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Gaussian matrix-product states for coding in bosonic communication channels Joachim Schäfer, Evgueni Karpov, and Nicolas J. Cerf QuIC, Ecole Polytechnique de Bruxelles, CP 165/59, Université Libre de Bruxelles, B-1050 Brussels, Belgium The communication capacity of Gaussian bosonic channels with memory has recently attracted much interest. Here, we investigate a method to prepare the multimode entangled input symbol states for encoding classical information into these channels. In particular, we study the usefulness of a Gaussian matrix-product state (GMPS) as an input symbol state, which can be sequentially generatedalthoughitremainsheavilyentangledforanarbitrarynumberofmodes. Weshowthatthe GMPScanachievemorethan99.9%oftheGaussiancapacityforGaussianbosonicmemorychannels with a Markovian or non-Markovian correlated noise model in a large range of noise correlation strengths. Furthermore, we present a noise class for which the GMPS is the exact optimal input symbol state of the corresponding channel. Since GMPS are ground states of particular quadratic Hamiltonians, our results suggest a possible link between the theory of quantum communication channels and quantum many-body physics. I. INTRODUCTION channelswithaMarkovianandnon-Markoviannoiseina 2 large region of noise correlation strengths. In Sec. III B, 1 weprovideaclassofnoisychannelsforwhichtheGMPS 0 Quantum communication channels are at the heart of is the exact optimal input state. Since the GMPS is as 2 quantuminformationtheory. Amongquantumchannels, well the ground state of particular quadratic Hamiltoni- n the bosonic Gaussian channels describe very common ans,thissuggestsadirectlinkbetweenthemaximization a physical links, such as the transmission via free space or J ofinformationtransmissioninquantumchannelsandthe optical fibers [1]. The fundamental feature of a quantum energyminimizationofquantummany-bodysystems. In 0 channelisitscapacity,whichisthemaximalinformation 3 Sec. III C, we also observe that the squeezing strengths transmissionrateforagivenavailableenergy. Thecapac- that are needed to realize the GMPS in an optical setup ity can be classical or quantum, depending on whether ] are experimentally feasible. Finally, our conclusions are h one sends classical or quantum information (here, we fo- provided in Sec. IV. p cus on the former). In previous works, it was shown - thatforcertainquantummemory channels,inparticular t n channelswithcorrelatednoise,theoptimalinputsymbol II. GAUSSIAN CAPACITY OF MEMORY a u state is entangled across successive uses of the channel; CHANNELS WITH CORRELATED NOISE see Refs. [2–11] and references therein. In general, such q [ multimode entangled states may be quite hard to pre- A. Gaussian bosonic channels pare, which motivates the present work. 2 v Inthispaper,weaddresstheproblemofimplementing Let us now consider an n-mode optical channel T(n), 0 the (optimal) input symbol state for Gaussian bosonic which can either be a bosonic additive noise channel or 0 channels with particular memory models. For this pur- a lossy bosonic channel. In the following, n single-mode 2 pose, we study the usefulness of the so-called Gaussian channel uses will be equivalent to one use of an n-mode 0 matrix-product state (GMPS) [12, 13] as an input sym- parallelchannel[16]. Eachmodej isassociatedwiththe . 