本講主要內(nèi)容1

QKD系統(tǒng)QBER

的計算(以退極化信道為例)2偏振態(tài)動力學(xué)3波色量子高斯信道4非馬爾可夫信道5主方程11QKD系統(tǒng)QBER的計算

(以退極化信道為例)[1]Chang-HuaZhu,Dong-XiaoQuan,FangZhang,Chang-XingPei,ImprovingKeyRateof

OpticalFiberQuantumKeyDistributionSystemBasedonChannelTomography,InternationalJournalofTheoreticalPhysics,February

2013,Vol.52,Issue

2,pp.596-6032So

Where,?,?,ando?arePaulimatrixes,

,andIf

theinitialpolarizationis|A〉=α|0)+β|1),inwhich,

then

thedensityoperatorof

input

state

isPim=|4)(A

。|退偏振信道Where

p

is

the

probability

with

which

the

state

is

depolarized,is

the

completely

mixed

state.For

any

state

p,Channelε:Pm→PoutThedepolarizingchannelmodelgiven

as3光量子計數(shù)率

(count/clickrate)t:the

transmission

rate

ofthe

quantum

channel

at

the

receiver

ηD:the

efficiency

of

the

detector?The

wholetransmissionrateof

thesystemt=t,t?70p.:the

dark

count

probabilitym:the

number

of

the

detectors

in

each

basis.?Thecountrate

tm=1-(1-t)"when

dark

count

is

not

taken

into

account;?The

count

rate

is

tn=tnma+mpa(1-tnm)when

dark

count

is

taken

into

accountewhen

dark

count

is

not

taken

into

account

the

mean

countrate

Rofweak

coherentpulse

atthe

receiver

i

n:the

photon

number

in

each

pulse,Poisson

distributed,α(aB/Km):the

fiber

loss

per

kilometer,?1(Km):the

fiberlength.?Then

the

transmission

rate

of

the

fiber

t,=10%μ:the

mean

photon

number

of

each

weak

coherent

pulse.?ThesiftedkeyrateRa=R-p.=(1-a“+mp2e=)p.Where,p.denotes

the

percent

of

qubit

used

to

sift

key.Qubit

error

comes

from

two

ways,one

is

the

incorrect

clicksofthe

detectors

due

to

channel

noise,the

other

is

the

dark

count

ofdetectors.Where,p

denotes

the

probability

with

which

photonsarrives

at

incorrect

detectors,zis

the

error

probability

fromdark

count,m=2.When

darkcountis

taken

into

accountQBER5偏振失配For

BB84protocol,{|0》,|1),|+)I-},input

density

operators

PmoPmnL,Pimand

Pin,the

output

densityoperators:Poco?Pm?3Pou?and

Pou?,the

corresponding

measurement

operatorsSo,

For

BB84QKD

system

in

which

polarization

codin

.The

sifted

key

rate

of

BB84

QKD

system,the

QBER

Q

can

be

given

as?The

probability

with

which

the

Bob

receives

bit“1”when

Alice

sends

bit“0”?The

probability

with

which

the

Bob

receives

bit“0”when

Alicesends

bit“1”?60.2p=0.01p=0.02p=0.05p=0.08p=0.100.12-0.10.08-0.060.04-0.02-40

60

80100

120Lengthofopticalfiber(Km)QBER0.18-0.16-0.14-QBER20Key

rateR=R(IAB-IAB)IABdenotesthemutualinformationbetweenAliceandBobIAB=1-H?(Q.)Q:the

estimated

QBER

of

the

sifted

keyH?(Q)is

the

binary

entropy

function,H?(Q?)=-Qlog?(Q?)-(1-Q)log?(1-Q).IAB=1+Qlog?(Q)+(1-Q)log?(1-Q)Symmetric

individual

attack,the

maximum

information

obtained

by

Eve

is

82偏振態(tài)動力學(xué)[2]ZHU

Chang-hua,PEI

Chang-xing,QUAN

Dong-xiao,CHEN

Nan,YI

Yun-hui,

PolarizationStateDynamicsof

SinglePhotonPulse

UnderStochasticPolarizationModeDispersion

forOpticalFiberQuantum

Channels,/abs/0908.43709Here,k2(a,z)=B2(@,z)·I+β。(a,z)5(a,z)·σ,β。(@,z)isthepropagationconstantwithoutbirefingenceeffet,andthevectorσ=0e?+O?e?+0?es,with

