Quantum memory receiver for superadditive communication using binary coherent states
Aleksandra Klimek, Michał Jachura, Wojciech Wasilewski, Konrad Banaszek
aa r X i v : . [ qu a n t - ph ] D ec August 10, 2018 Journal of Modern Optics jmo˙super
To appear in the
Journal of Modern Optics
Vol. 00, No. 00, 00 Month 20XX, 1–9
Quantum memory receiver for superadditive communicationusing binary coherent states
Aleksandra Klimek, Micha l Jachura, Wojciech Wasilewski, and Konrad Banaszek ∗ Wydzia l Fizyki, Uniwersytet Warszawski, ul. Pasteura 5, 02-093 Warszawa, Poland ( v5.0 released January 2015 ) We propose a simple architecture based on multimode quantum memories for collective readout of classi-cal information keyed using a pair coherent states, exemplified by the well-known binary phase shift keyingformat. Such a configuration enables demonstration of the superadditivity effect in classical communi-cation over quantum channels, where the transmission rate becomes enhanced through joint detectionapplied to multiple channel uses. The proposed scheme relies on the recently introduced idea to prepareHadamard sequences of input symbols that are mapped by a linear optical transformation onto the pulseposition modulation format [Guha, S.
Phys. Rev. Lett. , , 240502]. We analyze two versions ofreadout based on direct detection and an optional Dolinar receiver which implements the minimum-errormeasurement for individual detection of a binary coherent state alphabet. Keywords: quantum memory; optical communication; quantum measurement;
1. Introduction
One of the striking consequences of the quantum nature of physical systems is the impossibility todiscriminate perfectly their states that are non-orthogonal in terms of the scalar product betweenthe corresponding state vectors [1]. This fact has profound implications for secret communicationin the form of quantum key distribution protocols [2], but it also leads to non-trivial effects whentransmission of classical information is considered [3]. In optical communication, standard schemesto encode a stream of bits employ a pair of coherent states, e.g. the vacuum state and a coherentstate with a non-zero amplitude in the case of on-off keying (OOK), or two coherent states withequal amplitudes but opposite phases in binary phase shift keying (BPSK) [4]. When an energyconstraint is imposed in the above schemes, the error rate grows with the decreasing signal power, asthe two coherent states encoding the bit value become less and less distinguishable in the quantummechanical sense. An intriguing strategy to boost throughput in such a case is to employ collectivedetection of the received signal, which for very weak signals can even qualitatively enhance thescaling of the attainable transmission rate with the mean power. The fundamental reason behindthis enhancement is that a quantum measurement provides in general only partial knowledge aboutthe state of the measured system and collective detection of multiple elementary systems can bedesigned to reveal more relevant information [5, 6].An elegant scheme to achieve superadditivity for binary phase shift keyed signals has beenrecently described by Guha [7]. The basic idea is to prepare sequences of BPSK symbols that can bemapped using a linear optical circuit onto the pulse position modulation (PPM) format. This formatcan be read out using direct detection. Moreover, with the right choice of the sequence length [8–11]this strategy approaches in the leading order the capacity of a narrowband bosonic channel for low ∗ Corresponding author. Email: [email protected] 10, 2018 Journal of Modern Optics jmo˙super signal powers [12]. The purpose of the present contribution is to propose an implementation ofthe linear circuit processing BPSK sequences in a multimode quantum memory interface [13–15].The proposal is motivated by recent demonstrations of fully controllable linear transformationsbetween atomic spin coherences and optical fields [16, 17]. This approach would be well suitedto process sequences transmitted in a single spatial mode and encompassing multiple time bins.The presented scheme points to possible applications of quantum memories not only in quantuminformation processing, but also in future optical communication systems operated at the quantumlimit.This paper is organized as follows. First, in Sec. 2 we review the principle of BPSK and theattainable transmission rates in the low power regime. The strategy to achieve superadditivityusing sequences of BPSK symbols is summarized in Sec. 3. The proposal for the quantum memoryinterface to process BPSK sequences is described in Sec. 4. For short sequence lengths, we analyzein Sec. 5 possible gains from the application of a minimum-error Dolinar receiver at one of theoutput ports. Finally, Sec. 6 concludes the paper.
