Quantum Forbidden-Interval Theorems for Stochastic Resonance

0.32 MB·English

by  Mark M. Wilde

Quantum Forbidden-Interval Theorems for Stochastic Resonance
Mark M. Wilde1,2 and Bart Kosko1
∗
1Center for Quantum Information Science and Technology, Department of Electrical Engineering,
University of Southern California, Los Angeles, California 90089, USA and
2Hearne Institute for Theoretical Physics, Department of Physics and Astronomy,
Louisiana State University, Baton Rouge, Louisiana 70803
(Dated: February 2, 2008; Received;Revised; Accepted;Published)
We extend the classical forbidden-interval theorems for a stochastic-resonance noise benefit in a
nonlinear system to a quantum-optical communication model and a continuous-variable quantum
key distribution model. Each quantum forbidden-interval theorem gives a necessary and sufficient
condition that determines whether stochastic resonance occurs in quantum communication of clas-
8 sical messages. The quantum theorems apply to any quantum noise source that has finite variance
0 or that comes from the family of infinite-variance alpha-stable probability densities. Simulations
0 show the noise benefits for the basic quantum communication model and the continuous-variable
2 quantumkey distribution model.

n
PACSnumbers: 03.67.-a,03.67.Hk,42.50.Dv,05.45.Vx,05.45.-a
a
Keywords: stochasticresonance, quantum optics,alpha-stablenoise,quantum communication
J

1
2 Stochastic resonance (SR) occurs in a nonlinear sys- Threshold θ = 1.6, Signal value α = 1.1
x
tem when noise benefits the system [1, 2, 3, 4, 5]. SR
]
h can occur in both classical and quantum systems [4, 6]
p that use noise to help detect faint signals. The footprint 0.2
n
nt- osyfsSteRmispearfonromnmanocneotmoneaicsucruervdeeptehnadtsroensultthsewinhteennstihtye matio 0.15
qua ofofrtahequnaonisteumso-uorpctei.caFlicgoumrem1unshicoawtisonsuscyhstaenmSwRitshurbfoatche ual Infor 0.1
[ additive channel noise and squeezing noise. Mutual in- ut 0.05
formation measures the noise benefits in bits. M

1
0
v The classicalSR forbidden-intervaltheorems give nec- 4
3 −2
41 efistsainrytearnmdssoufffimciuetnutalcoinnfdoirtmioantsiofnor[7a,n8]SRwhneonistehebesnyes-- Channel noise 2− σ 10.5 1.5 1 Squ0eeze− n1oise − r
1
tem nonlinear is a threshold. The noise benefit turns
3
on whether the noise mean or location a lies in an in- FIG. 1: (Color online) Stochastic resonance in the basic
.
1 terval that depends on the threshold θ and the bipolar quantum-optical communication model with Gaussian noise.
0 subthreshold signals A and A: SR occurs if and only if The sender Alice encodes coherent states with amplitude
8 a / (θ A,θ+A) where A−<A<θ. This result holds A = 1.1. The receiver Bob decodes with threshold θ = 1.6.
0 fo∈rallfi−nite-variancenoise−andallinfinite-variancestable Thegraphshowsthesmoothedmutualinformationasafunc-
:
v noise. Butit guaranteesonly that someSR noise benefit tion of the standard deviation σ of the quantum Gaussian
Xi occurs in the system for the given choice of parameters. noise and the squeezing strength r for 100 simulation runs.
Each run generated 10,000 input-output signal pairs to esti-
SR stochastic learning algorithms[9] canthen searchfor
r mate the mutual information. The SR effect occurs because
a the optimal noise level. the channel noise mean µ = 0 and thus µ lies outside the
This paper generalizes the classical forbidden-interval forbidden interval (.5,2.7).
theorems to a quantum-optimal communication system
that uses squeezed light [10, 11]. The corresponding
quantum forbidden-interval theorems give similar nec-
tribution with thresholding [12, 13].
essary and sufficient conditions for a noise benefit but
include the strength of light squeezing as a parameter. ModelforQuantum-OpticalThresholding System—We
The quantum-optical system in Figure 1 produces SR first develop the basic quantum-optical communication
because the noise mean is zero and so does not lie in protocol. The first quantum forbidden-interval theorem
the system’s forbidden interval (.5,2.7). We also show applies to this communication model. We present the
thatmodifiedversionsofthequantumforbidden-interval sender Alice’s operations for encoding information, the
theorems hold in continuous-variable quantum key dis- effect of the noisy quantum channel on the state that
Alicetransmits,andthereceiverBob’sdetectionscheme.