,and

.I

is

the

unit

matrix.?頻域波動方程Fiberbiefringencevector:b(@,z),b(a,z)=b?e+b?e?+b?es,wheree,e?,e?areunitvectors

in

the

Stoke's

space,w

is

frequency,and

z

is

the

distance

from

input

end.Thebirefringence

effect

on

the

field

canbe

derived

from

wave

equation,which

givese10Weassumeb(a,z)=f(a)b(2),f(a)=γ@+so(a-a)+..Innon-dispersionmedium,onlythe

first

term

remains,that

is

f(の)=yw.?Where,E(a)is

the

amplitude

of

the

field

mode,A(a,z)is

the

fequency

spectrum

of

the瓊斯向量-2-dmionaJoeseco

1L.)Under

slowly-varying

approximation

we

obtainLet11雙折射向量模型Underthe

condition

oflinearbirefringence,let?b(z)=2b?(z)e+2b,(z)e?+0·E=2b?(z)e+2b,(z)?We

select

the

modelA1

proposed

by

Huang

and

Yevick,?and

variance

σ2.ε

corresponds

to

the

mean

fluctuation

magnitude

ofthe

stochastic

fiberbirefringence?0<ε<1.L,denotestheinversecoupling

strength

offiberand

equals

the

shortest

fiber

correlationlength

when

ε→

1.Here

variance

,whereWhe,8?(2)andg,(2)areWhiteGasianNoisewithmeanzeroL,is

the

beat

length.?hereC

。(o,z)andC?(a,z)arepolarizationamplitudeofelcticfield.Letφ(w)be

the

frequency

envelope

of

the

input

single-photon

wave

packet,whichfollows,thenthestatevectortakesthe

form:瓊斯向量和狀態(tài)的解4(z)》=??(a)|a)×[C?(@,z)|1]+C

。(a,z)|o]]do=??(a)|o)×A(a,z)do13The

partial

trace

ofp(z)forthefrequency

freedom?Ps(2)=t。[p(2)]=?do|p(の)2[4(0,2)±+(@,2)]A。>=α|0)+B|1),inwhich

|a2+|β2=1?保真度Thedensityoperatorp(z)canbegivenasThe

degree

ofpolarization(DOP)ofthe

pulsecan

be

given

as?DOP(2)={2#[p?(2)]-1}*um)=(?(a)la)da×|4),Pm=jdo|p()2|4,Y46lThe

fidelity

between

input

state

and

output

state

is14保真度分析結(jié)果0.950.90.80.750.75—Gaussianspectrum----Lorentzianspectrum

---Rectangularspectrum10

1520

30

35

Lengthofopticalfiber(m)rectangularLorentzianGaussian40

45

50Fidelity153波色量子高斯信道[3]朱暢華，裴昌幸，權(quán)東曉，陳南，易運(yùn)暉，

基于信道估計的自適應(yīng)連續(xù)變

量量子密鑰分發(fā)方法

，物理學(xué)報，Vol.58,N0.4,April,2009,pp.2184-218816玻色量子高斯信道令Pim表示輸入態(tài)的密度算子，信道將輸入態(tài)映射為輸出態(tài)Pou?Pm→Pou=C[Pi]若信道輸入為相干態(tài)|a),

則

Pm=la)(a|?A.Holevo

給出了玻色量子高斯信道的表達(dá)式：

?式中平移算符

D(z)=e-2a,a,a

分別為輸入態(tài)

|a)的生成算子和湮滅算子，,Nc

為信道噪聲的方差(即平均光子數(shù))。

?量子光學(xué)高斯態(tài)的密度算子為17玻色量子高斯信道(續(xù))處于高斯態(tài)的光脈沖的光子數(shù)服從泊松分布，平均光子數(shù)為N。?定