1 bol state for the Gaussian bosonic channel with addi- 0 tive noise [9, 11] and the lossy Gaussian bosonic channel annihilationaˆj andcreationaˆ†j operators,orequivalently 2 [7, 8, 10]. This translationary-invariant state is heav- with the pair of quadrature operators qˆj =(aˆj+aˆ†j)/√2 1 : ily entangled and can be generated sequentially, which and pˆj = i(aˆ†j −aˆj)/√2, which obey the canonical com- v happens to be crucial for its use as a multimode input mutation relation [qˆ,pˆ ] = iδ . By defining the vector i j ij Xi symbol state in the transmission via a Gaussian bosonic of quadratures Rˆ = (qˆ1,...,qˆn;pˆ1,...,pˆn)T, we can ex- r channel. The GMPS are known to be a useful resource press the displacement vector m = Tr[ρRˆ] of any state a for quantum teleportation protocols [14, 15], but, to our ρ, along with its covariance matrix (CM) knowledge, they have never been considered in the context of quantum channels. γ =Tr[(Rˆ m)ρ(Rˆ m)T] J/2, − − − In Sec. II, we give an overview of the method used to (cid:18) 0 I(cid:19) derive the Gaussian capacity of Gaussian bosonic mem- with J =i I 0 , ory channels, following our previous work [3, 9, 11]. Our − original results are presented in Sec. III, where we ad- where I is the n n identity matrix. In phase space, a × dress the use of GMPS in this context. In Sec. III A, we Gaussian state is defined as a state ρ having a Wigner showthattheGMPS,thoughnotbeingtheoptimalinput distributionthatisGaussian; hence,itisfullycharacter- state, is close-to-capacity achieving for Gaussian bosonic ized by its mean m and CM γ. 2 For the channel encoding, we consider a continuous Fromnowon,weconsidertheoptimalinputandmodula- alphabet, that is, we encode a complex number q +ip tion eigenvalue spectra in the limit of an infinite number instead of a discrete index into each symbol state. We ofchannelusesn ,soallmatricesmustbeexpanded →∞ encode a message of length n into a 2n-dimensional real to infinite dimensions, see Ref. [9]. vector r = (q ,q ,...,q ;p ,p ,...,p )T. Physically, this For an input energy n above a certain threshold 1 2 n 1 2 n encoding corresponds in phase space to a displacement n , the optimal eigenvalue spectra are linked via thr by r of the n-partite Gaussian input state defined by its a global quantum water filling solution [9], that is, meanm andCMγ . Themodulationofthemultipar- γq (x) = γp (x) = const., x where x is a contin- in in ∗ ∗ ∀ ∈A tite input state is taken as a (classical) Gaussian multi- uousspectralparameterwithinaspectraldomain and partiteprobabilitydensityp (r)withmeanm and γq,p (x)isthespectrumoftheq andpblocksoftheAopti- mod mod ∗ CM γ . The means of the input state m and clas- malmodulatedoutputCMγ =γq γp . Furthermore, mod in ∗ ∗ ∗ ⊕ sical modulation m can be set to zero without loss the optimal input state was determined as [9, 10] mod ofgeneralitybecausedisplacementsleavetheentropyin- (cid:115) variant; hence, they do not play any role in the capacity 1 γq,p(x) formulas defined in Sec. II B. The action of the chan- γiqn,p∗(x)= 2 γepn,qv(x), (4) nelT(n) isthusfullycharacterizedintermsofcovariance env matrices, that is, which corresponds more precisely to the spectrum of the γout =κγin+κ(cid:48)γenv, (1) q and p blocks of the optimal input CM, γi∗n. We remark γ =γ +κγ , that this holds for both the additive noise [9] and lossy out mod channel [10]. where γout and γ are the CM of the individual out- Inthefollowing,wewillconsidernoisemodels(seeSec. put and modulated output states, respectively. For II C) characterized by a CM with symmetric spectrum, κ=κ(cid:48) =1, Eq.(1)definesthebosonicGaussianchannel i.e., γq (x) = γp ( x), where is the size of with additive noise, where γenv is the CM of a (clas- the speencvtral domaeninv |A.