4718 ieee transactions on information … ieee transactions on information theory, vol. 59, no. 7,...

4718 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 59, NO. 7, JULY 2013

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

Abstract—Channel polarization is a phenomenon in which aparticular recursive encoding induces a set of synthesized channelsfrommany instances of a memoryless channel, such that a fractionof the synthesized channels becomes near perfect for data trans-mission and the other fraction becomes near useless for this task.Mahdavifar and Vardy have recently exploited this phenomenonto construct codes that achieve the symmetric private capacity forprivate data transmission over a degraded wiretap channel. In thispaper, we build on their work and demonstrate how to constructquantum wiretap polar codes that achieve the symmetric privatecapacity of a degraded quantum wiretap channel with a classicaleavesdropper. Due to the Schumacher–Westmoreland correspon-dence between quantum privacy and quantum coherence, wecan construct quantum polar codes by operating these quantumwiretap polar codes in superposition, much like Devetak’stechnique for demonstrating the achievability of the coherentinformation rate for quantum data transmission. Our schemeachieves the symmetric coherent information rate for quantumchannels that are degradable with a classical environment. Thiscondition on the environment may seem restrictive, but we showthat many quantum channels satisfy this criterion, includingamplitude damping channels, photon-detected jump channels,dephasing channels, erasure channels, and cloning channels. Ourquantum polar coding scheme has the desirable properties ofbeing channel-adapted and symmetric capacity-achieving alongwith having an efficient encoder, but we have not demonstratedthat the decoding is efficient. Also, the scheme may require en-tanglement assistance, but we show that the rate of entanglementconsumption vanishes in the limit of large blocklength if thechannel is degradable with classical environment.

Index Terms—Degradable quantum channels, quantum polarcodes, symmetric coherent information, symmetric private infor-mation.

I. INTRODUCTION

I N a seminal paper on quantum error correction, Shor setout the “goal of [defining] the quantum analog of the

Shannon capacity [41] for a quantum channel, and [finding]encoding schemes which approach this capacity” [42]. At thetime, it was not really clear how to define the quantum capacityof a quantum channel, but Shor’s quantum error correctioncode [42] gave some clues for constructing more general

Manuscript received November 05, 2011; accepted February 06, 2013. Dateof publication March 07, 2013; date of current version June 12, 2013. M. M.Wilde was supported by the MDEIE (Québec) PSR-SIIRI international collab-oration grant. S. Guha was supported by the DARPA Information in a Photon(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.2013.2250575

encoding schemes. Subsequently, several authors contributedincreasingly sophisticated quantum error correction codes [10],[18], [46] and others established a good definition of upperbounds on the quantum capacity of a channel [4], [5], [37],[38], culminating in some high-performing quantum error-cor-rection codes [25], [29], [30], [33] and random-coding-basedschemes for achieving the coherent information rate [38] of aquantum channel [11], [28], [43]. For some channels known asdegradable quantum channels [12], in which the channel to theenvironment is noisier than the channel to the intended receiver,the random-coding-based schemes from [11], [28], [43] achievetheir quantum capacity, due to the particular structure of thesechannels.In spite of the astounding progress in both quantum error

correction [13] and quantum Shannon theory [48], none ofthe high-performance codes constructed to date are provablycapacity-achieving, and none of the aforementioned schemesthat achieve the capacity are explicit (the proofs instead exploitrandomness to establish the existence of a code). Among theschemes that achieve the quantum capacity, perhaps Devetak[11] provides the most clear recipe to a quantum code designerinterested in constructing a capacity-achieving quantum code.His proof takes a cue from a certain security proof of quantumkey distribution [44] and the Schumacher–Westmoreland cor-respondence between quantum privacy and quantum coherence[39], by first establishing the existence of codes that achievethe private capacity of a quantum wiretap channel and thendemonstrating how to operate such a code in superpositionso that it achieves the quantum capacity. It is also clear thatthe structure of his codes bears some similarities with Calder-bank–Shor–Steane codes [10], [46].Along with the above developments, there have been im-

pressive breakthroughs in classical coding theory and informa-tion theory [14], one of which is Arikan’s recent work on polarcodes [2]. Arikan’s polar codes exploit a phenomenon known aschannel polarization, in which a particular recursive encodinginduces a set of synthesized channels from the original memo-ryless noisy channels. The synthesized channels are such thata fraction of them is perfect for data transmission, while theother fraction is completely useless, and the fraction that is per-fect is equal to the symmetric capacity of the original channel.The codes are channel adapted, in the sense that Arikan’s “polarcoding rule” establishes through which of the synthesized chan-nels the sender should transmit data, and this polar coding ruledepends on the particular channel being used. The codes arenear explicit and have the desirable property that both the en-coding and decoding are efficient (the complexity of each is

, where is the blocklength of the code).Arikan’s work might make us wonder whether it would be

possible to construct polar codes for transmitting quantum dataover general quantum channels, and the development in the