理

：

對

于

玻

色

量

子

高

斯

信

道

，

如

果

輸

入

為

高

斯

態(tài)

，

其

方

差

為N,則

經(jīng)

過

信

道

映

射

后的

輸出

態(tài)

也

為

高

斯

態(tài)，

且

其

方

差(

平均光子數(shù))變?yōu)镹+Nc。證明：?所以，輸出態(tài)也為高斯態(tài)，且其平均光子數(shù)變?yōu)镹+Nc。184非馬爾可夫信道[4]WeiLiu,ChanghuaZhu,LinxiZhang,ChangxingPei,PerformanceAnalysisof

Polarization

Coding

BB84

Quantum

Key

Distribution

System

Under

non-Markovian

Channel,Proceedingsof2017

International

Conference

on

Computer,Information

and

Telecommunication

Systems(CITS),Dalian,China,July21-23,201719其中v=t/2t是無量綱時間，A(V)=e?[cos(μV)+sin(μV)/μ]是阻尼諧振子，其頻率為μ?=

√

(4k?π,)2-1,且K2=a2+d(i≠j≠k),a是系統(tǒng)與外部環(huán)

境之間的耦合強(qiáng)度，t

是系統(tǒng)的特定頻率。?具有色噪聲的退偏振信道Kraus

算符形式：其中A=

√

(V)o?,A?=

√5(V)o?,A?=

√?(V)o?,A?=

√5(V)I是Kraus

算符，退偏振信道

20輸入量子態(tài)為

,α和β滿足|a2+IB2=1Pau=∑4PmA?保真度

：保真度210.70.52T=0.5-----

-=a?=a3=0.3-----

-a1=a?=a?=0.5——a?=a?=0.2,a?=5689100.5-0.4-0.3-0.210T=1——

a,=a?=a?=0.3——a?=a?=a?=0.5

——a,=a=0.2,a-58保真度結(jié)果保真度保真度0.922非馬爾科夫0.80.6-0.50.30.2-

0.5

1.52.5馬爾可夫和非馬爾可夫信道對比保真度230.50.45

0.40.35

0.30.250.20.15

0.10.0500

50

100

150Q

BER結(jié)果a?=a?=0.2,a?=1a?=a?=0.2,a?=2a?=a?=0.2,a?=5350

400450

500光纖長度(Km)QBER240.05-50

100

150

200

250300光纖長度(Km)QBER

結(jié)果T=0.8T=1T=1.5350

400

4505C0.5r0.450.40.350.30.250.20.150.1QBER25[5]https://www.weimer.itp.uni-hannover.de/fileadmin/weimer-group/Open2014/master.pdf[6]張永德，量子信息的物理原理，科學(xué)出版社

5主方程26We

can

define

a

superoperator

L

such

that

Cp=-i/[H,p].It

is

called

a

superoperatorbecause

it

is

an

object

that

acts

on

an

operator

and

results

in

a

new

operator.If

theHamiltonian

is

time-independent,we

may

formally

integrate

the

Liouville

von

Neumann

equation

and

obtainp(t)=exp(Ct)p(0)=V(t)p(0),

(2)where

V

is

another

superoperator

that

maps

the

density

matrix

from

its

initial

form

to

its

form

at

timet

and

therefore

is

called

a

dynamical

map.It

is

related

to

the

unitary

evolutionoperator

U(t)=exp(-iHt/左)according

to

V(t)p(0)=U(t)p(O)U(t).(3)超算子The

Liouville

von

Neumann

equation

isgivenby

(1)27Similar

to

the

case

of

a

closed

quantum

system,we

can

write

the

dynamical

map

of

anopen

quantum

system

as

an

exponential

of

the

generator

of

the

semigroup,V(t)=exp(Ct).(1