensemble of quantum states, is simply given a classical description. Or if Alice sends p , ψ which undergo some noise process of the form i i { | i} the state she wantsBobto haveinto the channelexpect- ρ (ρ), where is a quantum operation: What is the →S S ing it to be noiseless and only finds out about the noise best operation to perform in order to undo this noise as later,atwhichpoint shedecides to sendsomeadditional well as possible? In other words, we wish to find an op- classical information. eration suchthat p ψ ( (ψ ψ ))ψ is maxi- T i ih i|T S | iih i| | ii We start by describing how the most general possible mized. The second pPart is to determine the best way for scheme willwork. Itis wellknownthatthe mostgeneral Alicetodividetheinitialensembleinto2nsub-ensembles, evolutionwhicha quantum state can undergo(assuming given that an answer to the first part will determine for that if measurements are performed, their results are to Bob an operation to perform on each sub-ensemble. be averaged over) corresponds to a completely positive Unfortunately, even the first of these appears to be trace-preserving map [10]. In the Kraus representation, a difficult problem in itself. Some progress is made by this can be written as: Barnum and Knill in [11], but they are concerned with maximizing entanglement fidelity and their results are ρ−→ρ′ = MiρMi†, (1) only valid for ensembles of commuting density operators X i and so are not immediately useful for our problem. For the rest of this paper, we are less ambitious. We where consider only a very simple instance of the problem in M†M =I (2) which Alice sends just one cbit and the pure states she X i i sends are qubit states drawn from a distribution which i is uniform over the Bloch sphere. The noisy quantum and I is the identity. We refer to such a map as a quan- channel is a depolarizing channel, which acts as: tum operation. Both the noise experienced by the state I as it passes down the channel and Bob’s operation take ρ αρ+(1 α) , (3) this form. Now consider that the experiment described −→ − 2 above is repeated many times - each time, Alice sends a where 0 α 1. We will see that the solution even quantum state down the noisy channel, and n classical to this se≤emin≤gly trivial problem involves a surprising bits, and Bob performs some quantum operation. Fo- amount of structure - suggesting that relationships be- cus attention on those runs of the experiment in which tween classicaland quantum informationin generalmay the classical bit-string has a certain value, say k. We well be very intricate. mayaswellregardBobasperformingthesamequantum operation on each of these runs. We use the fact that a probabilistic mixture of quantum operations is itself III. ONE QUBIT AND ONE CBIT a quantum operation. So we can stipulate without loss of generality that which quantum operation Bob applies Consider the scenario in which Alice sends to Bob a depends deterministically on the values of the n cbits. pure state drawn from a uniform distribution over the We can also stipulate without loss of generality that Bloch sphere, which gets depolarized on the way, and a the values of the n cbits sent by Alice depend determin- single noiseless classical bit. From the above, we know isticallyonwhichquantumstatesheissending. Suppose, that Alice must divide the surface of the Bloch sphere to the contrary, that a particular quantum state deter- intotwosubsets,S0 andS1,whichcorrespondtothecbit minesonlyprobabilisticallythevaluesofthecbits. Then, taking the value ‘0’ or ‘1’. We must then find, in each insteadofregardingAlice andBobasusinga probabilis- case,theoptimalquantumoperationforBobtoperform, tic scheme, one might regard them as using one from given that the depolarized qubit lies in that particular severaldeterministic schemes, with certainprobabilities. subset. Butthentheaveragefidelityobtainedwillbetheaverage We begin byassuming thatAlice divides upthe Bloch over that obtained for each of the deterministic schemes sphere in the following fashion: andwe woulddo better simply to use whicheverofthese is the best. Assumption 1 For a general state, ψ , we have that It follows from the above that we lose no generality if ψ S0 iff ψ 0 2 cos2(β/2), w|heire 0 is some | i ∈ |h | i| ≥ | i we restrict ourselves to schemes which work as follows. fixed