Capacity of a bosonic memory channel with Gauss-Markov noise

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Schäfer, Joachim et al. “Capacity of a Bosonic Memory Channel with

Gauss-Markov Noise.” Physical Review A 80.6 (2009) : 062313. ©

2009 The American Physical Society

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http://dx.doi.org/10.1103/PhysRevA.80.062313

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American Physical Society

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Final published version

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http://hdl.handle.net/1721.1/65077

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Capacity of a bosonic memory channel with Gauss-Markov noise

Joachim Schäfer, David Daems, and Evgueni Karpov

QuIC, Ecole Polytechnique, Université Libre de Bruxelles, CP 165, 1050 Brussels, Belgium Nicolas J. Cerf

QuIC, Ecole Polytechnique, Université Libre de Bruxelles, CP 165, 1050 Brussels, Belgium

and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 共Received 8 July 2009; published 7 December 2009兲

We address the classical capacity of a quantum bosonic memory channel with additive noise, subject to an input energy constraint. The memory is modeled by correlated noise emerging from a Gauss-Markov process. Under reasonable assumptions, we show that the optimal modulation results from a “quantum water-filling” solution above a certain input energy threshold, similar to the optimal modulation for parallel classical Gauss-ian channels. We also derive analytically the optimal multimode input state above this threshold, which enables us to compute the capacity of this memory channel in the limit of an infinite number of modes. The method can also be applied to a more general noise environment which is constructed by a stationary Gauss process. The extension of our results to the case of broadband bosonic channels with colored Gaussian noise should also be straightforward.

DOI:10.1103/PhysRevA.80.062313 PACS number共s兲: 03.67.Hk, 89.70.Kn, 03.67.Bg, 03.67.Mn

I. INTRODUCTION

The growing importance of quantum communication mo-tivates a strong research activity on quantum channels. The classical capacity of a communication channel is defined as the supremum of the rate of classical information that can be transmitted via the channel. An important question for quan-tum channels is whether or not entangling the channel inputs may improve the transmission rate. The additivity conjecture proven to be true for some memoryless quantum channels 共see 关1兴 and references herein兲 implies that entanglement

does not improve the transmission rate. Although this con-jecture was disproved recently for another class of channels in Ref. 关2兴, it remains an interesting question in itself to

determine whether entanglement helps or not to achieve the capacity. In particular, it was shown for some channels with memory, where the environment exhibits a correlation be-tween subsequent uses of the channel, that the transmission rate can be increased by using entangled input states.

In the case of a quantum memory channel with discrete alphabet, the first studies of the classical capacity considered a depolarizing channel关3兴 and a quasiclassical depolarizing

channel关4兴 and showed the existence of a threshold on the

degree of memory above which entangled signals improve the transmission rate with respect to product states. Further studies关5,6兴 on a qubit channel with finite memory derived

bounds on the classical and quantum capacities. In Ref.关7兴,

the classical and quantum capacities were discussed in a gen-eral framework, where furthermore, a malicious third party may have control of the initializing memory. The case of general Pauli channels with memory was studied explicitly in Ref. 关8兴, where it was shown that the optimal states are

either product states or Bell states separated by a memory threshold. The behavior of channels with correlated error was connected to quantum phase transitions in many-body systems in Ref. 关9兴. The quantum capacity of a dephasing

channel was computed in Ref. 关10兴, where an enhancement

of the capacity with increasing memory was proven. For

higher dimensions, it has been shown that the capacity of qudit channels exhibits the same threshold phenomenon as Pauli qubit channels关11,12兴.

For a quantum channel with continuous alphabet, it was first shown for an additive bosonic channel and a lossy bosonic channel, respectively关13,14兴, that in the presence of

a memory, some degree of entanglement between the input states is necessary to achieve the capacity, which is in con-trast to the behavior reported for discrete quantum channels. Indeed, the optimal input states correspond to Einstein-Podolski-Rosen共EPR兲 states with finite squeezing which in-creases with the degree of memory, in contrary to either maximally or nonentangled states in the discrete case. A lossy bosonic channel with correlated environment and an input energy constraint has been studied关15–17兴. Lower and

upper bounds for the classical capacity were derived in Ref. 关14兴 and the capacity was calculated in Refs. 关15–17兴.

In this paper, we discuss a bosonic memory channel with additive noise modeled by a Gauss-Markov process. We present an extension of the model of Ref. 关13兴, where two

uses of a bosonic additive channel with correlated noise were treated, to the case of n channel uses with Gauss-Markov noise. We determine the classical capacity of the channel in the asymptotic limit.

In Sec.II, we recall the definition of the classical capacity of quantum channels and specify the model of the channel under an input energy constraint. SectionIIIis devoted to the treatment of a monomodal channel with phase-dependent noise where the quantum water-filling emerges as optimal solution. In Sec.IV, we introduce the method for finding the capacity and apply it to the Gauss-Markov memory. Under reasonable assumptions, we determine the optimal solution as a global water-filling solution and obtain the classical ca-pacity of the channel. SectionVtreats the capacity in several limiting cases: the limit of full correlations, the classical limit, the case of a modified thermal noise, and the transition from finite to infinite uses.