| −Furthermore,|Ath|e noise models sdiecsaclr)ibGinagusnsioainsem-inudltuicpeadrtidteispplraocbeambeinlittsy idnenpshitaysepesnpva(rce) feunlefirlglymnaxtxh{aγteqnivs(xre)q}uA=ireγdeqnvto(0f)u.lfiFlolrtthheisgcloabsea,ltqhueainntpuumt (see Ref. [11] for details). For κ = η and κ(cid:48) = 1 η, water filling solution and Eq. (4), for all x, is given by − with a beamsplitter transmittance η [0,1], Eq. (1) de- ∈ fines the lossy channel where γ stands for the CM of 1 the environment state (see Refe.nv[10] for details). Both n≥nthr ≡γiqn∗(0)+γeqnv(0)− 2 −N¯, (5) channels obey the physical energy constraint that reads Tr(γ + γ )/(2n) 1/2 = n, where n is the mean whereN¯ = 1 (cid:82) dxγq (x)standsfortheaddednoise in mod − x env photon number at the input. energy. Thro|Au|gho∈uAtthispaper,weonlyconsiderthecase above threshold, when n n . Then, the Gaussian thr ≥ capacity of the channel with additive noise is given by B. Gaussian capacity [11] In recent works, we found the Gaussian capacity (i.e., C = g(cid:0)n+N¯(cid:1) thecapacitywhenrestrictedtoGaussianinputstatesac- 1 (cid:90) (cid:18)(cid:113) 1(cid:19) cordingtotheusualGaussianchannelminimumentropy − dxg γoqu∗t(x)γopu∗t(x)− 2 , (6) conjecture) and optimal input encoding for the additive |A| x Gaussian channel with noise correlations between subse- ∈A quent uses of the channel modeled by the CM [9, 11] whereγoqu,pt∗(x)=γiqn,p∗(x)+γeqn,pv(x)accordingtoEq.(1). The function g(x) stands for the entropy of a ther- (cid:18) (cid:19) γ = γeqnv 0 (2) mal state with x photons. It is defined as g(x) = env 0 γepnv (x + 1)log(x+1) xlogx if x > 0, and g(x) = 0 if − x 0, where log(x) denotes the logarithm to base 2. where γeqnv and γepnv are commuting matrices of dimen- ≤Now, if one restricts the input states to independent sion n n. The absence of correlations between q and × coherent states in the case of global water filling, one p in Eq. (2) is generally considered to describe a natural may also define the coherent-state rate [11], which is noise. We found that the optimal input and modulation CM γ and γ are diagonal in the same basis as the given by Eq. (6) replacing γoqu,pt∗(x) by 1/2 + γeqn,pv(x). i∗n m∗od For the lossy channel the quantities given by Eqs. (5) noiseCMγ ,andhavethesameblockstructure. Thus, env γi∗n = γiqn∗⊕γipn∗ and γm∗od = γmq∗od⊕γmp∗od. In addition, abnydre(6p)laacisngwenll as thηen,cγoqh,per(exn)t-stat(e1rateη)aγrqe,po(bxt)aiannedd itmhepolipestimalinputstateispure,i.e.,det(2γi∗n)=1,which γiqn,p∗(x) → ηγiqn,p→∗(x). Neonvte tha→t all t−hese eexnvpressions also rely on the assumption that the Gaussian capacity γ =(cid:18)γiqn∗ 0 (cid:19). (3) of independent Gaussian channels is additive, see Ref. i∗n 0 14(γiqn∗)−1 [17]. 3 C. Noise models (a) TMS TMS TMS Let us introduce two different noise models which will r r r T T T beusedtomodeltheGaussianmemorychannels,namely a Markovian and non-Markovian model. l }{ r l }{ r 1. Markov additive noise 1 � 10 1 � 10 B B InRefs. [9,11],weconsideredaclassicalMarkovnoise 2 2 with variance N 0, given by M ≥ � (cid:18)M(φ) 0 (cid:19) GMPS γ =N , (7) i i+1 envM M 0 M( φ) − (b) where M(φ) is an n n matrix defined as M (φ) = ij × φ|i−j|, with the correlation parameter 0 φ < 1. Note 0 1} ≤ | i that M(φ) and M( φ) commute in the limit of an infi- − nite number of channel uses. In this limit, the spectra of � the quadrature blocks γeqn,pvM ≡NMM(±φ) are given by |0i S(�rB) 10 B 1 φ2 γeqn,pvM(x)=NM 1+φ2 −2φcos(x), x∈[0,2π], (8) |0i S(rB) 2 ∓ FIG. 1. (a) Optical scheme for the preparation of the Gaus- withtheupper(lower)signstandingfortheq(p)quadra- sianmatrix-productstate(GMPS),slightlymodifiedwithre- ture. By using Eq. (4), we find that the optimal input specttoRef. [13]. Here,TMSstandsforatwo-modesqueezed state is an infinite product of squeezed states. Then, vacuum state with squeezing r , while γ represents the T B when rotated back to its original basis, the optimal in- three-mode building block. Note that all TMS and three- put state becomes a multimode entangled state [9, 11]. modebuildingblockscouldeachbegeneratedbyasinglede- vice that is used repeatedly. One