AlicewantstoencodeamessagebitS usingquantum-
optical techniques. The protocol begins with Alice pos-
∗Electronicaddress: [email protected] sessing a vacuum mode. We describe our model in the
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2
Heisenbergpicture. Letxˆdenotetheposition-quadrature RandomvariablesXe r,ν ,andX areindependentbe-
− x H
operatorofAlice’svacuummodewherexˆ= aˆ+aˆ /√2 causerandomvariableXe rcomesfromthevacuumfluc-
† −
and aˆ is the annihilation operator for her vacuum mode tuationsofAlice’soriginalmode,becauseν isBob’sloss
x
(cid:0) (cid:1)
[11]. We consider only the position-quadrature opera- of knowledge due to the state’s propagation through a
tor’s evolution. Her vacuum state collapses to a zero- noisyquantumchannel,andbecauseX comesfromthe
H
mean 1/2-variance Gaussian random variable X if she vacuum contributions of non-ideal position-quadrature
measuresit withanidealposition-quadraturehomodyne homodyne detection. Let random variable N sum all
detector. Suppose that Alice does not measure it. Sup- noise terms:
poseinsteadthatshesendshermodethroughaposition-
quadrature squeezer. Suppose further that she can con- N √η Xe−r+νx + 1 ηXH. (4)
≡ −
trol the strength of squeezing with a squeezing param-
The density p (n)(cid:0)of random(cid:1)varipable N is
N
eter r. The position-quadrature squeezer is a unitary
otoprerxˆateovrolSvˆe(sr)un≡deerxpthners(cid:16)qaˆu2ee−ze(cid:0)raˆa†s(cid:1)2Sˆ(cid:17)†o(r[)1x1ˆ]Sˆ. (Hr)er=oxˆpee−rar-. pN(n)=(cid:16)p√ηXe−r ∗p√ηνx ∗p√1−ηXH(cid:17)(n) (5)
She encodes the random message bit S 0,1 by dis- wherep√ηXe−r(n)isthedensityofazero-meanηe−2r/2-
placing her state by α C if S = ∈1 {or b}y α if varianceGaussianrandomvariable,p√ηνx(n)istheden-
eSxp= 0α.aˆTheαdiaˆspl[a1c1e]m. Lenett∈αis abuenthitearcyonodpietiroantoarldDˆis−p(αla)c≡e- s(1ity oηf)√/2η-νvxa,ripa√n1c−eηGXaHu(snsi)anisrtahneddoemnsviatyriaobflae,zaenrod-medaen-
†− ∗ S − ∗
ment α =( 1)S+1α. Her operator xˆe r evolves under notes convolution. The density pN(n) is a convolution
the (cid:8)conSdition−al(cid:9)displacement Dˆ(α ) as− because random variables Xe−r, νx, and XH are inde-
S pendent. So Bob’s received signal using (3) and (4) is
Dˆ†(αS) xˆe−r Dˆ(αS)=xˆe−r+( 1)S+1αx (1) √η( 1)S+1αx + N. Bob thresholds the result of the
− −
non-ideal homodyne detection with a threshold θ to re-
where αx = Re(cid:0){α}.(cid:1)This equality gives the Heisenberg- trieve a random bit Y where
picture observable that corresponds to Alice’s mode be-
foreshesendsitoverthenoisychannel. Themessagebit Y u √η( 1)S+1αx+N θ (6)
S appears as a displacement in (1). ≡ − −
(cid:16) (cid:17)
Alice sends her mode to Bob over an additive noisy and u is the unit Heaviside step function defined as
bosonic channel [14] that adds a random displacement u(x) = 1 if x 0 and u(x) = 0 if x < 0. This final
ν C to its input state. The channel randomly dis- bit Y that Bob≥detects shouldbe the messagebit S that
pla∈ces any annihilation operator aˆ as Dˆ†(ν)aˆDˆ(ν) = Alice first sent.
aˆ+ν. This is the quantum-channel analogue to a clas- QuantumAlpha-StableNoise—Thenoiserandomvari-
sical continuous additive noisy channel [15]. The term able ν need not have a finite second moment or fi-
x
xˆe−r+( 1)S+1αx+νxistheHeisenberg-pictureposition- nite higher-order moments. Some researchersargue that
−
quadratureobservablethatcorrespondstothestatethat quantum-optical noise arises from a large number of in-
Bob receives after Alice sends her mode over the noisy dependent random effects and and that it is Gaussian
channel. Random variable νx = Re ν and corresponds because of the central limit theorem [17, 18]. But these
{ }
to the position-quadrature noise. random effects need not converge to a Gaussian random
BobdetectstheinformationthatAliceencodesbyper- variableeventhoughthey convergeto arandomvariable
forming position-quadrature homodyne detection with with a bell-curve density. The generalized central limit
inefficient photodetectors. We model this non-ideal ho- theorem states that all and only normalized stable ran-
modyne detection as a lossy transmission through a ma- domvariablesconvergeindistributiontoastable random
terial with linear absorption [16]. A beamsplitter with variable[19]. Soanimpulsive quantumnoisesourcemay
transmittivity η models the linear absorptive material. havea limiting alpha-stabledensity throughaggregation
Then the Heisenberg-picture observable after the lossy or directly through transformation as when the Cauchy
beamsplitter is density arises from the tangent of uniform noise.
Alpha-stablenoisemodelsdiversephysicalphenomena
√η ( 1)S+1αx+xˆe−r+νx + 1 ηxˆH (2) such as impulsive interrupts in phone lines, underwater
− −
where η is(cid:16)the quantum efficiency o(cid:17)f thpe homodyne de- acoustics, low-frequency atmospheric signals, and grav-
itational fluctuations [20]. The parameter α (different
tectionand xˆ is the positionquadratureoperatorof an
H from “coherent state” α) lies in (0,2] and parametrizes
input vacuum mode. Bob measuresthe position quadra-
the thickness of the curve’s tails. The curve’s tail thick-
ture observable and the state collapses to the random
ness increases as α decreases: α = 1 corresponds to the
variable
thick-tailed Cauchy random variable and α = 2 corre-
√η ( 1)S+1αx+Xe−r+νx + 1 ηXH. (3) sponds to the familiar thin-tailedGaussianrandomvari-
− − able. Parameter β is a skewness parameter such that
(cid:16) (cid:17) p
X is a zero-mean 1/2-variance Gaussian random vari- β = 0 gives a symmetric density. Parameter γ is a dis-
H
able that corresponds to the vacuum observable xˆ . persion parameter that acts like the variance because it
H
3
quantifiesthespreadorwidthofthealpha-stabledensity sulting theorem gives necessary and sufficient conditions
around its location parameter a. for the SR effect in CVQKD.
Quantum Forbidden-Interval Theorem—Theorem 1
The theorems have security implications for CVQKD
below shows that any finite-variance quantum noise
with thresholding. Suppose that I(A,B) is the mutual
or any infinite-variance alpha-stable noise produces the
information between Alice and Bob and that I(A,E) is
SR effect in our model. The theorem states that the
the mutual information between Alice and an attacker
SR effect occurs for finite-variance noise if and only
Eve. The SR effect influences the privacy condition
if the noise mean µ falls outside the forbidden in-
νx I(A,B)>I(A,E) [21] because it affects I(A,B).
