Classical capacity of quantum Gaussian codes without a phase reference: when squeezing helps
Marco Fanizza, Matteo Rosati, Michalis Skotiniotis, John Calsamiglia, Vittorio Giovannetti
CClassical capacity of quantum Gaussian codeswithout a phase reference: when squeezing helps
M. Fanizza, ∗ M. Rosati, † M. Skotiniotis, J. Calsamiglia, and V. Giovannetti NEST, Scuola Normale Superiore and Istituto Nanoscienze-CNR, I-56126 Pisa, Italy F´ısica Te`orica: Informaci´o i Fen`omens Qu`antics, Departament de F´ısica,Universitat Aut`onoma de Barcelona, 08193 Bellaterra (Barcelona) Spain
We study the problem of transmitting classical information using quantum Gaussian states in theabsence of a shared phase reference. This problem is relevant for long-distance communication in freespace and optical fiber, where phase noise is typically considered as a limiting factor. We considera family of phase-noise channels with a finite decoherence time, such that the phase-reference islost after m consecutive uses of the transmission line. The Holevo capacity of these channels isalways attained with photon-number encodings, challenging with current technology. Restricting toGaussian encodings, we show that the optimal encoding can always be generated by acting with aHaar-random passive interferometer on ensembles of product single-mode squeezed-coherent states.Hence for coherent-state encodings the optimal rate depends only on the total-energy distributionand we provide upper and lower bounds for all m , the latter attainable at low energies with on/offmodulation and photodetection. We generalize this lower bound to squeezed-coherent encodings,exhibiting for the first time to our knowledge an unconditional advantage with respect to anycoherent encoding for m = 1 and a considerable advantage with respect to its direct coherentcounterpart for m >
1. Finally, we show that the use of part of the energy to establish a referenceframe is sub-optimal even at large energies. Our results represent a key departure from the case ofphase-covariant Gaussian channels and constitute a proof-of-principle of the optimality of squeezingin communication without a phase reference.
Introduction.—
The possibility of mantaining ashared reference frame [1] between the sender andthe receiver is often assumed a priori in analyzingcommunication scenarios. This is the case, for exam-ple, in long-distance communication on optical fiberand in free space, where the information is encodedin quantum states of the electromagnetic field [2]and transferred through a quantum Gaussian chan-nel, such as attenuation or additive noise [3, 4]. Thesechannels belong to the set of phase-covariant Gaus-sian channels, for which coherent states are known tominimize the output entropy [5–8] and can be usedto attain the channel capacity, i.e., the maximumclassical-information transmission rate, provided thatthe sender and the receiver can mantain a phase refer-ence. In this work we consider a non-Gaussian mem-ory channel [9] for which complete decoherence takesplace after m subsequent uses of the transmission line,which effectively models the loss of a common phasereference. This model takes into account that anyphase-decoherence mechanism will realistically takeplace in a finite time T ; hence if the sender can pro-duce subsequent signals at time intervals of δt (cid:28) T ,a total of m = [ Tδt ] signals can be sent before theonset of decoherence [10] and we show that the ad-dition of squeezing can greatly enhance the commu-nication rate, surpassing any coherent-state commu-nication strategy, and thus provide a clear departurefrom the case of phase-covariant Gaussian channels.When phase-noise mechanisms are absent, a phasereference can be established before communication bysending an “idler” signal, e.g., a coherent state thatacts as a phase reference, so that all other signalsare phase-locked with respect to the idler or by ex-plicitly performing a phase-estimation procedure ona reference state [1, 11, 12]. However, such phase- alignment strategies can be seriously affected or evenentirely ruled out by phase diffusion [13–16]. Suchphase-noise mechanism is modelled by applying a suit-ably randomized phase shift at the receiver end, whichcan be extremely detrimental for common communi-cation methods based on the discrimination of a phaseimprinted on a coherent state [17–22]. Moreover,the case of complete phase loss describes the effec-tive channel seen by a photodetector, one of the mostcommon measurement devices in long-distance com-munication. Motivated by the application in commu-nication, several works observed an enhancement inestimation, discrimination and modulation-restrictedcommunication via the addition of squeezing to co-herent states [22–31].We first show that the energy-constrained channelcapacity is equal to that of the noiseless channel butit is attained by using photon-number encodings, stillchallenging with current technology. Hence we char-acterize the structure of optimal coherent-state encod-ings with unconstrained decoding, showing that thecalculation of the maximum information transmissionrate reduces to the optimization of a functional ofthe probability distribution of the total energy. Ourresult implies that the strategy of using part of theenergy for preparing a phase-reference state in onemode and using the other modes to communicate isin general suboptimal. In particular, at leading orderin the high-energy limit, we show that a coherent-state encoding is strictly better than any strategy in-volving a fixed reference mode, independently of thestate used as phase reference. Finally, we show thatsqueezed-coherent states can be used to attain a largercommunication rate with respect to