basis state corresponding to the point (0,0,1), or The ensemble is divided up into 2n sub-ensembles. Alice the north pole, on the Bloch sphere and 0 β π/2. ≤ ≤ uses the n classical bits to tell Bob which sub-ensemble Otherwise ψ S1. | i∈ 2 We conjecture that this assumption leads to an optimal belonging to S0, but this is not allowed (such an oper- scheme (it seems very likely, for example, that in the ation is not a contraction). Bob’s operation will in fact optimal scheme the sets S0 and S1 will be simply con- consistofatranslationinthez-directionandcontractions nected and unlikely that the optimal scheme will be less parametrized by γ and δ. It is clear geometrically that symmetric than the one presented). In the rest of this in the optimum scheme, V =I, where I is the identity. section, we derive the optimal quantum operations for Our aim is now, for fixed α, β and k, to find the opti- Bobto perform, in the cases that ψ S0 and ψ S1. mumvaluesofγ andδ,consistentlywiththeirdescribing | i∈ | i∈ It is helpful to write quantum operations in a differ- a genuine quantum operation (which, recall, must corre- ent way. Suppose that a general qubit density matrix is spondto a completely positive mapon the set ofdensity written matrices). Infact,onecanshowthatcompletepositivity implies that: 1 ρ= (I +~r.~σ), (4) 2 0 k 1, (8) where ~r is a real 3-vector and ~r 1. Then, from the ≤ ≤ | | ≤ 0 δ 1 k, (9) fact that a quantum operation is linear, we can write it ≤ ≤ − in the form [10]: and ~r ~r′ =A~r+~b, (5) −→ 0 γ √1 k. (10) where A is a real 3 3 matrix and~b is a real 3-vector. ≤ ≤ − × We have also automatically included the conditions that These conditions are necessary but not sufficient. The a quantum operation must be trace-preserving and pos- actual derivation of these conditions is unenlightening, itive. The condition of complete positivity imposes fur- so we do not reproduce it here. ther constraints on A and ~b. Of course we must also It is easy to see now that Bob’s best operation will be have that ~r′ 1 ~r′. Further, we can write A in the | | ≤ ∀ characterizedby setting γ =√1 k and δ =1 k. This form U.S, where U is orthogonal (i.e. a rotation) and S − − gives: is symmetric. So we can view a quantum operation as a deformation of the Bloch sphere along principal axes A=Diag(√1 k,√1 k,1 k) (11) determined by S, followed by a rotation, followed by a − − − translation. Suppose now that ψ S0. Bob performs an oper- and | i ∈ ation characterized by A and~b. From the symmetry of the problem, it follows that the fidelity obtained (aver- ~b=(0,0,k). (12) agedoverall ψ suchthat ψ S0)isunchangedifBob performs a di|ffeirent operat|ioin,∈characterized by A′ and In fact, remarkably, this corresponds to an already well ~b′, where A′ =O(θ).A.O(θ)T and~b′ =O(θ).~b and where known quantum operation, usually described as an ‘am- O(θ) is a rotation of angle θ about the z-axis. It follows plitudedampingchannel’. Amplitudedampingisusually from this that the fidelity is also unchanged if Bob per- studiedforitsphysicalrelevance-itcorrespondstomany forms an operation characterizedby A′′ and~b′′, where: naturalphysicalprocesses. For example, it may describe anatomcoupledtoasinglemodeofelectromagneticradi- 1 2π A′′ = dθ O(θ).A.O(θ)T (6) ation undergoing spontaneous emission, or a single pho- 2π Z0 ton mode from which a photon may be scattered by a beamsplitter [10]. This suggeststhat ourschemeshould and be easily implementable experimentally. 1 2π One can run through similar arguments for the case ~b′′ = dθ O(θ).~b. (7) 2π Z0 |ψi ∈ S1. Again, it turns out that Bob’s optimal operation is essentially an amplitude damping operation, This means that without loss of generality, we can re- exceptthatinthiscase,thevector~bwillpointinthe op- strict Bob to actions of the form ~r V.A.~r+~b, where posite direction i.e. Bob’s operation will involve a trans- → A = Diag(γ,γ,δ),~b = (0,0,k) and V is a fixed rotation lationofthe Blochspheredownwards,towardsthesouth about the z-axis. From the condition that ~r′ 1 ~r′, pole, as well as some contraction. For the rest of this | | ≤ ∀ we get γ , δ , k 1. Quantum operations are contrac- paper we calculate the optimum fidelity that Bob can | | | | | |≤ tions on the Bloch sphere. achieve for a given α. For fixed β, one can optimize Recall that the qubit which Bob receives has been de- over the value of k separately for the cases ψ S0 and | i∈ polarized. We can write its density matrix in the form ψ S1 (theoptimumvalueofk maysometimesbezero | i∈ ρ=1/2(I+α~r.