polar codes for classical-quantum channels

IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 2, FEBRUARY 2013 1175

Polar Codes for Classical-Quantum ChannelsMark M. Wilde, Member, IEEE, and Saikat Guha

Abstract—Holevo, Schumacher, and Westmoreland’s codingtheorem guarantees the existence of codes that are capacity-achieving for the task of sending classical data over a channelwith classical inputs and quantum outputs. Although they demon-strated the existence of such codes, their proof does not provide anexplicit construction of codes for this task. The aim of this paperis to fill this gap by constructing near-explicit “polar” codes thatare capacity-achieving. The codes exploit the channel polarizationphenomenon observed by Arikan for the case of classical channels.Channel polarization is an effect in which one can synthesize a setof channels, by “channel combining” and “channel splitting,” inwhich a fraction of the synthesized channels are perfect for datatransmission, while the other channels are completely useless fordata transmission, with the good fraction equal to the capacity ofthe channel. The channel polarization effect then leads to a simplescheme for data transmission: send the information bits throughthe perfect channels and “frozen” bits through the useless ones.The main technical contributions of this paper are threefold. First,we leverage several known results from the quantum informationliterature to demonstrate that the channel polarization effectoccurs for channels with classical inputs and quantum outputs.We then construct linear polar codes based on this effect, and theencoding complexity is , where is the blocklengthof the code. We also demonstrate that a quantum successive can-cellation decoder works well, in the sense that the word error ratedecays exponentially with the blocklength of the code. For this lastresult, we exploit Sen’s recent “noncommutative union bound”that holds for a sequence of projectors applied to a quantum state.

Index Terms—Channel combining, channel splitting, clas-sical-quantum polar code, non-commutative union bound,quantum successive cancellation decoder.

I. INTRODUCTION

S HANNON’S fundamental contribution was to establishthe capacity of a noisy channel as the highest rate at

which a sender can reliably transmit data to a receiver [1].His method of proof exploited the probabilistic method andwas thus nonconstructive. Ever since Shannon’s contribution,researchers have attempted to construct error-correcting codesthat can reach the capacity of a given channel. Some of themost successful schemes for error correction are turbo codesand low-density parity-check codes [2], with numerical results

Manuscript received November 01, 2011; accepted June 29, 2012. Date ofpublication September 13, 2012; date of current version January 16, 2013.M. M. Wilde was supported by the MDEIE (Québec) PSR-SIIRI internationalcollaboration grant. S. Guha was supported by the DARPA Information in aPhoton (InPho) program under Contract HR0011-10-C-0159.M. M. Wilde is with the School of Computer Science, McGill University,

Montreal, QC H3A 2A7, Canada (e-mail: [email protected]).S. Guha is with the Quantum Information Processing Group, Raytheon BBN

Technologies, Cambridge, MA 02138 USA (e-mail: [email protected]).Communicated by A. Ashikhmin, Associate Editor for Coding Techniques.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TIT.2012.2218792

demonstrating that these codes perform well for a variety ofchannels. In spite of the success of these codes, there is noproof that they are capacity achieving for channels other thanthe erasure channel [3].Recently, Arikan constructed polar codes and proved that

they are capacity achieving for a wide variety of channels [4].Polar codes exploit the phenomenon of channel polarization, inwhich a simple, recursive encoding synthesizes a set of chan-nels that polarize, in the sense that a fraction of them becomeperfect for transmission while the other fraction are completelynoisy and thus useless for transmission. The fraction of thechannels that become perfect for transmission is equal to thecapacity of the channel. In addition, the complexity of both theencoding and decoding scales as , where is theblocklength of the code. Arikan developed polar codes afterstudying how the techniques of channel combining and channelsplitting affect the rate and reliability of a channel [5]. Arikanand others have now extended the methods of polar coding tomany different settings, including arbitrary discrete memory-less channels [6], source coding [7], lossy source coding [3],[8], and the multiple access channel with two senders and onereceiver [9].All of the above results are important for determining both

