Table Of ContentCapacities of Quantum Channels for Massive Bosons and Fermions
Aditi Sen(De)1,2, Ujjwal Sen1,2, Bartosz Gromek3, Dagmar Bruß4, and Maciej Lewenstein1,2,∗
1ICFO-Institut de Ci`encies Fot`oniques, Parc Mediterrani de la Tecnologia, E-08860 Castelldefels (Barcelona), Spain
2Institut fu¨r Theoretische Physik, Universit¨at Hannover, D-30167 Hannover, Germany
3Department of Theoretical Physics, University of Lodz, ul. Pomorska 149/153, PL-90236 Lodz, Poland
4Institut fu¨r Theoretische Physik III, Heinrich-Heine-Universit¨at Du¨sseldorf, D-40225 Du¨sseldorf, Germany
Weconsiderthecapacityofclassicalinformationtransferfornoiselessquantumchannelscarrying
a finite average number of massive bosons and fermions. The maximum capacity is attained by
transferring the Fock states generated from the grand-canonical ensemble. Interestingly, the chan-
nel capacity for a Bose gas indicates the onset of a Bose-Einstein condensation, by changing its
qualitativebehavioratthecriticality,whileforachannelcarryingweaklyattractivefermions,itex-
hibitsthesignaturesof Bardeen-Cooper-Schrieffertransition. Wealso showthat fornoninteracting
6 particles, fermions are bettercarriers of information than bosons.
0
0
2 A communication channel carrying classical informa- densation (BEC), due to the lack of constraint on their
tion by using quantum states as the carriers of informa- number. Massive bosons, however, do exhibit BEC; we
n
tion, has been a subject of intensive studies. The fun- show in this Letter that the channel capacity of mas-
a
J damental result in this respect is the “Holevo bound” sive bosons indicates the onset of BEC, by changing its
8 [1] (see also e.g. [2, 3, 4]), obtained more than 30 years behavior from being concave with respect to tempera-
ago, which gives the capacity of such channels. An es- ture, to being convex. The bosons that we consider in
2
sential message carried by the Holevo bound is that at this Letter are noninteracting. Noninteracting fermions,
v
most n bits (binary digits) of classical information can however, do not exhibit any phase transition. Interact-
8
2 be sent via a quantum system of n distinguishable qubits ing fermions, on the other hand, exhibit the Bardeen-
0 (two-dimensionalquantumsystems). Howeverinrealistic Cooper-Schrieffer (BCS) transition, and as we show in
5 channels,wherethequantumsystemisusuallyofinfinite this Letter, the capacity of interacting fermionic chan-
0 dimensions,theHolevoboundpredictsinfinitecapacities. nels exhibits the onset of such transition. We obtain our
5
In realistic channels, it is therefore important to give a results by simulating a finite number of particles in the
0
physical constraint on the carriers of the information. channel,andnotthethermodynamicallimitofaninfinite
/
h
Information carried over long distances usually em- number of particles. We also show that for a wide range
p
ploys electromagnetic signals as carriers of information. ofpowerlawpotentials,includingtheharmonictrapand
-
t Capacities of such channels have been studied quite ex- therectangularbox,andformoderateandhightempera-
n
a tensively (see e.g. [2, 5, 6]). In this case, the physi- tures,thefermionsarebettercarriersofinformationthan
u cal constraint that is used to avoid the infinite capacity bosons, for the case of noninteracting particles.