~σ),where ~r =1. Ideally,Bobwouldlike implying that Bob’s best operation is to do nothing). | | anoperationwhichtakesα~r ~r,atleastforthosestates One can then optimize over β. → 3 IV. ACHIEVABLE FIDELITY 2. If Alice can send to Bob one cbit but cannot use a quantum channel, then the best obtainable fi- delityis3/4(AlicetellsBob‘upper’or‘lower’hemi- After the actionofthe depolarizingchannelandBob’s sphereandBobpreparesastatewhichisspinupor quantum operation, we have that: spin down accordingly). With our scheme we have sinθcosφ F¯ > 3/4 if α > 0. Thus the quantum channel is ~r = sinθsinφ  some use for any α>0. cosθ   3. There is a kink in the graph at α 0.54. Fur- √1 kα sinθcosφ ther numericalinvestigationsrevealw≈hythis is the ~r′ = √1−kα sinθsinφ . (13) case. Denote the optimum value of β (the angle −→ − (1 k)α cosθ+k which describes how Alice is dividing up the Bloch  −  sphere) by β . Below this value of α, we have opt The averagefidelity is given by: β = π/2. At α 0.54, β suddenly jumps to opt opt ≈ 1.1 and then decreases as α increases. This is 1 2π β ∼ F¯ = dφ dθ sinθ shown in Figure 2. 4π Z0 Z0 1 1+α√1 ksin2θ+α(1 k)cos2θ+kcosθ β 2(cid:16) − − (cid:17) opt 1 2π π 3 + dφ dθ sinθ 4π Z0 Zβ 2.5 1 1+α√1 k′sin2θ+α(1 k′)cos2θ k′cosθ , 2 2(cid:16) − − − (cid:17) 1.5 (14) 1 where k′ is Bob’s quantum operation parameter in 0.5 the case that ψ S1 and is defined so that A = α | i ∈ Diag(√1 k′,√1 k′,1 k′) and~b=(0,0, k′). 0.2 0.4 0.6 0.8 1 Optimi−zing over−k, k′ −and β numerically −leads to the FIG. 2. Optimal value of β plotted against α, graph shown in Figure 1 which shows the achievable fi- whereβ characterizeshowAlicedividesuptheBloch delityforadepolarizingchannelparametrizedbyα. Also sphere (see assumption 1). Note that the graph is shown on the graph is the fidelity obtained in the case disjoint at α≈0.54. that Alice sends no cbit and Bob performs no quantum operation. 4. Intheα< 0.54region,whereβ =π/2,wecan opt calculate F¯∼analytically yielding F¯ = 3 + 1 α2 . 4 33−2α Fidelity 1 5. As α 0, Bob’s operation tends towards a simple → ‘swap’operationwhichmapsallpointsintheBloch 0.9 spheretooneofthepolesdependingonwhichhemi- 0.8 sphere the qubit lies in. 0.7 6. Ifα< 0.72,thenF¯ < 0.872andAliceandBob, ∼ ∼ iftheycan,woulddobettertouseaprotocoldueto 0.6 Gisin in which Alice sends two noiseless cbits and α no quantum information [12]. 0.2 0.4 0.6 0.8 1 FIG. 1. Average fidelity obtained for a depolariz- ingchannelparametrizedbyα. Twocasesareshown: V. CONCLUSION (i) a cbit is sent and Bob performs an ‘amplitude damping’operation(solidline)and(ii)Bobperforms no operation (dashed line). We have consideredsituations in which Alice and Bob wishto usenoiselessclassicalinformationto offsetquan- tum noise - a kind of error correction. An important We finish this section by noting some features of the feature is that Alice possesses a classical description of graph. the quantum states she wishes to send. After consider- 1. As we might expect, our scheme always yields an ing these situations in generality, we turned to consider advantage when compared with doing nothing. a very specific scenario in which Alice sends one qubit 4 which passes through a depolarizing channel accompa- [1] C. H. Bennett, G. Brassard, C. Crepeau, et al., Phys. nied by a noiseless classical bit. We described a scheme Rev. Lett.70, 1895 (1993). which we conjecture is optimal which involves Alice di- [2] C. H. Bennett, D. P. DiVincenzo, P. W. Shor, et al., vidinguptheBlochsphereasinassumption1aboveand Phys. Rev.Lett. 87, 077902 (2001). Bob performing ‘amplitude damping’ operations. Our [3] A. K. Pati, Phys. Rev.A 63, 014302 (2001). results for this scheme were obtained by brute force. [4] H-K.Lo, Phys.Rev.A 62, 012313 (2000). 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I am grateful to Trinity College, Cambridge for sup- [11] H. Barnum and E. Knill, quant-ph/0004088. port, CERN for hospitality and to the European grant [12] N. Gisin, Phys.Lett. A 210, 157 (1996). EQUIPforpartialsupport. IwouldliketothankAdrian [13] C. Fuchs,Fortschr. Phys. 46, 535 (1998). Kent and Sandu Popescu for useful discussions. 5