Bell inequality tests using asymmetric entangled coherent states in asymmetric lossy environments

0.43 MB·English

by  Chae-Yeun Park

Bell inequality tests using asymmetric entangled coherent states in asymmetric lossy
environments
Chae-Yeun Park and Hyunseok Jeong
∗
Center for Macroscopic Quantum Control, Department of Physics and Astronomy,
Seoul National University, Seoul, 151-742, Korea
(Dated: April14, 2015)
We study an asymmetric form of two-mode entangled coherent state (ECS), where the two local
amplitudeshavedifferent values, for testing theBell-Clauser-Horne-Shimony-Holt (Bell-CHSH) in-
equality. We find that the asymmetric ECSs have obvious advantages over the symmetric form of
ECSsintestingtheBell-CHSHinequality. Wefurtherstudyanasymmetricstrategyindistributing
5 anECSoveralossy environmentandfindthatsuchaschemecansignificantlyincreaseviolation of
1 theinequality.
0
2 PACSnumbers: 03.65.Ud,03.67.Mn,42.50.Dv

r
p
A I. INTRODUCTION photon on-off detection and photon number parity de-
tection, respectively, for the Bell-CHSH inequality tests.

3
Entangled coherent states (ECSs) in free-traveling Section III is devotedto the study ofdecoherence effects
1
fields [1–3] have been found to be useful for various ap- onBellinequalityviolationswithanasymmetricstrategy
] plications such as Bell inequality tests [4–17], tests for to share ECSs. We then investigate effects of inefficient
h non-localrealism[18,19],quantumteleportation[20–24], detectors in Sec. IV, and conclude with final remarks in
p quantum computation [25–31], precision measurements Sec. V.
-
t [32–39], quantum repeater [40] and quantum key distri-
n
bution [41]. The ECSs can be realized in various sys-
a
u tems that can be described as harmonic oscillators and II. BELL INEQUALITY TESTS WITH
q numerousschemesfortheirimplementinghavebeensug- ASYMMETRIC ENTANGLED COHERENT
[ gested [1–3, 42–46]. An ECS in a free-traveling field was STATES

experimentally generated using the photon subtraction
2
v techniqueontwoapproximatesuperpositionsofcoherent In this paper, we are interested in a particular formof
7 states(SCSs)[47]. Aproof-of-principledemonstrationof two-mode ECSs
0 quantumteleportationusinganECSasaquantumchan-
8 nel was performed [48]. ECS± = (α1 α2 α1 α2 ) (1)
0 | i N± | i⊗| i±|− i⊗|− i
So far, most of the studies on ECSs have considered
0
01. spsyelinmtsumidtieevtserihtcaotvydepetechsoehoesfratemwneoce-vmadoluudeee.tsotSaipnthecosetwoEnhCelSroessstaehrseeilgnoecntaeelsrtaaimnllyg- fiwtheheldernemoro|m±dαaeliiiiz,aaatnrioedncNofa±hcet=roern[s2t.±sWta2eteexnspoo(t−fe2athαma21pt−liat2umαdpe22ls)i]t−u±1dα/e2isafαor1er
5
Bell-type inequalities [6], it would be worth investigat- and α2 are assumed to be real without loss of general-
1
ing the possibility of using an asymmetric type of ECS ity throughout the paper. The ECSs show noticeable
:
v to reduce decoherence effects. In fact, asymmetric ECSs properties as macroscopic entanglement when the am-
Xi can be used to efficiently teleport an SCS [27, 48] and plitudes are sufficiently large and these properties have
to remotely generate symmetric ECSs [49] in a lossy en- been extensivelyexplored[9, 15,51, 52]. It is worthnot-
r
a vironment. In addition to the asymmetry of the ECSs, ing that there are studies on other types of ECSs such
as a closely related issue, it would be beneficial to study as multi-dimensional ECSs [53], cluster-type ECSs [54–
strategies for distributing the ECSs over asymmetrically 56], multi-mode ECSs [57] and generalization of ECSs
lossy environments. In this paper, we investigate asym- with thermal-state components [9, 58] while we focus on
metric ECSs as well as asymmetric entanglement dis- two-mode ECSs in free-traveling fields in this paper.
