Attaining Classical Capacity Per Unit Cost of Noisy Bosonic Gaussian Channels

PHYSICAL REVIEW A 104, 022605 (2021)

Attaining classical capacity per unit cost of noisy bosonic Gaussian channels

Marcin Jarzyna * Centre for Quantum Optical Technologies, Centre of New Technologies, University of Warsaw, ulica Banacha 2c, 02-097 Warszawa, Poland

(Received 24 June 2020; accepted 10 May 2021; published 13 August 2021)

I show that classical capacity per unit cost of noisy bosonic Gaussian channels can be attained by employing a generalized on-off keying modulation format and a projective measurement of individual output states. This means that neither complicated collective measurements nor phase-sensitive detection is required to communi- cate over optical channels at the ultimate limit imposed by laws of quantum mechanics in the limit of low average cost.

DOI: 10.1103/PhysRevA.104.022605

Transmission of information lies at the backbone of count- indicates a diverging difference between the two scenarios less technologies. Utilizing the quantum nature of light brings [14]. For additive noise channels the maximum CPC allowed promise to increase information transmission rates beyond by laws of quantum mechanics is finite but still greater than what is attainable by all conventional means [1]. In particular, what can be attained by conventional receivers [15,16]. for a pure-loss channel in moderate- and low-power regimes, It is known that in order to saturate the classical capacity as measured by the average number of photons per time bin, or CPC of the pure-loss and most other physical channels, it na, conventional receiver architectures like homodyne and is in principle necessary to use collective measurements on a heterodyne measurements, known to be almost optimal in the large number of channel outputs [14,17–21], which is usually large-power regime, are vastly outperformed by the ultimate not feasible in practice. This is because of the superadditivity quantum limit on the transmission rate, known as classical of accessible information with respect to joint measurements; capacity [2,3]. The difference is of a qualitative nature as con- i.e., more information per channel use can be gained by conventional phase-sensitive detection schemes allow for rates sidering collective measurements on several channel outputs scaling linearly with the average number of photons per time than by detection of just a single output at a time [22,23]. bin ∼na in the small-na regime, whereas classical capacity Note that this is a different effect than superadditivity of the 1 scales as ∼na log in the leading order [4–6]. classical capacity [24] which is due to the possibility of using 2 na This discrepancy is even more evident when one looks input states that are entangled between subsequent channel at the capacity per unit cost (CPC), which quantifies the uses. Even coherent detection schemes operating on single maximum amount of information that can be transmitted symbols like the Kennedy [25]orDolinar[26] receivers, per single photon for a particular protocol, i.e., assuming a despite outperforming classical shot-noise-limited receivers, certain modulation and detection scheme [7]. CPC contains cannot attain the capacity limit and have been realized only as all the information about the behavior of the conventional proof-of-principle examples [27,28]. Another issue in reach- capacity in the regime of small power as the latter is just ing the optimal performance is that the required ensemble of given by the CPC multiplied by the photon flux. Importantly, input states is a continuous family of coherent states with a CPC indicates the maximum attainable value of the photon Gaussian prior distribution which may be problematic to pro- information efficiency (PIE) which is an important figure in duce in realistic applications. It is therefore crucial to identify many communication scenarios [5,8–12] and quantifies how measurement schemes and modulation protocols that attain much information can be transmitted per single photon for the CPC bound and can be realized by current or near-future a given protocol operating at a particular level of na.Inthe existing technology. case of a lossy channel conventional phase-sensitive schemes In this paper I show that a realistic single-symbol projective allow for a constant CPC whereas the quantum-limited CPC measurement is sufficient to asymptotically saturate CPC and is infinite [4]. Although direct detection allows to attain the thus also classical capacity of any noisy Gaussian quantum latter [13], the second-order term ∼ log log 1 appearing in channel in the low-cost limit. The optimal signal modulation 2 na the respective PIE and lacking in the quantum-limited PIE format is just a binary signal alphabet known as generalized on-off keying (OOK) and presented schematically in Fig. 1. It is composed from an empty time bin (vacuum state) and an *[email protected] infrequently sent signal in a coherent state. To my knowledge this is the first result in which it is shown how to attain clas- Published by the American Physical Society under the terms of the sical capacity of realistic communication channels without Creative Commons Attribution 4.0 International license. Further using a highly complicated receiver architecture employing distribution of this work must maintain attribution to the author(s) sophisticated quantum measurements. Note that some partial and the published article’s title, journal citation, and DOI. results on the attainability of CPC were obtained in Refs. [15]