simple coherent-strategies. Interestingly, at variance with what hap-pens in phase estimation procedures [27, 30, 32] a r X i v : . [ qu a n t - ph ] J un where improvements are typically obtained using sig-nals with super-Poissonian photon-number statistics,the advantage we report here is obtained by tradingsignals characterized by Poissonian photon-numberdistribution with sub-Poissionian squeezed-coherentstates. In the case m = 1 we show that there ex-ists a range of energies where an explicit on/off en-coding, employing vacuum and a squeezed coherentstate, plus photodetection is provably better than anycoherent-state encoding under the same energy con-straint, using recent upper bounds on the capacity ofthe classical Poisson channel [33, 34]. Our result vin-dicates the optimality of non-classical Gaussian lightin a physically-motivated communication context, inspite of the optimality of coherent-state encodings forcommon attenuator and additive-noise channels. Theenhancement of the communication rate carries overto the case m >
1, although the explicit on/off strat-egy is not able to surpass the coherent-state upperbound. However, on the basis of the strong improve-ment that squeezing gives to the coherent-state lowerbound, we conjecture that using squeezing is benefi-cial even for arbitrary values of m . The model.—
The channel we consider takes as in-put the state ˆ ρ of m bosonic oscillators (modes) andapplies a random phase-shift, identical for each mode,acting as the following completely-positive and trace-preserving map:Φ m (ˆ ρ ) := (cid:90) dθ π e i θ ˆ n ˆ ρe − i θ ˆ n = ∞ (cid:88) n =0 p n ˆ ρ n , (1)where ˆ n = (cid:80) mi =1 ˆ a † i ˆ a i is the total-photon-numberoperator and ˆ a i , ˆ a † i are the bosonic creation andannihilation operators of the i -th mode, such that[ˆ a i , ˆ a † j ] = δ i,j , ˆΠ n is the projector on the subspacewith total photon number n , p n := Tr (cid:104) ˆΠ n ˆ ρ (cid:105) andˆ ρ n := ˆΠ n ˆ ρ ˆΠ n /p n . Thus, although the absolute phasebetween the sender and the receiver is lost after trans-mission, i.e., there are no coherence terms betweenblocks with different total photon number, the chan-nel does preserve coherence in the relative degrees offreedom of ˆ ρ , i.e., inside each block with fixed photonnumber. In particular, Φ m commutes with the actionof energy-preserving Gaussian unitaries, i.e., m -modeinterferometers, which can be employed to encode in- formation. Channel capacity.—
The classical capacity ofa quantum channel, i.e., the maximum classical-information transmission rate attainable with ar-bitrary encoding and decoding operations, isgiven in general by a regularization C (Φ m , E ) =lim k →∞ k max E ( k ) χ (Φ ⊗ km , E ( k ) ) of the Holevo quan-tity [4], χ (Φ , { q ( x ) , ˆ ρ x } ) := S (cid:0)(cid:82) dx q ( x )Φ(ˆ ρ x ) (cid:1) − (cid:82) dx q ( x ) S (Φ(ˆ ρ x )). Here S ( · ) is the von Neumann en-tropy and the maximization is over all input ensem-bles E ( k ) = { q ( x ) , ˆ ρ ( k ) x } , with ˆ ρ ( k ) x being (possibly en-tangled) states of k copies of the input Hilbert space.For infinite-dimensional Hilbert spaces it is also nec-essary to constrain the energy of the signals in theoptimization, usually via the mean-energy constraint (cid:82) dx q ( x )Tr (cid:104) ˆ n ˆ ρ ( k ) x (cid:105) ≤ kE . Since Φ m leaves Fockstates invariant, one can employ an encoding which isdiagonal in this basis and attain the same rate of theidentity channel on m modes, which, at constrainedmean energy E per use, is C (Φ m , E ) = m g ( Em ), with g ( E ) = − E log E + (1 + E ) log(1 + E ). A detailedproof of this fact is found in the Supplemental Mate-rial (SM). Covariant encodings.—
Since photon-numberstates are hard to produce, one can be interestedin constraining the ensembles to more accessiblestates [35]. With the reasonable assumption thatenergy-preserving Gaussian unitaries are allowed,a drastic simplification in the optimization on anyfamily of states can be obtained by exploiting thesymmetry of the channel. We use the fact that theaverage on Haar-random energy-preserving Gaussianunitaries ˆ U (or passive interferometers PI), which actas the group U( m ) in phase space [36], completelydepolarizes the state in blocks of fixed total photonnumber n , which have dimension (cid:0) n + m − m − (cid:1) : (cid:90) U( m ) dU ˆ U ˆ ρ ˆ U † = ∞ (cid:88) n =0 p n ˆΠ n (cid:0) n + m − m − (cid:1) . (2)This is a consequence of Schur’s lemma applied onthe decomposition into irreducible representations ofU( m ) of the Hilbert space of m modes. The decompo-sition in turn can be understood as a consequence ofthe connection between coherent states of an infinite-dimensional system with spin-coherent states of finitedimension [37, 38], detailed in the SM.The rate achievable with an arbitrary ensemble E = { q ( x ) , ˆ ρ x } is then χ (Φ m , E ) ≤ (cid:34) S (cid:32)(cid:90) dx q ( x ) (cid:90) U( m ) dU Φ m ( ˆ U ˆ ρ x ˆ U † ) (cid:33) − (cid:90) dx q ( x ) (cid:90) U( m ) dU S (Φ m ( ˆ U ˆ ρ x ˆ U † )) (cid:35) = χ (Φ m , E Haar )(3)where the inequality follows from the concavity and unitary-invariance of the von Neumann entropy and thefact that ˆ U and Φ m commute, while in the last equality we defined E Haar as the ensemble obtained by applyinga Haar-random PI ˆ U to the states extracted from E . The inequality means that one can always restrict themaximization of the Holevo quantity to ensembles of the form E Haar , which are invariant under total-phaseshifts. When the states ˆ ρ x are pure and calling p ( x ) their total-photon-number distribution, and denoting theprobabilities as p ( x ) ( n ) = p ( x ) n := Tr (cid:104) ˆΠ n ρ ( x ) (cid:105) , by applying Eq. (2) the Holevo quantity reads