tribution strategies, and find their evident advantages Wecall ECS+ (ECS− )theeven(odd)ECSbecause
| i | i
over symmetric ones for testing the Bell-Clauser-Horne- it contains only even (odd) numbers of photons. When
Shimony-Holt(Bell-CHSH) inequality[50] under various α1 = α2, we call the states in Eq. (1) symmetric ECSs,
conditions. and otherwise they shall be called asymmetric ECSs. It
The remainder of the paper is organized as follows. isstraightforwardtoshowthatanECScanbegenerated
In Sec. II, we discuss the Bell-CHSH inequality tests us- by passing an SCS in the form of α α [59], where
ing asymmetric ECSs with ideal detectors. We consider α2 =α2+α2, through a beam sp|litite±r.|−A bieam splitter
1 2
with an appropriate ratio should be used to generate an
asymmetric ECS with desired values of α and α . In
1 2
this paper, asymmetric ECSs with various values of α
1
∗ [email protected] and α are compared for given values of α. The average
2
2
photonnumbersoftheECSsaresolelydependentonthe 2.14 (a)
values of α and are simply obtained as 2.12
2.1
x
α2(1 e 2α2) ma2.08
− |
n¯± =hECS±|nˆ|ECS±i= 1±∓e−2α2 (2) |BO22..0046
wherenˆ =nˆ +nˆ andnˆ isthenumberoperatorforfield 2.02
1 2 i
mode i. 2
0 1 2 3 4 5
n¯
+
A. Bell-CHSH tests with photon on-off
2.8
measurements
2.7 (b)
2.6
We first investigate a Bell-CHSH inequality test us- x2.5
a
ing photon on-off measurements with the displace- m2.4
|
O
ment operations. A Bell-CHSH inequality test requires B2.3
parametrized measurement settings of which the out- |2.2
comesaredichotomizedtobeeither+1or 1[50]. Inthe 2.1
simplest example of a two-qubit system, a−parametrized 2
1 1.5 2 2.5 3 3.5 4 4.5 5
rotation about the x axis followed by a dichotomized
n¯
measurement in the z direction is used. In our study, −
thedisplacementoperatorthatisknowntowellapproxi-
mate the qubit rotationfor a coherent-statequbit [7,26] FIG.1. (Coloronline)OptimizedBell-CHSHfunctionagainst
the average photon number for (a) ECS+ and (b) ECS−
is used for parametrization. The displacement operator | i | i
using photon on-off measurements. The solid curves indicate
can be implemented using a strong coherent field and a
Bell-CHSH functions optimized over the displacement vari-
beam splitter with a high transmissivity. Together with ′ ′
ables ξ1, ξ1, ξ2 and ξ2 for the symmetric ECSs (α1 = α2 =
the displacement operator, the photon on-off measure- α/√2). ThedottedcurvesindicatetheoptimizedBell-CHSH
mentisexperimentallyavailableusingcurrenttechnology functions with asymmetric ECSs. In the latter case, in addi-
withanavalanchephotodetector[60]inspiteofthe issue tion to thedisplacement variables, amplitudes α1 and α2 are
of the detection efficiency. We shall analyze the effects also optimized underthecondition Eq. (2).
of the limited detection efficiency in Sec. IV.
Thephotonon-offmeasurementoperatorformodeiis
defined as usetheBroyden-Fletcher-Goldfarb-Shanno(BFGS)algo-
rithm [61] throughout this paper.