2469-9926/2021/104(2)/022605(6) 022605-1 Published by the American Physical Society MARCIN JARZYNA PHYSICAL REVIEW A 104, 022605 (2021)

where ns = b[ρx] is the cost of state ρx and | || | = | p(y|x) D[p(y x) p(y 0)] y p(y x)log2 p(y|0) is the Kullback- Leibler divergence between distributions p(y|x) and p(y|0) = Tr([ρ0]y ). Note that maximization in Eq. (1)is taken over input symbols but not the input states since the set FIG. 1. A scheme presenting communication with generalized of the latter is considered fixed. If one adds optimization over OOK modulation. A coherent state signal or a zero-cost state rep- input state ensembles and measurements on top of Eq. (1) λ resented by an empty time bin are sent with probabilities and one obtains the classical capacity per unit cost (CCPC) Cquant 1 − λ, respectively. The states evolve through a Gaussian channel which quantifies the best possible PIE attainable for a given that changes the amplitude and adds noise. The detection stage im- quantum channel [15,16] and is equal to plements a projective measurement onto the eigenbasis of an output D[[ρ]||[ρ ]] of a zero-cost state which can be realized by placing an antisqueezing C = max 0 . (2) quant ρ=| | operation and a subsequent photon-number-resolving measurement. 0 0 ns Both CPC and CCPC are attained in the limit of vanishing average cost per channel use, na → 0[7,15,16]. Therefore, since and [16], in which it was shown respectively that binary or they are defined as maximum ratios of respective capacities pulse position modulation formats are enough; however, the per channel use and average cost, in the low-cost limit one optimal positive operator valued measures (POVMs) were obtains for the classical capacity C ≈ naCquant and similarly found either only in the trivial case of the noiseless channel for the regular capacity. Importantly, in the low-cost regime, or were collective. CCPC is exactly equal to the PIE of the optimal protocol For a communication setup described by a quan- saturating the classical capacity of the channel. tum channel with a given ensemble of input states A basic model of an optical communication channel is a ρx used with an input symbol probability distribution Gaussian bosonic channel. Gaussian channels describe var- p(x) and receiver performing a measurement described ious effects that are characterized by evolution quadratic in byaPOVM{y}, the maximal information transmission creation and annihilation operators of the system, such as rate is quantified by the mutual information I = H(Y ) − linear losses, thermal noise, phase-sensitive noise, or squeez- | =− | = ing [31]. They can be characterized as the most general type H(Y X), where H(Y ) y p(y)log2 p(y) and H(Y X ) − | | of operations that preserve Gaussian character of quantum x y p(x)p(y x)log2 p(y x) with a conditional probability distribution evaluated through the Born rule, p(y|x) = states on which they act. A general Gaussian channel can Tr([ρx]y ). Importantly, each use of one of the states ρx be specified by a real matrix X and a real symmetric and is usually assumed to possess some kind of cost, determined non-negative matrix Y which satisfy certain conditions [32] by a properly chosen cost function b[ρx], e.g., the energy of to ensure complete positivity and trace preservation by the the state. Mutual information optimized over the input sym- channel. The output state first moments and covariance matrix bol probability distributions is known as the capacity of the are given by channel and quantifies the best rate for a given communication d = Xd , V = XV XT + Y, (3) protocol, i.e., assuming particular modulation and detection out in out in schemes. Capacity further optimized over input states en- where din and Vin are respectively the first moments vector sembles and, possibly collective, measurements performed on and the covariance matrix of the input state. Any Gaussian many channel outputs return the maximal information trans- channel can be decomposed into a fiducial channel and pas- mission rate for the channel, known as the classical capacity sive (i.e., conserving the energy) Gaussian unitary operations C. Finding classical capacity is in general a formidable task; preceding and following the former [33]. For a channel speci- however, for a special class of channels that I will consider, fied by X and Y the fiducial channel is given by matrices known as Gaussian channels, this problem can be solved 2s = |η| 10, = e 0 , under the constraint of fixed average cost per channel use, XF η YF y −2s (4) n = p(x)b[ρ ][6,29,30]. 0sgn() 0 e a x x √ A convenient quantity to analyze communication in the where η = det X, y = det Y, and s can be interpreted as low-cost regime is CPC, which is given by the maximum intrinsic channel squeezing. The original channel matrices can ratio of the capacity and average cost per channel use. CPC be written as characterizes the maximum amount of information that can be T carried out per unit cost, i.e., the efficiency of communication X = MXF, Y = MYFM , (5) rather than just the information rate. In the case of Gaussian channels the usual cost figure is the average energy of the where M is some symplectic operation and denotes phase- state and I will focus on this scenario. For a quantum channel space rotation. The exact relations between matrices in this that allows one to use a zero-cost state, or in other words a decomposition and the original channel can be found in Ref. [33]. The parameter η can be interpreted as a characteris- vacuum ρ0 =|00|, the CPC can be expressed by a compact formula [7] tic transmission coefficient of the channel; note, however, that it can be a number with an absolute value larger than 1 which describes phase-conjugating and amplifying channels. D[p(y|x)||p(y|0)] The CCPC of any Gaussian channel is equal to CCPC of C = max , (1) class = x 0 ns its corresponding fiducial channel. This is because the phase-

