Structured optical receivers to attain superadditive capacity and the Holevo limit
SStructured optical receivers to attain superadditive capacity and the Holevo limit
Saikat Guha
Disruptive Information Processing Technologies group,Raytheon BBN Technologies, Cambridge, MA 02138, USA
When classical information is sent over a quantum channel, attaining the ultimate limit to channelcapacity requires the receiver to make joint measurements over long codeword blocks. For a pure-state channel, we construct a receiver that can attain the ultimate capacity by applying a single-shotunitary transformation on the received quantum codeword followed by simultaneous (but separable)projective measurements on the single-modulation-symbol state spaces. We study the ultimate limitsof photon-information-efficient communications on a lossy bosonic channel. Based on our generalresults for the pure-state quantum channel, we show some of the first concrete examples of codes andstructured joint-detection optical receivers that can achieve fundamentally higher (superadditive)channel capacity than conventional receivers that detect each modulation symbol individually.
When the modulation alphabet of a communicationchannel are quantum states, the Holevo limit is an upperbound to the Shannon capacity of the physical channelpaired with any receiver measurement. Even though theHolevo limit is an achievable capacity, the receiver ingeneral must make joint ( collective ) measurements overlong codeword blocks—measurements that can’t be real-ized by detecting single modulation symbols followed byclassical post processing. This phenomenon of a joint-detection receiver (JDR) being able to yield higher ca-pacity than any single-symbol receiver measurement, isoften termed as superadditivity of capacity.For the lossy bosonic channel, a coherent-state modu-lation suffices to attain the Holevo capacity, i.e., non-classical transmitted states do not yield any addi-tional capacity [1]. Hausladen et. al.’s square-root-measurement (SRM) [2], which in general is a positiveoperator-valued measure (POVM), applied to a randomcode gives us the mathematical construct of a receiverthat can achieve the Holevo limit. Lloyd et. al. [3] re-cently showed a receiver that can attain the Holevo ca-pacity of any quantum channel by making a sequenceof “yes/no” projective measurements on a random code-book. Sasaki et. al. [4], in a series of papers, showed sev-eral examples of superadditive capacity using pure-statealphabets and the SRM. However, the key practical ques-tions that remain unanswered are how to design modula-tion formats, channel codes, and most importantly, struc-tured realizations of Holevo-capacity-approaching JDRs.In this paper, (i) we show that the Holevo limit of apure-state channel is attained by a projective measure-ment, which can be implemented by a unitary operationon the quantum codeword followed by separable projec-tive measurements on the single-modulation-symbol sub-spaces, (ii) we translate our result into an optimal re-ceiver for the lossy bosonic channel, and (iii) we showconcrete examples of codes and receivers that yield su-peradditive capacity for optical binary-phase-shift keying(BPSK) signaling at low photon numbers. These, we be-lieve, are the first receiver realizations that can exhibitsuperaddivity, and can be tested using laboratory optics.
Attaining Holevo limit of a pure-state channel.