half of the TMS generated at time i is used immediately to generate the GMPS mode i, while the other half is sent to a delay line (to be used at 2. Non-Markovian noise time i+1). After two Bell measurements (represented by curlybrackets)involvingthetwoTMShalves(notedlandr) A non-Markovian channel noise model was considered and the two upper modes of γ (noted 1 and 1’) followed by B in Refs. [8, 10], given by appropriateconditionaldisplacements,thethirdmode(noted 2) of γ collapses into the GMPS mode i. (b) Optical setup B (cid:18)esΩ 0 (cid:19) of the three-mode building block γB that is used to generate γ =N , (9) envN N 0 e sΩ anearest-neighborcorrelatedGMPS.Here|0(cid:105)denotevacuum − modes,S(r )isaone-modesqueezerwithparameterr ,and B B the bold horizontal bars represent 50:50 beamsplitters. withN 1/2fortheconsideredlossychannel(N 0 N N for a non-≥Markovian additive noise channel), s R, a≥nd ∈ whereΩisan nmatrixdefinedasΩ =δ +δ . ij i,j+1 i+1,j The spectra of×the quadrature blocks γq,p N e sΩ III. GAUSSIAN MATRIX PRODUCT STATE envN ≡ N ± read Wenowaddressthequestionofhowtoopticallyimple- γq,p (x)=N e 2scos(x), x [0,2π], (10) menttheoptimalinputstates. Inthiscontext,weexam- envN N ± ∈ inetheso-calledGaussianmatrix-productstate(GMPS), withtheupper(lower)signstandingfortheq(p)quadra- whichisheavilyentangledjustastheoptimalinputstate, ture. In the case of a global water filling, it was shown has a known optical implementation, and can be gener- that the optimal input state [Eq. (4)] is also entangled ated sequentially. This state was first discussed in Ref. in the original basis [8, 10], as for the Markov additive [12] as the ground state of particular Hamiltonians of noise. harmonic lattices. In general, GMPS are constructed Since the optimal input state for both noise models by taking a fixed number of finitely or infinitely en- M exhibits multimode entanglement across the subsequent tangled two-mode squeezed vacuum states shared by ad- usesofthechannel, withn , itspreparationmaybe jacent sites, and applying an arbitrary 2 to 1 mode →∞ M a very challenging task. This is what we investigate in Gaussian operation on each site i. the next section. In what follows, we restrict our discussion to a pure, 4 translationally-invariant, one-dimensional GMPS, and, 3 furthermore, to a single finitely entangled two-mode squeezed (TMS) vacuum state per bond between adja- 2.85 cent sites ( = 1). We use the protocol introduced in M Ref. [13],depictedinaslightlymodifiedforminFig.1(a). 2.7 Each GMPS mode i is obtained by operating on a three- ] s mode entangled state (called “building block”; see Refs. bit2.55 [ [13, 18] for details) together with the shares (l and r) e at 2.4 of the two TMS vacuum states connecting site i to the R left and right sites, respectively. As shown in Fig. 1(a), 2.25 a first teleportation is performed by making a Bell measurement on modes l and 1, followed by a conditional 2.1 displacement on mode 1. A second teleportation then (cid:48) is made with a Bell measurement on modes r and 1, 0 0.1 0.2 0.3 0.4 0.5 0.6 (cid:48) φ followed by a conditional displacement on mode 2. The final state of mode 2 then reduces precisely to that of FIG.2. RatesofachannelwithadditiveMarkovnoise: Gaus- theithmodeofthedesiredGMPS.Wefocusnowonthe siancapacityC (solidline),GMPS-rateR (crosses)and GMPS mathematical description of the GMPS and its use as an coherent-staterateR (dashedline)vs. correlationφ,where coh input state, while we discuss its experimental realization from top to bottom N ={0.5,0.7,1}. We took n=5. M with single-mode squeezers in Sec. IIIC. 2.8 A. GMPS as approximating input state 2.7 The CM of the GMPS can be written as 2.6 γGMPS = 21(cid:18)C0−1 C0(cid:19), (11) bits]2.5 [2.4 