Oˆi(ξ)=Dˆi(ξ) nX∞=1|nihn|−|0ih0|!Dˆi†(ξ) (3) eFriagTg.he1e,pohwpohttioimlneiznFeudigm.Bb2eerllss-hCn¯oH±wSsfHohrofwu|EnCcatsSiyo±mnismaaergetraipicnrsettsheetnhtEeedCaSvins-
where Dˆi(ξ) = exp ξaˆ†i −ξ∗aˆi is the displacement op- bsoelcidomceurivnesorsdheorwttohemraexsiumltizsefotrhesyBmemllevtrioiclaEtiConSss.wThhilee
erator and n denohtes the Focik state. The correlation
| i the dottedcurvescorrespondtogeneralcases(asymmet-
function is defined as the expectation value of the joint
ric ones). The Bell-CHSH functions are numerically op-
measurement
timized over all displacement variables and amplitudes
underthe conditionofEq.(2). The Bellviolationsoccur
E (ξ ,ξ )= Oˆ (ξ ) Oˆ (ξ ) (4)
O 1 2 h 1 1 ⊗ 2 2 i both for the even and odd ECSs regardless of the values
ofn¯ thatareconsistentwiththeresultsinRef.[7]. The
and the Bell-CHSH function is
odd±ECS, ECS− , violates the inequality more than the
other one,| ECS+i for a given average photon number.
=E (ξ ,ξ )+E (ξ ,ξ )+E (ξ ,ξ ) E (ξ ,ξ ).
BO O 1 2 O 1′ 2 O 1 2′ − O 1′ (2′5) The Bell-CH| SH fuinction for state ECS− reaches up to
| i
2.743 while the maximum Bell-CHSH func-
|BO|max ≈
In any local realistic theory, the absolute value of the tion for |ECS+i is |BO|max ≈ 2.131. This is due to the
Bell-CHSH function is bounded by 2 [50]. factthatthe oddECSis maximallyentangledregardless
We calculate an explicit form of the correlation func- of the value of n¯ [22, 62] and the on-off measurement
tionE (ξ ,ξ )usingEqs.(3)and(4),ofwhichthedetails can effectively re−veal nonlocality of the ECSs when the
O 1 2
are presented in in Appendix A. We then find the Bell- amplitudes are small [7].
CHSH function using Eq. (5) and its absolute maxi- It is obvious from Fig. 1 that one can increase the
O
B
mum values over the displacement variables ξ , amount of violations by using the asymmetric ECSs for
|BO|max 1
aξ1′n,uξm2 earnicdalξ2′mtuoltgievtahreiarbwleitmhan¯x±im,iαza1tiaonndmαe2t.hoIdt,raeqnudirwees dceiffrtearienncreegaiopnpseaorfsn¯f±o.r In¯n th&e c1a.s4e3.of sTtahtise |dEiffCeSr−eni,cethiins
−
3
1.6 2.8 2.8
2.7 (a)
2.7
2.6
1.3
2.6 x2.5
a
m2.4
2.5 |Π
1 B 2.3
2 2.4 | 2.2
α 2.1
0.7
2.3 2
0 5 10 15 20 25
2.2
0.4 n¯
2.1 +
2.8
0.1 2
0.1 0.4 0.7 1 1.3 1.6 2.7 (b)
2.6
α
1 x2.5
a
m2.4
|Π
FIG. 2. (Color online) Optimized Bell-CHSH function B 2.3
| 2.2
|BO|max using photon on-off m−easurements as a function of 2.1
amplitudes α1 and α2 for ECS . For each point of α1 and
| i 2
α2, the Bell-CHSH function is optimized over the displace- 5 10 15 20 25
ment variables. The blue line indicates the states that show
the maximum Bell violations among the states for the same n¯
−
theaverage photon numbers.
FIG.3. (Coloronline) OptimizedBell-CHSHfunctionsusing
photon numberparity measurements against average photon
numberof (a) theeven ECS and (b) theodd ECS. The solid
curves show the optimization results for symmetric form of
the optimized Bell-CHSH function reaches its maximum ECS. The dotted curves are the results for general form of
value 0.053 for n¯ 2.24. For this value of n¯ , the ECSs. Like the on-off measurement case, we optimized the
symme≈tric ECS give−s≈B 2.135 for α =−α displacement variables and amplitudes for each setting.
O max 1 2
| | ≈ ≈
1.04 while the asymmetric ECS yields B 2.189
O max
| | ≈
forα 1.26andα 0.77. The evenECSalsoshowsa
1 2
small i≈ncrease of the v≈iolationwhen an asymmetric ECS B. Bell-CHSH tests with photon number parity
is used in place of the symmetric ECS for n¯ & 2.83. measurements
+
The maximum difference is 0.007 when n¯ 3.93.