χ (Φ m , E Haar ) = (cid:34) (cid:80) ∞ n =0 (cid:82) dx q ( x ) p ( x ) n log (cid:0) n + m − m − (cid:1) + H ( (cid:82) dx q ( x ) p ( x ) ) − (cid:82) dx q ( x ) H ( p ( x ) ) (cid:35) = mg ( Em ) − D ( (cid:82) dx q ( x ) p ( x ) || p th ) − (cid:82) dx q ( x ) H ( p ( x ) ) , (4)where D ( ·||· ) is the Kullback-Leibler diver-gence, H ( · ) is the Shannon entropy and p th ( n ) = (cid:0) n + m − m − (cid:1) (cid:16) EE + m (cid:17) n (cid:16) mE + m (cid:17) m is the total-photon-number distribution of the thermal state on m modes with average energy per mode Em . Fromthis expression it is intuitively apparent that stateswith peaked total-photon-number distribution arepreferable, since they make H ( p ( x ) ) smaller withoutnecessarily increasing D ( (cid:82) dx q ( x ) p ( x ) || p th ). Indeed,as mentioned above, the capacity of the channelis attained by a thermal ensemble of Fock states,which make this term zero. This fact is importantto understand why squeezed states can be betterthan coherent states, a fact that we show in the nextsections. Gaussian encodings.—
In the rest of the article wewill restrict to Gaussian encodings, which are eas-ily realizable in practice. Furthermore, thanks tothe concavity of the entropy and the fact that anyGaussian state can be written as a mixture of pureGaussian states, it is straightforward to further re-strict the optimization to pure states (see SM). We arethen left to consider ensembles of the form E HaarG := { q ( (cid:126)r, (cid:126)α ) dU, U | (cid:126)r, (cid:126)α (cid:105)} , which can be generated by pro-ducing a tensor product of m squeezed-coherent states | (cid:126)r, (cid:126)α (cid:105) := ˆ S ( (cid:126)r ) ˆ D ( (cid:126)α ) | (cid:105) ⊗ m , with ˆ D ( α i ) = exp( α i ˆ a † i − α ∗ i ˆ a i ) and ˆ S ( r i ) = exp( r i (ˆ a i − ˆ a † i )) the displacementand squeezing operators, with probability q ( (cid:126)r, (cid:126)α ) andthen acting with an m -mode Haar-random PI ˆ U . Con-sequently, the optimal Gaussian rate is obtained bymaximizing Eq. (4) over q ( (cid:126)r, (cid:126)α ) with the total-photon-number distribution of | (cid:126)r, (cid:126)α (cid:105) given in [39, 40]. Coherent-state communication rate.—
Let us nowset (cid:126)r = 0 above and restrict to coherent-state encod-ings. The effect of a phase-noise model encompassingour Φ m on the recovery of information encoded incoherent states has been studied in [18], where theauthors derived approximations to the Holevo infor-mation of coherent-state ensembles with fixed energyin the low-photon-number sector. Here we providea general expression using the fact that the total-photon-number distribution of a coherent state | (cid:126)α (cid:105) isa Poissonian P ( s ) with probabilities P ( s ) ( n ) = P ( s ) n := e − s s n n ! and depends only on its total energy s := | (cid:126)α | .Hence the optimization in Eq. (4) can be restricted toinput distributions on the total energy only, q ( s ), and the optimal coherent-state rate R coh (Φ m , E ) := max q ( s ) χ (Φ m , E Haarcoh )= max q ( s ) (cid:34) ∞ (cid:88) n =0 (cid:90) ds q ( s ) P ( s ) n log (cid:18) n + m − m − (cid:19) + H (cid:18)(cid:90) ds q ( s ) P ( s ) (cid:19) − (cid:90) ds q ( s ) H ( P ( s ) ) (cid:35) (5)is attained by producing a single-mode coherent stateof energy s with probability q ( s ) and distributing itwith a Haar-random PI.In the SM we derive the exact expression of upperand lower bounds R up , lowcoh (Φ m , E ) on the optimal ratefor all E and m . For m = 1, only the last two termsof Eq. (5) contribute and we recover the well-knowndiscrete-time classical Poisson channel with input dis-tribution q ( s ). Its capacity is still an open problem inclassical information theory for which only upper andlower bounds are known [33, 34, 41–43]. For m > m and the outputphoton-number distribution. For general m , we con-sider a lower bound obtained by employing an on/offmodulation at the encoding, sending a Haar-randompulse ˆ U (cid:12)(cid:12)(cid:12)(cid:112) E/p (cid:69) with some probability p and the vac-uum otherwise. For m = 1, this lower bound canbe always attained by an explicit scheme, employingon/off modulation with a fixed pulse (cid:12)(cid:12)(cid:12)(cid:112) E/p (cid:69) plusphotodetection (OOP), while for m > m , the upper bound inspired by [33] andthe lower bound read˜ R upcoh (Φ m , E )= E (log E − log log E + ψ ( m ) + 2 + log 13) + o ( E ) (6) R lowcoh (Φ m , E ) = E (log E − log log E + log m ) + o ( E ) . (7)Importantly, at this order the OOP strategy attainsthe lower bound for all m , without the need for Haar-randomization. Let us stress however that the boundfrom Eq. (6) is worse than the bound R upcoh (Φ m , E ) wegive in SM (inspired by [34]) except for extremely lowenergies. In Fig. 1 we plot the latter upper and lowerbounds per unit of channel capacity for several valuesof m , observing a general increasing trend of the ratewith m , while in Fig. 2a we compare the lower boundwith the OOP rate.On the other hand, in the large-energy regime theoptimal rate reads R coh (Φ m , E ) = ( m − /
2) log E + O (1) and it is attained by a Gamma distribution q ( s ) in the total energy, corresponding to a thermalcoherent-state ensemble. The leading term in the FIG. 1. Plot (log-linear scale) of several rates per unit ofchannel capacity C (Φ m , E ) vs. the average energy E forseveral values of m : upper (dashed lines) and lower (dot-dashed lines) bounds on the optimal coherent-state rate, R up , lowcoh (Φ m , E ), lower bound (continuous line) on the op-timal squeezed-coherent-state rate, R lowsq − coh (Φ m , E ). Theoptimal coherent-state rate lies in the shaded region. Notethat as m increases, the coherent-state rate becomes com-parable with the capacity. Moreover, squeezing alwaysprovides a notable advantage over simple coherent encod-ing and it can even surpass the coherent-state encodingupper bound for m = 1. channel capacity C (Φ m , E ) is m log E . The rate at-tainable with a thermal coherent-state ensemble plusphotodetection behaves like m log E [43]. If instead m is sent to infinity with E/m := k fixed, the rateper use of the transmission line, with a thermal en-semble, converges to C (Φ , k ). For proofs of these factssee SM. Squeezing enhances the communication rate.—
