The fidelity of general bosonic channels with pure state input

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by  Meisheng Zhao

The fidelity of general bosonic channels with pure state input
Meisheng Zhao,1, Tao Qin,1 and Yongde Zhang2,1
∗
1Department of Modern Physics, University of Science and Technology of China, Hefei 230026, People’s Republic of China
2CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, People’s Republic of China
(Dated: February 1, 2008)
We first derive for the general form of the fidelity for various bosonic channels. Thereby we give
thefidelity of different quantum bosonic channel, possibly with product input and entangled input
respectively,as examples. The properties of the fidelity are carefully examined.
PACSnumbers: 03.65.Ud,03.67.-a,89.70.+c
7
0 Quantumbosonicchannelsareaspecific typeofquan- where Wε = exp iεRT is the Weyl opera-
{ }
0 tum channels with continuous alphabet [1]. Due to their tor. Here ε is 2n dimensional real vector, R =
2 important applications to optical transmission in which (x ,x ,...,x ,p ,p ,...,p ). the state canbe obtainedas
1 2 n 1 2 n
n photons are employed to convey the information, large ρ = (2π)−n κ(ρε)W (ε)+d2nε. A Gaussian state have
a efforts have been devoted to studying the properties of a GaussianRWeyl-Wigner distribution and thus can be
J these kinds of channels. For example, the calculationfor written as below:

3 the quantum and classicalcapacities, entropy, or fidelity
2
has drawn much attention. Particularly, the fidelity can
beappliedtomeasurehowclosethe inputstatesandthe ρG =(2π)−n e−21εΓεT+iDεTW (ε)+d2nε
3 Z
output states of the quantum channels are. A fidelity
v in which the 2n 2n matrix Γ is the covariance
8 of unity implies identical quantum states while a fidelity matrix and 2n di×mensional real vector D are the
5 of zero implies orthogonal quantum states. To some ex- displacements with defination D = tr(R ρ),Γ =
i i i,j
0 tents, it evaluates how well the quantum channels pre-
Re (R D )(R D ) ,i,j = (1,2,...,n). The posi-
7 serve the transmitted information thus it is an essential h i− i j − j iρ
tivityofquantumstatesrequiresΓ+iσ >=0, σ = iσ ,
0 physical quantity in quantum information theory [2, 3]. ⊕ 2
6 and σ2 is the second Pauli matrix, So Γ>=0 is also re-
The evaluation of the fidelity for a class of bosonic
0 quired. When the state is pure, Γ>0 Sp(2n).
channels might be also relevant for quantum tasks such ⊆
/
h as continuous variables cloning and teleportation [3]. AgeneralGaussianchannelρ T(ρ)canbe defined
−→
p by its action on weyl operator [6, 7],
Specifically,herewestudy thefidelityofgeneralquan-
-
t tum bosonic Gaussian channels on occasions input is a
n
a pmuurletipGleausscseinaanrisotsatoef.quTahnetsuemboinsfoonrimcacthiaonnnterlasnrsempirsessieonnt, T(Wε)−→Wε+Ae−21εGεT
u
q thus render our study into a nontrivial one.
: A quantum system with n modes described by n pairs where A,G is 2n 2n real matrices, and G > 0 should
v ×
ofcanonicalcoordinates(x ,p )isaCCR(canonicalcom- be symmetric matrix. Then we obtain output state
i i i
X muterelation)system[4,5],andtheannihilationandcre-
ar a(xtiio,npi)opaecrcaotrodrinsg(atoi,xai+i=) √o1f2(tahie+sea+im)o,dpeis=ar√−e2ir(ealai−teda+it)o. ρout = (2π1)n Z e−21ε(AΓAT+G)εT+iDATεTW (ε)d2nε
For such a system, a state ρ characterized by its Weyl-
Wigner distribution,
Therefore,the input-output fidelity ofa generalGaus-
κ(ρ )=tr(ρW ) sian channel with Gaussian pure input is
ε ε
̥ = Tr(ρ ρ )
in out
= (2π)−n e−21ε(AΓAT+G)εT+iDATεTTr ρinw+(ε) d2nε
Z
(cid:0) (cid:1)
= (2π)−n e−12ε(AΓAT+Γ+G)εT+iD(AT−I)εTd2nε
Z
= 1 exp DAT D 1 DAT D T
det√AΓAT +Γ+G (cid:26)− − 2AΓAT +2Γ+2G − (cid:27)
(cid:0) (cid:1) (cid:0) (cid:1)
2
As Γ > 0 and G > 0, so the displacements D will not Anotherimportantexampleistheclassicalnoisechan-
increase the fidelity. nel for which A=I,G>0. For these channels displace-
We will present some examples in next steps. mentshavenoeffectonfidelity,whichisagoodproperty
The first important example of Gaussian channel is for process like cloning or teleportation. The Gaussian
the single-mode amplification channel, for which A = C-VcloneandalargeclassofC-Vteleportationbothcan
√η I2, G = (η 1) I2, I2 is the identity matrix of two justbedescribedasasinglemodeclassicalnoisechannel.
−
dimension. Amplificationchannelhasimportantapplica-
tion in C-V cloning process. For this kind of channel,we Wewillconsidertheoccasionabitmorecomplexwhen
can give the fidelity directly there exists correlated noise.
̥=(det (1+η)Γ+η 1) 1 Now memory arises in bosonic Gaussian channel. The
−
p − bosonicGaussianmemorychannelischaracterizedbythe
the maximum ̥=2/(3η 1) reaches at Γ=I/2. following map [8]
−
$:$(ρ )= d2β d2β q(β ,β )D(β ) D(β )ρ D+(β ) D+(β )
in 1 2 1 2 1 2 in 1 2
Z ⊗ ⊗
with and lower bound ̥Low
Mem
q(β1,β2)= 1 e−β+γN−1β
π2 γ 1
| N| ̥Low =
p Mem (N +1)2 N2
whereβ =[ (β ), (β ), (β ), (β )]T andγ isthe −
1 1 2 2 N
ℜ ℑ ℜ ℑ
covariance matrix of the noise quadratures
From the equations above, we can see that when the
N 0 xN 0
− memoryintensityxofthischannelbecomeslarger,thefi-
0 N 0 xN
γN = xN 0 N 0  delityalsobecomeslarger. Besides,thefidelitydecreases
 −0 xN 0 N  with growing noise variance N.
 