0018-9448/$31.00 © 2013 IEEE

WILDE AND GUHA: POLAR CODES FOR DEGRADABLE QUANTUM CHANNELS 4719

classical world most relevant for this task is due to Madhav-ifar and Vardy [31]. There, they established that a modificationof Arikan’s original polar coding scheme can achieve the sym-metric private capacity of a degraded classical wiretap channel.(In order to make this statement, the sender and receiver ac-tually require access to a small amount of secret key, but therate of secret key needed vanishes when the code’s blocklengthbecomes large.) Thus, with the Madhavifar–Vardy scheme forpolar coding over classical degraded wiretap channels [31], theDevetak scheme for operating a quantum wiretap code in su-perposition [11], and our recent work on polar codes for trans-mitting classical data over quantum channels [49], it should beevident that one could put these pieces together in order to con-struct polar codes for transmitting quantum data over degrad-able quantum channels.In this paper, we pursue this direction by constructing

polar codes that achieve a symmetric capacity for transmittingquantum data over particular degradable quantum channels.These channels should satisfy the property that encodingclassical data in some orthonormal basis at their input leadsto commuting states for the environment (essentially, the en-vironment becomes classical), and we clarify later why this isimportant in our construction. Many degradable channels fallinto this class, including amplitude damping channels [16],photon-detected jump channels [1], erasure channels [19],dephasing channels [11], and cloning channels [6] (channelsinduced by universal cloning machines [9], [17]). These noisychannels occur naturally in physical processes, with amplitudedamping modeling photon loss or spontaneous emission, thephoton-detected jump channel modeling the spontaneous decayof atoms with a detected photon emission [1], the erasurechannel being a different model for photon loss [15], thedephasing channel modeling random phase noise in super-conducting systems [7], and the cloning channel modelingstimulated emission from an atom [27], [32], [45]. Our codesare symmetric capacity-achieving for all of the above channels,and this follows from analyzing a quantum polar coding rulefor these channels.We summarize briefly how the construction works. First, we

consider a quantumwiretap channel with one classical input andtwo quantum outputs, one for the legitimate receiver (Bob) andthe other for the wiretapper (Eve). We demonstrate that thesechannels polarize in four different ways, based on whether thechannels are good or bad for the receiver or the wiretapper. Inorder to have strong security, we must guarantee that the badchannels for the wiretapper are in fact “really bad” in a pre-cise sense, and it is for this reason that we consider channelswith classical environment. (Interestingly, we know of quantumchannels where it is not clear to us how to ensure that they be-come “really bad,” and we prove some results in this directionin Appendix A.) The resulting coding scheme is to send the in-formation bits through the channels which are good for Bob andbad for Eve, “frozen” bits through the channels that are bad forboth, half of a secret key through channels which are bad forBob but good for Eve, and randomized bits through the chan-nels good for both. By an analysis similar to that of Madhavifarand Vardy [31], we can demonstrate that this scheme achievesthe symmetric private capacity of a degraded quantum wiretap

channel with classical environment, while the rate of secret keyrequired vanishes in the limit of large blocklength.The main idea for constructing quantum polar codes for

degradable quantum channels with classical environment is justto operate the quantum wiretap code in superposition and ex-ploit Arikan’s encoding with CNOT gates with respect to someorthonormal basis. This amounts to sending information qubitsthrough the channels good for the receiver Bob and bad for theenvironment Eve, frozen ancilla qubits through the channelsthat are bad for both, half of shared entanglement through thechannels that are bad for Bob but good for Eve, and superposedancilla qubits in the state through thechannels that are good for both. The resulting quantum polarcodes are entanglement assisted [8], but we can prove that theentanglement consumption rate required vanishes in the limitof large blocklength (in this regard, the codes here are similarto Hsieh et al.’s recent ones [24]). Operating the quantum suc-cessive cancellation decoder from [49] in a coherent fashion,followed by controlled “decoupling unitaries,” allows us toexploit the properties of the quantum wiretap polar code inorder to prove that the above quantum polar code performs well(we note that this decoder is similar to Devetak’s [11]). Theresulting quantum polar codes achieve the symmetric quantumcapacity of degradable channels with classical environmentwith an encoding circuit that has complexity . Thedecoding unfortunately remains inefficient, but further effortsmay lead to an efficient realization of a decoder.Recently, Renes et al. have independently constructed

quantum polar codes that have both an efficient encoding anddecoding, though they achieve the coherent information rateonly for Pauli channels [35]. We should clarify the ways inwhich their scheme is different from ours. First, they restricttheir construction to Pauli channels because they are consid-ering the effective classical channels induced in the amplitude(Pauli- ) and phase (Pauli- ) bases (not all channels, in-cluding some of the ones mentioned above, induce classicalchannels in complementary bases). As a result, they can directlyimport Arikan’s ideas because they are dealing with classicalchannels in complementary bases. An additional bonus is thatthey obtain an efficient decoder as well as an efficient encoder,essentially because their decoder is Arikan’s successive cancel-lation decoder implemented as an efficient unitary operation.Their codes require the assistance of shared entanglement, butthere are some channels for which they can prove that it is notrequired.Our scheme is different from theirs in several ways. First,

we have a quantum polar coding rule that is adapted to a givenquantum channel. In particular, the rule used for determining thegood or bad channels is based on a quantum parameter (fidelity).Also, we demonstrate polarization in terms of two quantum pa-rameters, the fidelity, and the Holevo information, by buildingon our earlier results in [49]. It is for this reason that our schemeis symmetric capacity-achieving for a wide variety of quantumchannels. Also, the first part of our decoder is a coherent ver-sion of the quantum successive cancellation decoder from [49],rather than one that is based directly on the classical decoder.Since we have not proven that the quantum successive cancel-lation decoder from [49] has an efficient implementation, the


coherent version of it in this study is certainly not efficient. Inspite of the inefficiency of the quantum successive cancellationdecoder, this decoder is needed in order to achieve the sym-metric quantum capacity of the channels considered here. Fi-nally, all the channels we consider here only require a vanishingrate of entanglement assistance, due to an argument similar to[31, Proposition 22] and the fact that they are degradable with aclassical environment.We structure this paper as follows. Section II reviews our

work from [49] on polar codes for classical-quantum channels.We present in Section III our scheme for private communi-cation over quantum wiretap channels that are degraded witha classical environment. Finally, we demonstrate how to con-struct quantum polar codes from quantum wiretap polar codesin Section IV. Section V concludes with a summary and someopen questions.