terval (θ α ,θ+α ). The noise location a replaces
x x
the noise−mean in the forbidden-interval condition for We first present the model for CVQKD from [12, 13]
infinite-variance noise. So adding noise in the form of without including the attacker Eve. Alice wants to send
squeezing noise, channel noise, and detector inefficiency a randomsecretbit S to Bob. Alice randomly sends one
noise canenhance the performance ofthe quantumcom- of four coherent states to Bob: α , iα , α , iα
municationsystem. Figure1showsasimulationinstance where α R+. Random bit S{=| i0 |if sihe|−senid|s− αi}
∈ |− i
of the if-part of Theorem 1. or iα and S = 1 if she sends α or iα . Bob
|− i | i | i
randomly measures the state’s position quadrature or
The theorem states that the mutual information
momentum quadrature. Alice and Bob communicate
I(S,Y) between sender and receiver tends to zero as
classically after Alice sends a large quantity of quan-
all noise parameters decrease to zero. The theorem as-
tum data to Bob. They divide the measurement re-
sumes that the input and output signals are statisti-
sultsinto“correct-basis”and“incorrect-basis.”Thedata
cally dependent so that I(S,Y) > 0 where I(S,Y) =
is correct-basis if Bob measures the position quadrature
p (s,y)log(p (s,y)/(p (s)p (y))) [15]. So
s,y S,Y S,Y S Y whenAlicesends α , α orifBobmeasuresthemo-
the SR effect occurs because the mutual information
{| i |− i}
P mentumquadraturewhenAlicesends iα , iα . The
I(S,Y) must increase from zero as we add noise to the
{| i |− i}
dataisincorrect-basisifitisnotcorrect-basis. Aliceand
system:what goesdownmustgoup. Westatetheparam-
Bob keep only correct-basis data. Let x R be the re-
etersforthefinite-variancecasewithoutparenthesesand
∈
sult of Bob’s measurement. Bob sets a threshold θ and
the parameters for the infinite-variance case with paren-
assigns a bit value Y where Y = 1 if x θ, Y = 0 if
theses.
≥
x θ, and Y = ε otherwise. Symbol ε represents an
≤ −
inconclusive result.
Theorem 1 Suppose the position quadrature ν of the
x
channel noise has finite variance σ2 and mean µ (dis- Our analysis below corresponds only to correct-basis
νx νx
persion γ and location a). Suppose the input signal’s data because this data is crucial for determining the
position quadrature α is subthreshold: α <θ. Suppose resulting performance of the protocol. We present the
x x
there is some statistical dependence between input signal analysisonlyfortheposition-quadraturebasiscase. The
S and output signal Y so that the mutual information same analysis holds for the momentum-quadrature case.
obeys I(S,Y) > 0. Then the quantum communication
We now present a Heisenberg-picture analysis of the
system exhibits the nonmonotone SR effect if and only if
abovemodelandincludestrategiesthattheattackerEve
the position quadrature of the noise mean (location) does
canemploy. Thefirstfewstepsbegininthesamewayas
not lie in the forbidden interval: µ / (θ α ,θ+α )
νx ∈ − x x the basic protocol above with Eve controlling the noisy
t(haat∈/I((θS,−Yα)x,θ0+aαsxσ)2). Th0e(nγonm0o)n,oatsonre SR ,effaencdtaiss channel. Then xˆe−r +(−1)S+1αx +νx is the position-
→ νx → → →∞ quadrature observable for the state that Eve possesses.
η 1.
→ She performs an amplifier-beamsplitter attack [22] by
first passing the state through a phase-insensitive linear
Proof. The finite-variance proof for sufficiency and ne-
amplifier with gain G 1 [23]. She then leaks a fraction
cessityfollowstherespectiveproofsin[7]and[8]ifweuse 1 η of the state th≥rough a beamsplitter so that Bob
E
p (n) as the noise density. The infinite-variance proof −
N receives the fractionη . The Heisenberg-picture observ-
E
followstherespectivestableproofsin[7]and[8]ifweuse
able that corresponds to Bob’s state is
p (n) as the noise density and if ν is an alpha-stable
N x
random variable. Only slight modifications of the proofs
accountforthehomodyneefficiencyη. SeeAppendixIA η Gxˆ + η (G 1)xˆ + 1 η xˆ (7)
E s E − E1 − E E2
for the finite-variance proof and Appendix IB for the
p p p
infinite-variance proof.
SR in Continuous-Variable Quantum Key Distribu- where xˆs = xˆe−r + (−1)S+1α + νx. Modes xˆE1 and
tion—The SR effect occurs in the continuous-variable xˆE2 are vacuum modes resulting from the amplifier and
quantum key distribution (CVQKD) scenario from [12, beamsplitter and correspond to zero-mean 1/2-variance
13]. ThisCVQKDmodelthresholdsacontinuousparam- Gaussian random variables upon measurement. Bob
etertoestablishasecretkeybetweenAliceandBob. We then measures the above operator by non-ideal position-
modify the formofthe aboveforbidden-intervaltheorem quadrature homodyne detection. It collapses to the ran-
to include the subtleties of the CVQKD model. The re- dom variable N +√η η G( 1)S+1α where N sums all
E B
−
4
noise terms Threshold θ = 1.6, Signal value α = 1.1
x
N ≡ ηEηBG Xe−r+νx + ηEηB(G−1)XE1+ 0.2
p (cid:0) ηB(1−(cid:1)ηE)pXE2 + 1−ηBXH, ation 0.15
m
ηB istheefficiencyofpBob’shomodynedetpection,andXH nfor 0.1
is a zero-mean 1/2-variance Gaussian random variable al I
that arises from homodyne detection noise. The density utu 0.05
p (n) of random variable N is M
N
0
pN(n)= pN(0,σ2)∗p√ηEηBGνx (n) (8) 4 2 −1 −2
where p (0,σ2) is th(cid:0)e density of a zero-m(cid:1)ean Gaussian Channel noise − γ 0.5 1.5 1 Sq0ueeze noise − r
randomNvariable with variance
ηB ηEGe−2r+ηE(G 1)+(1 ηE) +1 ηB /2 FIG. 2: (Color online) SR in continuous-variable quantum
− − − key distribution. Alice encodes coherent states with ampli-
(cid:0) (cid:0) (cid:1) (cid:1) tude A = 1.1 and Bob decodes with threshold θ = 1.6. The
and p is the density of √η η Gν . Bob de-
√ηEηBGνx E B x graph shows the smoothed mutual information as a function
codes with a threshold θ and gets a randombit Y where
ofthedispersionγ ofinfinite-variancequantumCauchynoise
and squeezing strength r for 100 simulation runs. We donot
1 : N +√ηEηBG( 1)S+1α θ include amplifier, beamsplitter, or photodetector inefficiency
− ≥
Y = 0 : N +√η η G( 1)S+1α θ . (9) noise. Each run generated 10,000 input-output signal pairs
 E B − ≤−
ε : else toestimatethemutualinformation. TheSReffectoccursbe-
cause the channel noise location a = 0 and so a lies outside
ProtagonistsAliceandBobandantagonistEveallplay theforbidden interval(−2.7,−.5)∪(.5,2.7).