+
≈ ≈
We now consider the photon number parity measure-
In order to further investigate the advantages of the ments with the displacement operators for both modes.
asymmetricECS,weplottheoptimizedBell-CHSHfunc- Here we use the displacement operator again in order to
tion |BO|max as a function of α1 and α2 for |ECS−i in approximateaparametrizedrotationforacoherent-state
Fig.2. Thebluelineindicatesthepointforthemaximum qubit in the Bell-CHSH test. The displaced parity mea-
Bell-CHSHfunctionforeachvalueofn¯ . Theunsmooth surement is given by
changeinthebluelineatα =α 0.7−7(n¯ 1.43)re-
1 2
smualtxsimfruommtvhaeluneusmaerreiccaolmopptairmedizawti≈iothnpchroacnegs−essw≈ohferreellaotceadl Πˆi(ξ)=Dˆi(ξ) ∞ |2nih2n|−|2n+1ih2n+1| Dˆi†(ξ)
parameters, i.e., the displacement variables and ampli- nX=0(cid:0) (cid:1) (6)
tudes. We notethatsuchcomparisonsamonglocalmax-
ima at a number of different parameter regions lead to
where i 1,2 denotes each mode and the correlation
unsmoothchangesinseveralplotsthroughoutthis paper ∈ { }
function is
[63]. Infact, the blue line splits to twosymmetric curves
from the point of n¯ 1.43, according to our numerical E (ξ ,ξ )= Πˆ (ξ ) Πˆ (ξ ) (7)
calculation(whichi−so≈bviousbecause α andα aresim- Π 1 2 h 1 1 ⊗ 2 2 i
1 2
plyinterchangeable),whileweplotonlyoneofthecurves with which the Bell-CHSH function Π can be con-
B
for convenience. structed using Eq. (5). We have obtained and presented
an explicit form of the correlation function in Appendix
The results in Figs. 1 and 2 shows that when the av- A.
erage photon number is relatively large, the asymmet- The optimized Bell-CHSH functions are plotted for
ric ECS outperforms the symmetric one in testing the varying n¯ using parity measurements in Fig. 3. In
Bell-CHSH inequality with photon on-off measurements contrast t±o the on-off measurement case, the opti-
and displacement operations. On the other hand, when mized Bell-CHSH functions increase monotonically to-
the averagephotonnumber is small, the symmetric ECS wardCirel’son’sbound [64], 2√2, for both evenand odd
gives larger Bell violations. ECSsasshowninthefigure. Theseresultsareconsistent
4
1.6 2.7
CHSH inequality violation using a general form of ECS
2.6 and compare it to the result with a symmetric ECS for
1.3
both on-off and parity measurements.
2.5
1
2.4
A. Symmetric and asymmetric strategies for
2
α
2.3 entanglement distribution
0.7
2.2
In the presence of photon loss, the time evolution of
0.4
2.1 a density operatorρ is described by the master equation
as [65]
0.1 2
0.1 0.4 0.7 1 1.3 1.6
∂ρ
=Jˆρ+Lˆρ, (8)
α
1 ∂τ
where τ is the interaction time, and the Lindblad super-
FIG. 4. (Color online) Optimized Bell-CHSH function for
ECS+ using photon number parity measurements. We op- operators Jˆ and Lˆ are defined as Jˆρ = γ iaˆiρaˆ†i and
|timizeithe Bell-CHSH function for amplitudes, α1 and α2, Lˆρ=−γ/2 i(aˆ†iaˆiρ+ρaˆ†iaˆi) with a decayPrate γ.
with all displacement variables, and find that the function is We consider two different strategies, i.e. symmetric
optimized when α1 =α2. (A)andasyPmmetric(B)ones,whendistributing anECS
over a distance to Alice and Bob as illustrated in Fig. 5.