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II. BOSONIC ADDITIVE CHANNEL

In order to send classical information through a quantum channel, one chooses an alphabet and associates its letters to quantum states ␳_iin, where index i labels the letters of the alphabet. The input states sent through the channel interact with the environment and thus are modified at the output. The action of the channel T is a completely positive, trace-preserving map acting on the input states␳_iin,

␳iout= T关␳iin兴. 共1兲

In the messages, each letter appears with a certain probability piso that the overall modulated input state is described as a

mixture ␳in₌

兺

i pi␳i in . 共2兲

By linearity of T, Eq.共1兲 determines as well the action of the

channel on the overall modulated input共2兲, that is, ␳

¯⬅

兺

i

pi␳iout= T关␳in兴, 共3兲

where we will refer to¯ as the overall modulated output. In␳ order for the overall modulated input to be physical, it has to obey the energy constraint

兺

i

piTr共␳i

in

a†a兲 ⱕ n¯, 共4兲 where n¯ is the maximum mean photon number per use of the channel and will be referred to as “input energy” in the following.

The classical capacity C共T兲 of the channel T represents the supremum on the amount of classical bits which can be transmitted per invocation of the channel via quantum states in the limit of an infinite number of channel uses. This quan-tity can be calculated with the help of the so-called one-shot capacity, defined as in关18兴, C₁共T兲 = sup ␳i in ,pi ␹, 共5兲

where the Holevo␹ quantity reads

␹= S

冉

兺

i piT关␳i in_兴

冊

−

兺

i piS共T关␳i in_兴兲, _共6兲

with the von Neumann entropy S共␳兲=−Tr共␳log␳兲 where log denotes the logarithm to base 2. The supremum in Eq.共5兲 is

taken over all ensembles of 兵pi,␳i

in_{其 of probability}

distribu-tions pi and pure input “letter” states␳iin 关18兴.

The term “one-shot” means that only one invocation of T is needed to calculate Eq.共5兲. Using this quantity, the

capac-ity C共T兲 defined as above may be evaluated in the following way. A number n of consecutive uses of the channel T can be equivalently considered as one parallel n-mode channel T共n兲, which is used only one time. Then the capacity C共T兲 is evaluated in the limit

␣ in

, where d2n_␣_{= dRe兵}_␣

1其dIm兵␣1其...dRe兵␣n其dIm兵␣n其 with Gaussian

distribution f共␣兲 =exp关−␣ T_␥ mod −1 _␣_兴 ␲n

冑

_det共_␥ mod兲 , 共12兲

centered at zero and characterized by the modulation covari-ance matrix ␥mod.

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As we are no longer dealing with probability distributions pibut with probability densities f共␣兲, the summations in the

formulae above are replaced by proper integrations. The ac-tion of T共n兲on an input state carrying a message␣reads as in 关13兴 T共n兲关␳_␣in兴 =␳_␣out =

冕

d2n␤fenv共␤兲D共␤n兲丢 ¯ 丢D共␤1兲␳␣inD†共␤1兲丢 ¯ 丢D†_共_␤ n兲, 共13兲 with d2n_␤_{= d Re}_兵_␤

1其d Im兵␤1其 ... d Re兵␤n其d Im兵␤n其, ␤=共Re兵␤1其, ... ,Re兵␤n其,Im兵␤1其, ... ,Im兵␤n其兲T,

and the displacement operator D共␤j兲=exp共␤jaˆj

†₋_␤

j

ⴱ_aˆ

j兲. The

displacement is applied according to the Gaussian distribu-tion of the environment

fenv共␤兲 = exp关−␤T_␥ env −1_␤_兴 ␲n

冑

_det共_␥ env兲 , 共14兲

with共classical兲 covariance matrix ␥env. If this matrix is not diagonal, then the environment introduces correlations be-tween the successive uses of the channel. These correlations model the memory of the channel.

Since we centered the distributions of the environment and modulation as well as the Wigner function of the input state around zero, the covariance matrices of the state carry-ing the message␣at the output of the channel␳_␣outand of the overall modulated output state ␳read, respectively,

␥out=␥in+␥env,

␥=␥out+␥mod. 共15兲

The one-shot capacity of such a system is C1共T共n兲兲 = sup

␥in,␥mod

␹n, 共16兲

with the Holevo␹ quantity which then reduces to

␹n= S共␳兲 − S共␳_␣

out_兲. _共17兲

In the case of a Gaussian state ␳, the von Neumann en-tropy can be expressed in terms of the symplectic eigenval-ues␯jof its covariance matrix

S共␳兲 =

兺

j g

冉

兩␯j兩 − 1 2

冊

, g共x兲 =

再

共x + 1兲log共x + 1兲 − x log x, x ⬎ 0 0, x = 0.