the limits on data transmission and methods for achievingthese limits on classical channels. The description of a classicalchannel arises from modeling the signaling alphabet, thephysical transmission medium, and the receiver measurement.If we are interested in accurately evaluating and reaching thetrue data-transmission limits of the physical channels, withan unspecified receiver measurement, and whose informationcarriers require a quantum-mechanical description, then itbecomes necessary to invoke the laws of quantum mechanics.Examples of such channels include deep-space optical chan-nels and ultra-low-temperature quantum-noise-limited RFchannels. Achieving the classical communication capacityfor such (quantum) channels often requires making collectivemeasurements at the receiver, an action for which no classicaldescription or implementation exists. The quantum-mechanicalapproach to information theory [10], [11] is not merely aformality or technicality—encoding classical information withquantum states and decoding with collective measurementson the channel outputs [12], [13] can dramatically improvedata transmission rates, for example, if the sender and receiverare operating in a low-power regime for a pure-loss opticalchannel (which is a practically relevant regime for long haulfree-space terrestrial and deep-space optical communication)[14], [15]. Also, encoding with entangled inputs to the channelscan increase capacity for certain channels [16], a superadditiveeffect which simply does not occur for classical channels.The proof of one of the most important theorems of quantum

information theory is due to Holevo [12] and Schumacher and

0018-9448/$31.00 © 2012 IEEE

1176 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 2, FEBRUARY 2013

Westmoreland [13] (HSW). They showed that the Holevo in-formation of a quantum channel is an achievable rate for clas-sical communication over it. Their proof of the HSW theorembears some similarities with Shannon’s technique (including theuse of random coding), but their main contribution was the con-struction of a quantummeasurement at the receiving end that al-lows for reliable decoding at the Holevo information rate. Sincethe proof of the HSW theorem, several researchers have im-proved the proof’s error analysis [17], and others have demon-strated different techniques for achieving the Holevo informa-tion [18]–[22]. Very recently, Giovannetti et al. proved that a se-quential decoding approach can achieve the Holevo information[23]. The sequential decoding approach has the receiver ask,through a series of dichotomic quantummeasurements, whetherthe output of the channel was the first codeword, the secondcodeword, etc. (this approach is similar in spirit to a classical“jointly typical” decoder [24]). As long as the rate of the code isless than the Holevo information, then this sequential decoderwill correctly identify the transmitted codeword with asymptot-ically negligible error probability. Sen has recently simplifiedthe error analysis of this sequential decoding approach (rathersignificantly) by introducing a “noncommutative union bound”in order to bound the error probability of quantum sequentialdecoding [25].In spite of the large amount of effort placed on proving that

the Holevo information is achievable, there has been relativelylittle work on devising explicit codes that approach the Holevoinformation rate.1 The aim of this paper is to fill this gap bygeneralizing the polar coding approach to quantum channels. Indoing so, we construct the first explicit class of linear codes thatapproach the Holevo information rate with asymptotically smallerror probability.The main technical contributions of this paper are as follows.1) We characterize rate with the symmetric Holevo infor-mation [27], [12], [10], [11] and reliability with the fi-delity [28], [29], [10], [11] between channel outputs cor-responding to different classical inputs. These parametersgeneralize the symmetric Shannon capacity and the Bhat-tacharya parameter [4], respectively, to the quantum case.We demonstrate that the symmetric Holevo informationand the fidelity polarize under a recursive channel trans-formation similar to Arikan’s [4], by exploiting Arikan’sproof ideas [4] and several tools from the quantum infor-mation literature [10]–[33].

2) The second contribution of ours is the generalizationof Arikan’s successive cancellation decoder [4] to thequantum case. We exploit ideas from quantum hypoth-esis testing [34]–[38] in order to construct the quantumsuccessive cancellation decoder, and we use Sen’s recent“noncommutative union bound” [25] in order to demon-strate that the decoder performs reliably in the limit of

1This is likely due to the large amount of effort that the quantum infor-mation community has put towards quantum error correction [26], which isimportant for the task of transmitting quantum bits over a noisy quantumchannel or for building a fault-tolerant quantum computer. Also, there mightbe a general belief that classical coding strategies would extend easily forsending classical information over quantum channels, but this is not the casegiven that collective measurements on channel outputs are required to achievethe Holevo information rate and the classical strategies do not incorporatethese collective measurements.

many channel uses, while achieving the symmetric Holevoinformation rate.

