Classical capacity of the lossy bosonic channel: the exact solution

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by  Vittorio Giovannetti

Classical capacity of the lossy bosonic channel: the exact solution
V. Giovannetti1, S. Guha1, S. Lloyd1,2, L. Maccone1, J. H. Shapiro1, and H. P. Yuen3
1Massachusetts Institute of Technology – Research Laboratory of Electronics
2Massachusetts Institute of Technology – Department of Mechanical Engineering
77 Massachusetts Ave., Cambridge, MA 02139-4307.
3Northwestern University – Department of Electrical and Computer Engineering,
2145 N. Sheridan Rd., Evanston, IL 60208-3118.
The classical capacity of the lossy bosonic channel is calculated exactly. It is shown that its
Holevoinformation isnotsuperadditive,andthatacoherent-stateencodingachievescapacity. The
capacity of far-field, free-space optical communications is given as an example.
PACSnumbers: 03.67.Hk,42.50.-p,89.70.+c,05.40.Ca
4
0
0 Aprincipalgoalofquantuminformationtheoryiseval- Classical capacity.– The classical capacity of a chan-
2
uating the information capacities of important commu- nel can be expressed in terms of the Holevo information

n nication channels. At present—despite the many efforts
a that have been devoted to this endeavor and the theo- χ(pj,σj) S( pjσj) pjS(σj), (1)
J retical advances they have produced [1]—exact capacity ≡ Xj −Xj
5
results are known for only a handful of channels. In this
1 where pj are probabilities, σj are density operators and
paper we consider the lossy bosonic channel, and we de- S(̺) Tr[̺log ̺] is the von Neumann entropy. Since
2 velop an exact result for its classical capacity C, i.e., it is n≡ot−known if2χ is additive, C must be calculated by
v the number of bits that it can communicate reliably per
maximizing the Holevo information over successive uses
2 channel use. The lossy bosonic channel consists of a col- of the channel, so that C =sup (C /n) with
1 n n
lection of bosonic modes that lose energy en route from
080 tshpeacteraonrsompittitcearltfiobtehretrreacnesivmeirs.siTonyp,iicnawlehxiachmpphleostaornesfareree Cn =pmj,aσxj χ(pj,N⊗n[σj]), (2)
3 employedtoconveytheinformation. Theclassicalcapac- where the states σ live in the Hilbert space ⊗n of n
j
0 ity of the lossless bosonic channel—whose transmitted successive uses of the channel and is the coHmpletely
/ N
h states arriveundisturbedatthe receiver—wasderivedin positive map that describes the channel [9]. In our case,
p [2, 3]. When there is loss, however, the received state istheHilbertspaceassociatedwiththebosonicmodes
- is in general different from the transmitted state, and Hused in the communication and is the loss map. Be-
t
n quantum mechanics requires that there be an accompa- cause is infinite dimensional,NC diverges unless the
n
a H
nyingquantumnoisesource. In[4]afirststeptowardthe maximization in Eq. (2) is constrained: here we assume
u
q capacity of such channels was given by considering only that the mean energy of the input state in each of the n
: separable encoding procedures. Here, on the contrary, realizationsofthechannelisafixedquantity . Formul-
v E
it is proven that the optimal encoding is indeed separa- timode bosonic channels, is given by , where
Xi ble. We obtain the value of C in the presence of loss k is the loss map for theNkth mode, whNichkcNakn be ob-
N
r when the quantum noise source is in the vacuum state, tained,tracingawaythevacuumnoisemodeb ,fromthe
a k
i.e., when it injects the minimum amount of noise into Heisenberg evolution
the receiver. Our derivation proceeds by developing an
a′ =√η a + 1 η b , (3)
upper bound for C and then showing that this bound k k k − k k
p
coincides with the lower bound on C reported in [5, 6]. with a and a′ being the annihilation operators of the
Ourupperboundresultsfromcomparingthecapacityof k k
input and output modes and 0 η 1 is the mode
k
the lossy channel to that of the lossless channel whose ≤ ≤
transmissivity (quantum efficiency).