Quantum entanglement has proven to be a valuable
where x is the correlation coefficient ranging from 0 to
resource which has wide applications in quantum infor-
1. When x = 1, the channel is with full memory; when
mation domain [9]. Its intrinsic nonlocal nature often
x = 0, the channel is memoryless.we can easily rewrite
enables it to have better performance to accomplish the
this channel in Heisenberg picture:
quantum information tasks.
$(W )=W eεγNεT When the input state to the bosonic memory chan-
ε ε
nel is entangled continuous variable state, more complex
Assume the input state is product coherent state
results are expected.
ρ = γ γ γ γ Assume the input state is bipartite entangled vacuum
in 1 2 1 2
| ih |
state
Then we have the following relations:
Γ= 12 ρin =S(γ)|00i= cos1hγetanhγa+1a+2 |00i
Finally the fidelity comes to be
1 In this case, we have Γ
̥ =
Mem
det√2Γ+γ
N
1
= (1) cosh2r 0 sinh2r 0
(N +1)2 N2x2 0 cosh2r − 0 sinh2r
− Γ= 
sinh2r 0 cosh2r 0
fidReleitmyehmabseurpinpgerthbaotuxndrḁnUgepspfrom 0 to 1, therefore the  − 0 sinh2r 0 cosh2r 
Mem
1
̥Upp = and
Mem (N +1)2
3
cosh2r+N 0 sinh2r xN 0
− −
0 cosh2r+N 0 sinh2r+xN
G=γ = 
N sinh2r xN 0 cosh2r+N 0
 − 0 − sinh2r+xN 0 cosh2r+N 
 
Consequently, the fidelity is With the presence ofmemory, the fidelity of the chan-
nel with product coherent input state increases with
1
̥= (2) growing memory intensity x. If the input state to the
1+N2+2Ncosh2r x2N2 2xNsinh2r bosonic memory channel is entangled squeezed vacuum
− −
state,thefidelityofthechannelalsoincreaseswithgrow-
Examining Eq. (2), we find that same principles hold:
ing squeeze parameter γ, which represents how highly
the more noisy the channel, the less the fidelity of the
entangled the state is.
channel. Besides, the fidelity of the channel increases
The studies here can be deepened by exploiting the
withlargersqueezeparameterr. Andstill,the fidelityof
fidelity of the channel with more complex input states.
the channel grows larger with increasing memory inten-
Our study is limited to finite uses of the bosonic chan-
sity x.
nels. The situation for the fidelity of the channel of infi-
In this paper, we study partially the fidelity in cases
nite uses of the bosonic channels is expected to be more
of bosonic Gaussian channel with a pure state input. In
interesting.
principle, the fidelity decreases with noise variance N.
When there is leakage of energy from the channel into The fidelity is a crucial physical quantity to evaluate
the environment, the fidelity drops off with passing time the quality of information transmission. We hope our
t. researchcan make this issue more clarified.
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