The complexity of the encoding part of our polar codingscheme is where is the blocklength of the code(the argument for this follows directly from Arikan’s [4]). How-ever, we have not yet been able to show that the complexity ofthe decoding part is (as is the case with Arikan’sdecoder [4]). Determining how to simplify the complexityof the decoding part is the subject of ongoing research. Fornow, we should regard our contribution in this paper as a moreexplicit method for achieving the Holevo information rate (ascompared to those from prior work [12], [13], [18]–[21]).Onemight naively think from a casual glance at our paper that

Arikan’s results [4] directly apply to our quantum scenario here,but this is not the case. If one were to impose single-symbol de-tection on the outputs of the quantum channels,2 such a proce-dure would induce a classical channel from input to output. Inthis case, Arikan’s results do apply in that they can attain theShannon capacity of this induced classical channel.However, the Shannon capacity of the best single-symbol de-

tection strategy may be far below the Holevo limit [14], [15].Attaining the Holevo information rate generally requires thereceiver to perform collective measurements (physical detec-tion of the quantum state of the entire codeword that may notbe realizable by detecting single symbols one at a time). Weshould stress that what we are doing in this paper is differentfrom a naive application of Arikan’s results. First, our polarcoding rule depends on a quantum parameter, the fidelity, ratherthan the Bhattacharya distance (a classical parameter). The polarcoding rule is then different from Arikan’s, and we would thusexpect a larger fraction of the channels to be “good” channelsthan if one were to impose a single-symbol measurement andexploit Arikan’s polar coding rule with the Bhattacharya dis-tance. Second, the quantum measurements in our quantum suc-cessive cancellation decoder are collective measurements per-formed on all of the channel outputs. Were it not so, then ourpolar coding scheme would not achieve the Holevo informationrate in general.We organize the rest of the paper as follows. Section II pro-

vides an overview of polar coding for classical-quantum chan-nels (channels with classical inputs and quantum outputs). Thisoverview states the main concepts and the important theorems,while saving their proofs for later in the paper. The main con-cepts include channel combining, channel splitting, channel po-larization, rate of polarization, quantum successive cancellationdecoding, and polar code performance. Section III gives moredetail on how recursive channel combining and splitting lead totransformation of rate and reliability in the direction of polariza-tion. Section IV proves that channel polarization occurs underthe transformations given in Section III (the proofs in Section IVare identical to Arikan’s [4] because theymerely exploit his mar-tingale approach). We prove in Section V that the performanceof the polar coding scheme is good, by analyzing the error prob-ability under quantum successive cancellation decoding. We fi-nally conclude in Section VI with a summary and some openquestions.

2For instance, all known conventional optical receivers are single-symbol de-tectors. They detect each modulated pulse individually, followed by classicalpostprocessing.

WILDE AND GUHA: POLAR CODES FOR CLASSICAL-QUANTUM CHANNELS 1177

II. OVERVIEW OF RESULTS

Our setting involves a classical-quantum channel with aclassical input and a quantum output :

where and is a unit trace, positive operator calleda density operator. We can associate a probability distributionand a classical label with the states and by writing thefollowing classical-quantum state [11]:

Two important parameters for characterizing any classical-quantum channel are its rate and reliability.3 We define the ratein terms of the channel’s symmetric Holevo informationwhere

is the quantum mutual information of the state ,defined as

and the von Neumann entropy of any density operatoris defined as

(Observe that the von Neumann entropy of is equal to theShannon entropy of its eigenvalues.) It is also straightforwardto verify that

The symmetric Holevo information is nonnegative by concavityof von Neumann entropy, and it can never exceed one if thesystem is a classical binary system (as is the case for the clas-sical-quantum state ). Additionally, the symmetric Holevoinformation is equal to zero if there is no correlation betweenand . It is equal to the capacity of the channel for trans-

mitting classical bits over it if the input prior distribution is re-stricted to be uniform [12], [13]. It also generalizes the sym-metric capacity [4] to the quantum setting given above.We define the reliability of the channel as the fidelity be-

tween the states and [10], [11], [28], [29]:

where is the nuclear norm of the operator :

Let denote the reliability of the channel :

The fidelity is equal to a number between zero and one, andit characterizes how “close” two quantum states are to one an-other. It is equal to zero if and only if there exists a quantum

3We are using the same terminology as Arikan [4].

measurement that can perfectly distinguish the states, and it isequal to one if the states are indistinguishable by any measure-ment [10], [11]. The fidelity generalizes the Bhattacharya pa-rameter used in the classical setting [4]. Naturally, we wouldexpect the channel to be perfectly reliable if andcompletely unreliable if . The fidelity also serves asa coarse bound on the probability of error in discriminating thestates and [37], [39].We would expect the symmetric Holevo information

if and only if the channel’s fidelityand vice versa: . The followingproposition makes this intuition rigorous, and it serves asa generalization of Arikan’s first proposition regarding therelationship between rate and reliability. We provide its proofin the Appendix.