冎

共18兲

The energy 共or mean photon number兲 constraint 共4兲 is given

by

1

2n关Tr共␥in兲 + Tr共␥mod兲兴 − 1

2= n¯ . 共19兲 We remark that in Eqs. 共17兲–共19兲, the quantities depend

only on the covariance matrices and not on a particular mes-sage ␣. Therefore, we can fully discuss the action of the channel solely in terms of covariance matrices.

III. MONOMODAL PHASE-DEPENDENT CHANNEL

First we study the monomodal channel T, where the input state undergoes a phase-dependent noise with covariance matrix ␥env=

冉

␥env共q兲 0 0 ␥_env共p兲

冊

, ␥env 共q兲_⫽_␥ env 共p兲_. _共20兲

The optimal solution of a channel with such a noise has not yet been fully studied. In Ref. 关20兴, the capacity of a

multi-access channel with such a noise was studied but the optimal input and modulation were not discussed in detail. It was shown关19兴 for a monomodal thermal channel that a coherent

input state which is modulated according to a Gaussian modulation achieves the capacity. We conjecture that this proof can be extended to the case of the noise given by Eq. 共20兲, i.e., that a Gaussian input state and a Gaussian

modu-lation remain optimal.

The general input and modulation covariance matrices read ␥in=

冉

␥in共q兲 ␥in共qp兲 ␥in共qp兲 ␥in共p兲

冊

, ␥mod=

冉

␥mod共q兲 ␥mod共qp兲 ␥mod共qp兲 ␥mod共p兲

冊

. 共21兲 We now determine the optimal␥in,␥modby solving equa-tion Eq.共16兲 with energy constraint 共19兲 and the requirement

to have a pure input, i.e.,

det␥in= 1

4. 共22兲

With these constraints, we can write out the total Lagrangian of the system, i.e.,

L = g

冉

冑

共␥in共q兲+␥env共q兲 +␥mod共q兲兲共␥in共p兲+␥env共p兲+␥mod共p兲兲 − 共␥in共qp兲+␥mod共qp兲兲2−

1 2

冊

− g

冉

冑

共␥in 共q兲₊_␥ env 共q兲_兲共_␥ in 共p兲₊_␥ env 共p兲_{兲 − 共}_␥ in 共qp兲_兲2₋1 2

冊

−␨共␥_in共q兲+␥_mod共q兲 +␥_in共p兲+␥_mod共p兲兲 −␶共␥_in共q兲␥_in共p兲兲, 共23兲

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where ␨ and ␶ are Lagrangian multipliers. We arrive at a system of eight equations where the solution reads as fol-lows. First, we find that

␥in共qp兲=␥mod共qp兲= 0, 共24兲

i.e., all three covariance matrices are diagonal in the same basis. Second, we obtain

␥in共q兲+␥env共q兲+␥mod共q兲 =␥in共p兲+␥env共p兲+␥mod共p兲 , 共25兲

which can be regarded as a “quantum water-filling”关27兴

so-lution and implies that the optimal output state␥is a thermal state 共see Fig. 1兲. This repartition of input energy for the

preparation of the quantum state and the classical modulation generalizes the situation of parallel classical channels with a joint energy constraint. Here, we have two channels: one for the q quadrature and one for the p quadrature. The eigenval-ues of the covariance matrix of the modulated output state read ␥ ¯共q兲=␥¯共p兲= n¯ +␥env 共q兲₊_␥ env 共p兲 2 + 1 2, 共26兲 which simply reflects the total energy of the system. Third, we obtain the optimal degree of squeezing of the input state

␥in共q兲 ␥in共p兲 = ␥env 共q兲 ␥env共p兲 . 共27兲

Thus, the squeezing at the input has to match exactly the phase dependence of the noise.

Since all matrices are diagonal, we conclude that the vari-ables ␥_in共q兲,␥_in共p兲,␥共q兲_mod,␥_mod共p兲 are the eigenvalues of ␥in,␥mod. Using the condition that the input has to be pure 共22兲, we

conclude from Eq. 共27兲 that ␥in共q兲= 1 2

冑

␥env共q兲 ␥env共p兲 , ␥_in共p兲=1 2

冑

␥env共p兲 ␥env共q兲 . 共28兲 Then, the modulation eigenvalues simply read

␥mod共q兲 =␮−␥out共q兲,

␥mod共p兲 =␮−␥out共p兲, 共29兲

where ␥_out共q,p兲=␥_in共q,p兲+␥_env共q,p兲 and where the quantum water-filling level ␮⬅␥¯共p兲=¯␥共q兲 is determined by Eq.共26兲.

Equa-tions 共28兲 and 共29兲 are depicted in Fig. 1. Note that both modulation eigenvalues 共29兲 have to be nonnegative to be

physical. This condition together with Eq. 共28兲 result in the

threshold on the input energy

n

¯ⱖ n¯_thr=1 2

冉

冑

max兵␥env共q兲,␥env共p兲其

min兵␥env共q兲,␥env共p兲其

+兩␥_env共q兲 −␥_env共p兲兩 − 1

冊

, 共30兲 below which the obtained solutions do not hold.