Inthis section we show that the addition of squeezingcan greatly enhance the coherent-state communica-tion rate in the presence of phase noise. Unlike thecase of coherent states, Eq. (5), if we set (cid:126)r (cid:54) = 0 above,the optimization of the prior cannot be reduced toone or few global variables and it becomes hard toprovide an upper bound on the optimal squeezed-coherent rate. Still, an achievable lower bound on thisrate, R lowsq − coh (Φ m , E ), can be obtained by generaliz-ing the on/off encoding: with probability p we send aHaar-random pulse ˆ U | (cid:126)r, (cid:126)α (cid:105) of energy Ep , otherwise thevacuum. Moreover, as for coherent states, for m = 1an explicit OOP strategy with a fixed pulse | (cid:126)r, (cid:126)α (cid:105) suf-fices to obtain the same rate. Numerical evidencesuggests to concentrate all the energy in one pulsebefore ˆ U , i.e., (cid:126)r = ( r, , · · · ,
0) and (cid:126)α = ( α, , · · · , α ∈ R and r >
0. Intuitively, this is aimed atreducing the photon-number variance as pointed outafter Eq. (4). The results can be found in the SM.In Fig. 1 we compare R lowsq − coh (Φ m , E ) with R up , lowcoh (Φ m , E ) per unit of capacity, as a function ofthe energy E and for several values of m . The plotshows that the use of squeezing provides a large in-crease of the communication rate for all m and E with respect to its direct counterpart R lowcoh (Φ m , E ).Importantly, for m = 1, there is also a small range FIG. 2. Plot (log-linear scale) of several rates per unity ofchannel capacity. (a/left) Coherent and squeezed-coherentlower bounds (continuous) and their achievable counter-parts with photodetection (dashed), for m = 2. (b/right)Gaussian coherent-state ensembles on all the m = 2 modesor with a fixed reference on one mode. of energies where R lowsq − coh (Φ m , E ) > R upcoh (Φ m , E ),proving that an explicit on/off strategy with fixedsqueezed-coherent pulse and photodetection is suffi-cient to surpass the Poisson capacity, a question thatcan be traced back to [24]. The deviation of OOPfrom the lower bound for m > o ( E ), both the lower bound R lowsq − coh (Φ m , E )and the squeezed-coherent OOP rate are equal to thecoherent-state lower bound. Communication with phase reference states.—
Theoptimality of covariant encodings makes strategieswith phase reference states suboptimal in principle,but it is still worth to compare them with covari-ant encodings. In the SM we calculate the expres-sion of the rate for an encoding which uses Ex en-ergy in the first mode to prepare a fixed phase ref-erence state and E (1 − x ) on the remaining m − m − E (1 − x ). Asymptotically at high energy, the leadingterm of this upper bound is ( m −
1) log E , indepen-dently of x and of the reference state, which is lessthan what one obtains with a thermal ensemble of co-herent states by log E . The comparison in the finiteenergy regime with encodings using a truncated phasestate | ψ (cid:105) = [2 xE + 1] − / (cid:80) xEn =0 | n (cid:105) and a thermal en-semble of coherent states is available in Fig. 2b. Thissuggests suboptimality of phase-reference strategies.On the other hand, we conjecture that, if the phasereference state has large energy variance, even cod-ing on the energy of the phase reference mode is notbeneficial, in light of the discussion after Eq. (4). Conclusions.—
We have analyzed the performanceof Gaussian encodings in the presence of phase-noisewith a finite decoherence time, such that m succes-sive signals can be sent before losing the phase refer-ence. This is a physically-motivated example of non-Gaussian channel, and we showed that good encod-ings make an intelligent use of the relative degrees offreedom, rather than trying to synchronize a commonphase. Indeed, phase synchronization schemes withquantum-enhanced phase estimation seem to be un-favored with respect to general coherent-state strate-gies, if the global energy cost is taken into account.Moreover, we showed that squeezing can greatly en-hance the communication rate, as an effect of reduc-ing the entropy of the total-photon-number distribu-tion. In particular for m = 1 we proved that, for thefirst time to our knowledge, an explicit strategy, al-ternating between the vacuum and a squeezed coher-ent state, together with photodetection, outperformsany coherent-state code. This is particularly inter-esting considering that it can be easily realized withcurrent technology, while the as-yet-unknown optimalcoherent-state rate will need in general the use of en-tangled measurements at the receiver side, which arestill challenging. We leave as an open questions: theoptimality of strategies employing non-zero squeez-ing among Gaussian states for any m and E ; thesub-optimality of ensembles using states with super-Poissonian statistics, which is good for phase syn-chronization, with respect to coherent-state strate-gies. Moreover, we did not consider the possibilityof sending entangled squeezed states across the chan-nel uses, which could in principle further enhance thecommunication rates due to superadditivity. Acknowledgments.—