II. REVIEW OF POLAR CODES FOR CLASSICAL-QUANTUMCHANNELS

We begin by providing a brief review of polar codes con-structed for classical-quantum channels [49]. There, we consid-ered channels with binary classical inputs and quantum outputsof the form:

where denotes the channel, , and is a densityoperator. The relevant parameters that determine channel per-formance are the fidelity and the sym-metric Holevo information

with and the von Neumann entropy. Channels with are nearly

noiseless and those with are near to being com-pletely useless. The fidelity generalizes the Bhattacharya dis-tance [2] in the sense that if the twostates and commute (i.e., if the channel is classical).In coding classical information for the above channel, we

consider copies of , such that the resulting channelis of the form

where is the length input and is the output state.We can extend Arikan’s idea of channel combining to this clas-sical-quantum channel, by considering the channels induced bya transformation on an input bit (row) vector :

where , with being a bit-reversal permuta-tion matrix that reverses the order of the bits and

This classical encoding is equivalent to a network of classicalCNOT gates and permutation operations that can be imple-mented with complexity (see [2, Figs. 1–3] and[49, Figs. 1 and 2]). We can also define the split channels fromthe above combined channels as

(1)

where

(2)

(3)

The interpretation of this channel is that it is the one “seen” bythe bit if all of the previous bits are available and if weconsider all the future bits as randomized. This motivatesthe development of a quantum successive cancellation decoder[49] that attempts to distinguish from by adap-tively exploiting the results of previous measurements and Hel-strom–Holevo measurements [22], [23] for each bit decision.Arikan’s polar coding rule is to divide the channels into

“good” ones and “bad” ones. Let and be areal such that . The polar coding rule divides thechannels as follows:

(4)

(5)

so that the channels in are the good ones and thosein are the bad ones. Observe that the quantum polarcoding rule involves the quantum channel parameter , ratherthan a classical one such as the Bhattacharya distance.The following theorem is helpful in determining what frac-

tion of the channels become good or bad [3].Theorem 1 (Convergence Rate): Let be a

random process with and satisfying

(6)

(7)

where is some positive constant. Letexist almost surely with . Then, for any

:

and for any ,

One can then consider the channel combining and splittingmentioned above as a random birth process in which a channel

is constructed from two copies of a previous one ac-cording to the rules in Section IV of [49]. One can then con-sider the process andprove that it is a bounded super-martingale by exploiting the re-lationships given in [49, Proposition 10]. From the convergenceproperties of martingales, one can then conclude that con-verges almost surely to a value in , and the probabilitythat it equals zero is equal to the symmetric Holevo information

. Furthermore, since the process satisfies the relations


in (6) and (7), the following proposition on the convergence rateof polarization holds.Theorem 2: Given a binary input classical-quantum channeland any :

One of the important advances in [49] was to establish thata quantum successive cancellation decoder performs well forpolar coding over classical-quantum channels. In this case, thedecoder is some positive operator-valued measure (POVM)

that attempts to decode the information bits reliably.In particular, we showed the following bound on the perfor-mance of such a decoder (by exploiting Sen’s “noncommutativeunion bound” [40]):

under the assumption that the sender chooses the informationbits according to a uniform distribution. Thus, by choosingthe channels over which the sender transmits the informationbits to be in and those over which she transmitsagreed upon frozen bits to be in , we obtain the fol-lowing bound on the probability of decoding error:

This completes the specification of a polar code for classical-quantum channels.We end this section by stating a lemma that will prove useful

for us.Lemma 3: Let and both be binary-input classical-

quantum channels, such that is a degraded version of ,in the sense that

where is the classical input to the channels and is some de-grading quantum channel from to . Letand denote the corresponding synthesizedchannels from channel combining and splitting. Then, isdegraded with respect to for all , and furthermore,we have that and .

Proof: This lemma follows straightforwardly from the def-inition in (1), and the fact that quantum mutual information andfidelity are monotone under quantum processing with the de-grading map [48].We can then observe from the above lemma and (4) that

if is degraded with respect to , the good channels forare a subset of those that are good for :

. Similarly, the following relationship holds as well:.

III. QUANTUM WIRETAP POLAR CODES

We now discuss how to construct polar codes that achievethe symmetric private information rate for a quantum wiretapchannel. The results in this section build upon those of Mah-davifar and Vardy in [31].

The model for a binary-input quantum wiretap channel is asfollows:

where and is a density operator on a tensorproduct Hilbert space . The legitimate receiver Bob has ac-cess to the system and the eavesdropper Eve has access to thesystem . Thus, Bob’s density operator is

and Eve’s density operator is

The quantum wiretap channel is degraded if there exists somequantum channel such that the following condition holds forall :

Let denote the channel to Bob:

(8)

and let denote the channel to Eve:

(9)

In order to make a statement about the strong security of aquantumwiretap polar code, we need to ensure that the channelsover which the sender is transmitting information bits to Bobshould be “really bad” for Eve. That is, it is not sufficient for thechannels to satisfy (5), but they should be divided as to whetherthey are poor for Eve according to the following stronger crite-rion:

Dividing the channels for Eve in this way makes it nearly im-possible for her to determine whether the sender transmits a zeroor one through these channels in the limit where becomeslarge. In what follows, we say that the channels inare “bad” for Eve while those inare “good” for Eve.It is again important for us to know what fraction of the

channels become bad for Eve in order to establishthat the quantum wiretap polar codes are symmetric ca-pacity-achieving—i.e., it would be good to have anothertheorem similar to Theorem 2 for this case. In order to havesuch a theorem, we would require a birth process that obeys theproperties in (6) and (7). Fortunately, in the case that andcommute, we have the following proposition.Proposition 4: Suppose that the states and for the bi-

nary-input classical-quantum channel commute. Then, forany :

where is the process .