Knowing all optimal eigenvalues of the system, we can easily determine the one-shot capacity defined by Eq.共16兲,

C₁= g

冉

¯ +n ␥env 共q兲₊_␥ env 共p兲 2

冊

− g共

冑

␥env 共q兲_␥ env 共p兲_{兲 . n¯ ⱖ n¯} thr. 共31兲 One notices that in the first term the algebraic mean of the noise 共␥_env共q兲+␥_env共p兲兲/2 and in the second term the geometric mean关␥_env共q兲␥_env共p兲兴−1/2appear. Comparing C1to the definition of

the Holevo quantity 共6兲, one can say that the first term

cor-responds to the entropy of a thermal state with a mean pho-ton number identical to the mean energy of the system and, roughly speaking, the second term corresponds to the aver-age of the entropies of two thermal states with means

␥env共q兲,␥env共p兲. In the case␥env共q兲=␥env共p兲= N, we recover the capacity

of a monomodal thermal channel C = g共n¯+N兲−g共N兲.

IV. GAUSS-MARKOV CHANNEL

Now we proceed with the central part of this paper: the treatment of the multimode channel as introduced in Eq.共13兲

with a Gauss-Markov noise.

A. Noise model

Assume a Gaussian-distributed real random vector Z that is generated by a Markov process

Zi=␾Zi−1+ Wi, 0ⱕ␾⬍ 1, 共32兲

where Wiare Gaussian distributed and identically and

inde-pendently distributed. We set without loss of generality the expectation values E共Zi兲=E共Wi兲=0. We request in addition

that the variance Var共Zi兲=N, which leads to Var共Wi兲

=共1−␾2_{兲N. Then we conclude that the matrix elements of}

the covariance matrix M for the Gauss-Markov process共32兲

read

Mij共␾兲 ⬅ Cov共Zi,Zj兲 = N␾兩i−j兩, 0ⱕ␾⬍ 1, 共33兲

where N苸R denotes the variance and ␾ is the nearest-neighbor correlation. Equation 共33兲 is a symmetric Toeplitz

matrix which is completely defined by its diagonal elements tk共M兲= N␾兩k兩= t−k共M兲共see Appendix A兲.

q p Quadratures Optimal eigenvalues γ_env(q) γ_env(p) γ_in(q) γ_in(p) γ_mod(q) γ_mod(p) µ

FIG. 1. 共Color online兲 Quantum water-filling solution for the monomodal channel. Optimal eigenvalues for both quadratures.

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In the following, we treat the channel given by Eq.共13兲

with a classical noise covariance matrix

␥env=

冉

M共␾兲 0

0 M共−␾兲

冊

, 共34兲

where M共␾兲 reads as in Eq. 共33兲. In other words, the noise of

the q 共p兲 quadratures results from a Gauss-Markov process with共anti-兲 correlated nearest neighbors.

B. Solution

In order to determine the capacity of the memory channel 共13兲, one needs to find the optimal input and modulation

covariance matrices which obey the energy constraint 共19兲.

One can treat equally this problem in the basis where the channel becomes memoryless provided that one can rotate the noise covariance matrix by a passive symplectic transfor-mation into the basis where it is diagonal. This is possible because a passive symplectic transformation does neither change the entropy nor the energy constraint of the system. For the class of additive bosonic channels, where the noise covariance matrix ␥env does not contain any cross correla-tions between the quadratures, i.e., has the form

␥env=

冉

␥env共q兲 0 0 ␥env共p兲

冊

, 共35兲

one can show that the transformation that diagonalizes the noise is symplectic only if the two block matrices ␥_env共q兲 and

␥env共p兲, respectively, commute共see Appendix A兲.

Applied to the Gauss-Markov noise␥envas defined in Eq. 共34兲, this condition requires that 关M共␾兲,M共−␾兲兴=0.

How-ever, for finite n, one can show that this does not hold and, as a consequence, ␥env cannot be diagonalized by a passive symplectic transformation. In order to overcome this prob-lem, we introduce another noise covariance matrix␥_env

⬘

that does have commuting block matrices关see Eq. 共35兲兴 and,

fur-thermore, an eigenvalue spectrum that asymptotically con-verges to the eigenvalue spectrum of ␥env. The capacity of this modified channel system and of the original one will be identical, since this quantity is found in the limit of an infi-nite number of channel uses.

We introduce the noise covariance matrix

is also diag-onal in this basis.

We now take the limit n→⬁. In Appendix B, we deter-mine the spectrum of M共␾兲 in this limit, which is identical to the asymptotic spectrum of M共C兲共␾兲. Thus, as the spectra of the two noise matrices ␥envand␥env

⬘

are now identical, the optimal input and modulation covariance matrices ␥in and

␥in

⬘

,共␥mod and␥mod

⬘

兲 coincide.