q problem, is an energy constraint. Due to the form of Suppose therefore that a sender (Alice) encodes the
:
v the Holevo bound, the ensemble that maximizes the ca- classical message i (occuring with probability p ) in the
i
i pacity, turns out to be the canonical ensemble (or the state ̺ , and sends it to a receiver (Bob). The channel
X i
microcanonical ensemble, depending on the type of the is noiseless, while ̺ can be mixed. To obtain informa-
i
r
a energy constraint) of statistical mechanics [5]. tion about i, Bob performs a measurement M (on the
In recent experiments, it has been possible to produce ensemble E = {pi,̺i}) to obtain the post-measurement
atomic waveguides in optical microstructures [7], or on ensemble {pi|m,̺i|m}, with probability qm. The infor-
an atom chip [8], that may serve as quantum channels mation gained by this measurement can be quantified
of macroscopic (or at least mesoscopic) length scales. by the mutual information IM(E) between the index i
Channels carrying massive particles have possibly fasci- and the measurement results m: IM(E) = H({pi}) −
nating applications in quantum information processing. PmqmH({pi|m}). Here H({ri}) = −Pirilog2ri is the
It is thus important to obtain their capacities. Since we Shannon entropy of a probability distribution {ri}. The
are dealing now with massive information carriers, it is accessible information Iacc(E) = maxMIM(E) is ob-
not enough to put an energy constraint only. Rather, it tained by maximizing over all possible M.
is natural to give a particle number constraint as well The Holevo bound gives a very useful upper bound
as an energy constraint. This, of course, hints at the on the accessible information for an arbitrary ensemble:
grand-canonical ensemble (GCE) of statistical mechan- I (E) ≤ χ(E) ≡ S(̺)− p S(̺ ). Here ̺ = p ̺ ,
acc Pi i i Pi i i
ics. Indeed, we show in this Letter, the ensemble that and S(η) = −tr(ηlog η) is the von Neumann entropy
2
maximizes the capacity of noiseless channels that carry of η. In a noiseless environment, the capacity of such
massive bosons or fermions, under particle number and an information transfer is the maximum, over all input
energy constraints,is GCE. Note that massless photons, ensemblessatisfyingagivenphysicalconstraint,oftheac-
despite being bosons, do not exhibit Bose-Einstein con- cessibleinformation. Itisimportanttoimposeaphysical
2
constraintonthe input ensembles,as arbitraryencoding
anddecoding schemesareincludedin the Holevobound. 2500(cid:13)
This has the consequence that the bound explodes for
2000(cid:13)
infinite dimensional systems: For an ensemble of pure
y(cid:13)
stateswith averageensemble state̺ψ, χ=S(̺ψ),which cit 1500(cid:13)
a
can be as large as log d, where d is the dimension of p
2 a 1000(cid:13)
C
the Hilbert space to which the ensemble belongs. If the
500(cid:13)
purestates areorthogonal,I =χ=log d, sothatthe
acc 2
capacity diverges along with its bound (see e.g. [2]). 0(cid:13)
0,0(cid:13) 0,5(cid:13) 1,0(cid:13) 1,5(cid:13) 2,0(cid:13) 2,5(cid:13) 3,0(cid:13)
To avoid this infinite capacity, one usually uses an en-
T/T(cid:13)
ergy constraint for channels that carry photons (see e.g. c(cid:13)
[2, 5, 6]). Suppose that the system is described by the
Hamiltonian H. Then the average energy constraint on FIG. 1: The channelcapacity, for thecase of noninteracting
a communication channel that is sending the ensemble bosons, plottedagainst T/Tc. Thelower curveisforthecase
E = {p ,̺ }, is tr(̺H) = E. Here ̺ is the average en- of100bosons,ina3Dboxwithpbc. For87Rbatomsinsuch
i i abox of volume1µm3, thermodynamical calculations predict
semble state, and E is the average energy available to
T ≈ 0.4µK. The upper curve is for 500 bosons in the same
the system. The capacity C of such a channel is then c
E trap. ThethermodynamicalestimationsofT arehigherthan
c
the maximum of I (E), over all ensembles, under the
acc thecorresponding values that we obtain (as indicated bythe
average energy constraint. Now Iacc(E) ≤ χ(E) ≤ S(̺), arrows).
andS(̺)ismaximized,underthesameconstraint,bythe
canonical ensemble (CE) corresponding to the Hamilto-
nian H, and energy E (see e.g. [9]). Moreover, this is
an ensemble of orthogonal pure states (the Fock states, implementation of such channels, this number is usually
or in other words number, states), so that C is also only at most moderately high.