Proposition 1: For any binary input classical-quantumchannel of the above form, the following bounds hold:

(1)

(2)

A. Channel Polarization

The channel polarization phenomenon occurs after synthe-sizing a set of classical-quantum channels

from independent copies of the classical-quantumchannel . The effect is known as “polarization” because afraction of the channels become perfect for data transmis-sion,4 in the sense that for the channels in thisfraction, while the channels in the complementary fraction be-come completely useless in the sense that in thelimit as becomes large. Also, the fraction of channels that donot exhibit polarization vanishes as becomes large. One caninduce the polarization effect by means of channel combiningand channel splitting.1) Channel Combining: The channel combining phase takes

copies of a classical-quantum channel and builds from theman -fold classical-quantum channel in a recursive way,where is any power of two: , . The zeroth levelof recursion merely sets . The first level of recursioncombines two copies of and produces the channel , de-fined as

(3)

where

Fig. 1 depicts this first level of recursion.The second level of recursion takes two copies of and

produces the channel :

(4)

4One cannot expect to transmit more than one classical bit over a perfect qubitchannel due to Holevo’s bound [27].


Fig. 1. Channel synthesized from the first level of recursion. Thick linesdenote classical systems while thin lines denote quantum systems (this is ourconvention for the other figures as well). The depicted gate acting on the channelinput is a classical controlled-NOT (CNOT) gate, where the filled-in circle actson the source bit and the other circle acts on the target bit. Its truth table is

.

Fig. 2. Second level of recursion in the channel combining phase.

where

so that

Fig. 2 depicts the second level of recursion.The operation in Fig. 2 is a permutation that takes

. One can then readily checkthat the mapping from the row vector to the channel inputsis a linear map given by with

The general recursion at the th level is to take two copies ofand synthesize a channel from them. The first part is

to transform the input sequence according to the followingrule for all :

The next part of the transformation is a “reverse shuffle”that performs the transformation:

The resulting bit sequence is the input to the two copies of.

The overall transformation on the input sequence is alinear transformation given by where

(5)

where

and is a permutation matrix known as a “bit-reversal” op-eration [4].2) Channel Splitting: The channel splitting phase consists

of taking the channels induced by the transformationand defining new channels from them. Let denote theoutput of the channel when inputting the bit sequence .We define the th split channel as follows:

(6)

where

(7)

(8)

We can also write as an alternate notation

These channels have the same interpretation as Arikan’s splitchannels [4]—they are the channels induced by a “genie-aided”quantum successive cancellation decoder, in which the thdecision measurement estimates given that the channeloutput is available, after observing the previous bitscorrectly, and if the distribution over is uniform. Thesesplit channels arise in our analysis of the error probability forquantum successive cancellation decoding.3) Channel Polarization: Our channel polarization theorem

below is similar to Arikan’s Theorem 1 [4], though ours appliesfor classical-quantum channels with binary inputs and quantumoutputs:

Theorem 2 (Channel Polarization): The classical-quantumchannels synthesized from the channel polarize, inthe sense that the fraction of indices for which

goes to the symmetric Holevo informationand the fraction for which goes tofor any as goes to infinity through powers

of two.The proof of the above theorem is identical to Arikan’s proof

with a martingale approach [4]. For completeness, we providea brief proof in Section IV.4) Rate of Polarization: It is important to characterize the

speed with which the polarization phenomenon comes into playfor the purpose of proving this paper’s polar coding theorem.We exploit the fidelity of the split channels in order tocharacterize the rate of polarization:

(9)

The theorem below exploits the exponential convergence re-sults of Arikan and Telatar [40], which improved upon Arikan’s


original convergence results [4] (note that we could also use themore general results in [41]):

Theorem 3 (Rate of Polarization): Given any clas-sical-quantum channel with , any ,and any constant , there exists a sequence of sets

with such that

Conversely, suppose that and . Then, for anysequence of sets with , thefollowing result holds:

The proof of this theorem exploits our results in Section IIIand [40, Th. 1].

B. Polar Coding

The idea behind polar coding is to exploit the polarizationeffect for the construction of a capacity-achieving code. Thesender should transmit the information bits only through thesplit channels for which the reliability parameteris close to zero. In doing so, the sender and receiver can achievethe symmetric Holevo information of the channel .1) Coset Codes: Polar codes arise from a special class of

codes that Arikan calls “ -coset codes” [4]. These -cosetcodes are given by the following mapping from the input se-quence to the channel input sequence :

where is the encoding matrix defined in (5). Suppose thatis some subset of . Then we can write the above

transformation as follows:

(10)

where denotes the submatrix of constructed fromthe rows of with indices in and denotes vector binaryaddition.Suppose that we fix the set and the bit sequence . The

mapping in (10) then specifies a transformation from the bit se-quence to the channel input sequence . This mapping isequivalent to a linear encoding for a code that Arikan calls a

-coset code where the sequence identifies thecoset. We can fully specify a coset code by the parameter vector

where is the length of the code,is the number of information bits, is a set that identifies theindices for the information bits, and is the vector of frozenbits. The polar coding rule specifies a way to choose the indicesfor the information bits based on the channel over which thesender is transmitting data.2) Quantum Successive Cancellation Decoder: The spec-

ification of the quantum successive cancellation decoder iswhat mainly distinguishes Arikan’s polar codes for classical

channels from ours developed here for classical-quantumchannels. Let us begin with a -coset code with parametervector . The sender encodes the information bitvector along with the frozen vector according to thetransformation in (10). The sender then transmits the encodedsequence through the classical-quantum channel, leading toa state , which is equivalent to a state upto the transformation . It is then the goal of the receiver toperform a sequence of quantum measurements on the statein order to determine the bit sequence . We are assumingthat the receiver has full knowledge of the frozen vectorso that he does not make mistakes when decoding these bits.Corresponding to the split channels in (6) are the fol-

lowing projectors that can attempt to decide whether the inputof the th split channel is zero or one:

where denotes the square root of a positive operator ,denotes the projector onto the positive eigenspace of a

Hermitian operator , and denotes the projection ontothe negative eigenspace of . After some calculations, we canreadily see that

(11)

(12)

where

The above observations lead to a method for a successivecancellation decoder similar to Arikan’s [4], with the followingdecoding rule:

ifif

where is the outcome of the following th measure-ment on the output of the channel (after measurementshave already been performed):

We are assuming that the measurement device outputs “0” if theoutcome occurs and it outputs “1” otherwise. (Note

that we can set if the bit is a frozen bit.)

The above sequence of measurements for the whole bit stream


corresponds to a positive operator-valued measure (POVM)where

The above decoding strategy is suboptimal in two regards.First, the decoder assumes that the future bits are unknown (andrandom) even if the receiver has full knowledge of the futurefrozen bits (this suboptimality is similar to the suboptimality ofArikan’s decoder [4]). Second, the measurement operators formaking a decision are suboptimal as well because we choosethem to be projectors onto the positive eigenspace of the dif-ference of the square roots of two density operators. The op-timal bitwise decision rule is to choose these operators to bethe Helstrom–Holevo projector onto the positive eigenspace ofthe difference of two density operators [34], [35]. Having ourquantum successive cancellation decoder operate in these twodifferent suboptimal ways allows for us to analyze its perfor-mance easily (though, note that we could just as well have usedHelstrom–Holevo measurements to obtain bounds on the errorprobability). This suboptimality is asymptotically negligible be-cause the symmetric Holevo information is still an achievablerate for data transmission even for the above choice of measure-ment operators.3) Polar Code Performance: The probability of error

for code length , number of informa-tion bits, set of information bits, and choice for thefrozen bits is as follows:

where we are assuming a particular choiceof the bits in the sequence of projectors

and the

convention mentioned before that if is a

frozen bit. We are also assuming that the sender transmits theinformation sequence with uniform probability . Theprobability of error averaged over all choicesof the frozen bits is then

(13)

One of the main contributions of this paper is the followingproposition regarding the average ensemble performance ofpolar codes with a quantum successive cancellation decoder:

Proposition 4: For any classical-quantum channel withbinary inputs and quantum outputs and any choice of ,the following bound holds:

Thus, there exists a frozen vector for each suchthat

4) Polar Coding Theorem: Proposition 4 immediately leadsto the definition of polar codes for classical-quantum channels:

Definition 5 (Polar Code): A polar code for is a -cosetcode with parameters where the information setis such that and

We can finally state the polar coding theorem for classical-quantum channels. Consider a classical-quantum channeland a real number . Let

with the information bit set chosen according to the polar codingrule in Definition 5. So is the block error proba-bility for polar coding over with blocklength , rate , andquantum successive cancellation decoding averaged uniformlyover the frozen bits .

Theorem 6 (Polar Coding Theorem): For any clas-sical-quantum channel with binary inputs and quantumoutputs, a fixed , and , the block errorprobability satisfies the following bound:

The polar coding theorem above follows as a straightforwardcorollary of Theorem 3 and Proposition 4.