022605-2 ATTAINING CLASSICAL CAPACITY PER UNIT COST OF … PHYSICAL REVIEW A 104, 022605 (2021) space rotation in Eq. (5) does not change the energy of input ation M in Eq. (5) which gives eventually states and the symplectic transformation M can be always ρ = Uˆ Sˆ(r)Dˆ (γ )ρ Dˆ †(γ )Sˆ†(r)Uˆ † . (12) undone by incorporating a proper unitary transformation at M y M the channel output. It is known that for any Gaussian channel Importantly, any one-mode symplectic transformation can be CCPC is saturable in the low-cost limit by the generalized realized by a combination of two phase-space rotations and OOK modulation format [15]. For general quantum Gaussian ˆ = ˆ ˆ ˆ squeezing. Therefore, one can write UM Uθ2 S(z)Uθ1 , where channels it was shown in Refs. [15,16] that CCPC is equal to ˆ θ Uθi denotes the unitary rotation by phase i and z is the squeezing introduced by M. =|η|ω + 1 , Cquant max log2 1 (6) The POVM that I will consider is a projective measurement nb onto the eigenbasis of an output of the zero-cost state. In the where the parameter η is defined in Eq. (4) and nb and case of Gaussian channels with average number of photons 2r ωmax = e are respectively the average thermal energy and per channel use constraint the latter is the vacuum state’s the squeezing of the output state of the fiducial channel if the output, given by Eq. (12) with γ = 0. The measurement is input was in a vacuum state: therefore given by projections onto squeezed number states † † † † = Uˆθ Sˆ(z)Uˆθ Sˆ(r)|kk|Sˆ (r)Uˆ Sˆ (z)Uˆ . It can be experi- k 2 1 θ1 θ2 1 |η| |η| mentally realized by a photon-number-resolving measurement n + = + ye2s + ye−2s , (7) b 2 2 2 preceded by an antisqueezing operation in the right direc- tion or by a proper combination of phase-space rotations and |η| + 2s squeezing (see Fig. 1). 2r 2 ye ωmax = e = . (8) For the coherent state with amplitude α the measurement |η| + −2s 2 ye statistics is given by a conditional probability distribution [34] k 2 2 Importantly, for any Gaussian channel with additive noise − |γ | |γ | nb + p(k|α) = e nb 1 L − , (13) CCPC has a finite value, meaning that in the low-cost regime k+1 k (nb + 1) nb(nb + 1) classical capacity is equal to C ≈ n C. Therefore, it is enough a γ to show that a receiver attains the CCPC to show its opti- where Lk denotes the kth Laguerre polynomial and is given α mality also from the point of view of the actual capacity per by Eq. (10). In the generalized OOK modulation format can α = α = channel use. take√ only two values: 0 0 for the vacuum state and s I will consider the generalized OOK modulation format, ns for the signal state. Plugging Eq. (13) into Eq. (1), the capacity per unit cost is