We encode classical information using a Q -ary modu-lation alphabet of non-orthogonal pure-state symbols in S ≡ {| ψ (cid:105) , . . . , | ψ Q (cid:105)} . Each channel use constitutes send-ing one symbol. We assume that the channel preservesthe purity of S , thus taking the states {| ψ q (cid:105)} to be thoseat the receiver. The only source of noise is the physicaldetection of the states. Assume that the receiver detectseach symbol one at a time. Channel capacity is given bythe maximum of the single-symbol mutual information, C = max { p i } max (cid:110) ˆΠ (1) j (cid:111) I (cid:16) { p i } , (cid:110) ˆΠ (1) j (cid:111)(cid:17) bits/symbol , (1)where the maximum is taken over priors { p i } over the al-phabet and a set of POVM operators (cid:110) ˆΠ (1) j (cid:111) , 1 ≤ j ≤ J on the single-symbol state-space. The measurement ofeach symbol produces one of J possible outcomes, withconditional probability P ( j | i ) = (cid:104) ψ i | ˆΠ (1) j | ψ i (cid:105) . To achievereliable communications at a rate close to C , forwarderror-correction will need to be applied on the discretememoryless channel with transition probabilities P ( j | i ).In other words, for any rate R < C , there exists a se-quence of codebooks C n with K = 2 nR codewords | c k (cid:105) ,1 ≤ k ≤ K , each codeword being an n -symbol tensorproduct of states in S , and a decoding rule, such thatthe average probability of decoding error (guessing thewrong codeword), ¯ P ( n ) e = 1 − K (cid:80) Kk =1 Pr(ˆ k = k ) → n → ∞ . In this ‘Shannon’ setting, optimal decod-ing is a maximum likelihood (ML) decision, which can inprinciple be pre-computed as a table lookup (see Fig. 1).We define C n as the maximum capacity achievable withmeasurements that jointly detect up to n symbols. Thefact that joint detection allows for C n + m > C n + C m , (or C n > C ) is referred to as superadditivity of capacity. TheHolevo-Schumacher-Westmorland (HSW) theorem says, C ∞ ≡ max { p i } S (cid:32)(cid:88) i p i | ψ i (cid:105)(cid:104) ψ i | (cid:33) = lim n →∞ C n , (2)the Holevo bound, is the ultimate capacity limit, where S (ˆ ρ ) = − Trˆ ρ log ˆ ρ is the von Neumann entropy, and a r X i v : . [ qu a n t - ph ] J a n FIG. 1: Classical communication system, shown here fora BPSK alphabet. If the receiver uses symbol-by-symboldetection, maximum capacity = C bits/symbol. If theDetection+Demodulation block is replaced by a general n -input n -output quantum measurement, maximum capacity= C n bits/symbol. Superadditivity: C ∞ > C n > C , where C ∞ is the Holevo limit. The joint-detection structure shownachieves Holevo limit for BPSK over a lossy bosonic channel. that C ∞ is achievable with joint detection over long code-word blocks. Calculating C ∞ however, doesn’t requirethe knowledge of the optimal receiver measurement. Inother words, if we replaced the detection and demodu-lation stages in Fig. 1 by one giant quantum measure-ment, then for any rate R < C ∞ , there exists a se-quence of codebooks C n with K = 2 nR codewords | c k (cid:105) ,1 ≤ k ≤ K , and an n -input n -output POVM overthe n -symbol state-space (cid:110) ˆΠ ( n ) k (cid:111) , 1 ≤ k ≤ K , suchthat the average probability of decoding error, ¯ P ( n ) e =1 − K (cid:80) Kk =1 (cid:104) c k | ˆΠ ( n ) k | c k (cid:105) →
0, as n → ∞ . Theorem 1:
For a pure-state channel, a projective mea-surement can attain C ∞ , and can be implemented as aunitary transformation on the codeword followed by aparallel set of separable single-symbol measurements. Proof:
Consider a codebook C with K = 2 nR code-words | c k (cid:105) , 1 ≤ k ≤ K , each codeword being an n -symboltensor product of states in S . The SRM (which in gen-eral is a POVM) on a random codebook can achieve theHolevo capacity [2]. However, Helstrom showed that theminimum probability of error (MPE) measurement fordiscriminating K pure states with the least average prob-ability of error is a K -element projective measurement onthe span of those K states [7]. By definition, the MPEmeasurement on C must achieve a lower average probabil-ity of error in discriminating the codewords in C than theSRM. Hence, the MPE measurement is capacity achiev-ing. In other words, given any reliable communication FIG. 2: Photon information efficiency (bits per received pho-ton) as a function of mean photon number per mode, ¯ n . threshold on the decoding error rate, P th , there exists acodebook C of a long enough length n , such that the MPEmeasurement on C described by the projectors (cid:110) ˆΠ ( n )MPE (cid:111) with ˆΠ ( n )MPE ,k = | w k (cid:105)(cid:104) w