aroleinthe SReffectinAlice andBob’scommunication
of a secret key. Alice can add Heisenberg noise in the
form of squeezing. Eve can add channel, amplifier, and
leakage noise in her attack. Bob can add photodetec-
tor inefficiency noise. The modified quantum forbidden-
alsomaynotappearrealisticbecausetheirproofrequires
interval theorem characterizes this interplay and gives a
infinite squeezing in the limit and thus requires infinite
necessary and sufficient condition for the SR effect. Fig-
energy. But the theorems guarantee that the SR effect
ure 2 shows a simulation instance of the if-part of the
occursforsomefinitesqueezing. Thesimulationsinboth
theorem.
figuresdisplaythefullnonmonotoneSRsignatureforex-
Theorem 2 Suppose the channel noise position quadra- perimentally plausible squeezing values and for realistic
ture has finite variance σ2 and mean µ (dispersion γ channel noise levels.
νx νx
andlocationa). Supposetheinputsignal’samplitudeαis
subthreshold: α<θand α> θ. Supposethereissome Forbidden interval theorems may hold for more com-
statistical dependence be−tween −input signal S and output plex quantum systems. The quantum systems in this
signal Y so that the mutual information obeys I(S,Y)> paper use noisy quantum processing to produce a
0. Then the quantum key distribution system exhibits the mutual-information benefit between two classical vari-
nonmonotoneSReffectifandonlyiftheposition quadra- ables. Other systems might use noise to enhance the
ture of the noise mean (location) does not lie in the for- fidelityofthecoherentsuperpositionofaquantumstate.
bidden interval: µ / ( θ α, θ+α) (θ α,θ+α) The performance measure would be the coherent infor-
(a / ( θ α, θ+νxα∈) −(θ− α,−θ+α)).∪The−nonmono- mation [24] because it corresponds operationally to the
ton∈e S−R e−ffect−is that I∪(S,Y−) 0 as σ2 0 (γ 0), capacity of a quantum channel [25]. The coherent infor-
as r , as G 1, as η →1, and aνsxη→ 1. → mation also relates to the quantum channel capacity for
E B
→∞ → → → sending private classical information [25]. This suggests
Proof. TheproofmethodfollowstheproofofTheorem1 further connections between SR and QKD and the po-
usingp (n)in(8). Theproofrequiresthreecasesrather tential for new learning algorithms that can locate any
N
thantwobecauseofthedifferencesbetweenCVQKDand noise optima.
thebasicmodel. SeeAppendix ICforthefinite-variance
proofandAppendixIDfortheinfinite-varianceproof. The authors thank Todd A. Brun, Igor Devetak,
Conclusion—Theorems 1 and 2 guarantee only that Jonathan P. Dowling, and Austin Lund for helpful
the nonmonotone SR effect occurs. They do not give discussions. MMW acknowledges support from NSF
theoptimalcombinationofchannelnoise,squeezing,and GrantCCF-0545845,theHearneInstituteforTheoretical
photodetector inefficiency noise. Nor do they guarantee Physics,ArmyResearchOffice,andDisruptiveTechnolo-
a large increase in mutual information. The theorems gies Office.
5
I. APPENDIX Consider the case where y =1.
θ+√ηαx
A. Proof of Theorem 1 (Finite Variance)
p (10) p (11)= p (n) dn
Y S Y S N
| | − | | −Zθ−√ηαx
The proofs for sufficiency and necessity follow the re-
So the result follows if
spective proof methods in [7] and [8] if we use (5) as the
noise density. θ+√ηαx
Let us calculate the four conditional probabilities p (n) dn 0 (16)
N
→
pY|S(0|0), pY|S(0|1), pY|S(1|0), pY|S(1|1). Zθ−√ηαx
p (00) as σ2 0, as r , and as η 1. Suppose the mean
Y|S | νx → →∞ →
=Pr u √η(−1)S+1αx+N −θ | S =0 µνx ∈/ (θ−αx,θ+αx)
=Prn√(cid:16)η( 1)S+1α +N θ <0(cid:17) S =0o by hypothesis. We ignore the zero-measure cases where
x
− − |
µ =θ α or µ =θ+α .
=Prn √ηα +N <θ =Pr N <θ+√ηoα νx − x νx x
x x
{− } { }
θ+√ηαx Case 1: Suppose first that µ < θ α . So µ +α <
= p (n) dn (10) νx − x νx x
N θ and thus
Z−∞
The other conditional probabilities follow from similar √η(µ +α ) µ +α <θ
νx x ≤ νx x
calculations:
for any η (0,1]. Pick
θ √ηαx ∈
−
p (01)= p (n) dn (11)
Y|S | Z−∞ N ǫ= 12(θ−√ηαx−√ηµνx)>0.