E
reached for this ensemble [5]. The channel capacity de- Noninteracting bosons. Here, nb = 1/(eβ(εi−µb) −1),
i
pends solelyonthe averageensemblestate,whichinthis and the channel capacity (in bits) is given by (see e.g.
optimalcaseis the canonicalequilibrium(thermal) state [12]) Cbe = − nblog nb−(1+nb)log (1+nb) .
̺eq = exp(−βH)/Z, where β = 1/kBT, with kB being HereµbEi,Ns the chePmiic(cid:0)alipote2ntiial. Forgivienav2eragepia(cid:1)r-
the Boltzmann constant, and T the absolute tempera-
ticle number N and absolute temperature T, one uses
ture. Z = tr(exp(−βH)) is the partition function. For the energy constraint to find µb for that case. The en-
given E, T is given by E = − ∂ (log Z). As a result
∂β e ergy is then given by the average particle number con-
of particle number nonconservation, noninteracting pho- straint, and the capacity by Cbe . Let us consider the
tons and hence their capacity do not exhibit signatures E,N
case when the trap is a 3D-box of volume L3 with peri-
of a condensation. However, effectively interacting pho-
odic boundary condition (pbc), so that the energy levels
ton fluids and photon condensation effects are possible, are 2π2~2(n2 +n2 +n2), n ,n ,n = 0,±1,.... Here
in principle, by using nonlinear cavities, in which case mL2 x y z x y z
m is the mass of the individual particles in the trap.
thephotonsmayacquireaneffectivemass(seee.g. [10]).
InFig. 1,weplotCbe vs. T/T ,fordifferentN. Here
In the case of channels that carry massive particles, it E,N c
is natural to impose the additional constraintof average Tc =(2π~2/mkB)(N/2.612L3)2/3 isthecriticaltempera-
particlenumber. SupposethatAlicepreparesN particles ture,asobtainedinthethermodynamical (largeN)limit.
in a trap, and transfers them to Bob. Let the trap have Thecapacitychangesitsshapefrombeingconcavetobe-
energy levels ε , and let n be the average occupation ing convex with respect to temperature, at the onset of
i i
number of the i-th level. Then the conservation of the a BEC. The thermodynamical estimation of Tc is higher
satvreariangteopfaartfiixcleednauvmerbaegrereenaedrsgyP, fionria=giNve,naenndertghyeEco,nis- itnhganfoorugrrovwaliunegsNof.TScufcohridnidffiecraetniotnNo,fwaigthapthheasgaaplsorebdeuecn-
n ε =E. The channelcapacityC ofsuchachan- obtainedpreviously(seee.g. [11]). Wehavecheckedthat
nPeli isi tihe maximum of I (E), overEa,lNl ensembles that the predicted approximate gap in Ref. [11], is in agree-
acc
satisfy these two constraints. Under these constraints, ment with our calculations for N = 1000. For lower N
the von Neumann entropy of the average state of the however,the predictionis nolongervalid,as expectedin
system is maximized by GCE. Again the ensemble ele- Ref. [11]. Note that the capacity increases with increas-
mentsarepureandorthogonal(Fockstates),whencethe ing N, and it has the same qualitative behavior for a 3D
channel capacity is reached by the same ensemble. box without pbc, as well as for a harmonic trap.