averageinput energy matches the averageoutput energy
The main result of this paper is that the capacity of
constraint for the lossy case [7]. This argument is anal-
the lossy bosonic channel, in bits per channel use, is
ogous to the derivation of the classical capacity of the
erasure channel [8]. The lower bound comes from cal- C =max g(η N ), (4)
k k
culating the Holevo information for appropriately coded Nk Xk
coherent-stateinputs. Thus,becausethe twobounds co-
whereg(x) (x+1)log (x+1) xlog xandwherethe
incide,wenotonlyhavethecapacityofthelossybosonic ≡ 2 − 2
maximizationisperformedonthemodalaveragephoton-
channel,but we also knowthat capacitycanbe achieved
number sets N that satisfy the energy constraint
by transmitting coherent states. { k}
~ω N = , (5)
k k
E
Xk
2
(ω is the frequency of the kth mode). Discussion.– Some important consequences derive
k
We derive Eq. (4) by giving coincident lower and up- from our analysis. First, capacity is achievedby a single
per bounds for C. The right-hand side of Eq. (4) was use of the channel (n = 1) employing random coding—
shown, in [6], to be a lower bound for C by generaliz- factorized over the channel modes—on coherent states
ing the narrowband analysis of [5]. This expression was as shown in Eq. (7). This means that, at least for
obtained from Eq. (2) by calculating χ for n = 1 under this channel, entangledcodewords are not necessaryand
the following encoding: in every mode k we use a mix- thatthe Holevoinformationis notsuperadditive. Notice
ture of coherent states µ weighted with the Gaussian that the lossy bosonic channel can accommodate entan-
k
| i
probability distribution glement among successive uses of the channel, as well
as entanglement among different modes in each channel
pk(µ)=exp[ µ2/Nk]/(πNk). (6) use. Surprisingly, neither of these two strategies is nec-
−| |
essaryto achievecapacity. Nor is it necessaryto use any
This corresponds to feeding the channel the input state
non-classical state, such as a photon number state or a
squeezed state, to achieve capacity; classical (coherent
̺= dµpk(µ) µ k µ , (7) state) light is all that is needed. Classical light suffices
Z | i h |
Ok because the loss map simply contracts coherent-state
N
codewords in phase space toward the vacuum state. Co-
which is a thermal state that contains no entanglement
herentstatesretaintheirpurityinthisprocess,andhence
or squeezing. The right-hand side of Eq. (4) is also an
thenon-positivepartoftheHolevoinformation—thesec-
upper bound for C. To see that this is so, let p¯ , σ¯
j j
ond term of the right-hand side of Eq. (1)—retains its
be the optimal encoding on n uses of the channel, which
maximum value of zero. Despite the preceding proper-
givesthe capacityC of Eq.(2). The definition ofχ and
n
ties,quantumeffects arerelevantto communicationover
the subadditivity of the von Neumann entropy allow us
the lossy bosonic channel. For example, our proof does
to write
not exclude the possibility of achieving capacity using
n
quantum encodings, and such encodings may have lower
C 6S( ⊗n[σ¯])6 S( [̺(l)]), (8)
n N Nk k error probabilities, for finite-length block codes, than
Xl=1Xk those of the capacity-achieving coherent state encoding.