2. Binary phase shift keying
Any two coherent states can be mapped via a unitary linear optical transformation onto a pair withthe same mean photon number but opposite phases. This transformation can be realized using abeam splitter with transmission approaching one and an auxiliary coherent field [18]. Moreover,if both the states are equiprobable such a pair minimizes the mean energy for a fixed separationbetween the complex amplitudes of the coherent states, characterizing their distinguishability.Therefore in the following we will restrict our attention to this special case, commonly known inoptical communication under the acronym BPSK. In simple terms, bits are encoded in the signof the complex amplitude ± α of coherent pulses, prepared with the same mean photon number¯ n = | α | in each use of the channel. For large mean photon numbers ¯ n , the two coherent states | α i and | − α i describing the pulses are almost orthogonal and the bit value can be read out witha negligible error using e.g. homodyne detection. Readout becomes less trivial in the regime oflow mean photon numbers, when ¯ n ≪
1, as the quantum mechanical scalar product between thetwo coherent states used for communication is then substantially nonzero, |h α | − α i| = e − n , andtherefore they cannot be distinguished with certainty [19].The usefulness of a communication scheme for classical information transmission can be char-acterized with mutual information, which describes the strength of correlations between systempreparations at the channel input and measurement results at the channel output. Importantly,mutual information provides the upper limit on the attainable transmission rate for a given com-munication scheme [20]. When two equiprobable quantum states are used as preparations and thephysical systems transmitted in consecutive channel uses are measured individually, the optimaldetection strategy is to apply the minimum-error measurement described by Helstrom [21, 22].From the information theoretic point of view, such a scheme is described by a binary symmetricchannel with the error rate given by ε (¯ n ) = 12 (1 − p − |h α | − α i| ) = 12 (1 − p − e − n ) . (1)In the above expression we explicitly used the two coherent states constituting the BPSK alphabet.For a binary symmetric channel representing individual detection mutual information reads: I ind = 1 − H (cid:0) ε (¯ n ) (cid:1) ≈ β ¯ n, (2)where H ( x ) = − x log x − (1 − x ) log (1 − x ) is the binary entropy measured in bits. The second2 ugust 10, 2018 Journal of Modern Optics jmo˙super approximate expression in Eq. (2) results from expanding mutual information up to the linear termin ¯ n and is valid in the regime ¯ n ≪
1, with the proportionality constant equal to β = 2 / ln 2 ≈ . χ , which is defined mathematicallyas the difference between the von Neumann entropy S ( · ) of the average state emerging from thechannel and the average entropy of individual output states [25]. In the case of BPSK modulation,because individual states remain pure after transmission, the Holevo quantity is equal to the entropyof the statistical mixture of the two coherent states: χ = S (cid:0) | α i h α | + | − α i h− α | (cid:1) = H (cid:0) (1 − |h α | − α i| ) (cid:1) ≈ ¯ n log n . (3)The last expression, specifying the leading term in the expansion when ¯ n ≪
1, shows that comparedto individual measurements, collective detection enables a qualitative change in the scaling ofattainable information with ¯ n . Furthermore, the Holevo quantity calculated in Eq. (3) approachesasymptotically for ¯ n →
3. Hadamard sequences
Although general strategies to construct collective measurements approaching the Holevo quantityhave been given theoretically [26], the challenge is to design joint detection schemes that could beimplemented in practice using viable components. For BPSK modulation, a very elegant scalablescheme for sequence lengths L equal to integer powers of two has been described by Guha [7]. Thebasic idea, shown schematically in Fig. 1, is to select from all 2 L combinations of BPSK symbolsonly L sequences that correspond to rows of a Hadamard matrix of dimension L . Hadamardmatrices are symmetric with binary entries ±