It is important to stress here that the channel capaci- Fermions are better carriers of information than
ties that we derive in this Letter are all for the case of a bosons. For spin-s noninteracting fermions, nf =
i
given finite average number of particles in the trap, and g/(eβ(εi−µf) + 1), where g = 2s + 1, and µf is
notinthethermodynamiclimit. Thisisbecauseinareal the fermion chemical potential. The channel capac-
3
ity (in bits) is given by (see e.g. [12]) Cfd =
E,N
nf nf nf nf 700(cid:13)
−g i log i +(1− i )log (1− i) . Again, for
Pi(cid:16) g 2 g g 2 g (cid:17) 600(cid:13)
given N and T, one obtains µf from average particle
500(cid:13)
number conservation, which then gives the capacity. y(cid:13)
The bosons that we have considered in this Letter cit 400(cid:13)
a
p 300(cid:13)
are spinless. To make a fair comparison of the capaci- a
C
ties, we consider “spinless”, i.e. polarized fermions with 200(cid:13)
g = 1. Let us start with bosons, and perform the high 100(cid:13)
temperature expansion. First, we expand the fugacity 0(cid:13)
zb = eβµb in powers of N/S1, where Sk = Pie−kβεi, 0,0(cid:13) 0,5(cid:13) 1,0(cid:13) 1,5(cid:13) 2,0(cid:13) 2,5(cid:13) 3,0(cid:13) 3,5(cid:13)
and find the coefficients from the average particle num-
ber conservation. We use this expansion to find an ex- FIG.2: Wecomparethechannelcapacityforthecaseof100
pansion of the nb’s, which in turn is substituted in the (noninteracting) spinless fermions (uppercurve)with that of
i
formula for Cbe . The same calculation is done for 100bosons(lowercurve). Thetrapisa3Dboxwithpbc. For
E,N
fermions. We perform the calculation up to the third thelowercurve,thehorizontalaxisisT/Tc,whilefortheup-
order, and find that CEbe,N = (P3i=1αbi(N/S1)i)log2e+ pisetrhceuFrveer,miittiesmTp/eTrfa,tuwrhe.ere Tf =(~2/2mkB)(6π2N/gL3)2/3
βb(N/S )log (N/S ) + βb(N/S )2log (N/S ), whereas
1 1 2 1 2 1 2 1
Cfd =( αf(N/S )i)log e+βf(N/S )log (N/S )+
E,N Pi i 1 2 1 1 2 1
βf(N/S )2log (N/S ). The coefficients of first order
2 1 2 1
perturbation are equal: αb1 = αf1 = S1 + D1. In spinless bosons trapped in a 3D box with pbc.
the next order, they differ by a sign: αb = −αf =
2 2 Note that we have found that the capacities for a 3D
S /2 − S D /S + D . The third order perturbation
2 2 1 1 2 box without pbc are lowerthan those with pbc, both for
coefficients are again equal: αb3 = αf3 = −3S2+S3/3+ bosons and fermions. The capacity for a harmonic trap,
2S22/S1+(2S22−S1S3)D1/S12−2S2D2/S1+D3. Also,β1b = withthesamecharacteristiclengthscale,hasahigherca-
β1f =−S1, β2b =0, β2f =2S2; here Dk =Piβεie−kβεi. pacity(bothforbosonsaswellasforfermions)thanthat
To find out the potentials and dimensions for which for a 3D box with pbc. More importantly, the capacities
this perturbation technique is systematic, we consider for the case of fermions do not show any signatures of
uniform power law potentials, such as V = rγ, in criticality, as expected.
a d-dimensional Cartesian space of (x ,...,x ), with
1 d Interacting fermions. Until now, we have been deal-
r = x2+...+x2. We calculate S and D , replac-
p 1 d k k ing with the case of noninteracting bosons and fermions.
ing sums by integrations with density of states ρ(ε) =
Although a system of noninteracting massive bosons ex-
ddpddxδ(p2/2m+V −ε), the latter integration being
hibits condensation, this is not the case for noninteract-
R
overthe phasespace. Onemaythencheckthatthe tech-
ing fermions. A systemofinteracting fermions,however,
niqueissystematicwhen1/γ+1/2>1/d. Thisincludes,
can exhibitCooperpairing,and,consequently,superfluid
e.g., the 3D and 2D harmonic potentials, for which we BCS transition (see e.g. [12]). It is therefore interesting
also performed the summations directly, and obtained
tosee whethersucha“condensation”canbe observedin
the same results. Note that 0 = βb < βf, whereas
2 2 the capacity of a channel transmitting trapped interact-
αb =S (1−βhεi +βhεi ) (where hεi stands for the
2 2 β 2β β ing fermions. We consider here a 3D box (of volume
average energy with Boltzmann probabilities e−βεi/S1) L3) with pbc, within which N fermions are trapped.