M. F. and V. G. acknowledgesupport from PRIN 2017 “Taming complexity withquantum strategies”. M. R., M. S. and J. C. acknowl-edge support from the Spanish MINECO, projectFIS2016-80681-P with the support of AEI/FEDERfunds; the Generalitat de Catalunya, project CIRIT2017-SGR-1127. This project has received fundingfrom the European Union’s Horizon 2020 research andinnovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreement No 845255. M. S. also ac-knowledges support from the Baidu-UAB collabora-tive project Learning of Quantum Hidden MarkovModels. ∗ [email protected] † [email protected][1] S. D. Bartlett, T. Rudolph, and R. W. Spekkens,Rev. Mod. Phys. , 555 (2007), arXiv:0610030[quant-ph].[2] C. M. Caves and P. D. Drummond, Rev. Mod. Phys. , 481 (1994).[3] A. S. Holevo and V. Giovannetti, Reports Prog. Phys. , 046001 (2012), arXiv:1202.6480.[4] A. S. Holevo, Quantum Systems, Channels, Informa-tion (De Gruyter, Berlin, Boston, 2012).[5] V. Giovannetti, S. Guha, S. Lloyd, L. Maccone, J. H.Shapiro, and H. P. Yuen, Phys. Rev. Lett. , 4(2004), arXiv:0308012 [quant-ph].[6] A. Mari, V. Giovannetti, and A. S. Holevo, Nat.Commun. (2014), 10.1038/ncomms4826.[7] V. Giovannetti, R. Garc´ıa-Patr´on, N. J. Cerf,and A. S. Holevo, Nat. Photonics , 796 (2014),arXiv:1312.6225.[8] V. Giovannetti, A. S. Holevo, and R. Garc´ıa-Patr´on,Commun. Math. Phys. , 1553 (2015).[9] F. Caruso, V. Giovannetti, C. Lupo, and S. Mancini,Rev. Mod. Phys. , 1203 (2014), arXiv:1207.5435.[10] An alternative is to send a finite number of signals in parallel using different frequency slots.[11] J. H. Shapiro and S. R. Shepard, Physical Review A(1991), 10.1103/PhysRevA.43.3795.[12] D. W. Berry and H. M. Wiseman, Phys. Rev. Lett. , 5098 (2000).[13] J. P. Gordon and L. F. Mollenauer, Opt. Lett. ,1351 (1990).[14] K. H. Wanser, Electron. Lett. , 53 (1992).[15] L. C. Sinclair, F. R. Giorgetta, W. C. Swann, E. Bau-mann, I. Coddington, and N. R. Newbury, Phys. Rev.A - At. Mol. Opt. Phys. , 023805 (2014).[16] M. G. Genoni, S. Olivares, and M. G. Paris,Phys. Rev. Lett. (2011), 10.1103/Phys-RevLett.106.153603, arXiv:1012.1123.[17] S. Olivares, S. Cialdi, F. Castelli, and M. G. A. Paris,Phys. Rev. A - At. Mol. Opt. Phys. , 1 (2013),arXiv:1305.4201.[18] M. Jarzyna, K. Banaszek, and R. Demkowicz-Dobrza´nski, J. Phys. A Math. Theor. , 275302(2014).[19] A. Klimek, M. Jarzyna, V. Lipi, K. Banaszek,M. G. A. Paris, V. Lipi´nska, A. Klimek, K. Banaszek,and M. G. A. Paris, Opt. Express , 1693 (2016),arXiv:1509.00009.[20] H. Adnane, B. Teklu, and M. G. A. Paris, J. Opt.Soc. Am. B , 2938 (2019), arXiv:1909.07138.[21] M. T. DiMario, L. Kunz, K. Banaszek, and F. E.Becerra, npj Quantum Inf. (2019), 10.1038/s41534-019-0177-4, arXiv:1907.12515.[22] G. Chesi, S. Olivares, and M. G. Paris, Phys. Rev.A , 032315 (2018), arXiv:1710.09577.[23] H. P. Yuen and J. H. Shapiro, IEEE Trans-actions on Information Theory (1978),10.1109/TIT.1978.1055958.[24] B. E. Saleh and M. C. Teich, Phys. Rev. Lett. ,2656 (1987).[25] A. Vourdas and J. R. Da Rocha, J. Mod. Opt. ,2291 (1994).[26] H. P. Yuen (2004) arXiv:0109054 [quant-ph].[27] A. Monras, Phys. Rev. A - At. Mol. Opt. Phys. ,033821 (2006).[28] H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwa-sawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura,D. W. Berry, T. C. Ralph, H. M. Wiseman, E. H.Huntington, and A. Furusawa, Science (80-. ). ,1514 (2012).[29] G. Cariolaro, R. Corvaja, and G. Pierobon, PhysicalReview A - Atomic, Molecular, and Optical Physics(2014), 10.1103/PhysRevA.90.042309.[30] C. Sparaciari, S. Olivares, and M. G. A. Paris, J.Opt. Soc. Am. B , 1354 (2015).[31] M. Jarzyna, V. Lipi´nska, A. Klimek, K. Banaszek,and M. G. A. Paris, Optics Express (2016),10.1364/oe.24.001693, arXiv:1509.00009.[32] C. M. Caves, Phys. Rev. D , 1693 (1981).[33] L. Wang and G. W. Wornell, IEEE Trans. Inf. Theory , 4299 (2014).[34] M. Cheraghchi and J. Ribeiro, (2018),arXiv:1801.02745.[35] Note that, although the channel is additive, i.e., itscapacity is attained with product encodings over dif-ferent uses, superadditivity may arise due to the con-strained input. For simplicity, we will restrict to prod-uct Gaussian encodings in the paper.[36] A. Serafini and G. Adesso, J. Phys. A Math. Theor. , 8041 (2007).[37] A. M. Perelomov, Commun. Math. Phys. , 222(1972), arXiv:0203002 [math-ph].[38] W. M. Zhang, D. H. Feng, and R. Gilmore, Rev. Mod. Phys. , 867 (1990).[39] H. P. Yuen, Phys. Rev. A , 2226 (1976).[40] J. J. Gong and P. K. Aravind, Am. J. Phys. , 1003(1990).[41] A. Martinez, J. Opt. Soc. Am. B , 739 (2007).[42] A. Lapidoth, L. Wang, J. H. Shapiro, andV. Venkatesan, in IEEE Conv. Electr. Electron. Eng.Isr. Proc. (2008) pp. 654–658, arXiv:0810.3564.[43] A. Lapidoth and S. M. Moser, IEEE Trans. Inf. The-ory , 303 (2009). [44] R. J. Glauber, Phys. Rev. , 2766 (1963),arXiv:Phys. Rev. Letters 10, 84 (1963).[45] A. Serafini, Quantum Continuous Variables: APrimer of Theoretical Methods (CRC Press, 2017).[46] M. Hayashi,
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Classical capacity of the phase noise channel