where σ¯ p¯ σ¯ and [̺(l)] is the reduced density This is certainly true for the lossless case. In particu-
≡ j j j Nk k lar, it was already known that C can be achieved with a
operator ofPthe kth mode in the lth realization of the
number-state alphabet [2, 3]; our work shows that there
channel, which is obtained from ⊗n[σ¯] by tracing over
N is also a coherent-state encoding that achieves capacity
all the other modes and over the other n 1 channel
− forthiscase. [Thetwoproceduresemploythesameaver-
realizations. The first inequality in Eq. (8) comes from
ageinputstate,Eq.(7)]. However,the probabilityofthe
bounding C by the amount of information that can be
n
receiver confusing any two distinct finite-length number
transmitted through a lossless channel with input state
state codewords is zero in the lossless case, whereas it is
⊗n[σ¯], viz., the output of the lossy channel with opti-
N positive for all pairs of finite-length coherent-state code-
malinputstateσ¯ [7]. NowletN(l) betheaveragephoton
k words. The lossless case also provides an example of the
number for the state ̺(kl); {Nk(l)} must satisfy the energy possible role of quantum effects at the receiver: the op-
constraint (5) for all l [10]. Moreover, the loss will leave timal coherent-state system uses a classical transmitter,
only η N(l) photons, on average, in the corresponding butitsdetectionstrategy,canbehighlynon-classical[9].
k k
output state [̺(l)]. This implies that Incontrast,theoptimalnumber-statesystemfortheloss-
Nk k less channel requires a non-classical light source, but its
S( [̺(l)])6g(η N(l)), (9) receiver uses simple modal photon counting.
Nk k k k How well can we approachthis capacity using conven-
where the inequality follows from the fact that the term tionaldecodingprocedures? Usingthecoherent-stateen-
on the right is the maximum entropy associated with codingofEq.(7)witheitherheterodyneorhomodynede-
states that have η N(l) photons on average [3, 11]. In- tection, the amount of information that can be reliably
k k
troducing Eq. (9) into (8), we obtain the desired result transmitted is
n I =max ξlog (1+η N /ξ2), (11)
Cn 6 g(ηkNk(l))6nmax g(ηkNk), (10) Nk Xk 2 k k
Xl=1Xk Nk Xk
where ξ = 1/2 for homodyne and ξ = 1 for hetero-
where the maximizationis performed overthe sets N dyne, and where, as usual, the maximization must be
k
{ }
that satisfy Eq. (5). Because Eq. (10) holds for any n, performed under the energy constraint (5). Equation
weconcludethattheright-handsideof(4)isindeedalso (11) has been obtained by summing over k the Shannon
an upper bound for C. capacitiesfortheappropriatedetectionprocedure[3]. In
3
general I < C: heterodyne or homodyne detection can- providealowerboundonC in[6]. Inorderto showthat
notbeusedtoachievethecapacity. However,heterodyne the right-hand side of Eq. (15) is also an upper bound,
is asymptotically optimal in the limit of large numbers consider the lossless broadbandchannel in which the av-
of photons in all modes, N for all k, because erage input power is equal to η , viz., the average out-
k
→ ∞ P
g(x)/log (x) 1 as x . put power of the lossy channel. According to [2], the
2 → →∞
The capacity expression C can be simplified by us- capacityofthischannelis( πη /3) /ln2,whichcoin-
P T
ing standard variational techniques to perform the con- cideswiththeright-handsipdeofEq.(15). Thereasoning
strained maximization in Eq. (4), yielding [6] givenaboveforthesingle-modecasenowimpliesthatthe
broadbandlosslesschannel’scapacitycannotbelessthan
C = g(ηkNk(β)) , (12) thatofthebroadbandlossychannel,thuscompletingthe
Xk proof.
where N (β) is the optimal photon number distribution
k
1/η
k
N (β)= , (13)
k eβ~ωk/ηk 1
−
with β being a Lagrange multiplier that is determined bits)
(
through the constraint on averagetransmitted energy.
In the following sections we calculate the capacities 2(cid:25) !T
C
of some bosonic channels. The first two examples help
clarify the derivation of Eq. (4); the last is a realistic
model of frequency-dependent lossy communication, on
whichwealsoevaluatetheperformanceofhomodyneand
heterodyne detection.