∞
p (10)= p (n) dn (12)
Y S N
| | Zθ+√ηαx So θ−√ηαx−ǫ=√ηµx+ǫ. Then
∞
pY S(11)= pN(n) dn (13) θ+√ηαx
| | Zθ−√ηαx pN(n) dn
Proof (Sufficiency). Assume that 0 < p (s) < 1 to Zθ−√ηαx
S ∞
avoid triviality when pS(s) = 0 or 1. I(S,Y) = 0 if ≤ pN(n) dn
and only if S and Y are statistically independent [15]. Zθ−√ηαx
We show that S and Y are asymptotically independent: ∞
p (n) dn
I(S,Y) 0 as σ2 0, as r , and as η 1. The ≤ N
following→definitioνnxh→olds for th→e p∞roofs that fo→llow: Zθ−√ηαx−ǫ
∞
p (n) dn
σ2 η e 2r/2+σ2 +(1 η)/2 (14) ≤ N
≡ − νx − Z√ηµνx+ǫ
Waserneed t,oasnhdowas(cid:0)tηhat p1Yf|Sor(ys|s,(cid:1))y = p0Y,(1y). aCsoσnν2sxide→r a0n, ==PPrr{NN ≥µ√+ηµǫνx +ǫ}
→ ∞ → ∈ { } { ≥ }
algebraicmanipulationusingthelawoftotalprobability: =Pr N µ ǫ
{ − ≥ }
Pr N µ ǫ
pY(y)= pY S(y s) pS(s) (15) ≤ {| − |≥ }
| | σ2
s
X
=p (y 0) p (0)+p (y 1) p (1) ≤ ǫ2
Y S S Y S S
| | | |
=pY|S(y|0) pS(0)+pY|S(y|1) (1−pS(0)) Spo t(h0e0)resuplt f(o0llo1w)s w0haesnσ2µνx <0, aθs r− αx ,baecnaduases
= pY|S(y|0)−pY|S(y|1) pS(0)+pY|S(y|1) ηY|S1.| − Y|S | → νx → →∞
→
We can show(cid:0) by a similar method(cid:1)that
Case 2: Suppose next that µ > θ + α so that
p (y)= p (y 1) p (y 0) p (1)+p (y 0) νx x
Y Y|S | − Y|S | S Y|S | µνx −αx >θ >0. Choose √η large enough so that
So p (y) (cid:0) p (y 1) and p (cid:1)(y) p (y 0) as
pY S(Yy 1) →pY SY(y|S0)| 0. CoYnsider→theYc|aSse |where √η >θ/(µνx −αx)
y =| 0.| − | | →
So √η(µνx −αx)>θ. Pick
θ+√ηαx
p (00) p (01)= p (n) dn 1
Y|S | − Y|S | Zθ−√ηαx N ǫ= 2(√ηµνx −θ−√ηαx)>0.
6
So θ+√ηαx+ǫ=√ηµνx −ǫ. Then √ηµνx +ǫ.
θ+√ηαx p (00)
pN(n) dn Y|S |
Zθ−√ηαx = θ+√ηαxp (n) dn
θ+√ηαx N
pN(n) dn Z−∞
≤ θ+√ηαx ǫ
Z−∞ − p (n) dn
θ+√ηαx+ǫ ≥ N
pN(n) dn Z−∞
≤ √ηµνx+ǫ
Z−∞ = p (n) dn
√ηµνx−ǫ N
pN(n) dn Z−∞
≤
∞
Z−∞ =1 pN(n) dn
=Pr N √ηµ ǫ −
{ ≤ νx − } Z√ηµνx+ǫ
=Pr N µ ǫ =1 Pr N √ηµ +ǫ
{ ≤ − } − { ≥ νx }
=Pr N µ ǫ =1 Pr N µ+ǫ
{ − ≤− } − { ≥ }
Pr N µ ǫ =1 Pr N µ ǫ
≤ {| − |≥ } − { − ≥ }
σ2 1 Pr N µ ǫ
≥ − {| − |≥ }
≤ ǫ2 σ2
1
≥ − ǫ2
So p (00) p (01) 0 asσ2 0,as r , and
as ηY|S1 w| he−n µY|S>|θ+→α . Thuνsx → →∞ →1
→ νx x
µνx ∈/ (θ−αx,θ+αx) saismσilν2axr→ly f0o,raspr →(1∞1,)a.nPdicaks aηn→y µ1. We(pθrovαe t,hθe+reαsu)lt.
Then µ > θY|Sα| and µ + ανx>∈θ >−0.x Suppoxse
isasufficientconditionforthenonmonotoneSReffectto νx − x νx x
occur. thatη is largeenoughso that√η >θ/(µνx +αx). Then
Proof (Necessity). The system does not exhibit the √η(µνx +αx) > θ and √ηµνx +√ηαx −θ > 0. Pick
nseonnsmeotnhoattoIn(eSS,YR)eiffsemctaixfimµνuxm∈a(sθσ−ν2xα→x,θ0+, aαsxr) →in t∞he, ǫ√=ηµ21νx(cid:0)√−ηǫµ.νx +√ηαx−θ(cid:1) > 0 so that θ−√ηαx+ǫ =
and as η 1. I(S,Y) H(Y) = H(S) as σ2 0,
→ → νx →
as r → ∞, and as η → 1. Assume that 0 < pS(s) < pY|S(1|1)
1 to avoid triviality when p (s) = 0 or 1. We show
S ∞
that H(Y) H(S) and H(Y S) 0 as σ2 0, as = pN(n) dn
r , an→d as η 1. It is|ma→ximum inνxth→is limit Zθ−√ηαx
→ ∞ →
becauseI(S,Y)=H(Y) H(Y S)andI(S,Y) H(S) ∞
p (n) dn
− | ≤ N
bythedataprocessinginequalityforaMarkovchain[15]. ≥
Consider the conditional entropy H(Y S): Zθ−√ηαx+ǫ
| ∞
= p (n) dn
N
H(Y S) Z√ηµνx−ǫ
| √ηµνx−ǫ
=− pY,S(y,s)log2pY|S(y|s) =1− pN(n) dn
Xs,y Z−∞
=1 Pr N √ηµ ǫ
= p (s) p (y s)log p (y s) (17) − { ≤ νx − }
− S Y|S | 2 Y|S | =1 Pr N µ ǫ
s y − { ≤ − }
X X
=1 Pr N µ ǫ
− { − ≤− }
Suppose for now that p (y s) 1 or 0 for all s,y 1 Pr N µ ǫ
H{0,(1Y}Sa)s σν20xb→yin0s,peacstirYng|→S(17|∞)a,n→adnadppaslyiηng→1lo1g. 1T=he∈n0 ≥1− σ2{| − |≥ }
| → 2 ≥ − ǫ2
and0log 0=0by L’Hoˆspital’srule. Soweaimtoprove
2 1
that each of the conditional probabilities vanish or ap- →
proach 1 in the above limit if µ (θ α ,θ+α ).