In the case of strategy A, photon loss occurs symmetri-
with previous studies [6, 7]. As explained in Ref. [7], the cally for both modes of the ECS during the decoherence
reasonthat the Bell-CHSH violation increases monoton- time τ. On the other hand, in strategy B, an ECS is
ically in the case of the parity measurement but not in first generated in the location of Alice and one mode of
the case of the on-off measurement can be explained as the ECS is sent to Bob at a distance. In the latter case,
follows. When the average photon number of the sym- photon losses occur only in one of the field modes but
metric ECS is sufficiently large, it can be represented the decoherencetime becomes2τ. Assuming azerotem-
as a maximally entangled two-qubit state in a 2 2 peraturebath anda decayrateγ for both cases,a direct
⊗
Hilbert space spanned by the even and odd SCS basis calculation of the master equation leads to the states
(α α ) and the displacement operator well ap-
| i ± | − i
pthreoxvimiolaatteisonthaeppqruobaicthreostathtieonm.axTihmeurmefovrael,uien, 2th√is2,liwmiitth, ρ±A(α1,α2,t)= N±2 |√tα1ih√tα1|⊗|√tα2ih√tα2|
the parity measurement that perfectly discriminates be- e−2(1−nt)(α21+α22)
±
tween the even and odd SCS. In contrast, the photon
√tα √tα √tα √tα
number on-offmeasurementcannotcause Bell violations | 1ih− 1|⊗| 2ih− 2|
for large amplitudes because the vacuum weight in the (cid:2)+ √tα1 √tα1 √tα2 √tα2
|− ih |⊗|− ih |
state almost vanishes in the limit of α 1 [7].
≫ + √tα √tα √tα √t(cid:3)α ,
Asimpliedbytheapparentoverlapsbetweenthe cases 1 1 2 2
|− ih− |⊗|− ih− |
of symmetric ECSs (solid curve) and those of the asym- o(9)
metric ECSs(dotted), ournumericalinvestigationsshow
thatunlikethecasewithon-offmeasurements,theasym- ρ±B(α1,α2,t)= N±2 |α1ihα1|⊗|tα2ihtα2|
mureet4ricshEoCwSssthdeoonpottimshizoewdaBneyll-lCarHgeSrHBfeulnlcvtiioolnatoiofnEs.CFSi+g- ±e−2(1−nt2)α22 |α1ih−α1|⊗|tα2ih−tα2|
| i
for amplitudes α1 and α2. The blue line in the figure in- +|−α1ihα1|⊗(cid:2) |−tα2ihtα2|
dicates the states with which the maximum violations
+ α α tα t(cid:3)α (10)
are obtained for the same averagephoton numbers. The |− 1ih− 1|⊗|− 2ih− 2|
figure shows that the asymmetry of the amplitudes does o
forstrategiesA andB,respectively,where t=e γτ. For
not improve the amount of violations in the case of the −
parity measurement. We conjecture that the symmetric convenience, we define the normalized time r = 1 t
−
which has the value of zero when τ = 0 and increases
structure of the parity measurement is closely related to
this result. to 1 as τ increases to the infinity. If the cross terms
in Eqs. (9) and (10) vanish, the states become classical
mixtures of two distinct states and quantum effects gen-
III. BELL-CHSH INEQUALITY TESTS UNDER erally disappear. We observe that the cross term in ρ±A
DECOHERENCE EFFECTS is proportional to e−2(1−t)(α21+α22) and it is proportional
to e−2(1−t2)α22 in ρ±B. This implies that one may reduce
Inthis section,weconsiderdecoherenceeffects onBell decoherence effects using strategy B by making the am-
nonlocality tests using an ECS. We will study the Bell- plitude smaller for the mode in which loss occurs (i.e.,
5
(a) τ τ (a)1.6 2.06
(cid:36)(cid:79)(cid:76)(cid:70)(cid:72) (cid:54)(cid:82)(cid:88)(cid:85)(cid:70)(cid:72) (cid:37)(cid:82)(cid:69) 2.05
1.3
2.04
τ 1
(b)
2 2.03
(cid:36)(cid:79)(cid:76)(cid:70)(cid:72) (cid:54)(cid:82)(cid:88)(cid:85)(cid:70)(cid:72) (cid:37)(cid:82)(cid:69) α
0.7
2.02
FIG.5. SchemestodistributeanECSinasymmetricway(a) 0.4
2.01
and in an asymmetric way (b) strategies. (a) In strategy A,