In this limit, the spectrum of ␥env as introduced in Eq. 共34兲 becomes continuous and reads for the two quadrature

blocks

␥env共q兲共x兲 =␥env共p兲共␲− x兲 = ␭共M兲共x兲, 共40兲

where ␭共M兲共x兲 is the spectrum of M共␾兲 given by Eq. 共B7兲,

i.e.,

␭共M兲_{共x兲 = N} 1 −␾2

1 +␾2− 2␾cos共x兲, 共41兲 where x苸关0,␲兴 is the spectral parameter.

For each individual channel, we know from Eq.共25兲 that

if its allocated input energy is sufficient, then the optimal overall output state is a thermal state, i.e.,

␥in共q兲共x兲 +␥env共q兲共x兲 +␥mod共q兲 共x兲 =␥in共p兲共x兲 +␥env共p兲共x兲 +␥mod共p兲共x兲.

共42兲 As in the monomodal case, each input has to match the an-isotropy of the corresponding noise, i.e.,

␥in共q,p兲共x兲 = 1 2

冑

␥env共q,p兲共x兲 ␥env共p,q兲共x兲 . 共43兲

In order to check whether the optimal input state is en-tangled in the original basis 关i.e., the basis where the noise covariance matrix is introduced in Eq. 共34兲兴, we use the

transformation Q given by Eq. 共A4兲 for each quadrature

, 共45兲

where n¯ is the mean maximum photon number constraint defined in Eq.共4兲.

From classical information theory, we know that the amount of classical information sent through a system of parallel channels is maximized by a water-filling solution 关21兴. By Eq. 共42兲 we confirm that this holds for the optimal

modulation spectrum, i.e.,

␥mod共q,p兲共x兲 =␮gl−␥out共q,p兲共x兲, 共46兲

where ␥_out共q,p兲共x兲=␥_in共q,p兲共x兲+␥_env共q,p兲共x兲 is the 共quantum兲 output spectrum and␮_glis now a global quantum water-filling level over all channels, such that

1

␲

冕

0 ␲

dx␥_mod共q,p兲共x兲 = 共1 −␩兲n¯. 共47兲

The solution is graphically represented in Fig. 2. One con-cludes that the input energy threshold which determines whether the modulation 共46兲 is positive over the whole

do-main x苸关0,␲兴 is given by 共1 −␩兲n¯ ⱖ␥out共q兲共0兲 − 1 ␲

冕

0 ␲ dx␥_out共q兲共x兲. 共48兲 Using the definition in Eq. 共45兲 and the property of the

en-vironment 1 ␲

冕

dx␥env共q,p兲共x兲 = N, 共49兲 we define n ¯_thr共␾,N兲 =

冉

1 +␾ 1 −␾− 1

冊冉

N + 1 2

冊

, 共50兲 i.e., the minimum input energy needed to modulate the whole channel spectrum.

If we assume that for given␾, N, n

¯ⱖ n¯_thr共␾,N兲, 共51兲 then共see Fig.2兲, using Eqs. 共45兲 and 共49兲, we determine the

global quantum water filling as

␮gl=共1 −␩兲n¯ + 1 ␲

冕

0 ␲ dx␥_out共q兲共x兲 = n¯ + N + 1 2, 共52兲 which corresponds to the overall modulated output variance of each of the channels, i.e.,

␥

¯共q兲共x兲 =␥¯共p兲共x兲 =␮_gl= n¯ + N +1

2, 共53兲 where ␥¯共q,p兲共x兲 correspond to the spectra of the covariance matrix ␥ as defined in Eq. 共15兲. With the definition of the

symplectic spectra for the quantum output, the total output, and the noise in the asymptotic limit, that is,

␯out共x兲 =

冑

␥out共q兲共x兲␥out共p兲共x兲, ␯

¯共x兲 =

冑

␥¯共q兲共x兲␥¯共p兲共x兲,

␯env共x兲 =

冑

␥env共q兲共x兲␥env共p兲共x兲, 共54兲

we are now ready to determine the capacity of the channel above threshold for an infinite number of uses: we take the limit of Eq.共16兲 where the sum tends to an integral over the

whole domain, i.e.,

C = lim n_→⬁ 1 n␥insup,␥mod ␹n = 1 ␲

冕

0 ␲ dx

再

g

冉

¯␯共x兲 −1 2

冊

− g

冉

␯out共x兲 − 1 2

冊

冎

= g共n¯ + N兲 − 1 ␲

冕

0 ␲ dxg共

冑

␥_env共q兲共x兲␥_env共p兲共x兲兲共55兲 =g共n¯ + N兲 − 1 ␲

冕

0 ␲ dxg关␯_env共x兲兴, 共56兲 where␯env共x兲 is the asymptotic symplectic spectrum of␥env. We conclude that the capacity is given by the difference of the entropy of a thermal state with mean photons n¯ + N and, roughly speaking, the mean entropy of the environment. This is a generalization of the one-shot capacity found for the monomodal channel found in Eq.共31兲.