e where C is a n n circulant symmetric matrix. In at × R2.3 Ref. [12], it was proven that the correlations of a one- dimensional GMPS decay exponentially. Therefore, in 2.2 the limit n → ∞, the spectrum of C−1 reduces (up to a 2.1 change of variance) to the spectrum of M(φ) [21], that is, 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 s 1 λ(C−1)(x) γq (x) 2 ≡ GMPS (cid:18) 1 φ2 (cid:19) FIG. 3. Rates of a channel with additive non-Markovian =N˜ − in +∆ ,(12) noise: Gaussian capacity C (solid line), GMPS-rate RGMPS 1+φ2in−2φincos(x) (crosses) and coherent-state rate Rcoh (dashed line) vs. cor- with x [0,2π], N˜ 0, 0 φ < 1, ∆ R, and relation s, where from top to bottom NN ={0.5,0.7,1}. We ∈ ≥ ≤ in ∈ took n=5. the additional condition ∆N˜ 1/2 ensuring that the ≥ − spectrum corresponds to a quantum state. By com- paring the spectrum of Eq. (12) with the optimal in- Since the noise spectra defined in Sec. IIC satisfy the put spectra [Eq. (4)] for the noise models of Eqs. (7) same symmetry, it is intuitively clear that this type of and (9), one can directly verify that the optimal input GMPS is the most suitable state for these noise mod- state is not a GMPS. However, one may use the GMPS els. The optical setup for the three-mode building block as an approximation of the optimal input state for both that generates this nearest-neighbor GMPS is depicted thesenoisemodels. Bycalculatingthetransmissionrates in Fig. 1(b). More details on it are provided in Sec. III for noise models [Eqs. (7) and (9)] with the GMPS C. as input state [using Eq. (6) and replacing γoqu,pt∗(x) by From Eq. (13) and the fact that the GMPS used as an γq,p (x)+γq,p(x)], we find numerically that the high- input is a pure state, i.e., γq (x)γp (x)=1/4, x, GMPS env GMPS GMPS ∀ est transmission rate is achieved for a GMPS with near- we find that N˜ = (1+φ2 )/(1 φ2 ) and N˜∆ = 1/2. est neighbor correlations γGMPS,n.n. [13]. We find that Thus,thenearestneighboirncorre−lateidn GMPShasqu−adra- among all GMPS given by Eq. (12), which can be gen- ture spectra erated with the setup defined in Fig. 1, only the GMPS 1+φ2 1 withnearestneighborcorrelationshasasymmetricspec- γq,p (x)= in , (14) trum, that is GMPS,n.n. 1+φ2in∓2φincos(x) − 2 γq (x)=γp (π x). (13) with the upper (lower) sign for the q (p) quadrature. GMPS,n.n. GMPS,n.n. − 5 Weconfirmthesamebehaviorforthelossychannelwith 3.5 non-Markovian noise, as shown in Fig. 4 for different beamsplitter transmittances η. Theoptimalinputcorrelationsφ forbothnoisemod- 3 ∗in elsareapproximatelygivenbyφ/2ands/2,respectively, ] as can be seen in Fig. 5(a) and Fig. 5(b). This can be s bit verified as follows. Since the quantum water filling solu- [2.5 e tion holds for the GMPS with nearest neighbor correla- Rat tions, its rate is given by Eq. (6) replacing γoqu,pt∗(x) by γq,p (x)+γq,p(x). In order to find the optimal φ 2 GMPS,n.n. env in itissufficienttominimizeonlythesecondterminEq.(6) as only this term depends on φ . This term is a definite in integral of a function whose primitive is not expressed in 1.5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 s terms of elementary functions and φin. However, if the integrandas afunctionofparameter φ canbeproperly in minimized for all values of the variable of integration x FIG. 4. Rates of a lossy channel with non-Markovian noise: theintegralwillalsobeminimized. Inordertoverifythis GaussiancapacityC(solidline),GMPS-rateR (crosses) GMPS and coherent-state rate R (dashed line) vs. correlation possibility we take the first derivative of the integrand coh s, where from top to bottom η = {0.5,0.7,0.9}. We took and set it to zero. This leads to the following relation: N =1 and n=5. N γq (x) (1+φ 2+2φ cosx)2 env = ∗in ∗in . (16) φ*in rin φ*in rin