equal to shown schematically in Fig. 1. The input message is encoded in a series of time bins, each of which can be either empty ∞ √ √ − λ = 1 | 1 − | , with probability 1 or can carry a coherent state with Cclass p(k ns )log2 H[p(k ns )] ns p(k|0) an average energy ns with corresponding probability λ.The k=0 average cost per channel use of such an input ensemble is (14) given by na = λns. The phase-space rotation appearing in =− the decomposition in Eq. (5) changes just the phase of the where H[p(x)] x p(x)log2 p(x) is the Shannon entropy input state and thus can be neglected without loss of generality of distribution p(x). The first term in the bracket in Eq. (14)is by tuning the phase of the coherent state properly. For input equal to in a coherent state with amplitude α the output of the fiducial ∞ √ 1 channel of a general single-mode Gaussian channel is given p k| n ( s )log2 | by a density matrix = p(k 0) k 0 ρ = ˆ |η|α ˆ ρ ˆ† ˆ † |η|α , 1 D( )S(r) nb S (r)D ( ) (9) = +|γ |2 + + + , (nb s )log2 1 log2(1 nb) (15) ˆ · , ˆ · nb where D( )S( ) are displacement and squeezing operators ∞ nk γ ρ = b | | where s is the displacement defined in Eq. (10) evaluated and n = k+1 k k is a thermal state with the √ b k 0 (nb+1) 2 for αs = ns. Since |γs| =|η|nsωmax in Eq. (10), for large average energy nb and I have chosen the phase of α such that ns the expression in Eq. (15) is in the leading order equal the state is aligned with the position quadrature. |η| ω + 1 to ns max log2(1 ), which is exactly the CCPC of the The order of the squeezing and displacement operators in nb channel multiplied by the cost of the signal state. The remain- Eq. (9) can be exchanged by the rule [34] ing term in the bracket in Eq. (14) is upper bounded by r √ Dˆ ( |η|α)Sˆ(r) = Sˆ(r)Dˆ (γ ),γ= |η|αe . (10) 2 H[p(k| ns )] g(nb +|γs| ), (16) Note that since r for a fiducial channel is defined to be a real where g(x) = (x + 1)log (x + 1) − x log (x) is the entropy γ α 2 2 parameter, the phase of depends only on the phase of , of a thermal state with an average energy x. This is because the which I set to zero. Therefore the output state in Eq. (9) can right-hand side is the maximal possible entropy for any dis- be written as 2 tribution with a fixed expected value k=nb +|γs| . Since |γ |2 ∼ ρ = Sˆ(r)Dˆ (γ )ρ Dˆ †(γ )Sˆ†(r). (11) s ns in Eq. (10) in the large-ns limit the expression on y |γ |2 + the right-hand side of Eq. (16) is equal to log2 s O(1). The state in Eq. (11) undergoes then evolution through the Plugging these results into Eq. (14) and going with signal cost unitary transformation UˆM corresponding to symplectic oper- to infinity, ns →∞, one obtains that the term corresponding