0 0 1 2 3 4 5 Spectral parameter x Output spectrum, distributed energy π 3/4π π/2 π/4 modulation energy per mode (1−η)n

FIG. 2. 共Color online兲 Global quantum water-filling solution. Output spectra ␥_out共q兲共x兲 and␥_out共p兲共x兲 shown by the solid decreasing curve and dashed increasing curve vs spectral parameter x. The solid bar represents the global quantum water-filling level␮gl; the

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V. LIMITING CASES

In this section, we analyze the capacity 共56兲 in several

limiting cases, such as the case of the classical limit, the limit of full correlations, the case of a symmetric noise model, and the transition from a finite to an infinite number of channel uses.

A. Classical limit

In the classical limit, we increase the input energy and the noise while keeping the signal-to-noise ratio constant. From the limiting behavior of g共x兲,

lim

x→⬁关g共x兲 − log共x兲兴 = 0, 共57兲

we conclude that we have to replace the g共x兲 functions in Eq. 共56兲 by logarithms. The integral term can be simplified with

the help of Ref.关22兴

1

␲

冕

0 ␲

dx log关

冑

␥env共q兲共x兲␥env共q兲共x兲兴 = log关N共1 −␾2兲兴. 共58兲

Thus, we conclude that

lim n ¯,N→⬁,n¯/N=cC⬅ Ccl= log

冋

1 1 −␾2

冉

1 + n ¯ N

冊

册

. 共59兲 Since g共x兲⬍log共x兲, we reach this limit from below and hence, for finite n¯ , N, the classical capacity of the bosonic channel is always smaller than the capacity of the classical Gaussian additive channel. This is the expected result be-cause for the bosonic channel 共for ␾⫽0兲, a certain amount of energy is needed to prepare the input squeezed state, which is not the case for the classical channel. In the classi-cal limit, the relative fraction of energy used for squeezing vanishes to zero and therefore the capacities coincide. In Fig.

3, we plotted the capacity C and relative fraction ␩ versus the noise parameter N. For each ␾, we fixed the signal-to-noise ratio n¯/N⬅n¯_thr共␾, N

⬘

兲/N

⬘

, with N

⬘

⬅1 which guaran-tees to be above threshold ∀ N⬎1. The capacities all tend to their classical limit 共59兲, while all fractions ␩ decrease toward zero. Interestingly, for fixed signal-to-noise ratio, we find that ␩共␾= 0.7兲⬎␩共␾= 0.9兲. This is due to the fact that n

¯_thr共␾= 0.7, 1兲Ⰶn¯_thr共␾= 0.9, 1兲. Although we need more in-put energy to match the higher correlation, its cost with re-spect to the total energy n¯ is lower.

B. Full correlations

We now investigate the limit of full correlations, i.e., ␾ →1. Since the first term in Eq. 共56兲 is independent of␾, we take only the limit of the integral term 共where we drop the constant N inside the eigenvalue functions for now兲. For the integrand, we find that

lim

␾→1g

冉

1 −␾2

冑

共1 +␾2_兲2_{− 4}_␾2_cos2_x

冊

→ 0, 0 ⬍ x ⬍␲.

共60兲 At the borders x =兵0,␲其, the integrand is equal to one for arbitrary ␾. As this contribution is however infinitesimally

small, it is easy to show that the integral in Eq.共56兲 vanishes

and we conclude that lim

␾→1C = g共n¯ + N兲. 共61兲

This result is in contrast to the result in the classical limit 共59兲, for which

lim

␾→1Ccl→ ⬁. 共62兲

Note that Eq.共61兲 is valid only when the waterfilling

condi-tion共50兲 is satisfied. However, as␾gets close to one while n is fixed, at some point this condition will be violated. In order to satisfy this condition with increaseing␾one has to increase n according to 共50兲, so that 共61兲 becomes correct

only as an asymptotic limit, with the right hand side diverg-ing as␾goes to infinity. Nevertheless, we conjecture that for fixed n the capacity of the quantum channel will be finite in the limit ␾→1, because even in the absence of noise the capacity is finite as it is equal to g共n兲. Hence, unlike in the classical case, for our quantum channel there is no diverging benefit from increasing correlations. This conjecture will be studied further in a forthcoming paper.

C. Symmetric correlations

In this section, we consider a modified, symmetric envi-ronment with same correlations in both quadratures

␥env,s=

冉

M共␾兲 0

0 M共␾兲

冊

. 共63兲

As both quadrature blocks have now identical spectra, we treat a set of independent thermal channels when ␥env,s is

0 5 10 15 20 25 30 35 40 45 50 0 0.01 0.02 0.03 0.04 Fraction of input energy for squeezing η 0 5 10 15 20 25 30 35 40 45 500 2 4 6 8 Noise parameter N Capacity C

FIG. 3. 共Color online兲 Fraction of input energy used for squeez-ing ␩ and capacity C in bits per use vs noise parameter N. The increasing curves correspond to the capacities; the decreasing to␩. Gray horizontal lines correspond to the classical limit C_cl of each capacity. For all graphs, the dotted, dashed, and solid line corre-spond to ␾=0.4,0.7,0.9. For each ␾, we set ∀ Nⱖ1:n¯/N = n¯_thr共␾,N⬘兲/N⬘, with N⬘⬅1.