k | (where {| w (cid:105) , . . . , | w K (cid:105)} forma complete ortho-normal (CON) basis for span( C )), canattain a probability of decoding error P ( n ) e = 1 − (1 /K ) (cid:80) Kk =1 |(cid:104) c k | w k (cid:105)| ≤ P th . Now, let us define theMPE measurement on the states in S as ˆΠ (1)MPE ≡{| m q (cid:105)(cid:104) m q |} , where {| m (cid:105) , . . . , | m Q (cid:105)} is a CON basis ofthe single-symbol state space H ≡ span( S ). Let us de-fine the Kronecker-product CON basis of the n -symbolstate space H n = H ⊗ n , M ≡ {| m (cid:105) , . . . , | m Q n (cid:105)} , where | m q (cid:105) = | m q (cid:105)| m q (cid:105) . . . | m q n (cid:105) , where q k ∈ [1 , . . . , Q ], ∀ k . Let us define another CON basis of the n -symbol state space H n by extending {| w (cid:105) , . . . , | w K (cid:105)} —the vectors describing the MPE measurement on C —as W ≡ {| w (cid:105) , . . . , | w K (cid:105) , | w K +1 (cid:105) , . . . , | w Q n (cid:105)} . Fi-nally, let us extend the codebook C into the set of all Q n length- n sequences of modulation symbols C E ≡{| c (cid:105) , . . . , | c K (cid:105) , | c K +1 (cid:105) , . . . , | c Q n (cid:105)} , and express each | c k (cid:105) in both CON bases M and W , | c k (cid:105) = (cid:80) Q n j =1 (cid:104) m j | c k (cid:105)| m j (cid:105) , | c k (cid:105) = (cid:80) Q n j =1 (cid:104) w j | c k (cid:105)| w j (cid:105) , where (cid:104) m j | c k (cid:105) ≡ ( U M ) kj and (cid:104) w j | c k (cid:105) ≡ ( U W ) kj are ( k, j ) th elements of the unitarymatrices U M and U W . The bases M and W are uni-tarily equivalent. Thus, the MPE measurement on C is equivalent to first applying a unitary U on the code-word followed by the projective measurement describedby M , which is essentially n parallel (and separable)MPE measurements (cid:110) ˆΠ (1)MPE (cid:111) , on the single-symbol sub-spaces. The unitary U is given by (in the basis M ), U = Q n (cid:88) k =1 Q n (cid:88) j =1 u ∗ jk | m j (cid:105)(cid:104) m k | ; u jk = (cid:0) U − W U M (cid:1) jk . (3) Superadditive optical receivers.
Consider a single-mode lossy bosonic channel (such as a far-field single-spatial-mode free-space-optical (FSO) channel), wheredata is modulated using a succession of pulses (orthogo-nal temporal modes) with mean received photon num-ber ¯ n per mode, where each temporal mode (pulse)carries one modulation symbol. The Holevo capacityis given by, C ult (¯ n ) = g (¯ n ) = (1 + ¯ n ) log (1 + ¯ n ) − ¯ n log ¯ n bits/symbol , which is attained using a coherent-state modulation, i.e., non-classical modulation statescan’t get any higher capacity [1]. Since pure loss pre-serves coherent states (with linear amplitude attenua-tion), it suffices to define capacity as a function of themean photon number per received mode ¯ n , and the resultsderived above for a pure-state channel applies. Achiev-ing the Holevo limit requires an optimal codebook andjoint measurement on long codeword blocks. At high¯ n , symbol-by-symbol heterodyne detection asymptoti-cally achieves C ult (¯ n ). The low photon number (¯ n (cid:28) n [12]. Thereis no fundamental upper bound to the PIE; however,higher PIE necessitates lower ¯ n . Furthermore, binarymodulation and coding is sufficient to meet the Holevolimit at low ¯ n . Specifically, the binary-phase-shift key-ing (BPSK) alphabet S ≡ {| α (cid:105) , | − α (cid:105)} , | α | = ¯ n , is theHolevo-optimal binary modulation at ¯ n (cid:28)
1. Dolinarproposed a structured receiver that realizes the binaryMPE projective measurement on an a pair of coherentstates using single photon detection and coherent opti-cal feedback [5]. If the Dolinar receiver (DR) is used todetect each symbol, the BPSK channel is reduced to aclassical binary symmetric channel (BSC) with capacity C = 1 − H ( q ) bits/symbol, q = [1 − √ − e − n ] /
2. Thisis the maximum achievable capacity when the receiverdetects each symbol individually, which includes all con-ventional (direct-detection and coherent-detection) re-ceivers. The PIE C (¯ n ) / ¯ n caps out at 2 / ln 2 ≈ . n (cid:28)
1. Closed-form expressions and scal-ing behavior of C n , the maximum capacity achievablewith measurements that jointly detect up to n symbols,for n ≥ C ∞ (¯ n ) = H ([1+ e − n ] / C (¯ n ) / ¯ n and C ∞ (¯ n ) / ¯ n ,shown in Fig. 2. It is interesting to reflect on the pointshown by the orange circle (at 10 bits/photon) in Fig. 2,which says that for a 1 . µ m far-field FSO system operat-ing at 1 GHz modulation bandwidth, the laws of physicspermit reliable communications at 0 .