Proof: The proof proceeds along similar lines as Theorem3.15 in [26]. Since the states and commute, they are ef-fectively classical, and the fidelities in reduce to the clas-sical Bhattacharya parameters. It is then possible to show thatthis process satisfies

(10)

(11)

The first relation follows from [26, Lemma 3.16], and the secondfollows from [26, Lemma 2.16]. We can then rewrite the aboveconditions as follows:

(12)

(13)

Defining by , it is now clear the processsatisfies the conditions in (6) and (7). Since we know thatconverges almost surely to a random variable taking

values in with , it followsthat converges almost surely to with

. The process then satisfies all the requirementsneeded to apply Theorem 1, so that

which in turn, from the relation ,implies that

giving the statement of the proposition.It is worthwhile to discuss why we specialized the above

proposition to the case where the states and are com-muting. First, as we demonstrate in Appendix B, there are manyexamples of natural quantum channels for which this condi-tion holds, including amplitude damping channels, photon-de-tected jump channels, dephasing channels, erasure channels,and cloning channels. Thus, the quantum wiretap polar codingscheme in this section and the quantum polar coding schemein the next section work well for these channels. On the otherhand, there exist quantum wiretap channels for which the crit-ical inequality in (10) does not hold. For example, Appendix Ademonstrates a violation of the inequality whenever the statesand are pure and such that . So the

scheme given in this section does not necessarily achieve thesymmetric private capacity for such channels because it is notclear how to guarantee that the fraction of bad channels for Eveis equal to .We can now establish our scheme for a quantumwiretap polar

code. We divide the set into four different subsets:

Observe that , , , and form a partition of becausethey are all pairwise disjoint and . Thus,the set consists of channels that are good for Bob and bad forEve, has the channels that are bad for both, has the channelsthat are good for Eve and bad for Bob, and has the channelsthat are good for Eve and good for Bob. The Mahdavifar–Vardycoding scheme is then straightforward.1) Send the information bits through the channels in .2) Send the frozen bit vector through the channels in .3) Send randomized bits through the channels in .4) We suppose that Alice and Bob have access to a secret keybefore communication begins. Alice inputs her half of thesecret key into the channels in .1 Mahdavifar and Vardydemonstrated that the fraction tends to zero inthe limit [31], and a slight modification of theirargument demonstrates that the codes constructed herehave the same property. This implies that the rate of secretkey needed to ensure reliability and strong security forthis coding scheme vanishes and is thus negligible in theasymptotic limit (we require the strong security criterionfor when we produce quantum polar codes from quantumwiretap polar codes).

The following theorem guarantees that the rate of thequantum wiretap polar code is equal to the symmetric privateinformation.Theorem 5: For the quantum wiretap polar coding scheme

given above, a degraded quantum wiretap channel with andas defined in (8) and (9), with having a classical output,

and for sufficiently large , its rate converges tothe symmetric private information:

Proof: We just need to determine the size of the set .From basic set theory, we know that

Consider that

1This is a slight variation of the Mahdavifar–Vardy coding scheme that en-sures both reliability and strong security. Mahdavifar and Vardy were incon-clusive about the reliability of their coding scheme because they were unableto make any statements about the reliability of the channels in . This minorvariation with the addition of a secret key ensures security and reliability be-cause a secret key is a hybrid of a frozen bit and a randomized bit. It is similarto a frozen bit in that its value is available to Bob and thus he does not need todecode the bit channels with secret key input. It is similar to a randomized bitfrom the assumption that its value is uniform and unknown to Eve.


So it follows that

In the limit as becomes large, we know from Proposition 4that

and from Theorem 2 that

Finally, we later show that . The statementof the theorem then follows.The following theorem demonstrates that the quantum

wiretap polar coding scheme has strong security:Theorem 6: For the quantum wiretap polar coding scheme

given above, a degraded quantum wiretap channel with andas defined in (8) and (9), with having a classical output,

and for sufficiently large , it satisfies the following strong se-curity criterion:

Proof: Consider that

The first equality is from the chain rule for quantum mutualinformation and by defining to be the indices in preceding. The second equality follows from the assumption that the bitsin are chosen uniformly at random. The first inequality isfrom quantum data processing. The third equality is from thedefinition of the synthesized channels . Continuing, wehave

The first inequality is from [49, Proposition 1]. The final in-equality follows from the definition of the set .

We also know that the code has good reliability, in the sensethat there exists a POVM such that

This POVM is the quantum successive cancellation decoder es-tablished in [49]. The quantum successive cancellation decoderoperates exactly as before, but it needs to decode both the infor-mation bits in and the randomized bits in . It also exploitsthe frozen bits in and the secret key bits in to help withdecoding.Finally, we can prove that the rate of secret key bits required

by the scheme vanishes in the limit as becomes large.Proposition 7: For the quantum wiretap polar coding scheme

given above, a degraded quantum wiretap channel with and, as defined in (8) and (9), with having a classical output,

the rate vanishes as becomes large:

Proof: This result follows by an argument similar to thatfor Proposition 22 in [31], but we need to modify it slightly.We prove that the sets , , and arepairwise disjoint (note that we define the set as in(4), but with respect to the channel ). It then follows that

(14)

So we prove that these sets are disjoint. First, consider thatand are disjoint by definition because is formedfrom an intersection with . Next, observe that forsufficiently large , and are disjointby definition. Observe that and are disjoint be-cause

which follows from being a degraded version ofand Lemma 3 (also, is the complement of

). By applying Proposition 4, we know that

and from Theorem 2, we know that

Thus, the statement of the proposition follows from (14) and theabove asymptotic limits.