νx ∈ − x x
Considerfirstp (00). Pickanyµ (θ α ,θ+α ). as σ2 0,as r ,and as η 1. So p (00)
Then θ + α Y|Sµ | > 0 and θ ν>x ∈µ − αx . Thxen 1,p νx(→11) 1,→p ∞(10) 0,→and p (0Y1|S) | 0 →as
θ21 >θ+√√ηη(αµxνx−x−−√αηµνxx)νxfor>an0ysoη th∈at(0θ,ν+1x].−√ηPαxixck−ǫǫ == σnoν2xtYd→|Sisp0l,|aayst→rhe→no∞nYm,|Saonndo|taosn→eηS→R 1effaencdt.Yth|Se sy|ste→m does
(cid:0) (cid:1)
7
B. Proof of Theorem 1 (Infinite Variance) Proof (Necessity). Choosea (θ αx,θ+αx). Then
∈ −
The proofs for sufficiency and necessity follow the re-
spective stable proof methods in [7] and [8] if we use (5)
as the noise density and if ν is an alpha-stable random
x
variable.
The characteristic function ϕ (ω) of an alpha-stable
νx
noise source with density p (n) is the following:
νx
ϕ (ω)=
νx
απ
α
exp iaω γ ω 1+iβsign(ω)tan (18)
− | | 2
n (cid:16) (cid:16) (cid:17)(cid:17)o
whereαisthecharacteristicexponentandβisaskewness
parameter. The characteristic function of p (n) is as
N
follows
ϕ (ω)
N
θ+√ηαx
=(cid:16)ϕ√ηXeη−er ·2ϕrω√2ηνx ·ϕ√1−ηXH(cid:17)(ω) (1 η)ω2 pY|S(0|0)=Z−θ∞ pN(n) dn (27)
−
=exp − 4 ϕνx(√ηω)exp − −4 = pN(n+√ηαx) dn (28)
(cid:26) (cid:27) (cid:26) (cid:27)
Z−∞
ηe 2r+1 η ω2 θ
−
=ϕνx(√ηω)exp(−(cid:0) 4 − (cid:1) ) (19) → δ(n−a+αx) dn (29)
Z−∞
θ+αx
from (5) and the convolution theorem. = δ(n a) dn=1 (30)
−
Proof (Sufficiency). Take the limit of the character- Z−∞
istic function ϕ (ω) as γ 0, as squeezing parameter as γ 0, as r , and as η 1 (31)
N → → →∞ →
r , and as homodyne efficiency η 1 to obtain the
fo→llow∞ing characteristic function. → pY S(11)= ∞ pN(n) dn (32)
| | Zθ−√ηαx
r ,lγim0,η 1ϕN(ω)=exp{iaω} (20) = ∞pN(n √ηαx) dn (33)
→∞ → → Zθ −
The probability density p (n) then approaches a trans- ∞
N δ(n a α ) dn (34)
x x
lated delta function → − −
Zθ
∞
r→∞,lγi→m0,η→1pN(n)=δ(n−a) (21) =Zθ−αxδ(n−ax) dn=1 (35)
as γ 0, as r , and as η 1 (36)
The conditional probability difference obeys: → →∞ →
θ+√ηαx
p (00) p (01)= p (n) dn (22)
Y S Y S N
| | − | | Zθ−√ηαx
θ+αx
p (n) dn (23)
N
≤
Zθ−αx
Pick a / (θ α ,θ+α ). Consider the following limit:
x x
∈ −
lim p (00) p (01) (24)
Y S Y S
r ,γ 0,η 1 | | − | |
→∞ → →
θ+αx
lim p (n) dn (25)
N
≤r→∞,γ→0,η→1Zθ−αx
θ+αx
= δ(n a) dn=0 (26)
−
Zθ−αx
because a / (θ α ,θ+α ).
x x
∈ −
8
C. Proof of Theorem 2 (Finite Variance) apsY|GS(y|s1)→forpsY,(yy) as0σ,1ν2x.→W0e,daosnro→t c∞on,saidseηr p→1,(yanεd)
Y S
The mean µ and variance σ2 of noise random variable becau→se the proba∈b{ility}pS(ε) is zero and so the|pro|b-
N are µ=√ηEηBGµνx and ability pY (ε) is also zero. Consider the expansion in
(15) using the law of total probability. The expansion
σ2 =η η Gσ2 + is the same even when including symbol ε because ε
E B νx has zero probability: p (ε) = 0. So p (y) p (y 1)
ηB ηEGe−+2r(1+ηEηE()G−1) +1−ηB /2. (37) aanndd ppY(y()11→) pYp|S(y(|10S0))as p0Y.|SC(o0n|0si)dY−erptYhe|→Sc(0a|sY1e)|Sw→he|r0e
(cid:18) (cid:18) − (cid:19) (cid:19) Y S Y S
y =0. | | − | | →
We compute the six conditional probabilities: p (00),
Y S
p (01), p (10), p (11), p (ε0), and p | (ε|1).
Y|S | Y|S | Y|S | Y|S | Y|S | −θ+√ηEηBGα
p (00) p (01)= p (n) dn (44)
Y S Y S N
pY|S(0|0) | | − | | Z−θ−√ηEηBGα
=Pr N + ηEηBG( 1)S+1α θ S =0 Consider the case where y =1.
− ≤− |
n p o
=Pr ηEηBGα+N θ θ+√ηEηBGα
− ≤−
p (11) p (10)= p (n) dn (45)
=PrnNp<−θ+ ηEηBGα o Y|S | − Y|S | Zθ−√ηEηBGα N
nθ+√ηEηBGαp o So the result follows if both of the above conditional
−
= pN(n) dn (38) probability differences vanish as σν2x → 0, as r → ∞,
Z−∞ as ηE,ηB 1, and as G 1. Suppose the mean
→ →
The other conditional probabilities follow from similar µνx ∈/ (−θ−α,−θ+α)∪(θ−α,θ+α) by hypothesis.
reasoning: We ignore the zero-measure cases where µνx = θ − α,
µ =θ+α, µ = θ+α, or µ = θ α.