canbe explicitly evaluatedby using the density of states The fermions behave like an ideal Fermi-Dirac gas, ex-
ρ(ε). As a result, we obtain αb ≤0≤αf, implying that cept when pairs of them with equal and opposite mo-
2 2
Cbe < Cfd for a large range of sufficiently high tem- mentum, and opposite spin components have kinetic en-
E,N E,N
perature, and for power law potentials that satisfy the ergy within an interval ∆ǫ on either side of the Fermi
same condition as systematicity. We have surface. In that case, the pairs experience a weak at-
Theorem. For power law potential traps (with power traction. The Hamiltonian can then be written as H =
γ and dimension d), and for sufficiently high tempera- t (a† a + a† a ) + V a† a† a a ,
P~k ~k ~k,↑ ~k,↑ ~k,↓ ~k,↓ P~k,~l ~k,~l ~k,↑ −~k,↓ −~l,↓ ~l,↑
ture,thecapacityoffermions isbetterthanthatofbosons where t =~2k2/2m, whereas V vanishes except when
~k ~k,~l
when 1/γ+1/2>1/d. |µ − ~2k2/2m| ≤ ∆ǫ and |µ − ~2l2/2m| ≤ ∆ǫ, in
This includes, e.g., the harmonic trap in 2D and 3D,
which case, V = −V < 0. Using mean field ap-
the 3D rectangular box, and the 3D spherical box. The ~k,~l 0
proximation,the averageoccupationnumber inthis case
theoremholdsforquitemoderateT,sinceweworkupto
turns out to be N = (1 − (ǫ /E )tanh(βE /2))/2,
the order (N/S )3. Numerical simulations show indeed ~k ~k ~k ~k
1
where ǫ = t − µ, E = ǫ2 +|∆ |2. Here ∆ =
that it holds also for low temperatures, as seen, e.g., in ~k ~k ~k q ~k ~k ~k
Fig. 2 for 100 spinless fermions and same number of ∆(T), when |ǫ | ≤ ∆ǫ, and vanishing otherwise. This
~k
4
one after finding the temperature difference.
90(cid:13)
Toconclude,wehaveconsideredtheclassicalcapacities
80(cid:13) C(cid:13) of noiseless quantum channels carrying a finite average
number of massive bosons or fermions. We have shown
y (C)(cid:13) 70(cid:13) 0(cid:13) 1(cid:13) that the capacities are attained on the grand-canonical
pacit 60(cid:13) ensemble of statistical mechanics. Capacity of a channel
a
C carryingbosonsindicatestheonsetofBose-Einsteincon-
50(cid:13)
densation, by changing its behavior from being concave
40(cid:13) to convex with respect to the temperature, at the tran-
0,00(cid:13) 0,05(cid:13) 0,10(cid:13)
sition point. Also the signature of the onset of Bardeen-
T/T(cid:13)
f(cid:13) Cooper-Schrieffer transition can be observed for weakly
interacting fermions. We show analytically that for non-
FIG. 3: Channel capacity of 100 interacting fermions in a interacting particles, fermionic channels are better than
3D box trap with pbc, against T/Tf. The capacity is ini- the bosonic ones, in a wide variety of cases.
tially convex, and then becomes concave, for higher T’s, as
WeacknowledgesupportfromtheDFG(SFB407,SPP
illustrated in theinset.
1078,SPP1116,436POL),theAlexandervonHumboldt
Foundation,theSpanishMECgrantFIS-2005-04627,the
ESF ProgramQUDEDIS, and EU IP SCALA.
is the so-called “gap”, given by the equation, ∆ =
~l
V (∆ /E )tanh(βE /2), where the summation runs
0P~k ~k ~k ~k
only for an energy interval ∆ǫ about the Fermi surface.
Note thatthe occupationnumbersinthis casearediffer-
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5
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