1, and their rows (or equivalently columns) formmutually orthogonal vectors [27]. Collective detection of such
Hadamard words is facilitated by anobservation that rescaling a Hadamard matrix by 1 / √ L yields an orthogonal matrix which can bein principle implemented as a linear optical circuit. Because of the orthogonality property, eachHadamard word fed into the circuit will generate a non-zero pulse only in one output port of thecircuit, different for each sequence, while all other ports will remain dark.The above scheme effectively converts Hadamard BPSK words into the well known pulse positionmodulation (PPM) format, in which information is encoded in the position of a single pulse in thetotal number of L otherwise empty bins. The most obvious strategy to read out the position ofthe pulse is to employ direct detection. Assuming ideal, unit-efficiency photon counting detectorswithout dark counts, either the position of the pulse is identified unambiguously, or this informationis erased if no count is generated for any bin. From the information theoretic perspective sucha communication scheme corresponds to the well-known erasure channel [20], for which mutualinformation per one bin reads: I PPM = pL log L. (4)where p is the probability of detecting the position of the pulse. In our case, because all L BPSKstates interfere constructively at one output port of the Hadamard circuit producing a pulse with3 ugust 10, 2018 Journal of Modern Optics jmo˙superFigure 1. An exemplary superadditive communication scheme using the BPSK for-mat for the sequence length L = 8. The sender prepares sequences of BPSK symbolswith ± signs defined by rows of a Hadamard matrix. At the receiver side, the sym-bols are interfered using a linear circuit described by a Hadamard matrix rescaledby 1 / √ L . This maps the BPSK sequences onto the pulse position format where onlyone bin contains a pulse carrying the energy of the entire sequence. The position ofthe pulse identifies unambiguously the received sequence. the mean photon number L ¯ n , the probability p is given by p = 1 − e − L ¯ n . (5)Expanding the above expression up to the first order yields p ≈ L ¯ n , which implies that I PPM ≈ ¯ n log L. (6)This value is higher than the Helstrom limit for individual detection I ind ≈ β ¯ n when L > β ≈ . L = 8.It is worth to emphasize that the simple formula in Eq. (6) is valid only for L ¯ n ≪
1, as for largermean photon numbers the probability p saturates at one. The exact expression given in Eq. (4) hasa well defined maximum as a function of L , which can be approximately identified by expanding p up to the second order in L ¯ n . Mutual information I PPM evaluated at this maximum has theexpansion in ¯ n ≪ I PPM ≈ ¯ n log n − ¯ n log ln 1¯ n . (7)On the other hand, the capacity of a narrowband bosonic channel is given up to the second orderas ¯ n log (1 / ¯ n ) + ¯ n/ ln 2 for low signal powers and it coincides with the Holevo quantity for BPSKmodulation found in Eq. (3). It is seen that although the leading orders of both expressions arethe same, the first order corrections exhibit different behaviour.4 ugust 10, 2018 Journal of Modern Optics jmo˙superFigure 2. A quantum memory interface for converting Hadamard sequences of BPSKsymbols into the PPM format shown schematically for L = 8 sequence length. Thehorizontal axis represents the time flow. Arriving pulses interact with initially un-occupied memory modes depicted as horizontal lines. Black diagonal bars indicatebeamsplitter-type interactions with the π phase shift introduced for transmissions inboth directions and reflections from shaded sides. Horizontal and vertical bars are ad-ditional π phase shifts. Fractions labelling bars indicate power reflection coefficients.Unlabeled diagonal bars correspond to perfect reflections. At the output the memorymodes are read out using direct detection. The case when one of the detectors isreplaced by the Dolinar receiver is discussed in Sec. 5.