In this section we show that the classical capacity of Φ m is C (Φ m , E ) = m g ( Em ). Since the entropy ismaximized, under an average-energy constraint, by a thermal state [2], it is straightforward to see that for anensemble E ( k ) with energy constraint kEχ (Φ ⊗ km , E ) ≤ S (ˆ ρ th ( kE )) = mk (cid:88) j =1 g ( E j ) = k m g ( Em ) , (8)where the first inequality follows from discarding negative terms in the Holevo quantity and using the fact thatfor the relative entropy 0 ≤ D (ˆ ρ || ˆ ρ th ( kE )) = S (ˆ ρ th ( kE )) − S (ˆ ρ ) whenever Tr [ˆ ρ ˆ n ] = E , with equality if andonly if ˆ ρ = ˆ ρ th ( kE ). ˆ ρ th ( kE ) is a thermal state of mk modes with total average energy kE , which can alwaysbe written as tensor product of single-mode thermal states with average energies E j = E/m . Moreover, theentropy of each single-mode thermal states is g ( E j ) := ( E j + 1) log( E j + 1) − E j log E j . Finally, the upperbound of Eq. (8) is achievable using an ensemble of tensor-product Fock states on m modes, which are pure andinvariant under the action of Φ m , with a thermal probability distribution of total average energy E , so that theaverage output state of the channel is exactly ˆ ρ th ( E ). Hence we conclude that C (Φ m , E ) = m g ( Em ). Moreover,the same arguments above apply to any phase-noise channel with arbitrary phase distribution, provided thatthe phase-shift is identical on each mode, so that their capacity is given by the same expression. Note that thisis the same rate attainable by m uses of the identity channel with average energy per mode Em , implying that,if the sender and receiver can produce and detect Fock states, then Φ m is essentially noiseless. Decomposition into irreducible representations of U( m ) In this section we determine the decomposition into irreducible representations of U( m ) of the Hilbert spaceof m modes. Since coherent states are an overcomplete set, we can first restrict to study the action of U( m ) oncoherent states and then straightforwardly extend the result to arbitrary bosonic states via the decompositionˆ ρ = (cid:90) d m (cid:126)α P ρ ( (cid:126)α ) | (cid:126)α (cid:105) (cid:104) (cid:126)α | , (9)where P ρ ( (cid:126)α ) is the Glauber-Sudarshan P -representation [44, 45] of the m -mode bosonic state ˆ ρ .We will make use of a crucial property that connects coherent states of an infinite-dimensional system withspin-coherent states of finite dimension [37, 38]. First, note that an m-mode coherent state can be decomposedas | (cid:126)α (cid:105) = ∞ (cid:88) n =0 (cid:113) P ( s ) n | ψ n ( (cid:126)α ) (cid:105) , (10)where s := | (cid:126)α | the mean energy of the state, P ( s ) n := e − | (cid:126)α | | (cid:126)α | n √ n ! , is a Poissonian distribution, and ˆΠ n | (cid:126)α (cid:105) = (cid:113) P ( s ) n | ψ n ( (cid:126)α ) (cid:105) . Explicitly | ψ n ( (cid:126)α ) (cid:105) = (cid:88) (cid:80) mi =1 n i = n (cid:115)(cid:18) n { n i } (cid:19) m (cid:89) i =1 u n i i | (cid:126)n (cid:105) where we have introduced the multi-mode Fock state | (cid:126)n (cid:105) = | n (cid:105) ⊗ · · · ⊗ | n m (cid:105) , and (cid:126)u := (cid:126)α | (cid:126)α | . Now observe thateach | ψ n ( (cid:126)α ) (cid:105) lives in a finite-dimensional subspace and it can be mapped to the state of n copies of a m -levelsystem state with coefficients (cid:126)u : ( m (cid:88) i =1 u i | i (cid:105) ) ⊗ n = (cid:88) (cid:80) mi =1 n i = n m (cid:89) i =1 u n i i (cid:88) σ ∈ S n U ( σ ) (cid:12)(cid:12)(cid:12) (cid:126)n ( m ) (cid:69) ∼ = | ψ n ( (cid:126)u ) (cid:105) , (11) where (cid:12)(cid:12) (cid:126)n ( m ) (cid:11) is the tensor-product state (cid:12)(cid:12) (cid:126)n ( m ) (cid:11) = | , · · · , (cid:124) (cid:123)(cid:122) (cid:125) n , · · · m, · · · , m (cid:124) (cid:123)(cid:122) (cid:125) n m (cid:105) , with n i repetitions of the i -thbasis element, U ( σ ) is a permutation of the m -level systems and the isomorphism is defined on the basis ofpermutation-symmetric states (cid:0) n { n i } (cid:1) − / (cid:80) σ ∈ S n U ( σ ) (cid:12)(cid:12) (cid:126)n ( m ) (cid:11) → | (cid:126)n (cid:105) . Finally, thanks to this mapping, the actionof an energy-preserving Gaussian unitary ˆ U corresponding to U ∈ U( m ) in phase space, can also be written asˆ U | (cid:126)α (cid:105) = | U (cid:126)α (cid:105) = ∞ (cid:88) n =0 (cid:113) P ( s ) n ˆ d ( n,m ) U | ψ n ( (cid:126)u ) (cid:105) , (12)where ˆ d ( n,m ) U is the image of U with respect to the irreducible representation of U( m ) on the permutation-symmetric subspace of n m -level systems. This is enough to conclude that each block with total photonnumber n hosts the irreducible representation of U( m ) corresponding to the Young diagram of one row oflength n , which has dimension (cid:0) n + m − m − (cid:1) [46].By Schur’s lemma it then follows that the Haar average decoheres blocks with different total photon numbersand, inside each block with fixed total photon number, it acts as a U( m )-twirling: (cid:90) U( m ) dU ˆ U | (cid:126)α (cid:105) (cid:104) (cid:126)α | ˆ U † = ∞ (cid:88) n =0 P ( s ) n (cid:90) U( m ) dU ˆ d ( n,m ) U | ψ n ( (cid:126)u ) (cid:105) (cid:104) ψ n ( (cid:126)u ) | ˆ d ( n,m ) † U = ∞ (cid:88) n =0 P ( s ) n ˆΠ n (cid:0) n + m − m − (cid:1) . (13)This result can then be applied to each coherent-state term in the decomposition of Eq. (9), obtaining Eq. (2)of the main text. Pure-state ensembles are always optimal among Gaussian encodings
Consider an ensemble comprising general Gaussian states of the form ˆ ρ G = ˆ U ˆ S ( (cid:126)r ) ˆ D ( (cid:126)α )ˆ ρ th ˆ D † ( (cid:126)α ) ˆ S † ( (cid:126)r ) ˆ U † ,where ˆ ρ th is an m -mode thermal state , ˆ U is an m -mode PI and ˆ D ( (cid:126)α ), ˆ S ( (cid:126)r ) are the tensor product of single-mode displacement operators ˆ D ( α i ) = exp( α i ˆ a † i − α ∗ i ˆ a i ) and squeezing operators ˆ S ( r i ) = exp( r i (ˆ a i − ˆ a † i )),respectively. Now recall that any thermal state can be decomposed as a mixture of coherent states withGaussian weights, i.e., ˆ ρ th = (cid:82) d m (cid:126)β p G ( (cid:126)β ) | (cid:126)β (cid:105)(cid:104) (cid:126)β | [45] and hence every Gaussian state ˆ ρ G can be writtenas a mixture of pure Gaussian states with Gaussian weight. Then for any mixed-state Gaussian ensemble E G := { q x , ˆ ρ G ( x ) } , respecting the mean-energy constraint, one can consider an equivalent pure-state Gaussianensemble ˜ E G := { q x p G ( (cid:126)β | x ) , ˆΨ G ( (cid:126)β, x ) } , comprising all the pure states ˆΨ( (cid:126)β, x ) = (cid:12)(cid:12)(cid:12) ψ ( (cid:126)β, x ) (cid:69) (cid:68) ψ ( (cid:126)β, x ) (cid:12)(cid:12)(cid:12) , with (cid:12)(cid:12)(cid:12) ψ ( (cid:126)β, x ) (cid:69) = ˆ U x ˆ S ( (cid:126)r x ) ˆ D ( (cid:126)α x ) (cid:12)(cid:12)(cid:12) (cid:126)β (cid:69) , that take part in the decomposition of some ˆ ρ G ( x ), with proper weights. Thenby the equivalence of these two ensembles and the concavity of the entropy we obtain, for any channel Φ actingon m bosonic modes, χ (Φ , E G ) = S (cid:18)(cid:90) dx q ( x )Φ(ˆ ρ G ( x )) (cid:19) − (cid:90) dx q ( x ) S (Φ(ˆ ρ G ( x ))) ≤ S (cid:18)(cid:90) dx q ( x ) p G ( (cid:126)β | x )Φ( ˆΨ( (cid:126)β, x )) (cid:19) − (cid:90) dx q ( x ) p G ( (cid:126)β | x ) S (cid:16) Φ( ˆΨ G ( (cid:126)β, x )) (cid:17) = χ (Φ , ˜ E G ) . (14)This implies that, when optimizing the Holevo quantity over Gaussian encodings, one can always restrict topure states. General bounds on the coherent-state optimal rate
An upper bound on Eq. (5) can be easily obtained by bounding the first term and the last two terms separately.For the latter we employ a recent upper bound on the capacity of the Poisson channel with average-energyconstraint E [34]: f ( E ) := E log (cid:32) (cid:0) e γ (cid:1) E + 2 E e γ E + 2 E (cid:33) + log (cid:32) √ e (cid:32)(cid:114) e γ ) E + 2 E E − (cid:33)(cid:33) ≥ max q ( t ) (cid:20) H (cid:18)(cid:90) ds q ( s ) P ( s ) (cid:19) − (cid:90) ds q ( s ) H ( P ( s ) ) (cid:21) , (15)where γ is the Euler-Mascheroni constant. For the first term in Eq. (5) instead we observe that (cid:80) ∞ n =0 P ( s ) n log (cid:0) n + m − m − (cid:1) is a concave function of s. Indeed its second derivative evaluates to d d s { ∞ (cid:88) n =0 P ( s ) n log (cid:18) n + m − m − (cid:19) } = s − ∞ (cid:88) n =0 P ( s ) n (( s − n ) − n ) log (cid:18) n + m − m − (cid:19) =+ s − ∞ (cid:88) n =0 P ( s ) n ( s log (cid:18) n + m − m − (cid:19) + s ( n + 1) log (cid:18) n + mm − (cid:19) − s (2 s + 1) log (cid:18) n + mm − (cid:19) )= ∞ (cid:88) n =0 P ( s ) n (log (cid:18) n + m − m − (cid:19) + log (cid:18) n + m + 1 m − (cid:19) − (cid:18) n + mm − (cid:19) ) ≤ ∞ (cid:88) n =0 P ( s ) n m (cid:88) i =1 log ( n + i )( n + i + 2)( n + i + 1) < P ( s ) n n α = sP ( s ) n − n α − , note that log (cid:0) x + m − m − (cid:1) = (cid:80) mi =1 log x + ii , and use the factthat log x ( x +2)( x +1) < xx +1 . Therefore, by applying again Jensen’s inequality onthe integral in s , as well as recalling that (cid:82) dp ( s ) s = E , we obtain the following inequality: (cid:90) dp ( s ) ∞ (cid:88) n =0 P ( s ) n log (cid:18) n + m − m − (cid:19) ≤ ∞ (cid:88) n =0 P ( E ) n log (cid:18) n + m − m − (cid:19) , (17)Hence the optimal coherent-state rate can be upper bounded for all E and m by R upcoh (Φ m , E ) := f ( E ) + ∞ (cid:88) n =0 P ( E ) n log (cid:18) n + m − m − (cid:19) . (18)Out of all the upper bounds originating from the Poisson channel, that of Eq. (18) appears to be the tightestone for almost all energies and it is the one we employ in Fig. 1. In particular, its low-energy expansion reads R upcoh (Φ m , E ) = E log 1 E + E ( c + ψ ( m )) + o ( E ) , (19)where c = e
12 + γ √ − ≈ . γ is the Euler-Mascheroni constant and ψ ( m ) is the Digamma function, whichbehaves as ψ ( m ) = log m + O ( m ) for large m . Still, at extremely low energies (10 − ÷ − ) and for all m , onecan use another upper bound on the Poisson channel capacity [33] which provides the following asymptotically-tighter upper bound for the rate of our channel:˜ R upcoh (Φ m , E ) = E log 1 E + E ( − log log 1 E + 2 + log 13 + ψ ( m )) + o ( E ) . (20)An achievable lower bound is instead provided by a corresponding one for the Poisson channel. In thefollowing we will adapt an on/off modulation that attains the Poisson channel capacity with unconstraineddecoding at the leading order in E and in general provides a good lower bound for E (cid:46) | (cid:105) with probability 1 − p and otherwise a Haar-random coherent pulse ˆ U | α (cid:105) ⊗ | (cid:105) ⊗ m − of energy | α | = Ep . Following the same reasoning applied in the main text to obtain the optimal coherent-state rate, it isstraightforward to see that the net effect of this encoding is that of inducing an on/off total energy distributionin Eq. (5), i.e., { q (0) = 1 − p, q ( Ep ) = p } . The corresponding rate is R lowcoh (Φ m , E, p ) := p ∞ (cid:88) n =1 P ( E/p ) n log (cid:32) n + m − m − (cid:33) + h (cid:16) − p + pP ( E/p )0 (cid:17) + ∞ (cid:88) n =1 h (cid:16) pP ( E/p ) n (cid:17) − p ∞ (cid:88) n =0 h (cid:16) P ( E/p ) n (cid:17) = p ∞ (cid:88) n =1 P ( E/p ) n log (cid:32) n + m − m − (cid:33) + h (cid:16) − p + pP ( E/p )0 (cid:17) + (cid:16) − P ( E/p )0 (cid:17) h ( p ) − ph (cid:16) P ( E/p )0 (cid:17) , (21) where we have defined h ( x ) = − x log x and used the property h ( xy ) = xh ( y ) + yh ( x ). One can then maximizethis function over p ∈ [0 ,