266 Gbps with only3 . A two-symbol superadditive JDR—
Some examplesof superadditive codes and joint measurements havebeen reported [4, 6], but no structured receiver de-signs. An ensemble (a (2 , ,
1) inner code [13]) con-taining three of the four 2-symbol BPSK states, S ≡{| α (cid:105)| α (cid:105) , | α (cid:105)| − α (cid:105) , | − α (cid:105)| α (cid:105)} , with priors (1 − p, p, p ), FIG. 3: A two-symbol JDR that attains ≈ .
5% higher capac-ity for BPSK than the best single-symbol (Dolinar) receiver. ≤ p ≤ .
5, can attain, with the best 3-element pro-jective measurement in span(S ), up to ≈ .
8% highercapacity that C [6]. Since this is a Shannon capacityresult, a classical outer code with codewords comprisingof sequences of states from S will be needed to achievethis capacity I > C . Using the MPE measurement on S (which can be analytically calculated [7], unlike thenumerically optimized projections in [6]), I /C ≈ . , ,
1] code (a beamsplitter) followed bytwo separable single-symbol measurements (in this case,a single-photon detector (SPD), and a DR) (see Fig. 3),can attain I /C ≈ . S attain C ,since the single-shot measurement that maximizes the ac-cessible information in S could in general be a 6-elementPOVM [8]. However, Theorem 1 proves that as the sizeof the inner code n → ∞ , a projective measurement onthe codebook that involves an n -mode unitary followedby a DR-array is capable of attaining C ∞ , without anyadditional outer code (see Fig. 1). An n -symbol superadditive JDR— A (2 l − , l , l − )BPSK Hadamard code, with ¯ n -mean-photons BPSKsymbols, has the same geometry (mutual inner-products)and thus is unitarily equivalent to the (2 l , l , l − ) pulse-position-modulation (PPM) code with 2 l ¯ n -mean-photonPPM pulses. The former is slightly space-efficient , sinceit achieves the same equidistant distance profile, but withone less symbol. Consider a BPSK Hadamard code de-tected by a 2 l -mode unitary transformation (with oneancilla mode prepared locally at the receiver, in the | α (cid:105) state) built using ( n log n ) / n = 2 l -element SPD-array,as shown (for n = 8) in Fig. 4(a). The beamsplitterarray is reminiscent of the fast Walsh Hadamard trans-form, if we recall that each 50-50 beamsplitter imple-ments an order-2 FFT. The beamsplitters ‘unravel’ theBPSK codebook into a PPM codebook, separating outthe photon energies in spatially separated bins, makingpossible discriminating the codewords using a SPD array.This receiver design is a more ‘natural’ choice for spatialmodulation, across, say orthogonal spatial modes of anear-field FSO channel. The ancilla mode at the receivernecessitates a local oscillator phase locked to the receivedpulses, which is hard to implement. Since the number FIG. 4: (a) The BPSK (7 , ,
4) Hadamard code is unitarilyequivalent to the (8 , ,
4) pulse-position-modulation (PPM)code via a Walsh transform built using twelve 50-50 beam-splitters. (b) Bit error rate plotted as a function of ¯ n . of ancilla modes doesn’t scale with the size of the code,we can append the ancilla mode to the transmitted code-word, so that the received ancilla can serve as a pilot tonefor our interferometric receiver. The Shannon capacityof this code-JDR pair—allowing coding over the erasureoutcome (no clicks registered at any SPD element)—is I n (¯ n ) = (log K/K )(1 − exp( − d ¯ n )) bits/symbol. InFig. 2, we plot the envelope, max n I n (¯ n ) / ¯ n (the greendotted plot), as a function of ¯ n . This JDR not only at-tains a much higher superadditive gain than the n = 2JDR we describe above, it doesn’t need phase trackingand coherent optical feedback like the DR. Note that n -ary PPM signaling also achieves the capacity I n (¯ n ) witha SPD receiver, albeit with a much higher ( × l ) peakpower as compared to BPSK. However, Theorem 1 saysthat the receiver construct shown in Fig. 1 is capable ofbridging the rest of the gap to the Holevo limit (i.e., theblue plot in Fig. 2) in conjunction with an optimal BPSKcode.