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diagonalized. We recover from Eq.共43兲 immediately that in

the asymptotic limit, the optimal input spectra read

␥in,s共q,p兲共x兲 = 1 2

冑

␥env,s共q,p兲共x兲 ␥env,s共p,q兲共x兲 =1 2, ∀ x, 共64兲 i.e., the overall optimal input state is a set of coherent states and entanglement does not improve the transmission rate. This is in agreement with previous investigations of the two-mode two-model discussed in 关13兴. Recently, the same

observa-tion was made independently in Ref.关23兴. D. From finite to infinite number of uses

If one applies the method introduced in Sec. IV B to a noise covariance matrix, where the two quadrature block ma-trices 关see Eq. 共35兲兴 commute for finite n, then the optimal

transmission rate reads as in Eq.共56兲, where the integral is

replaced by a sum term normalized by n,

R共n兲= g共n¯ + N兲 −1 n

兺

k=1

n

g共

冑

␥_env,k共q兲 ␥_env,k共p兲 兲, 共65兲

where ␥_env,k共q兲 ,␥_env,k共p兲 denote the eigenvalue spectra for the given environment.

Although for finite n the two block matrices M共␾兲 and

M共−␾兲 of the introduced Gauss-Markov noise 共34兲 do not

commute, we investigate R共n兲as it converges with increasing channel uses n to the capacity 共56兲, where ␥_env,k共q兲 ,␥_env,k共p兲 are the spectra of M共␾兲,M共−␾兲 and numerically obtained. In Fig.4, we plot R共n兲for fixed N, various correlation strength

␾, and a fixed input energy n¯, which is above threshold for the strongest ␾. In addition, we denoted by gray bars for each ␾ the asymptotic capacity given by Eq. 共56兲. In the

plotted region, we observe that R共n兲indeed converges to the capacity C.

VI. CONCLUSIONS

The classical capacity of a multimode channel with Gauss-Markov noise was found under certain assumptions above a certain input energy threshold. By diagonalizing the noise covariance matrix, one was led to treat first the capac-ity of a one-mode phase-dependent channel.

In the one-mode case, we have shown that above the threshold, the optimal quantum input state is a squeezed state where the squeezing matches the anisotropy of the noise. Furthermore, we found that the optimal overall modulated output state is a thermal state and therefore the classical modulation is determined by a quantum water-filling level.

For the multimode channel with Gauss-Markov noise, the optimal input and modulation were discussed in the asymptotic limit. Above an energy threshold, the optimal in-put eigenvalue of each channel is similar to the monomodal solution determined by the anisotropy of the noise of the channel. The optimal modulation was found to be given by a global quantum water filling, which lead us to the conclusion that the overall modulated output state is the same thermal state for all channels or equivalently all channel uses. When we rotated the covariance matrix of the overall input state back to the original basis, we confirmed that the first mode is in a thermal state and hence the total state is entangled.

Finally, several limit cases were discussed. First, the clas-sical limit of the channel was discussed and we showed that the quantum expression tends to the classical expression. Second, we argued that in the limit of full correlations, the capacity of the quantum model stays finite whereas it is di-verging for its classical counterpart. In the case of a symmet-ric noise model, we recovered that the optimal input is given by a set of coherent states. Last, we depicted the asymptotic behavior of a function which represents the optimal trans-mission rate for finite uses for commuting quadrature block matrices and observe that it tends to the capacity for infinite channel uses.

In addition, we remark that the proposed solution method can be applied more generally to an environment which is constructed by a stationary Gauss processes because these processes are characterized by symmetric Toeplitz matrices which commute asymptotically 共see Appendix A for more details兲. This means that for any pair of Toeplitz matrices

T , T

⬘

replacing M共␾兲,M共−␾兲 in Eq. 共34兲, the capacity

共above threshold兲 reads as in Eq. 共56兲 where the noise

spec-tra are replaced by the specspec-tra of T , T

⬘

.

ACKNOWLEDGMENTS

N.J.C. thanks Jeffrey H. Shapiro for useful discussions. J.S. thanks Raúl Garcia-Patrón for helpful comments to the one-mode channel solution. We thank Oleg V. Pilyavets for stimulating comments. We acknowledge financial support from the EU under project COMPAS, from the Belgian fed-eral program PAI under project Photonics@be, and from the Brussels Capital region under the projects CRYPTASC and Prospective Research for Brussels program.

APPENDIX A: TOEPLITZ AND CIRCULANT SYMMETRIC MATRICES

In this section, we specify the symplectic transformation that diagonalizes the noise ␥_env

⬘

共36兲 for finite dimension n

and␥env 共34兲 for n tending to infinity.