IV. QUANTUM POLAR CODES

From such a scheme for private classical communication overa quantum wiretap channel with classical environment, we canreadily construct a quantum polar code achieving the coherent


information of a degradable quantum channel by exploiting De-vetak’s ideas for quantum coding [11] and the recent quantumsuccessive cancellation decoder from [49]. First, recall that aquantum channel is specified by a completely positive trace-preserving map (we consider quantum channels with qubit in-puts in this work). Any suchmap has a dilation to a larger systemin which the dynamics over a tensor product space are unitary,i.e., it holds that

where is the isometric extension of the channel .The complementary channel is the map obtained by tracingover Bob’s system

Such a realization makes the quantum coding setting analogousto the quantum wiretap setting. We also define the symmetriccoherent information of the channel as

where the entropies result from sending half of a maximally en-tangled Bell state through the input of the channel. It is straight-forward to verify that

Our strategy for achieving the coherent information is to op-erate the quantum wiretap polar code in superposition (just asDevetak does [11]).We can similarly specify a quantum polar code by the param-

eter vector where these parameters areall the same as in the quantum wiretap polar code. The encoderis a coherent version of Arikan’s encoder where we replace clas-sical CNOT gates with quantum CNOT gates that act as followson a two-qubit state:

(15)

It is important to choose the orthonormal basis for the CNOT tobe the one such that the induced states for the environment com-mute. That is, consider the following induced classical-quantumchannel for the environment Eve:

where the basis is the same as in (15). The scheme in thissection works at the claimed rates if and commute (sothat we can exploit the result in Proposition 4). We prove inAppendix B that many important channels satisfy this criterion.We can now state our quantum polar coding theorem.Theorem 8 (Quantum Polar Coding): For any degradable

qubit-input quantum channel with classical environment,

there exists a quantum polar coding scheme for entanglementgeneration that achieves the symmetric coherent information,in the sense that the fidelity between the input entanglementand the generated entanglement is equal to ,where is the blocklength of the code and is some realsuch that . The scheme may require entanglementassistance, but the entanglement consumption rate vanishes inthe limit of large blocklength.

Proof: We assume that our task is merely to generate en-tanglement between Alice and Bob.2 Alice begins by preparingBell states locally on her side of the channel. Also, we assumethat Alice and Bob share a small number of ebits before commu-nication begins. We have the following structure for our code.1) Alice sends half of the locally prepared Bell states throughthe channels in .

2) Alice sends the frozen ancilla qubits through the chan-nels in .

3) Alice sends states through thechannels in (these are bits frozen in the Hadamardbasis).

4) Alice sends her shares of the ebits through the channels in.

Thus, the state before it is input to the encoder is as follows:

where Alice possesses both shares of ,Alice possesses and the superposed state

, and Alice and Bob share the entan-

gled state . We can also write the abovestate as

and we furthermore require Alice to apply some gates that re-alize some relative phases so that the above state becomes

(It is possible to realize these phases with only linear overheadin the encoding. Also, we specify how to choose these phaseslater.) Alice then applies a coherent version of Arikan’s CNOTencoder [2], leading to the following encoded state:

2The task of entanglement generation is equivalent to the task of quantumcommunication if forward classical communication from sender to receiver isavailable. Furthermore, forward classical communication does not increase thecapacity of a quantum channel [4], [48].


where are the entanglement-assisted quantum code-words, given by

with the coherent Arikan CNOT encoder acting only onAlice’s registers. Observe that the above state is encoded ina Calderbank–Shor–Steane code [10], [46] (modulo the rela-tive phases) because the inputs are either information qubits,ancillas in a fixed state or , ancillas in a state , orshares of ebits, and furthermore, the encoder consists of justSWAP gates and CNOT gates. Alice transmits the registercontaining the states over the quantum channel, leadingto the following state:

where

Bob then applies the following coherent quantum successivecancellation decoder to his systems and his half of the en-tanglement :

(16)

The POVM elements are the same as those

in the incoherent quantum successive cancellation decoder from[49]. Placing them in the above operation allows this first part ofthe decoder to be an isometry and for the code to thus operate insuperposition. We claim that the following state has fidelity

with the state after Bob applies the above coherentquantum successive cancellation decoder:

(17)

where we specify the phases later. Indeed, consider thefollowing states:

Their overlap is as follows:

We can then consider the average overlap:

where the last inequality follows from the performance of thecorresponding quantum wiretap polar code. Thus, there mustexist some phases and such that the state in (17) hasa large overlap with the state resulting from the action of thecoherent quantum successive cancellation decoder in (16) (thisfollows from [11, Lemma 4]). The state resulting from the co-herent quantum successive cancellation decoder is then close tothe state in (17), which we repeat below:

For a particular value of , the above state is


Tracing over Bob’s systems gives the following state:

We know from the quantum wiretap polar code that the fol-lowing property holds:

Thus, from the fact that relative entropy upper bounds the tracedistance (see [48, Th. 11.9.2]), we know that

where

Thus, it follows that

and from the relationship between trace distance and fidelity[48], we have that

So, for each , consider that Bob possesses the purification ofthe state or and by Uhlmann’s theorem [47], [48],there exist isometries such that

for all , where we observe that is a purification of, defined as

Thus, Bob’s final task is to apply the controlled unitary on hisstate

leading to the following state

Fig. 1. Ratio of the true quantum capacity to the symmetric coherent infor-mation as a function of the channel parameter for both the amplitude dampingchannel and the photon-detected jump channel (see Appendix B for definitionsof these channels). Observe that the ratio is close to 1 for many values of thechannel parameters.