III. RECURSIVE CHANNEL TRANSFORMATIONS

This section delves into more detail regarding recursivechannel combining and channel splitting. Recall the channelcombining in (3)–(5) and the channel splitting in (6). Theseallowed for us to take independent copies of a clas-sical-quantum channel and transform them into thesplit channels . We show here how to break


Fig. 3. Channels and induced from channel combining and channelsplitting. The channel with input is induced by selecting the bituniformly at random, passing both and through the encoder, and thenthrough the two channel uses. The channel with input is induced byselecting uniformly at random, copying it to another bit (via the classicalCNOT gate), sending both and through the encoder, and the outputs arethe quantum outputs and the bit .

the channel transformation into a series of single-step trans-formations. Much of the discussion here parallels Arikan’sdiscussion in [4, Sec. II and III].We obtain a pair of channels and from two inde-

pendent copies of a channel by a single-step trans-formation if it holds that

where

(14)

Also, it should hold that

where

(15)

We use the following notation to denote such a transformation:

Additionally, we choose the notation and so thatdenotes the worse channel and denotes the better channel.Fig. 3 depicts the channels and .Thus, from the above, we can write

because, by the definition in (6), we have

We can actually write more generally

(16)

which follows as a corollary to

Proposition 7: For any , , and , itholds that

(17)

(18)

with defined in (6).Proof: The proof of the above proposition is similar to the

proof of Arikan’s Proposition 3 [4].

We can justify the relationship in (16) by observing that (17)and (18) have the same form as (14) and (15) with the followingsubstitutions:

A. Transformation of Rate and Reliability

This section considers how both the rate and relia-bility evolve under the general transformation in (16).All proofs of the results in this section appear in the Appendix.

Proposition 8: Suppose that forsome channels satisfying (14) and (15). Then the following rateconservation and polarizing relations hold:

(19)

(20)

We can conclude from the above two relations that

The following proposition states how the reliability evolvesunder the channel transformation:

Proposition 9: Suppose for somechannels satisfying (14) and (15). Then

(21)

(22)

(23)

By combining (21) with (22), we observe that the reliabilityonly improves under a single-step transformation:

The above propositions for the single-step transformationlead us to the following proposition in the general case.


Proposition 10: For any classical-quantum channel ,, , and , the local transformation in (16)

preserves rate and improves reliability in the following sense:

(24)

(25)

Channel splitting moves rate and reliability “away from thecenter”:

The reliability terms satisfy

(26)

(27)

and the cumulative rate and reliability satisfy

(28)

(29)

The above proposition follows directly from Propositions7–9. The relations in (28) and (29) follow from applying (24)and (25) repeatedly.

IV. CHANNEL POLARIZATION

We are now in a position to prove Theorem 2 on channel po-larization. The idea behind the proof of this theorem is identicalto Arikan’s proof of his Theorem 1 in [4]—with the relation-ships in Propositions 8 and 9 already established, we can readilyexploit the martingale proof technique. Thus, we only provide abrief summary of the proof of Theorem 2 by following the pre-sentation in [3, Ch. 2].Consider the channel . Let denote an -bit bi-

nary expansion of the channel index and let .Then we can construct the channel by combining twocopies of according to (17) if or by com-bining two copies of according to (18) if .We repeatedly construct all the way from until with theabove rule.Arikan’s idea was to represent the channel construction as a

random birth process in order to analyze its limiting behavior.In order to do so, we let be a sequence of i.i.d.uniform Bernoulli random variables, where we define each overa probability space . Let denote the trivial -field.Also, let denote the -fields that the random vari-ables generate. We also assume that

. Let and let denote a se-quence of operator-valued random variables that forms a treeprocess where is constructed from two copies of ac-cording to (17) if and according to (18) if .

The output space of the operator-valued random variableis equal to . We are not really concerned with thechannel process but more so with the fidelities

and Holevo information . Thus, we cansimply analyze the limiting behavior of the two random pro-cesses and

. By the definitions of the random vari-ables and , it follows that

We then have the following lemma.

Lemma 11: The sequence is a boundedsuper-martingale, and the sequence is abounded martingale.

Proof: Let be a particular realization of therandom sequence . Then, the conditional expectationsatisfies

where the second equality follows from the definition ofand and Proposition 10. The proof

for similarly follows from the definitions and Proposition10. The boundedness condition follows because ,

for any classical-quantum channel with binaryinputs and quantum outputs.

We can now finally prove Theorem 2 regarding channelpolarization. Given that is a bounded martingale and

is a bounded super-martingale, the limitsand converge almost surely and in to therandom variables and . The convergence implies that

as . By the definition of theprocess , it holds that with probability , sothat

It then follows that as , whichin turn implies that . We conclude that

almost surely. Combining this result with Propo-sition 1 proves that almost surely. Finally, we havethat because isa martingale.