νx νx − νx − −
θ √ηEηBGα
− −
pY|S(0|1)= pN(n) dn (39) Case1: Supposefirstthatµνx <−θ−α. Soµνx+α<−θ
pY|S(1|0)=ZZθ−+∞∞√ηEηBGαpN(n) dn (40) ǫeavned=r t√hu1η2sE−η√BθηG−Eη√>BηGE−(ηµθB/νGx(µ+ανx−α+)√α≤ηE)ηµ=BνxGθ+µ/ν|xαµνx<>+−αθ|0..wPheiScnko-
pY S(11)= ∞ pN(n) dn (41) −θ−√ηE(cid:0)ηBGα−ǫ=√ηEηBGµνx +ǫ. T(cid:1)hen
| | Zθ−√ηEηBGα θ+√ηEηBGα
−
p (n) dn
N
pY S(ε0)=1 pY S(00) pY S(10) Z−θ−√ηEηBGα
| | − | | − | | ∞
θ+√ηEηBGα pN(n) dn
= pN(n) dn (42) ≤Z−θ−√ηEηBGα
Z−θ+√ηEηBGα ∞
p (n) dn
N
≤
p (ε1)=1 p (01) p (11) Z−θ−√ηEηBGα−ǫ
Y|S | − Y|S | − Y|S | ∞ p (n) dn
θ √ηEηBGα ≤ N
= − p (n) dn (43) Z√ηEηBGµνx+ǫ
N
Z−θ−√ηEηBGα =Pr N ≥ ηEηBGµνx +ǫ
Proof (Sufficiency). We follow the proof method of =PrnN µp+ǫ o
Theorem 1 with some modifications. Note that the { ≥ }
=Pr N µ ǫ
conditions ηE,ηB 1 and G 1 constrain the way { − ≥ }
in which we take b≤oth of their≥limits to one. We give Pr N µ ǫ
≤ {| − |≥ }
the constraint on the values that the root of their σ2
product √ηEηBG may take for any given value of the ≤ ǫ2
noise mean µ . Assume that these constraints are im-
νx
plicitwhenconsideringthelimitintheproofsthatfollow. So the conditional probability difference in (44) van-
ishes as σ2 0, as r , as η ,η 1, and as
νx → → ∞ E B →
Assume that 0 < p (s) < 1 to avoid triviality when G 1 when µ < θ α. We now prove that the
S → νx − −
p (s) = 0 or 1. I(S,Y) = 0 if and only if S and Y are conditional probability difference in (45) vanishes when
S
statisticallyindependent[15]. WeshowthatS andY are µ < θ α. It follows that µ < θ α if µ <
asymptotically independent: I(S,Y) 0 as σ2 0, as νθx α.−So−µ +α < θ and thus ν√xη η G−(µ +ναx)
→ νx → − − νx E B νx ≤
r , as η 1, and as G 1. We need to show that µ +α < θ for any √η η G 0 because µ +α < 0.
→∞ → → νx E B ≥ νx
9
Pick ǫ = 1 θ √η η Gα √η η Gµ > 0. So as r , as η ,η 1,and asG 1. So µ +α<θ
2 − E B − E B νx →∞ E B → → νx
θ−√ηEηBGα(cid:0) −ǫ=√ηEηBGµνx +ǫ. Then(cid:1) µif −θ++αα<<θµwνxhe<nevθe−r √α.η TηhuGs √<ηEθ/ηBµG(µ+νxα+. αP)ic≤k
θ+√ηEηBGα ǫνx= 1 θ √η η Gα √Eη Bη Gµ |>νx0. |So θ
p (n) dn 2 − E B − E B νx −
N √η η Gα ǫ=√η η Gµ +ǫ.
Zθ−√ηEηBGα E B(cid:0) − E B νx (cid:1)
∞ θ+√ηEηBGα
≤ pN(n) dn pN(n) dn
Zθ−√ηEηBGα Zθ−√ηEηBGα
≤Zθ−∞√ηEηBGα−ǫpN(n) dn ≤Zθ−∞√ηEηBGαpN(n) dn
≤Z√∞ηEηBGµνx+ǫpN(n) dn ≤Zθ−∞√ηEηBGα−ǫpN(n) dn
=Pr N ≥ ηEηBGµνx +ǫ ∞ pN(n) dn
≤
=PrnN µp+ǫ o Z√ηEηBGµνx+ǫ
{ ≥ }
=Pr{N −µ≥ǫ} =Pr N ≥ ηEηBGµνx +ǫ
Pr N µ ǫ =PrnN µp+ǫ o
≤ {| − |≥ } { ≥ }
σ2 =Pr N µ ǫ
{ − ≥ }
≤ ǫ2 Pr N µ ǫ
≤ {| − |≥ }
So the conditional probability difference in (45) van- σ2
ishes as σν2x → 0, as r → ∞, as ηE,ηB → 1, and ≤ ǫ2
as G 1 when µ < θ α and with constraint
→ νx − − So the conditionalprobability difference in (45) vanishes
√η η G>θ/ µ +α.
E B | νx | as σν2x → 0, as r → ∞, as ηE,ηB → 1, and as G → 1
when θ +α < µ < θ α and with the constraint
CWaesefi2rs:t Spurpopvoesethnaetxtthtehatco−ndθit+ionαal<prµoνbxab<ilitθy−dαif-. √ηEηB−G≤min(θ/ν|xµνx +α−|,θ/|µνx −α|).
ference in (44) vanishes as σ2 0, as r , as
η ,η 1, and as G 1ν.x →So µ α→>∞θ if Case 3: Suppose next that µνx > θ + α so that
−Eθ +Bα→< µνx < θ − α.→Thus √ηEηνBxG−(µνx −−α) ≥ µprνoxb−abαilit>y θdiff>er0e.ncWe einfi(r4st4)prvoavneishtheastatsheσ2condit0io,naasl
µPνiθcxk+−ǫ√αη=η>21Gθ−α−θ+√wǫηh=Eenη√BevηGerαη+√Gη√µEηηEBηGBǫG.<µTνhxeθn/>|µνx0.−αS|o. rif→µνx∞>, aθs+ηEα,.ηBTh→us1,µaνxnd>a−s Gθ+→α1a.nSdoµµννxνxx−>→α−>θ+−αθ
− E B(cid:0) E B νx − (cid:1) aPnicdk√ǫηE=ηB1G(√µνηx −η αG)µ> +−θθ fo√r ηanηy √GηαEηB>G0≥. S0o.