4. Quantum memory implementation
In many commonly used optical communication links, e.g. fibres operating at telecom wavelengths,pulse sequences are transmitted in a single spatial mode. In this case, collective measurementsdescribed in the preceding section need to be implemented over multiple time bins. This requiressynchronization of individual incoming pulses at the receiver while retaining mutual phase relations.One possible solution would be to employ fast optical switches and delay lines to equalize pulsearrival times before the Hadamard circuit. An alternative is to use quantum memories to transformcoherently [28] the incoming pulses into the PPM format. Within this approach the Hadamardcircuit can be implemented piecewise with the incoming pulses using beamsplitter-type operationsbetween light pulses and quantum memory modes [16, 17, 29].A natural decomposition of the Hadamard circuit in this implementation is the triangular formof a general linear optical transformation discussed by Reck et al. [30]. Its explicit form is shownschematically in Fig. 2 in the case of L = 8 time bins. The first pulse is mapped onto a quantummemory mode. The l th pulse, l = 2 , . . . , L , goes through l quantum memory modes, as symbolizedby vertical lines in Fig. 2. In each memory mode a transformation combining the incoming lightwith the already stored excitation is driven by suitable control fields [16]. The final L th memory,empty so far, is driven so as to store all incoming light. At the end, the L quantum memory modescontain the received sequence converted into the PPM format. Detection of the excitations stored inthe memories can be achieved for example by mapping their contents back onto light and counting5 ugust 10, 2018 Journal of Modern Optics jmo˙super optical photons in the standard manner.
5. Hybrid detection
Using Hadamard words constructed from BPSK symbols as described in Sec. 3, superadditivity inmutual information can be demonstrated for at least L = 8 time bins. On the other hand, a verysimple hybrid scheme has been proposed for L = 2 bins, where two consecutive pulses are interferedon a 50 /
50 beam splitter with output ports monitored by a Dolinar receiver and a photon countingdetector [7]. With the right choice of probabilities of input sequences, the relative enhancement inmutual information is 2 . .
8% found numerically by optimizing jointtwo-system measurements [31]. We will now discuss generalization of the hybrid scheme to morethan two pulses.The basic idea is to supplement the set of Hadamard words by a sequence −− . . . − . This sequenceprovides a non-zero pulse at the same output port of the Hadamard circuit as ++ . . . +, but with theopposite phase. We will assume that this port is monitored by a Dolinar receiver, and both sequencesare prepared with equal probabilities (1 − λ ) /
2, where 0 ≤ λ ≤
1. The remaining Hadamard wordsare sent with identical probabilities λ/ ( L − . . . + and − − . . . − are identified with an error ε ( L ¯ n ), because the mean total photon number in the entire sequence is L ¯ n . Any other Hadamard word generates either measurement result on the Dolinar receiver withthe same probability 1 /
2, i.e. no information is obtained.Mutual information for the above communication scheme can be cast in the following form: I = 1 L { (1 − λ )[1 − H ( ε ( L ¯ n ))] + λp log ( L −
1) + H ( λp ) − λ H ( p ) } . (8)The overall multiplicative factor 1 /L stems from rescaling mutual information per one time bin.Within curly brackets, three contributions can be identified. The first one, given by 1 − H ( ε ( L ¯ n ))is mutual information for a binary symmetric channel with the error rate ε ( L ¯ n ) corresponding to aminimum-error measurement on sequences + + . . . + and − − . . . − . This contribution enters withthe weight 1 − λ , which is the overall probability of preparing either sequence. The second term, p log ( L − L − p . This channel describes situation when any other Hadamard sequence is used, whichoccurs with the overall probability λ . Finally, the combination of the last two terms, H ( λp ) − λ H ( p ),specifies mutual information for the so-called Z channel with a binary set of input symbols, whenone symbol used with probability λ is identified correctly with the probability p , whereas in theremaining 1 − p fraction of cases it gives the same result as the second symbol, used with theprobability 1 − λ . In optical communication, such a channel describes on-off keying where either