V. PERFORMANCE OF POLAR CODING

We can now analyze the performance under the above suc-cessive cancellation decoding scheme and provide a proof ofProposition 4. The proof of Theorem 6 readily follows by ap-plying Proposition 4 and Theorem 3.


First recall the following “noncommutative union bound” ofSen (see [25, Lemma 3]):

(30)which holds for projectors and a density operator.5 We begin by applying the above inequality to(defined in (13)):

where the second equality follows from our convention thatif is a frozen bit and the second inequality

follows from concavity of the square root. Continuing, we have

where we define

The first equality follows from exchanging the sums. Thesecond equality follows from expanding the sum and normal-

5We say that Sen’s bound is a “noncommutative union bound” becauseit is analogous to the following union bound from probability theory:

, whereare events. The analogous bound for projector logic would

be , if we thinkof as a projector onto the intersection of subspaces. The abovebound only holds if the projectors are commuting (choosing

, , and gives a counterexample). If theprojectors are noncommuting, then Sen’s bound in (30) is the next best thingand suffices for our purposes here.

ization . The third equality follows from bringing thesum inside the trace. Continuing

The first equality is from the definition in (8). The secondequality is from exchanging sums. The third equality is fromthe fact that

Continuing

The first equality is from the observations in (11) and (12) andthe definition in (6). The final inequality follows from [37,Lemma 3.2] and the definition in (9). This completes the proofof Proposition 4.We state the proof of Theorem 6 for completeness. Invoking

Theorem 3, there exists a sequence of sets with sizefor any and such that

and thus

This bound holds if we choose the set according to the polarcoding rule because this rule minimizes the above sum by defi-nition. Theorem 6 follows by combining Proposition 4 with thisfact about the polar coding rule.

VI. CONCLUSION

We have shown how to construct polar codes for chan-nels with classical binary inputs and quantum outputs, andwe showed that they can achieve the symmetric Holevo in-


formation rate for classical communication. In fact, for aquantum channel with binary pure state outputs, such as abinary-phase-shift-keyed coherent-state optical communica-tion alphabet, the symmetric Holevo information rate is theultimate channel capacity [15], which is therefore achieved byour polar code [42]. The general idea behind the constructionis similar to Arikan’s [4], but we required several technicaladvances in order to demonstrate both channel polarization atthe symmetric Holevo information rate and the operation ofthe quantum successive cancellation decoder. To prove thatchannel polarization takes hold, we could exploit several resultsin the quantum information literature [10], [11], [30]–[33] andsome of Arikan’s tools. To prove that the quantum successivecancellation decoder works well, we exploited some ideasfrom quantum hypothesis testing [34]–[38] and Sen’s recent“noncommutative union bound” [25]. The result is a near-ex-plicit code construction that achieves the symmetric Holevoinformation rate for channels with classical inputs and quantumoutputs. (When we say “near-explicit,” we mean that it stillremains open in the quantum case to determine which synthe-sized channels are good or bad.) Also, several works have nowappeared on polar coding for private classical communicationand quantum communication [43]–[47], most of which use theresults developed in this paper.One of the main open problems going forward from here is to

simplify the quantum successive cancellation decoder. Arikancould show how to calculate later estimates by exploitingthe results of earlier estimates in an “FFT-like” fashion, andthis observation reduced the complexity of the decoding to

. It is not clear to us yet how to reduce the com-plexity of the quantum successive cancellation decoder becauseit is not merely a matter of computing formulas, but rathera sequence of physical operations (measurements) that thereceiver needs to perform on the channel output systems. Ifthere were some way to perform the measurements on smallersystems and then adaptively perform other measurements basedon earlier results, then this would be helpful in demonstrating areduced complexity.Another important open question is to devise an efficient

construction of the polar codes, something that remains anopen problem even for classical polar codes. However, therehas been recent work on efficient suboptimal classical polarcode constructions [48], which one might try to extend to polarcodes for the classical-quantum channel. Finally, extending ourcode and decoder construction to a classical-quantum channelwith a nonbinary (M-ary) alphabet remains a good open linefor investigation.