1] to obtain the best lower bound R lowcoh (Φ m , E ) := max p R lowcoh (Φ m , E, p ). In particular,at low energies the maximum is attained for p = E log E , see [33] and the only term in the sum of order O ( E )is at n = 1, that is pP ( E/p )1 log m (cid:39) E log m . Hence in this limit the lower bound attains the upper boundEq. (20), as reported in the main text. Moreover, note that at low energies one can attain this rate at order O ( E ) by explicit on/off modulation plus photodetection (OOP), that sends independently on each mode afixed coherent pulse of energy Epm with probability p and the vacuum otherwise [33] (see also [25] for a lessperforming generalized PPM strategy). The rate for this strategy is immediately obtained from the on/offstrategy for the m = 1 case.At high energies, we check the rate achievable with a thermal ensemble of coherent states. An ensemble forwhich the average state is the thermal state can be obtained by encoding with probability distribution givenby a Gamma distribution p ( s ) = (cid:0) mE (cid:1) m e − sE/m s m − ( m − , indeed (cid:90) dsp ( s ) P ( s ) n = (cid:18) n + m − m − (cid:19) (cid:18) EE + m (cid:19) n (cid:18) mE + m (cid:19) m = p th n . (22)For this distribution one needs to evaluate only the average output entropy term. From the well known factthat H ( P ( s ) ) ≤ log 2 πe ( s + ) [43, 47], and from Jensen inequality, we have (cid:90) ∞ dsp ( s ) H ( P ( s ) ) ≤ (cid:90) ∞ dsp ( s ) 12 log 2 πe ( s + 112 ) ≤
12 log 2 πe ( E + 112 ) . (23)We then obtain a rate R th = mg ( E/m ) − (cid:90) ∞ dsp ( s ) H ( P ( s ) ) ≥ mg ( E/m ) −
12 log 2 πe ( E + 112 ) = ( m −
12 ) log E + O (1) . (24)On the other hand, by fixing E/m = k and sending m to infinity, we get a rate per use of the transmissionline which approaches the identity channel capacity with energy constraint k : R th m = g ( E/m ) − (cid:90) ∞ dsp ( s ) H ( P ( s ) ) ≥ g ( k ) − m log 2 πe ( km + 112 ) = C (Φ , k ) + O (cid:18) log mm (cid:19) . (25) The squeezed-coherent encoding
The photon-number distribution of a coherent squeezed state ˆ S ( r ) ˆ D ( α ) | (cid:105) , r ∈ R and α ∈ C , is given by p ( n | r, α ) = | c ( n | r, α ) | , where [39, 40] c ( n | r, α ) = ( n ! cosh( r )) − (cid:18)
12 tanh( r ) (cid:19) n/ H n (cid:104) α sinh(2 r ) − / (cid:105) exp (cid:20) − | α | −
12 tanh( r ) α (cid:21) (26)and H n ( γ ) is the Hermite polynomial of order n . Taking α ∈ R , the average energy of the state is E + = cosh(2 r ) + e − r α . Substituting for α we then obtain c ( n | r, E ) = ( n ! cosh( r )) − (cid:18)
12 tanh( r ) (cid:19) n/ H n (cid:34)(cid:115) (2 E + 1 − cosh(2 r )) e r r ) (cid:35) exp (cid:20) − (2 E + 1 − cosh(2 r )) e r r )) (cid:21) . (27) An achievable rate using these states for the encoding is obtained via the following on/off modulation: E p,(cid:126)r,(cid:126)α,U = { (1 − p ) | (cid:105) (cid:104) | ⊗ m , p dU ˆ U ˆ S ( (cid:126)r ) ˆ D ( (cid:126)α ) | (cid:105) (cid:104) | ⊗ m ˆ D ( (cid:126)α ) † ˆ S ( (cid:126)r ) † ˆ U † } , (28)where the vacuum state is sent with probability (1 − p ), while a pulse is sent with probability p . The latteris generated by a product of displacements and single-mode squeezing with fixed parameters on each mode, (cid:126)r, (cid:126)α ∈ R m , followed by a Haar-random passive Gaussian unitary ˆ U on the m modes. All the parameters p, (cid:126)r, (cid:126)α are chosen so as to satisfy an average-energy constraint for the ensemble and the total photon numberdistribution is Q ( (cid:126)r,(cid:126)α ) , with probabilities Q ( (cid:126)r,(cid:126)α ) ( n ) = Q ( (cid:126)r,(cid:126)α ) n = (cid:88) (cid:80) i n i = n m (cid:89) i =1 p ( n i | r i , α i ) . (29)Following the same reasoning leading to (21), the rate achievable with this encoding is R lowsq − coh = p ∞ (cid:88) n =1 Q ( (cid:126)r,(cid:126)α ) n log (cid:18) n + m − m − (cid:19) + h (cid:16) − p + pQ ( (cid:126)r,(cid:126)α )0 (cid:17) + (cid:16) − Q ( (cid:126)r,(cid:126)α )0 (cid:17) h ( p ) − p h (cid:16) Q ( (cid:126)r,(cid:126)α )0 (cid:17) . (30)0 Communicate with phase reference
Consider now the scenario where Alice and Bob use a fraction of the total available energy xE to prepare asingle mode state suitable for estimating the phase of the channel and (1 − x ) E is the average energy of theensemble of coherent states on the remaining m − | ψ (cid:105) ⊗ | (cid:126)α (cid:105) , with | (cid:126)α (cid:105) = ⊗ mi =2 | α i (cid:105) , (cid:104) ψ | ˆ n | ψ (cid:105) = xE , (cid:104) (cid:126)α | ˆ (cid:80) mi =2 ˆ n i | (cid:126)α (cid:105) = | α | . Since Φ m commutes with energy-preserving Gaussianunitaries on the last m − χ phcoh (Φ m , E, x ) = (cid:34) S (cid:18)(cid:90) dp ( (cid:126)α )Φ m ( | ψ (cid:105) (cid:104) ψ | ⊗ (cid:90) dU ˆ U | (cid:126)α (cid:105) (cid:104) (cid:126)α | ˆ U ) (cid:19) − (cid:90) dp ( (cid:126)α ) (cid:90) dU S (Φ m ( | ψ (cid:105) (cid:104) ψ | ⊗ ˆ U | (cid:126)α (cid:105) (cid:104) (cid:126)α | ˆ U )) (cid:35) (31)= (cid:34) S (cid:32)(cid:90) dp ( (cid:126)α )Φ m ( | ψ (cid:105) (cid:104) ψ | ⊗ ∞ (cid:88) n =0 P ( | α | ) n ˆΠ ( m − n (cid:0) n + m − m − (cid:1) ) (cid:33) − (cid:90) dp ( (cid:126)α ) S (Φ m ( | ψ (cid:105) (cid:104) ψ | ⊗ | (cid:126)α (cid:105) (cid:104) (cid:126)α | )) (cid:35) . (32)where ˆΠ ( m − n is the projector on the space of m − n . The first term isΦ m ( | ψ (cid:105) (cid:104) ψ | ⊗ ∞ (cid:88) n =0 P ( | α | ) n ˆΠ ( m − n (cid:0) n + m − m − (cid:1) ) = ∞ (cid:88) l =0 q ( xE ) l | l (cid:105) (cid:104) l | ⊗ ∞ (cid:88) n =0 P ( | α | ) n ˆΠ ( m − n (cid:0) n + m − m − (cid:1) , (33)where q ( xE ) n = Tr (cid:104) ˆΠ n | ψ (cid:105) (cid:104) ψ | ⊗ | (cid:105) (cid:104) | (cid:105) , and the second term can be computed by noting thatTr (cid:104) ˆΠ n | ψ (cid:105) (cid:104) ψ | ⊗ | (cid:126)α (cid:105) (cid:104) (cid:126)α | (cid:105) = (cid:80) nl =0 q ( xE ) l P ( | α | ) n − l . Therefore, denoting q ( E ) the probability distribution such that q ( E ) ( n ) = q ( E ) n and P ( s,E ) the probability distributions such that P ( s,E ) ( n ) = P ( s,E ) n = (cid:80) nl =0 q ( E ) l P ( s ) n − l , the rateis χ phcoh (Φ m , E, x ) = S [ q ( xE ) ] + S [ (cid:90) ∞ ds p ( s ) P ( s ) ] (34)+ ∞ (cid:88) n =0 (cid:90) ∞ ds p ( s ) P ( s ) n log (cid:18) n + m − m − (cid:19) − (cid:90) ∞ ds p ( s ) S [ P ( s,xE ) ] (35)Using a coherent state as reference, and coding with a thermal ensemble for m − x , χ phcoh (Φ m , E, x ) = ( m −
1) log E + 12 log E −