Theorem 1 applies readily to any higher-order mod-ulation format (
Q >
2, required to achieve capacity athigher ¯ n ), where the DR in the JDR structure must bereplaced by an extension of the DR that discriminates Q modulation symbols at their MPE limit (not known yet).Finally, note that in the proof of Theorem 1 , the MPEmeasurement on the single-symbol space ˆΠ (1)MPE , was justa convenient choice; any other projective measurementwould’ve worked. In the optimal JDR for BPSK there-fore, the DRs can be replaced by an array of Kennedy re-ceivers (which applies a coherent shift by − α followed by SPD—a lot simpler than the DR), since it also performsa projective measurement on span( S ). In Fig. 4(b), weplot the bit error rates P b ( E ) as a function of ¯ n for un-coded BPSK, and the [255 , , coding gain nowhas two components, a (classical) coding gain, and anadditional joint-detection gain .A great deal is known about binary codes that achievelow bit error rates on the BSC at ¯ n very close to theShannon limit [9]. It would be interesting to see howclose to the Holevo limit can these same codes perform,when paired with their respective quantum MPE mea-surements. It will be useful to design codes with symme-tries that allow them to approach Holevo capacity, withthe unitary U of the JDR in Fig. 1 realizable via a sim-ple network of beamsplitters, phase shifters, two-modesqueezers, and Kerr non-linearities (which form a uni-versal set for realizing an arbitrary multimode bosonicunitary [10]) along with a low-complexity outer code, ifat all. The fields of information and coding theory havehad a unique history. Even though many of its ultimatelimits were determined in Shannon’s founding paper [11],it took generations of magnificent coding theory research,to ultimately find practical capacity-approaching codes.Even though realizing reliable communications on an op-tical channel close to the Holevo limit might take a while,it certainly does seem to be in the visible horizon.This work was supported by the DARPA Informationin a Photon program, contract [1] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H.Shapiro, H. P. Yuen, Phys. Rev. Lett. , 027902 (2004).[2] P. Hausladen, R. Jozsa, B. Schumacher, M. Westmore-land, W. K. Wootters, Phys. Rev. A , 3 (1996).[3] S. Lloyd, V. Giovannetti, L. Macconne,arXiv:1012.0106v1 [quant-ph], (2010).[4] M. Sasaki, K. Kato, M. Izutsu, O. Hirota, Phys. Rev. A , 159 (1998).[5] S. J. Dolinar, Ph.D. thesis, MIT (1976).[6] J. R. Buck, S. J. van Enk, C. A. Fuchs, Phys. Rev. A ,032309, (2000).[7] C. W. Helstrom, Quantum Detection and EstimationTheory , Academic, New York, (1976).[8] P. Shor, arXiv:0206058v3 [quant-ph], (2003).[9] G. D. Forney,
Concatenated Codes , MIT Press, Cam-bridge, MA, (1966).[10] S. Sefi, P. Loock, arXiv:1010.0326v1 [quant-ph], (2010).[11] C. E. Shannon,