photon losses occur in both partswith thesame rate and for
0.1 2
timeτ. (b)InstrategyB,anECSisgeneratedinthelocation
0.1 0.4 0.7 1 1.3 1.6
of Alice and one of the field modes is sent to Bob. Photon
α
losses then occur only one of thetwo modes for time 2τ. 1
(b)1.6 2.16
2.14
2.14 1.3
2.12 ρ+ 2.12
B
2.1 ρ+ 2.1
A 1
2.08
max 2.06 α2 2.08
|O 2.04 0.7 2.06
B
| 2.02
2.04
2 0.4
1.98 (a) 2.02
1.96
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 2
0.1 0.4 0.7 1 1.3 1.6
r
α
1
2.8
2.7 ρ−
B FIG. 7. (Color online) Optimized Bell-CHSH values against
2.6 ρ−
2.5 A α1 and α2 (a) in strategy A and (b) in strategy B using on-
x off measurements for the odd ECSs. The normalized time is
a 2.4
m r = 0.2 for both (a) and (b). In (a), only one side of the
|O 2.3
B blue curvesis displayed. Since thecorrelation function has a
| 2.2
symmetry over the exchange of the two modes, the maximal
2.1 (b)
violation points also exist in the opposite side relative to the
2
1.9 centerlineof α1 =α2. Strategy Bgenerally shows thelarger
violations of the Bell-CHSH inequality than strategy A. The
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
optimizingvalueofα2 forthesamevalueofα1 with strategy
r B is always smaller than that with strategy A.
FIG. 6. (Color online) Optimized Bell-CHSH function
O max against thenormalized timer for(a)evenECSsand
|B | B. Bell-CHSH tests with on-off measurements
(b) odd ECSs using on-off measurements. The dotted curves
under photon loss effects
indicatetheoptimizedresultsforstrategyAforentanglement
distributionandthesolidcurvescorrespondtotheresultsfor
strategy B explained in the main text. Amplitudes α1, α2 We first consider Bell-CHSH tests with on-off mea-
togetherwithdisplacementvariablesareallnumericallyopti-
surementsincomparingstrategiesAandBregardingro-
mizedtofindthemaximumvaluesoftheBell-CHSHfunction
bustness to the decoherence effects. An explicit form of
forgivenr. Theoptimizingvaluesofα1 andα2 arefoundbe- the correlation functions calculated using Eqs. (9) and
tween0.4 and1.4, whererelativelysmaller valuescorrespond
(10) can be found Appendix A. We numerically opti-
to large values of r. The dot-dashed horizontal line indicates
mize the corresponding Bell-CHSH function, ,
theclassical limit, 2. O max
|B |
over all displacement variables and amplitudes α and
1
α for given r. The numerically optimized Bell-CHSH
2
functions against the normalized time are presented for
bothρ±A andρ±B inFig.6wherethedecreaseofviolations
by making the field mode sent to Bob in Fig. 5 to have due to the decoherence effects is apparent. The optimiz-
the smalleramplitude). Suchastrategywasalsoapplied ing values of amplitudes α and α are found between
1 2
tothetele-amplificationprotocol[48]andthedistributed 0.4 and 1.4, where relatively smaller values correspond
generation scheme for ECSs [49]. to large values of r. We observethat strategy B leads to
6
2.9 (a)1.6 2.016
2.8 (a) ρ+
2.7 ρB+ 2.014
2.6 A 1.3
2.012
B|Πmax 222...345 1 2.01
| 2.2 2 2.008
α
2.1
0.7
2 2.006
1.9
2.004
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
2.002
r
0.1 2
2.9
2.8 (b) ρ− 0.1 0.4 0.7 1 1.3 1.6
B
22..67 ρ−A α1
max 22..45 (b)1.6 2.12
B|Π 2.3
| 2.2 2.1
1.3
2.1
2
2.08
1.9 1
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
2 2.06
r α
0.7
2.04
FIG.8. (Coloronline)OptimizedBell-CHSHfunctionagainst
normalized decoherence time r for photon parity measure- 0.4 2.02
ment(a)forevenECSand(b)foroddECS.Thedottedcurves
showtheoptimizedresultsforstrategyAandthesolidcurves 0.1 2
showtheresultsforstrategyB.Thedot-dashedlineindicates 0.1 0.4 0.7 1 1.3 1.6
theclassical limit 2.