The desired, ideal state is as follows:

and, after a straightforward calculation, the overlap between theactual state and the desired state is equal to

Bob’s final move is to discard his register in , leading to astate close to the following entangled state shared between Aliceand Bob:

Putting everything together, we have a scheme that generatesentanglement with a fidelity equal to

where the error results from two parts in the protocol: the firsterror is from the reliability of the quantum wiretap polar codeand the second is from the strong security of the quantumwiretap polar code.The above scheme achieves the symmetric coherent informa-

tion rate and has an efficient encoder, but it is unclear to us rightnow how to improve the efficiency of the two-step decoder. If


one were able to improve the efficiency of the quantum succes-sive cancellation decoder, this would lead to an efficient imple-mentation of the first part of the decoding. Improving the effi-ciency of the second part might be possible by taking into ac-count the particular structure of these quantum polar codes.As a final point, we should note that the symmetric coherent

information is equal to the quantum capacity of the quantum era-sure channel, the quantum dephasing channel, and the universalcloning machine channel. This is not the case for the amplitudedamping channel and the photon-detected jump channel, but theratio between the true quantum capacity and the symmetric co-herent information is close to one for these channels (see Fig. 1).

V. CONCLUSION

We constructed polar codes for transmitting classical dataprivately over a quantum wiretap channel and for transmittingquantum data over a quantum channel. The rates achievable arerespectively equal to the symmetric private information and thesymmetric coherent information. In both settings, we requirethat the channel to the environment be effectively classical inorder to guarantee that the rates are as claimed. The codes ex-ploit the channel polarization phenomenon, first observed byArikan in the classical setting [2] and later in [49] for clas-sical-quantum channels.Two of the most important problems left open in this paper

are to determine an efficient quantum decoder and to find otherchannels outside of the class discussed here for which the sym-metric coherent information is achievable. Progress on the firstquestion is outlined in [50], though the main question of an effi-cient decoder still remains open. The recent work of Wilde andRenes resolves the second question [36], [51], [52], by adaptingan earlier coding scheme of Renes and Boileau [34] to the polarcoding setting.One might question whether we should call the codes

developed here and in [35] “quantum polar codes,” if thecriterion for a quantum polar code is that it be symmetriccapacity-achieving, channel-adapted, and possessing efficientencoders and decoders. The codes constructed here are sym-metric capacity-achieving and channel-adapted, but do nothave efficient decoders. The codes from [35] have efficientencoders and decoders, but they are capacity-achieving andchannel-adapted only for Pauli channels. Though, both con-structions certainly take advantage of the channel polarizationeffect, which gives the code construction its name. Solving theopen problems posed here would certainly lead to quantumpolar codes that possess all desiderata.

APPENDIX AFIDELITY OF PURE STATE CQ CHANNELS REMAINS INVARIANT

UNDER CHANNEL COMBINING

In the theorem below, we demonstrate that the fidelity for the“worse” channel [49] is equal to the fidelity of the originalchannel if the original channel is a classical-quantum statewith pure state outputs. The implication is that the inequality in(10) fails for this case.

Theorem 9: Suppose we have two pure states andsuch that the classical-quantum channel outputs if zero

is input and if one is input. The fidelity between these twopure states is as follows:

The states arising from channel combining in the worse direc-tion [49] are as follows:

Then, the following relation holds:

Proof: Recall that theUhlmannfidelity is equalto themaximumsquaredoverlap betweenpurifications of and,where themaximumisoverallpurifications.Sincebothof the

above states are rank two, it suffices to consider a 2-D purifyingsystem for both. Furthermore, we can fix one purification whilevarying the other one. So, one purification of is as follows:

and a varying purification of is as follows:

where we can set

Our goal is to maximize the overlap over all le-

gitimate choices of and . The overlap is asfollows:


We set (with ) so that we have

By choosing an appropriate and such thatand , this demonstrates that the result

holds for any pure states and .

APPENDIX BEXAMPLES OF CHANNELS WITH CLASSICAL ENVIRONMENT

We prove in this section that several important degradablechannels have a classical environment, in the sense that theclassical-quantum channel induced by inputting classical or-thonormal states at the input leads to commuting output states.That is, we would like to prove that for severalimportant channels, where

is the complementary channel, and is some or-thonormal basis.We begin with the amplitude damping channel. The comple-

ment of an amplitude damping channel with damping parameterhas the following action on a qubit input [48]:

where , is a complex number such that theinput matrix is positive, and the matrix representations are withrespect to the computational basis . (The complementis effectively an amplitude damping channel with damping pa-rameter .) The result follows by observing that

Consider the photon-detected jump channel from [1], [20].The authors of [20] demonstrated that the complement of thischannel is as follows:

So it is again clear that the computational basis suffices to makethe environment outputs commute.The complement of an erasure channel with erasure param-

eter is just an erasure channel with erasure parameter [48]:

where is some erasure symbol orthogonal to the space of .Thus, any basis suffices to demonstrate that the states for theenvironment commute.A qubit dephasing channel with parameter has the following

form:

where is one of the Pauli operators. The complement of thischannel has the following form for a pure state input [48]:

Thus, choosing the input basis to be the one for which acts asa “bit-flipping” operator leads to commuting outputs for the en-vironment. For example, if , then the basis is ,while if , the basis is .Finally, the complement of the cloning channel is available in

[20]. Due to the covariance of the cloning channel and its com-plement, inputting any orthonormal basis leads to the followingstates on the output:

(18)

(19)

where is the number of clones, , andthe states form an orthonormal basis. Thus, the environ-mental states commute for these channels.

ACKNOWLEDGMENT

The authors are grateful to S. Hamed Hassani for helpful dis-cussions concerning Theorem 3 in [21] and Omar Fawzi andPatrick Hayden for useful discussions.

REFERENCES[1] G. Alber, T. Beth, C. Charnes, A. Delgado, M. Grassl, and M.