Narrowband channel.– Consider the narrowband
P=P0
channel in which a single mode of frequency ω is em-
ployed. In this case, Eq. (12) becomes
FIG. 1: Capacities of the far-field free-space optical chan-
η nel as a function of the input power P (in the plot P ≡
C = g(cid:18)~ωE(cid:19) , (14) 2π~c2L2/(AtAr)). The solid curve is the capacity C fr0om
Eq. (16), the other two curves are the capacities I from
where N = /(~ω) is the average photon number at the Eq. (18) achievable with coherent states and heterodyne de-
E
input. Equation (14) was conjectured in [5], where it tection (dashed curve) or coherent states and homodyne de-
tection (dottedcurve). Notethat theheterodynedetection I
was given as a lower bound on C. The following simple
approaches theoptimal capacity C in thehigh-power limit.
argument shows that g(ηN) is also an upper bound for
C. Consider the lossless channel that employs ηN pho-
Far-field, free-space optical communication.– Con-
tons on average per channel use. Its capacity is given
sider the free-space optical communication channel in
bymax S(̺),wherethemaximizationisperformedover
̺
input states ̺ with mean energy ′ = η~ωN [12]. The which the transmitter and the receiver communicate
E through circular apertures of areas A and A that are
maximum, computed through variational techniques, is t r
separated by an L-m-long propagation path. At fre-
g(ηN) [3, 11]. The lossless channel cannot have a lower
quency ω there will only be a single spatial mode in the
capacity than the lossy channel, because both have the
transmitter aperture that couples appreciable power to
same average received energy, and the set of receiver
the receiver aperture when the Fresnel number D(ω)
density operators achievable over the lossy channel is a
≡
A A (ω/2πcL)2 satisfies D(ω) 1, [14]. This is the far-
proper subset of those achievable in the lossless system t r
≪
field power transfer regime at frequency ω, and D(ω) is
[7]. This implies that g(ηN) is an also upper bound on
the transmissivityachievedby the optimalspatialmode.
C and hence equal to C.
Abroadbandfar-fieldchannelresultswhenthetransmit-
Frequency-independent loss.– Now consider a broad-
terandreceiverusetheoptimalspatialmodesatfrequen-
band channel with uniform transmissivity, η = η, that
k
employsa setoffrequenciesω =k δω for k N. Inthis cies up to a critical frequency ωc, with D(ωc) 1. In
k ≪
∈ thiscaseweuseη =D(ω )inEq.(12),andthecapacity
case, Eq. (12) gives [13] k k
C becomes [13]
√η π
C = ln2r3P~T , (15) C = ωcT y0dxg 1 , (16)
2πy Z (cid:18)e1/x 1(cid:19)
where =2π/δωisthetransmissiontime,and = / 0 0 −
T P E T
is the averagetransmitted power. Equation(15) was de- where y is a dimensionless parameter inversely propor-
0
rivedforthelosslesscase(η =1)in[2]andwasshownto tional to the Lagrangemultiplier β, which is determined
4
from the power constraint energydevotedtothe transmissionis bounded. Interest-
ingly, quantum features of the signals (such as entangle-
2π~c2L2 y0 dx 1 ment or squeezing) are not required to achieve capacity,
= . (17)
P AtAr Z0 x e1/x−1 because an optimal coherent-state encoding exists. At
the decoding stage, however,quantum effects might still
Although C is proportional to the maximum frequency
be necessary (e.g., in the form of joint measurements on
ω , this factor cannot be increased without bound, for
c the output)asstandardhomodyneandheterodynemea-
fixed transmitter and receiver apertures, because of the
surements are not optimal, except for the high power
far-fieldassumption. Figure1 plotsC versus obtained
P regime where heterodyne detection is asymptotically op-
from numerical evaluation of Eqs. (16) and (17).
timal. The focus of this paper has been the lossy chan-
nel with minimal (vacuum-state) noise. A more general
treatment would include non-vacuum noise, and would
allow for amplification.
!
This work was funded by the ARDA, NRO, NSF, and
=
!)
by ARO under a MURI program.
D(
S
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c 0 0
T −
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quantity(2)yieldsthevonNeumannentropyoftheinput
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state, which is a subadditivequantity.
highpowerlevelsheterodyneprevailsthankstoitsband-
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