α
1
FIG. 9. (Color online) Using parity measurement for even
larger violations than strategy A for r & 0.07 when we
use the even ECS ρ+. For the smaller value of r, strat- ECSs, we optimize the Bell-CHSH function against α1 and
α2 for r = 0.1 (a) in strategy A and (b) in strategy B. All
egyAgivesslightlylargerviolationswherethedifference displacement variables are optimized for given α1 and α2.
is up to . 0.001. On the other hands, the odd ECS ρ
−
showslargerviolationsforstrategyBthanstrategyAre-
gardlessofthevalueofr. AsexplainedearlierinSec.IIA, egy A. In strategy A (Fig. 7(a)), the maximum value of
the discontinuities of the first derivative at r 0.09 for the Bell-CHSH function is 2.054 for α 0.81 and
ρ+ and r 0.07 for ρ+ in Fig. 6(a) emerge≈from the α 0.74. On the other ≈hands, the ma1xi≈mum value
nAumerical o≈ptimizationBprocess where local maxima for fo2r s≈trategy B (Fig. 7(b)) is 2.145 for α 0.76 and
1
different parameter regions are compared. α 0.50. Noting that 2.0≈is the maximu≈m value of
2
≈
The Bell violations of the odd ECSs for varying α the Bell function by a local realistic theory, the abso-
1
and α for r = 0.2, numerically optimized for the dis- lute value of the Bell function with strategy B, 2.145, is
2
placement variables, are shown in Fig. 7. Figure 7(a) significantly larger than that of strategy A, 2.054. It is
clearly shows the asymmetric behavior of the optimized a remarkable advantage of using asymmetric ECSs with
Bell-CHSH function under strategy A. Figure 7(b) is for the asymmetric distribution scheme in testing the Bell-
the case of asymmetric decoherence (strategy B). For a CHSH inequality.
givenaveragephotonnumber,theoptimizingvalueofα
2
instrategyBissmallerthanthatofstrategyA.Itshows
that an asymmetric ECS can reduce the asymmetric de- C. Bell-CHSH tests with photon number parity
coherenceeffects by lesseningthe amplitude of the mode measurement under photon loss effects
which decoherence occurs and adjusting the other mode
which is free fromdecoherence. These results are consis- Figure 8 presents the numerically optimized Bell-
tent with previous studies for Bell inequality tests using CHSH function with parity measurements against
Π
hybrid entanglement [66, 67] and quantum teleportation the normalized t|iBme|r (see Appendix A for details). For
[48, 68], where the amplitude of the mode that suffers boththeevenandoddECSs,strategyBgivesignificantly
decoherence should be kept small in order to optimize larger violations over strategy A. The optimizing values
Bell violations or teleportation fidelities. ofα andα approachinfinityforr 0. Thisisbecause,
1 2
→
It is important to note from Fig. 7 that strategy B when we use the photon number parity measurements
shows significantly larger violation than that of strat- and the photon loss is absence in one mode, the opti-
7
mized Bell-CHSH functions increase monotonically with
repect to the amplitude of that mode. However, we are (cid:39) (cid:54)(cid:82)(cid:88)(cid:85)(cid:70)(cid:72) (a)
interestedinvaluesthatcanbeobtainedwithinarealistic
range(e.g. n¯ <5). WethusplotinFig.9theoptimiza-
tion results fo±r the even ECSs under a normalized deco- (cid:39) (cid:54)(cid:82)(cid:88)(cid:85)(cid:70)(cid:72) (b)
herencetime(r =0.1)forvaryingα andα . Incontrast
1 2
tothecaseofon-offmeasurements,asshowninFig.9(a),
theasymmetryoftheamplitudesoftheECSdoesnotim- FIG.10. (Coloronline)Schematicsfor(a)displacedmeasure-
provethe amountof Bellviolations for strategyA where ment underdecoherence in a part of the entangled state and
(b)displacedmeasurementusinganinefficientdetector. Only
decoherence occurs symmetrically. Similar to the results
one part of theBell inequality test is shown.