Mussinger, “Stabilizing distinguishable qubits against spontaneousdecay by detected-jump correcting quantum codes,” Phys. Rev. Lett.,vol. 86, pp. 4402–4405, May 2001.

[2] E. Arikan, “Channel polarization: A method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,”IEEE Trans. Inf. Theory, vol. 55, no. 7, pp. 3051–3073, Jul. 2009.

[3] E. Arikan and E. Telatar, “On the rate of channel polarization,” in Proc.Int. Symp. Inf. Theory, Seoul, Korea, Jun. 2009, pp. 1493–1495.

[4] H. Barnum, E. Knill, and M. A. Nielsen, “On quantum fidelitiesand channel capacities,” IEEE Trans. Inf. Theory, vol. 46, no. 4, pp.1317–1329, Jul. 2000.

[5] H. Barnum, M. A. Nielsen, and B. Schumacher, “Information trans-mission through a noisy quantum channel,” Phys. Rev. A, vol. 57, no.6, pp. 4153–4175, Jun. 1998.

[6] K. Bradler, “An infinite sequence of additive channels: The classicalcapacity of cloning channels,” IEEE Trans. Inf. Theory, vol. 57, no. 8,pp. 5497–5503, Aug. 2011.

[7] F. Brito, D. P. DiVincenzo, R. H. Koch, and M. Steffen, “Efficientone- and two-qubit pulsed gates for an oscillator-stabilized Josephsonqubit,” New J. Phys., vol. 10, no. 3, pp. 033027-1–033027-33, 2008.

[8] T. A. Brun, I. Devetak, and M.-H. Hsieh, “Correcting quantum errorswith entanglement,” Science, vol. 314, no. 5798, pp. 436–439, Oct.2006.


[9] V. Buzek and M. Hillery, “Universal optimal cloning of arbitraryquantum states: From qubits to quantum registers,” Phys. Rev. Lett.,vol. 81, pp. 5003–5006, Nov. 1998.

[10] A. R. Calderbank and P. W. Shor, “Good quantum error-correctingcodes exist,” Phys. Rev. A, vol. 54, no. 2, pp. 1098–1105, Aug. 1996.

[11] I. Devetak, “The private classical capacity and quantum capacity of aquantum channel,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 44–45,Jan. 2005.

[12] I. Devetak and P. W. Shor, “The capacity of a quantum channel forsimultaneous transmission of classical and quantum information,”Commun. Math. Phys., vol. 256, pp. 287–303, 2005.

[13] S. J. Devitt, K. Nemoto, and W. J. Munro, “Quantum error correctionfor beginners,” arXiv:0905.2794, May 2009.

[14] G. D. Forney and D. J. Costello, “Channel coding: The road to channelcapacity,” Proc. IEEE, vol. 95, no. 6, pp. 1150–1177, Jun. 2007.

[15] R. M. Gingrich, P. Kok, H. Lee, F. Vatan, and J. P. Dowling, “All linearoptical quantum memory based on quantum error correction,” Phys.Rev. Lett., vol. 91, pp. 217901-1–217901-4, Nov. 2003.

[16] V. Giovannetti and R. Fazio, “Information-capacity descriptionof spin-chain correlations,” Phys. Rev. A, vol. 71, no. 3, pp.032314-1–032314-12, Mar. 2005.

[17] N. Gisin and S. Massar, “Optimal quantum cloning machines,” Phys.Rev. Lett., vol. 79, pp. 2153–2156, Sep. 1997.

[18] D. Gottesman, “Stabilizer codes and quantum error correction,” Ph.D.dissertation, California Inst. Technol., Pasadena, CA, USA, 1997.

[19] M. Grassl, T. Beth, and T. Pellizzari, “Codes for the quantum erasurechannel,” Phys. Rev. A, vol. 56, no. 1, pp. 33–38, Jul. 1997.

[20] M. Grassl, Z. Ji, Z. Wei, and B. Zeng, “Quantum-capacity-ap-proaching codes for the detected-jump channel,” Phys. Rev. A, vol. 82,pp. 062324-1–062324-5, Dec. 2010.

[21] S. H. Hassani and R. Urbanke, “On the scaling of polar codes: I. Thebehavior of polarized channels,” in Proc. IEEE Int. Symp. Inf. Theory,Austin, TX, USA, Jun. 2010, pp. 874–878.

[22] C. W. Helstrom, “Quantum detection and estimation theory,” J. Statist.Phys., vol. 1, pp. 231–252, 1969.

[23] A. S. Holevo, “An analog of the theory of statistical decisions in non-commutative theory of probability,” Trudy Moscov Mat. Obsc., vol. 26,pp. 133–149, 1972.

[24] M.-H. Hsieh, W.-T. Yen, and L.-Y. Hsu, “High performance entangle-ment-assisted quantum LDPC codes need little entanglement,” IEEETrans. Inf. Theory, vol. 57, no. 3, pp. 1761–1769, Mar. 2011.

[25] K. Kasai, M. Hagiwara, H. Imai, and K. Sakaniwa, “Quantum errorcorrection beyond the bounded distance decoding limit,” IEEE Trans.Inf. Theory, vol. 58, no. 2, pp. 1223–1230, Feb. 2012.

[26] S. B. Korada, “Polar codes for channel and source coding,” Ph.D.dissertation, Ecole Polytechnique Federale de Lausanne, Lausanne,Switzerland, Jul. 2009.

[27] A. Lamas-Linares, C. Simon, J. C. Howell, and D. Bouwmeester, “Ex-perimental quantum cloning of single photons,” Science, vol. 296, pp.712–714, 2002.