γepnv(x) (1+φ∗in2−2φ∗incosx)2 0.35 1.4 0.4 1.5 (a) (b) 0.3 1.2 As it happens in the general case, there is no unique pa- 0.32 1.2 0.25 1 rameter φ which satisfies Eq. (16) for all x. Neverthe- 0.2 0.8 0.24 0.9 less, it is ∗ipnossible to obtain an approximating equality 0.15 0.6 0.16 0.6 by neglecting the quadratic and higher order terms in 0.1 0.4 0.08 0.3 the noise spectra given by Eqs. (7) and (9) and in the 0.05 0.2 0 0 0 0 right-hand side of Eq. (16), i.e. 0 0.2 0.4 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 φ s 1+2αcos(x) 1+4φ cos(x) ∗in (17) FanIGd.co5r.reOsppotnimdianlginspquueteczoinrgrelratio(ndaφsh∗ined(sloinlied,lriingeh,tlaefxtisa)xviss). 1−2αcos(x) ≈ 1−4φ∗incos(x) in correlation (φ or s) for (a) the channel with additive Markov where α = φ for the Markovian noise and α = s for noise,wherethecrossesdepictφ/2;(b)thechannelwithnon- the non-Markovian noise, respectively. This is a valid Markoviannoise(lossyandadditive),wherethecrossesdepict approximation taking into account that φ <1 and can s/2. We took for both plots N =N =1 and n=5. ∗in M N be satisfied by a unique parameter φ for all x. Namely, ∗in we find the simple relations φ φ/2 and φ s/2, as ∗in ≈ ∗in ≈ verified in Fig. 5(a) and Fig. 5(b), respectively. Therefore, when looking for the optimal transmission rate, one has to optimize only over the parameter φ . in In order to satisfy the global water filling solution for B. GMPS as exact optimal input state theGMPS,wereplaceγiqn∗(0)byγGqMPS,n.n.(0)inEq.(5), whichleadstoamodifiedinputenergythresholddepend- Although we have seen that the GMPS is not the op- ing on φ , that is, in timal input state for the noise models introduced in Sec. nGthMrPS =nthr−[γiqn∗(0)−γGqMPS,n.n.(0)]. (15) IgIivCe,nibtyis possible to do better. Indeed, for all noises Aswerequirethattheinputenergyn≥nGthMrPS,Eq.(15) γenv =(Nenv Nenv) (C−1 C), (18) imposes an upper bound on φ . ⊕ × ⊕ in In Figs. 2-4, we plot the rates obtained for the GMPS where N is an n n matrix that commutes with C env with the spectrum given by Eq. (14) calculated via a given in Eq. (11), the×GMPS is the exact optimal input maximization over φin, which we denote as RGMPS. In state, that is Fig. 2, we observe that for the channel with additive iMnatrhkeovplnoottiseed(r7e)g,ioRnG,MRPS is c/loCse>-to0-.c9a9p9a.ciFtoyratchheieavdindgi-; γi∗n ≡γGMPS, n≥nGthMrPS, (19) GMPS tive channel with non-Markovian noise (9), we conclude wherenowtriviallynGMPS =n . Thisisadirectresult thr thr from Fig. 3 that the GMPS serves as a very good re- that can be deduced from the shape of the CM γ GMPS source as well; in the plotted region, R /C >0.999. andthefactthattheCMoftheoptimalinputstate(given GMPS 6 by Eqs. (3) and (4)) is diagonalized in the same basis as with θ = I I, where I is the n n identity matrix ⊕− × the CM of the noise. [22], Furthermore, as already mentioned, GMPS are known n tobegroundstatesofparticularquadraticHamiltonians 1(cid:77) Γ = diag t,t , [12]. More precisely, γGMPS is the CM of the ground t 2 { } state of the translationary invariant Hamiltonian, given i=1 n (cid:18) (cid:19) in natural units by 1(cid:77) u u 0 0 ΓT = ,   wt 2 0 0 u u i=1 − − (24) 1 (cid:88) (cid:88) Hˆ = 2 pˆ2i + qîVijqˆj, (20) Γ = 1(cid:77)2n (cid:18)w v(cid:19), i i,j ww 2 v w i=1 where qî and pî are the position and momentum opera- 1 Γ = γ (r ) γ ( r ), tors of an harmonic oscillator at site i and the potential TMS TMS T TMS T 2 ⊕ − matrix is simply given by V =C2, where C is defined in Eq. (11). where γTMS(r) ArealisticexampleforanoiseoftheshapeofEq. (18)   is given by the CM of the (Gaussian) state of the system ch(r) 0 0 0 sh(r) ··· ··· ··· defined in Eq. (20), i.e., a chain of coupled harmonic os-  0 ch(r) sh(r) 0 0 0   ··· ··· .  