0 10 20 30 40 50 60 70 80 90 100 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 Nr. of channel uses n R (n ) , Capacity C

FIG. 4. Function R共n兲and capacity C in bits per use vs number of channel uses n. The dotted, dashed, dashed-dotted, and solid curve correspond to R共n兲with␾=0,0.4,0.55,0.7. For each ␾, the corresponding capacity is shown by a gray horizontal line. For all plots, we took N = 1 and n¯ = 7.5⬎n¯thr共␾=0.7,N=1兲.

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A quadratic matrix T with dimension n⫻n is called a Toeplitz matrix关24兴 if

Tij= ti−j共T兲, 共A1兲

where t_i−j共T兲is a series of real numbers and i , j = 1 , . . . , n. The matrix in Eq. 共A1兲 belongs to the Wiener class if 兵tk共T兲其 is

absolutely convergent, that is,

兺k=−⬁ ⬁兩tk共T兲兩⬍⬁. Then the Fourier series

n − 1 2 2␲共n − 1兲 n

冊

冣

. 共A4兲

For even n, the same pattern holds, except for a row n−1/2共1,−1,1, ... ,−1兲 at i=n/2. The diagonal matrix

QTT共C兲Q converges with increasing n to D, where

D = diag共d1,d2, . . . ,dn兲, 共A5兲

where for odd n,

d1= fT共0兲,

d2j= d2j+1= fT共2␲kj/n兲, 共A6兲

dn= fT共␲兲 for even n, where j=1,2, ... ,共n−1兲/2, and fT共x兲

as defined in Eq. 共A2兲. In addition, if a Toeplitz matrix T

belongs to the Wiener class, then QTTQ also converges to D.

This means that T共C兲 and T can be asymptotically diagonal-ized in the same basis and hence that all Toeplitz matrices 共which belong to the Wiener class兲 asymptotically commute. Now we show that the transformation that diagonalizes

␥env

⬘

as defined in Eq.共36兲 is indeed symplectic. A transfor-mation S is called symplectic if

SJST= J, 共A7兲

where J is the commutation matrix defined in Eq.共10兲. For a

covariance matrix␥ of the shape

␥=

冉

␥q 0 0 ␥p

冊

, 共A8兲

a rotation R that diagonalizes␥is of the shape

R =

冉

Uq 0

0 Up

冊

. 共A9兲

Therefore, in order for R to be a symplectic transformation, we find the requirement that

Uq

T

Up= UqUp

T

=1, 共A10兲

where Uq, Up are orthogonal transformations. For the noise

共36兲, both quadrature blocks ␥q,␥p are diagonalized by the

same orthogonal transformation Uq= Up= Q and thus R

cor-responds to a passive symplectic transformation.

APPENDIX B: SPECTRUM OF A GAUSS-MARKOV COVARIANCE MATRIX

Our goal in the following is to determine the spectrum of

M共␾兲 defined in Eq. 共33兲. As this matrix generates a Markov

process, one intuitively expects M−1_共_␾_{兲 to be three diagonal,}

such that only nearest-neighbor terms appear in the exponen-tial of the resulting Gaussian distribution. Therefore, we con-sider a matrix

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V = 1 N 1 +␾2 1 −␾2

冢

1 − ␾ 1 +␾2 0 0 ¯ − ␾ 1 +␾2 1 − ␾ 1 +␾2 0 ¯ 0 − ␾ 1 +␾2 1 − ␾ 1 +␾2 ¯ 0 0 − ␾ 1 +␾2 1 ] ]

冣

, 共B1兲

where 0⫽N苸R,0ⱕ兩␾兩⬍1. We observe that

兺

k=−⬁ ⬁ 兩tk共V兲兩 = 1 N 1 +␾2 1 −␾2

冉

1 + 2 兩␾兩 ␾2_{+ 1}

冊

⬍ ⬁, 共B2兲

and therefore obtain the Fourier series

fV共x兲 = 1 N 1 +␾2 1 −␾2

冉

1 + 2 兩␾兩 ␾2_{+ 1}cos共x兲

冊

, 共B3兲

where the right-hand side of the latter was obtained by using Eq. 共A2兲. From 关24兴, we state that if the spectrum of a

Toeplitz matrix T is strictly positive, then the inverse T−1_is

asymptotically a Toeplitz matrix, with diagonals

tk共T −1_兲 = 1 2␲

冕

−␲ ␲ dxe −ikx fT共x兲 , n→ ⬁. 共B4兲 Inserting Eq.共B3兲 in the latter leads to

tk共V

−1_兲

= N␾兩k兩 共B5兲 or equivalently, we found that

lim

n→⬁

V→ M−1共␾兲. 共B6兲

Hence, in the limit of infinite number of rows and columns, we found the inverse of M共␾兲. We determine the spectrum of

V by using Ref.关26兴, take its inverse, and therefore receive

the spectrum of M共␾兲 in the limit of infinite channel uses ␭共M兲_{共x兲 = N} 1 −␾2

1 +␾2− 2␾cos共x兲, 共B7兲 with the spectral parameter x苸关0,␲兴.

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