APPENDIX

Proof of Proposition 1: The first bound in (1) follows fromHolevo’s characterization of the quantum cutoff rate (see [32,Proposition 1]). In particular, Holevo proved that the followinginequality holds for all :

where the entropy on the LHS is with respect to a classical-quantum state

By setting , the alphabet , and the distributionto be uniform, we obtain the bound

where the last line follows from

The other inequality in (2) follows from [33, eq. (21)]. Inparticular, they showed that

where the binary entropy. Combining this with the following

observation that holds for all gives the secondinequality:

Proof of Proposition 8: These follow from the same lineof reasoning as in the proof of Arikan’s Proposition 4 [4]. Weprove the first equality. Consider the mutual information

By the chain rule for quantum mutual information [11], we have

The inequality follows because


Thus

because [10], [11], [30]. We then have

and the inequality follows.

Proof of Proposition 9: We begin with the first equality.Consider that

The first two equalities follow by definition. The third equalityfollows from the multiplicativity of fidelity under tensor productstates [10], [11]:

The fourth equality follows from the following formula thatholds for the fidelity of classical-quantum states:

We now consider the second inequality. The fidelity also hasthe following characterization as the minimum Bhattacharyaoverlap between distributions induced by a POVM on the states[10], [11], [31]:

So

Let denote the POVM that achieves the minimum for:

Then, the POVM is a particular POVM that can tryto distinguish the states and . We then have

Making the assignments

the above expression is equal to

We can then exploit Arikan’s inequality in [4, Appendix D] tohave


The inequality follows from concavity offidelity and its multiplicativity under tensor products [10], [11]:

The inequality follows from the relationand the fact that .

ACKNOWLEDGMENT

We thank David Forney, MIT for suggesting us to try polarcodes for the quantum channel. We also thank Emre Telatar,EPFL for an intuitive tutorial on channel polarization at ISIT2011.

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MarkM.Wilde (M’99) was born in Metairie, Louisiana, USA. He received theB.S. degree in computer engineering from Texas A&MUniversity, College Sta-tion, Texas, in 2002, the M.S. degree in electrical engineering from Tulane Uni-versity, New Orleans, Louisiana, in 2004, and the Ph.D. degree in electrical en-gineering from the University of Southern California, Los Angeles, California,in 2008.Currently, he is a Postdoctoral Fellow at the School of Computer Science,

McGill University, Montreal, QC, Canada and will start in August 2013 asan Assistant Professor in the Department of Physics and Astronomy and theCenter for Computation and Technology at Louisiana State University. He haspublished over 60 articles and preprints in the area of quantum informationprocessing and is the author of the text “Quantum Information Theory,” to bepublished by Cambridge University Press. His current research interests are inquantum Shannon theory and quantum error correction.Dr. Wilde is a member of the American Physical Society and has been a re-

viewer for the IEEE TRANSACTIONS ON INFORMATION THEORY and the IEEEInternational Symposium on Information Theory.

Saikat Guha was born in Patna, India, on July 3, 1980. He received the Bach-elor of Technology degree in Electrical Engineering from the Indian Institute ofTechnology (IIT) Kanpur, India in 2002, and the S.M. (Master of Science) andPh.D. degrees in Electrical Engineering and Computer Science from the Mass-achusetts Institute of Technology (MIT), Cambridge, MA in 2004 and 2008,respectively.He is currently a Senior Scientist with Raytheon BBN Technologies, Cam-

bridge, MA, USA. Over the last four years, he has participated in several DoDfunded research programs at BBN, and is currently a principal investigator forthe DARPA Information in a Photon program, investigating attaining the ulti-mate information-theoretic limits of free-space optical communications as per-mitted by the laws of quantum physics. He is also a lead theorist on a project onnetwork communication theory, jointly funded by the US-ARL and UK-MoD.He represented India at the 29th International Physics Olympiad at Reykjavik,in July 1998, where he received the European Physical Society award for the ex-perimental component. He was a co-recipient of a NASA Tech Brief Award in2010, awarded by the NASA Inventions and Contributions Board, for his workon the phase-conjugate receiver for Gaussian-state quantum illumination. Hehas published several journal and conference articles, and holds three patents.His current research interest surrounds the application of quantum informa-tion and estimation theory to fundamental limits of optical communication andimaging. He is also interested in classical and quantum error correction, networkinformation and communication theory, and quantum algorithms.Dr. Guha is a member of the American Physical Society. He has served as

a reviewer for the IEEE TRANSACTIONS ON INFORMATION THEORY, the IEEETRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, the IEEE JOURNALOF SELECTED TOPICS IN QUANTUM ELECTRONICS, the IEEE International Sym-posium on Information Theory (ISIT) and the IEEE International Conferenceon Computer Communications (INFOCOM).

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