cillators at finite temperature T. We assume the system  .   0 sh(r) ch(r) 0 0 .  tobedescribedbyacanonicalensemble,thusthedensity  . ··· ··· .   . .  matrix of the oscillators is given by the Gibbs-state  . 0 0 ch(r) sh(r) 0 .  exp( βHˆ) = ... 0 0 sh(r) ch(r) 0 ··· ... , ρG = Tr[exp−( βHˆ)], (21)  ... ... ... ... ... ... ·.·.·. ...  −   where β =1/T. The CM γG of the Gaussian state ρG is  0 ... ... ... ... 0  givenbyEq.(18)withNenv =I+[2exp(βC) I]−1 (see sh(r) 0 0 0 ch(r) Ref.[19]fordetails),whereindeed[N ,C]=−0. There- ··· ··· ··· env fore, if we assume the noise of the channel to result from where ch(r) = cosh(2r) and sh(r) = sinh(2r), respec- a chain of coupled harmonic oscillators at finite temper- tively. ature T, that is, γ = γ , then the GMPS with CM WeobservethatthenearestneighborcorrelatedGMPS env G γ is both the ground state of the system given by requiresonlyonesqueezingparameterr togeneratethe GMPS B Eq. (20) and the exact optimal input state for n n . three-mode building block of Fig. 1(b). Furthermore, thr ≥ we can use finitely entangled TMS vacuum states with squeezing r . For simplicity, we set r = r r [23] T B T in C. Experimental realization and plot in Fig. 5 the squeezing strength neede≡d to gen- eratetheoptimalinputcorrelationφ fordifferentnoise ∗in Let us finally discuss the required optical squeezing correlations. For the Markov noise, in the plotted region strength to realize the optimal input correlation φ∗in for the required correlation does not exceed φ∗in,max ≈ 0.3, both noise models. We first present the mathematical which can be realized by r 1.08 (about 9.4 dB in,max ≈ description of the three-mode building block that gener- squeezing). For the non-Markovian noise, the required ates the GMPS with nearest neighbor correlations. The correlation does not exceed φ 0.4, which cor- ∗in,max ≈ CM of this building block is given by [18] responds to r 1.18 (about 10.2 dB squeezing). in,max ≈ This shows that the required squeezing values for the w v u 0 0 0  presented setup could be realized with accessible non- v w u 0 0 0   linear media for a realistic assumption of noise correla- 1u u t 0 0 0  γ =  , (22) tions(thesemaximalsqueezingvalueshaverecentlybeen B 20 0 0 w v u  −  realized experimentally, see, e.g., Ref. [20]). 0 0 0 v w u − 0 0 0 u u t − − (cid:112) IV. CONCLUSIONS with w = (t+1)/2, v = (t 1)/2 and u = (t2 1)/2, − − where t 1. The optical scheme for the three mode ≥ We have demonstrated that a one-dimensional Gaus- building block is depicted in Fig. 1(b), where S(r ) is B sian matrix-product state, a multimode entangled state a one-mode squeezer with parameter r such that t = B which can be prepared sequentially, can serve as a very cosh(2r ) [18]. The resulting CM of the n-mode pure B goodapproximationtotheoptimalinputstateforencod- GMPS is given by [13] inginformationintoGaussianbosonicmemorychannels. γ =Γ ΓT (Γ +θΓ θ) 1Γ , (23) The fact that the GMPS can be prepared sequentially is GMPS t− wt ww TMS − wt 7 crucial because it makes the channel encoding feasible, tuminformationtheoryandquantumstatisticalphysics. progressively in time along with the subsequent uses of the channel. For the analyzed channels and noise models, the GMPS achieves more than 99.9% of the Gaus- ACKNOWLEDGMENTS sian capacity and may be experimentally realizable as the required squeezing strengths are achievable within J.S. is grateful to Antonio Ac´ın, Jens Eisert, and present technology. Furthermore, we have introduced Alessandro Ferraro for helpful discussions, and acknowl- a class of channel noises, originating from a chain of edges a financial support from the Belgian FRIA foun- coupled harmonic oscillators at finite temperature, for dation. 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