apulse or an empty bin are sent in each channel use, and an ideal photon counting detector withoutdark counts is used at the output.In our case the probability p of a detector click is given by Eq. (5). Assuming that p ≪ H ( λp ) − λ H ( p ) ≈ λp log λ . (9)It is worth noting that the formula on the right hand side is formally equivalent to mutual informa-tion for the pulse position modulation format with 1 /λ input words specified in Eq. (4). In order tosimplify calculations, in Eq. (8) we will expand up to linear terms in ¯ n the expressions for p ≈ L ¯ n ugust 10, 2018 Journal of Modern Optics jmo˙superFigure 3. The ratio I / I ind of mutual information per bin for collective detectioncompared to the optimal individual detection case evaluated in Eq. (2). Solid linesdepict asymptotic results given in Eq. (6) for direct detection (gray solid line, redonline) and in Eq. (10) for hybrid detection (light gray solid line, orange online), with L treated as a continuous parameter. Numerical results based on the exact expressionsfor the error probability in Eq. (1) and the count probability in Eq. (5) are shownfor ¯ n = 2 × − (filled symbols) and ¯ n = 2 × − (empty symbols) in the case ofdirect detection (squares) and hybrid detection (circles). The dashed lines serve asguides to the eye. All sequence lengths L ≤
32 for which Hadamard matrices existhave been included in the calculations. and 1 − H ( ε ( L ¯ n )) ≈ βL ¯ n . After applying these approximations it is easy to find the optimal valueof λ , which taking into account the constraint 0 ≤ λ ≤ n ≪ I = ¯ n (cid:18) β + L − e β ln 2 (cid:19) , if L < e β + 1¯ n log ( L − , if L ≥ e β + 1 (10)It is seen that for large L we recover the expression for ( L − . . . + and − − . . . − at all. In these cases directdetection scheme yields higher mutual information. However, enhancement is possible for shortsequence lengths, as shown in Fig. 3 depicting the ratio I / I ind . In the plots, we used two values ofthe mean photon number: ¯ n = 2 × − when the asymptotic expression given in Eq. (10) is hardlydistinguishable within the resolution of the graph from numerical results, and ¯ n = 2 × − , whichallows us to identify deviations from the asymptotics with the increasing mean photon number. Itis seen that for larger ¯ n the superadditivity effect diminishes. In the case of direct detection one cannotice that mutual information I PPM approaches a maximum with the increasing sequence length L , which is simply a result of the saturation of the count probability p defined in Eq. (5).7 ugust 10, 2018 Journal of Modern Optics jmo˙super
6. Conclusions
We described theoretically a construction of a collective receiver for BPSK signal based on beam-splitter type transformations between incoming light pulses and quantum memory modes. Sucha receiver can be used to demonstrate the superadditivity effect in classical communication overa quantum channel, with enhancement most strongly pronounced in the low-power limit. An in-teresting extension of the presented work may be to go beyond a sequence of time bins and toconsider mixed time-frequency encodings within the available spectral bandwidth which could alsobe handled by architectures based on quantum memories [32].Multimode interference underlying the superadditivity of the presented receiver relies on perfectphase and amplitude matching between interfering pulses. A recent study suggests that the collec-tive BPSK detection scheme based on Hadamard words may be robust against moderate levels ofphase noise [33]. One should also take into account unequal losses induced by beam splitter opera-tions and finite lifetime of excitations stored in memory modes. The simplest strategy to mitigatethis would be to introduce additional attenuation in order to ensure proper contributions fromindividual input pulses to the output ports of the receiver. In this case, attainable mutual informa-tion calculated in Eqs. (6) and (10) would need to be multiplied by the overall power transmissioncoefficient, which diminishes the superadditivity effect.
7. Acknowledgements
We would like to thank J. Nunn for his encouragement to write up this contribution. We acknowl-edge insightful discussions with U. Andersen, F. E. Becerra, Ch. Marquardt, and M. G. A. Paris.This research was supported in part by the EU 7th Framework Programme projects SIQS (GrantAgreement No. 600645) and PhoQuS@UW (Grant Agreement No. 316244).
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