022605-3 MARCIN JARZYNA PHYSICAL REVIEW A 104, 022605 (2021)

7 is expected since in the small-na regime the capacity is low. From a practical point of view, this time may be decreased by 6 increasing the bandwidth of the link. The detection scheme I proposed above assumed no de- 5 pendence on the average cost. If such dependence is allowed 4 one may propose an even simpler binary detection scheme that still allows to attain CCPC asymptotically. Consider a thresh- 3 (th) = k(th) old detector with two-component POVM 0 k=0 k (th) = − (th) (th) 2 and 1 1 0 . The threshold value k can be freely adjusted by the receiver to the average cost of the incoming 1 signal. With such a choice, one obtains two outcomes with Normalized capacity per unit cost 0 respective probabilities 0.1 0.5 1 5 10 50 100 k(th) Cost of output signal state p0(α) = p(k|α), p1(α) = 1 − p0(α), (18) k=0 FIG. 2. Capacity per unit cost normalized to channel transmis- |α |η| |η| where p(k ) are defined in Eq. (13). Plugging the above sion as a function of the cost of the output signal state ns for distribution evaluated for the signal state and the zero-cost generalized OOK modulation and projective measurement onto ap- state into Eq. (1), one gets propriately squeezed number state (solid curves), threshold detector (dotted curves), and the ultimate quantum bound given by classical 1 √ C(th) = h [p ( n )] capacity per unit cost (dashed lines). Results for phase-insensitive n 2 0 s = = . = . s channels are nb 1 (black), nb 0 1 [red (dark grey)], nb 0 01 [orange (gray)], while the phase-sensitive channel with n = 0.1and √ √ b + 1 + 1 , r = ln 2/2 ≈ 0.34 is depicted by yellow (light gray) curves. p0( ns )log2 p1( ns )log2 p0(0) p1(0) to Eq. (16) vanishes and the capacity per unit cost is equal to (19) the quantum bound in Eq. (6), i.e., where h2(x) denotes the binary entropy function. The first term in Eq. (19) has to be smaller than the entropy on the = =|η|ω + 1 . left-hand side of Eq. (16) due to the data processing in- Cclass Cquant max log2 1 (17) nb equality and thus gives a vanishing contribution to CPC for Note that the result in Eq. (17) is asymptotic; i.e., it applies large ns. The distribution in Eq. (13) converges to a Gaussian in the limit of vanishing average cost per time bin n → 0, distribution in the large-ns limit and its variance is equal to a =|γ |2 + + + since this is the limit in which capacities per unit cost can Var (k) s (1 2nb) nb(nb 1). Therefore if one takes (th) = − |γ |2 > be attained. However, if one chooses an average cost per k (1 ) s with any 0 one can always find a > time bin at the output |η|n as a figure instead, the conclu- coherent state cost√ value n1 such that for any ns n1 one a (th) < − =|γ |2 + sions remain unchanged; i.e., the capacity per unit cost is has k E(k) c Var (k), where E(k) s nb is the still equal to Eq. (17) but normalized to the transmission |η| expectation value of the distribution in Eq. (13) and c is an and it is saturated for small output average cost per time bin arbitrary constant. This√ means that for sufficiently large ns the probability p(k| n ) takes non-negligible values only for |η|na → 0. The CPC in Eq. (17) diverges logarithmically with s √ √ k > k(th). As a consequence, p ( n ) ≈ 1 and p ( n ) ≈ 0, decreasing additive noise nb meaning that for channels that do 1 s 0 s which gives not introduce any additive noise, such as pure-loss and lossless k(th) + 1 1 channels, there is no limit on the attainable CPC. This was (th) ≈ + C log2 1 observed in Refs. [4,13,14,35] where it was shown that direct ns nb detection together with generalized OOK in principle allows − |γ |2 + attaining unbounded CPC for lossy channels. = (1 ) s 1 + 1 , log2 1 (20) Figure 2 shows classical capacity per unit cost normalized ns nb to the channel transmission |η| obtained for the discussed where I have plugged the expression for p1(0) from Eq. (18). communication strategy as a function of the cost of the output Since Eq. (20)isvalidforany >0, CPC has to converge to signal state. It is seen that respective capacities converge to the ultimate quantum limit, Eq. (2), with ns →∞. It is seen the asymptotic values given by the ultimate quantum limit, in Fig. 2 that the threshold detector also allows for saturation Eq. (6), both for phase-sensitive and phase-insensitive chan- of the quantum CPC bound, although the necessary signal nels. In the considered range of phase-insensitive noise nb = cost ns is much larger than for projection onto zero-cost state 0.01–1 the saturation happens around the signal output cost output eigenvectors. Unlike the previous case, the CPC for the |η|ns ≈ 10 which confirms the assumption of a necessary threshold detector is not a monotonic function of ns because large signal state cost. The latter is always possible since of the discrete nature of the threshold k(th). na = λns and the probability of sending a coherent state can In conclusion I have showed that the ultimate quantum be freely chosen. It is seen in Fig. 2 that for full saturation bound on the information transmission rate per unit of energy of the CCPC it is necessary to send strong coherent states of a general noisy bosonic Gaussian channel can be asymp- ns →∞, very rarely λ → 0. Note that this means that the to- totically attained by the generalized OOK modulation and a tal transmission time of a message may be large, although this projective measurement on the individual channel outputs in