−θ+√ηEηBGα 2 E B νx − E B
p (n) dn θ+√η η Gα+ǫ=√η η Gµ ǫ. Then
N − E B (cid:0) E B νx − (cid:1)
Z−θ−√ηEηBGα θ+√ηEηBGα
−θ+√ηEηBGαp (n) dn − pN(n) dn
≤Z−∞ N Z−θ−√θη+E√ηBηGEηαBGα
−θ+√ηEηBGα+ǫpN(n) dn ≤ − pN(n) dn
≤Z−∞ Z−∞θ+√ηEηBGα+ǫ
√ηEηBGµνx−ǫpN(n) dn ≤ − pN(n) dn
≤Z−∞ Z−√∞ηEηBGµνx−ǫ
=Pr N ≤ ηEηBGµνx −ǫ ≤ pN(n) dn
=PrnN µp ǫ o Z−∞
{ ≤ − } =Pr N η η Gµ ǫ
=Pr N µ ǫ ≤ E B νx −
{ − ≤− } =PrnN µp ǫ o
Pr N µ ǫ
{ ≤ − }
≤ {| − |≥ }
σ2 =Pr{N −µ≤−ǫ}
≤ ǫ2 Pr N µ ǫ
≤ {| − |≥ }
σ2
So the conditionalprobability difference in (44) vanishes
as σ2 0, as r , as η ,η 1, and as G 1 ≤ ǫ2
νx → → ∞ E B → →
when θ+α<µ <θ α. Wenowprovethatthecondi- So the conditional probability difference in (44) van-
tional−probabilityνxdiffer−ence in (45) vanishes as σ2 0, ishes as σ2 0, as r , as η ,η 1, and as
νx → νx → → ∞ E B →
10
G 1 when µ > θ + α. We lastly prove that the θ+√η η Gα ǫ=√η η Gµ +ǫ.
→ νx − E B − E B νx
conditional probability difference in (45) vanishes as
σ2 0, as r , as η ,η 1, and as G 1
νx → → ∞ E B → → pY S(00)
when µ > θ + α. So √η η G(µ α) > θ | |
νx E B νx − θ+√ηEηBGα
whenever √η η G > θ/(µ α). Pick −
ǫ = 1 √η Eη BGµ θ √η ηνxG−α > 0. So = pN(n) dn
2 E B νx − − E B Z−∞
θ+√ηEηB(cid:0)Gα+ǫ=√ηEηBGµνx −ǫ. Th(cid:1)en −θ+√ηEηBGα−ǫ
p (n) dn
N
≥
Z−∞
√ηEηBGµνx+ǫ
= p (n) dn
N
Z−∞
∞
=1 p (n) dn
θ+√ηEηBGα − N
p (n) dn Z√ηEηBGµνx+ǫ
N
Zθ−√ηEηBGα =1−Pr N ≥ ηEηBGµνx +ǫ
θ+√ηEηBGαp (n) dn =1 PrnN µp+ǫ o
≤ N − { ≥ }
Z−∞ =1 Pr N µ ǫ
θ+√ηEηBGα+ǫ − { − ≥ }
1 Pr N µ ǫ
pN(n) dn ≥ − {| − |≥ }
≤Z−∞ 1 σ2
√ηEηBGµνx−ǫ ≥ − ǫ2
p (n) dn
≤ N 1
Z−∞ →
=Pr N η η Gµ ǫ
≤ E B νx − as σ2 0, as r , as η ,η 1,
=PrnN µp ǫ o and νaxs G→ 1. We→pro∞ve the reEsultB sim→ilarly
{ ≤ − } →
=Pr N µ ǫ for pY S(ε1). We show that pY S(01) 0 and
{ − ≤− } p (1|1) | 0 so that p (ε1) |1. P|ick →any µ
≤Pσr2{|N −µ|≥ǫ} (α−Y>|θS−|αθ.,−→√θη+αη).G(µThe+nY|αSµ)νx>| >→θ−aθnd−√αη aηndGµµννxx ++∈
− E B νx − E B νx
≤ ǫ2 √η η Gα+θ > 0 whenever √η η G < θ/ µ +α.
PicEk ǫB= 1 √η η Gµ +√η ηEGBα+θ >|0νsxo tha|t
2 E B νx E B
θ √η η Gα+ǫ=√η η Gµ ǫ.
− − E B(cid:0) E B νx − (cid:1)
p (01)
Y S
| |
θ √ηEηBGα
Sotheconditionalprobabilitydifferencein(45)asσν2x → = − − pN(n) dn
0, as r , and as η ,η 1 when µ > θ + α
→ ∞ E B → νx Z−∞
µanνxd w∈/it(h−tθh−e cαo,n−stθr+ainαt)√∪η(Eθη−BGα,>θ+θ/α()µνixs−aαsu).ffiTciheunts ≤ −θ−√ηEηBGα+ǫpN(n) dn
condition for the nonmonotone SR effect to occur with Z−∞
the given constraints on the product √ηEηBG. = √ηEηBGµνx−ǫp (n) dn
N
Z−∞
=Pr N η η Gµ ǫ
Proof (Necessity). We prove that the SR effect does ≤ E B νx −
not occur when µ ( θ α, θ+α) (θ α,θ+α). =PrnN µp ǫ o
νx ∈ − − − ∪ − { ≤ − }
=Pr N µ ǫ
Case 1: Suppose first that µ ( θ α, θ+α). { − ≤− }
νx ∈ − − − Pr N µ ǫ
We prove with a similar Chebyshev bound that the con- ≤ {| − |≥ }
ditional probabilities p (00) 1 and p (ε1) 1 σ2
Y S Y S
as σ2 0, as r | ,| as→η ,η | 1,|and→as ≤ ǫ2
νx → → ∞ E B →
G 1. Then the mutual information I(S,Y) ap-
→
proaches its maximum H(S) as all noise vanishes. Pick any µ ( θ α, θ+α). Then µ <
νx ∈ − − − νx
Consider p (00). Pick any µ ( θ α, θ+α). θ + α and θ < α µ . √ηG(µ α) <
Then θ +Y|αS >| µ and α µνx ∈>−θ. −The−n θ > −θ and √η η Gµ −+ √νxη η Gα νθx − > 0
− νx − νx − − − E B νx E B −
√η η G(µ α) whenever √η η G > θ/ α µ . whenever √η η G > θ/ α µ . Pick ǫ =
PicEkǫB= 1 νxθ−+√η η Gα √ηEηBGµ >|0−sothνxa|t 1 √η η GµE B+√η η Gα| −θ ν>x| 0 so that θ +
2 − E B − E B νx 2 − E B νx E B − −
(cid:0) (cid:1) (cid:0) (cid:1)