[28] S. Lloyd, “Capacity of the noisy quantum channel,” Phys. Rev. A, vol.55, no. 3, pp. 1613–1622, Mar. 1997.

[29] Z. Luo, “Quantum error correcting codes based on privacy amplifica-tion,” arXiv:0808.1392, Aug. 2008.

[30] D. J. C. MacKay, G. Mitchison, and P. L. McFadden, “Sparse graphcodes for quantum error-correction,” IEEE Trans. Inf. Theory, vol. 50,no. 10, pp. 2315–2330, Oct. 2004.

[31] H. Mahdavifar and A. Vardy, “Achieving the secrecy capacity ofwiretap channels using polar codes,” IEEE Trans. Inf. Theory, vol. 57,no. 10, pp. 6428–6443, Oct. 2011.

[32] P. W. Milonni and M. L. Hardies, “Photons cannot always be repli-cated,” Phys. Lett. A, vol. 92, no. 7, pp. 321–322, Nov. 1982.

[33] D. Poulin, J.-P. Tillich, and H. Ollivier, “Quantum serial turbo-codes,”IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2776–2798, Jun. 2009.

[34] J. M. Renes and J.-C. Boileau, “Physical underpinnings of privacy,”Phys. Rev. A, vol. 78, pp. 032335-1–032335-12, Sep. 2008.

[35] J. M. Renes, F. Dupuis, and R. Renner, “Efficient quantum polarcoding,” Phys. Rev. Lett., vol. 109, no. 5, p. 050504, Aug. 2012.

[36] J. M. Renes and M. M. Wilde, “Polar codes for private and quantumcommunication over arbitrary channels,” arXiv:1212.2537, Dec. 2012.

[37] B. Schumacher, “Sending entanglement through noisy quantum chan-nels,” Phys. Rev. A, vol. 54, no. 4, pp. 2614–2628, Oct. 1996.

[38] B. Schumacher andM. A. Nielsen, “Quantum data processing and errorcorrection,” Phys. Rev. A, vol. 54, no. 4, pp. 2629–2635, Oct. 1996.

[39] B. Schumacher and M. D. Westmoreland, “Quantum privacy andquantum coherence,” Phys. Rev. Lett., vol. 80, no. 25, pp. 5695–5697,Jun. 1998.

[40] P. Sen, “Achieving the Han-Kobayashi inner bound for the quantuminterference channel by sequential decoding,” arXiv:1109.0802, Sep.2011.

[41] C. E. Shannon, “Mathematical theory of communication,” Bell Syst.Tech. J., vol. 27, pp. 379–423, 1948.

[42] P. W. Shor, “Scheme for reducing decoherence in quantum computermemory,” Phys. Rev. A, vol. 52, pp. R2493–R2496, Oct. 1995.

[43] P.W. Shor, “The quantum channel capacity and coherent information,”in Lecture Notes, MSRI Worksh. Quant. Comput., 2002.

[44] P. W. Shor and J. Preskill, “Simple proof of security of the BB84quantum key distribution protocol,” Phys. Rev. Lett., vol. 85, no. 2,pp. 441–444, Jul. 2000.

[45] C. Simon, G. Weihs, and A. Zeilinger, “Optimal quantum cloning viastimulated emission,” Phys. Rev. Lett., vol. 84, no. 13, pp. 2993–2996,Mar. 2000.

[46] A. M. Steane, “Error correcting codes in quantum theory,” Phys. Rev.Lett., vol. 77, no. 5, pp. 793–797, Jul. 1996.

[47] A. Uhlmann, “The transition probability in the state space of a algebra,”Rep. Math. Phys., vol. 9, no. 2, pp. 273–279, 1976.

[48] M. M. Wilde, “From classical to quantum Shannon theory,”arXiv:1106.1445, Jun. 2011.

[49] M. M. Wilde and S. Guha, “Polar codes for classical-quantum chan-nels,” IEEETrans. Inf. Theory, vol. 59, no. 2, pp. 1175–1187, Feb. 2013.

[50] M. M. Wilde, O. Landon-Cardinal, and P. Hayden, “Towards efficientdecoding of classical-quantum polar codes,” arXiv:1302.0398, Feb.2013.

[51] M. M. Wilde and J. M. Renes, “Polar codes for private classical com-munication,” in Proc. 2012 Int. Symp. Inf. Theory Appl., Honolulu, HI,USA, Oct. 2012, pp. 745–749.

[52] M.M.Wilde and J.M. Renes, “Quantum polar codes for arbitrary chan-nels,” in Proc. 2012 Int. Symp. Inf. Theory, Boston, MA, USA, Jul.2012, pp. 334–338.

Mark M. Wilde (M’99) was born in Metairie, Louisiana, USA. He receivedthe Ph.D. degree in electrical engineering from the University of SouthernCalifornia, Los Angeles, California, in 2008. He is a Postdoctoral Fellow atthe School of Computer Science, McGill University and will start in August2013 as an Assistant Professor in the Department of Physics and Astronomyand the Center for Computation and Technology at Louisiana State University.His current research interests are in quantum Shannon theory, quantum opticalcommunication, quantum computational complexity theory, and quantum errorcorrection.

Saikat Guha (M’09) was born in Patna, India, on July 3, 1980. He received thePh.D. degree in Electrical Engineering and Computer Science from the Mass-achusetts Institute of Technology (MIT), Cambridge, MA in 2008. He is cur-rently a Senior Scientist with Raytheon BBN Technologies, Cambridge, MA,USA. His current research interest surrounds the application of quantum infor-mation and estimation theory to fundamental limits of optical communicationand imaging. He is also interested in classical and quantum error correction,network information and communication theory, and quantum algorithms.

4718 ieee transactions on information … ieee transactions on information theory, vol. 59, no. 7,...

Documents