022605-4 ATTAINING CLASSICAL CAPACITY PER UNIT COST OF … PHYSICAL REVIEW A 104, 022605 (2021) the form of projection onto squeezed number states. As an I thank K. Banaszek, L. Kunz, W. Zwolinski,´ and M. implication the classical capacity of noisy Gaussian channels Lipka for insightful discussions. This work is part of the in the low-cost limit can also be saturated by the considered project “Quantum Optical Communication Systems” carried protocol. This is a qualitatively different situation than in the out within the TEAM programme of the Foundation for Polish pure-loss case in which there appears a nonvanishing gap be- Science co-ﬁnanced by the European Union under the Euro- tween the classical channel capacity and what can be achieved pean Regional Development Fund. with individual measurements.

[1] C. M. Caves and P. D. Drummond, Quantum limits on bosonic [16] D. Ding, D. S. Pavlichin, and M. M. Wilde, Quantum channel communication rates, Rev. Mod. Phys. 66, 481 (1994). capacities per unit cost, IEEE Trans. Inf. Theory 65, 418 (2019). [2] A. S. Holevo, The capacity of the quantum channel with general [17] P. Hausladen, R. Jozsa, B. Schumacher, M. Westmoreland, and signal states, IEEE Trans. Inf. Theory 44, 269 (1998). W. K. Wootters, Classical information capacity of a quantum [3] B. Schumacher and M. D. Westmoreland, Sending classical channel, Phys.Rev.A54, 1869 (1996). information via noisy quantum channels, Phys. Rev. A 56, 131 [18] M. Sasaki, K. Kato, M. Izutsua, and O. Hirota, A demonstration (1997). of superadditivity in the classical capacity of a quantum chan- [4] S. Guha, Structured Optical Receivers to Attain Superadditive nel, Phys. Lett. A 236, 1 (1997). Capacity and the Holevo Limit, Phys. Rev. Lett. 106, 240502 [19] C. H. Bennett, C. A. Fuchs, and J. A. Smolin, Entanglement- (2011). enhanced classical communication on a noisy quantum channel, [5] M. Takeoka and S. Guha, Capacity of optical communication in in Quantum Communication, Computing, and Measurement, loss and noise with general quantum Gaussian receivers, Phys. edited by O.Hirota, A. S.Holevo, and C. M.Caves (Springer, Rev. A 89, 042309 (2014). Berlin, 1997), pp. 79–88. [6] V. Giovannetti, R. García-Patrón, N. J. Cerf, and A. S. Holevo, [20] M. Sasaki, K. Kato, M. Izutsu, and O. Hirota, Quantum chan- Ultimate classical communication rates of quantum optical nels showing superadditivity in classical capacity, Phys. Rev. A channels, Nat. Photonics 8, 796 (2014). 58, 146 (1998). [7] S. Verdu, On channel capacity per unit cost, IEEE Trans. Inf. [21] S. Lloyd, V. Giovannetti, and L. Maccone, Sequential Projective Theory 36, 1019 (1990). Measurements for Channel Decoding, Phys. Rev. Lett. 106, [8] M. L. Stevens, D. O. Caplan, B. S. Robinson, D. M. Boroson, 250501 (2011). and A. L. Kachelmyer, Optical homodyne PSK demonstration [22] A. S. Holevo, On capacity of a quantum communication chan- 1 of 1.5 photons per bit at 156 mbps with rate- 2 turbo coding, nel, Probl. Peredachi Inf. 15, 3 (1979). Opt. Express 16, 10412 (2008). [23] A. Peres and W. K. Wootters, Optimal Detection of Quantum [9] S. Dolinar, K. M. Birnbaum, B. I. Erkmen, and B. Moision, Information, Phys.Rev.Lett.66, 1119 (1991). On approaching the ultimate limits of photon-efficient and [24] M. B. Hastings, Superadditivity of communication capacity bandwidth-efficient optical communication, in 2011 Interna- using entangled inputs, Nat. Phys. 5, 255 (2009). tional Conference on Space Optical Systems and Applications [25] R. S. Kennedy, A near-optimum receiver for the binary coherent (ICSOS) (IEEE, Piscataway, NJ, 2011). state quantum channel, MIT Res. Lab. Electron. Q. Prog. Rep. [10] S. Dolinar, B. I. Erkmen, B. Moision, K. M. Birnbaum, and 108, 219 (1973). D. Divsalar, The ultimate limits of optical communication [26] S. Dolinar, An optimum receiver for the binary coherent state efficiency with photon-counting receivers, in 2012 IEEE Inter- quantum channel, MIT Res. Lab. Electron. Q. Prog. Rep. 111, national Symposium on Information Theory Proceedings (IEEE, 115 (1973). Piscataway, NJ, 2012), pp. 541–545. [27] R. L. Cook, P. J. Martin, and J. M. Geremia, Optical coherent [11] N. Chandrasekaran and J. H. Shapiro, Photon information ef- state discrimination using a closed-loop quantum measurement, ficient communication through atmospheric turbulence - part Nature (London) 446, 774 (2007). I: Channel model and propagation statistics, J. Lightwave [28] F. E. Becerra, J. Fan, G. Baumgartner, J. Goldhar, J. T. Kosloski, Technol. 32, 1075 (2014). and A. Migdall, Experimental demonstration of a receiver beat- [12] W. Zwolinski,´ M. Jarzyna, and K. Banaszek, Range dependence ing the standard quantum limit for multiple nonorthogonal state of an optical pulse position modulation link in the presence of discrimination, Nat. Photonics 7, 147 (2013). background noise, Opt. Express 26, 25827 (2018). [29] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H. Shapiro, [13] M. Jarzyna, P. Kuszaj, and K. Banaszek, Incoherent on-off and H. P. Yuen, Classical Capacity of the Lossy Bosonic Chan- keying with classical and non-classical light, Opt. Express 23, nel: The Exact Solution, Phys. Rev. Lett. 92, 027902 (2004). 3170 (2015). [30] J. Schafer, E. Karpov, O. V. Pilyavets, and N. J. Cerf, Clas- [14] H. W. Chung, S. Guha, and L. Zheng, Superadditivity of sical capacity of phase-sensitive Gaussian quantum channels, quantum channel coding rate with finite blocklength joint mea- arXiv:1609.04119. surements, IEEE Trans. Inf. Theory 62, 5938 (2016). [31] C. Weedbrook, S. Pirandola, R. Garcia-Patron, N. J. Cerf, T. C. [15] M. Jarzyna, Classical capacity per unit cost for quantum chan- Ralph, J. H. Shapiro, and S. Lloyd, Gaussian quantum informa- nels, Phys.Rev.A96, 032340 (2017). tion, Rev. Mod. Phys. 84, 621 (2012).

022605-5 MARCIN JARZYNA PHYSICAL REVIEW A 104, 022605 (2021)

[32] A. S. Holevo and R. F. Werner, Evaluating capacities of bosonic [34] P. Marian and T. A. Marian, Squeezed states with thermal noise. Gaussian channels, Phys. Rev. A 63, 032312 (2001). I. Photon-number statistics, Phys. Rev. A 47, 4474 (1993). [33] J. Schäfer, E. Karpov, R. García-Patrón, O. V. Pilyavets, and [35] J. Chen, J. L. Habif, Z. Dutton, R. Lazarus, and S. Guha, N. J. Cerf, Equivalence Relations for the Classical Capacity Optical codeword demodulation with error rates below the stan- of Single-Mode Gaussian Quantum Channels, Phys. Rev. Lett. dard quantum limit using a conditional nulling receiver, Nat. 111, 030503 (2013). Photonics 6, 374 (2012).

022605-6