oftheon-offmeasurements,theasymmetryoftheampli-
tudes in ECSs increases the amount of Bell violation for
strategy B (Fig. 9(b)). In this case, the maximum value (a) 1 (b) 1
2.7 2.7
of optimized Bell-CHSH function is 2.146 for
Π max 0.95 0.95
α andα 0.46. A similarva|lBue| ≈ 2.131 2.55 2.55
1 2 Π max
→∞ ≈ |B | ≈
can be obtained for α1 =2 and α2 0.44 where the av- 0.9 2.4 0.9 2.4
≈
erage photon number is given by n+ ≈ 4.19. This value η0.85 2.25 η0.85 2.25
ismuchlargerthanthatofstrategyA,whichonly shows
Π max 2.014 for α1 =α2 0.44. 0.8 2.1 0.8 2.1
|B | ≈ ≈
2.01
0.75 2.0005 0.75 2.01
0.75 0.8 0.85 0.9 0.95 1 0.75 0.8 0.85 0.9 0.95 1
IV. EFFECTS OF DETECTION INEFFICIENCY η η
 
(c) 1 (d) 1
2.3 2.3
Herewe investigatethe effects ofdetectioninefficiency 0.95 2.2 0.95 2.2
in the Bell-CHSH inequality tests. An imperfect pho- 2.1 2.1
todectectorwithefficiencyη canbemodeledusingaper- 0.9 0.9
2.01
fectphotodetectortogetherwithabeamsplitteroftrans- η η 2.01
0.85 0.85
missivity √η in front of it [69]. Meanwhile, decoherence
by photonloss (Eqs.(9) and(10)) inthe entangledstate 2.0001
0.8 0.8 2.0001
canalsobe modeled usinga beamsplitter. The only dif-
ference between the effect of detection inefficiency and 0.75 0.75
0.75 0.8 0.85 0.9 0.95 1 0.75 0.8 0.85 0.9 0.95 1
that of photon loss in the entangled state is the order of
η η
 
photon loss and displacement as shown in Fig. 10. We
compare these two cases in Appendix B where the re-
FIG. 11. (Color online) Numerically optimized Bell-CHSH
sults show that the two cases lead the same Bell-CHSH function for all displacement variables and amplitudes using
function except for the coefficients of the displacement imperfect photodetectors. The detection efficiencies are η1
variables which disappear during the optimization pro- for mode 1 and η2 for mode 2. The results in (a) and (b)
cess. show theoptimized Bell-CHSH functions for displaced on-off
We have numerically investigated both cases of on-off measurement using the odd ECSs. The results are restricted
and parity measurements with limited efficiencies, η for tothesymmetricECSin(a)whiletheyareoptimizedforany
1
mode 1 and η for mode 2, for both even and odd ECSs. values of α1 and α2 with the asymmetric ECS in (b). The
As we find th2at the even ECS is better for on-off mea- optimizingvaluesofα1andα2for(a)and(b)arebetween0.4
to1.4. Resultswithparitymeasurementsareshownin(c)for
surementsandtheoddECSisbetterforparitymeasure-
theevensymmetricECSand(d)fortheevenasymmetricECS
ments,asalreadyimpliedinFigs. 6and8,wepresentthe
withanyvaluesofα1 andα2. Thetwouppermostlinesin(c)
two corresponding cases in Figure 11. It shows that the and(d)indicate2.4and2.5,respectively. Whenη1 =η2 =1,
on-off measurement scheme generally gives much larger the maximum violations of the Bell-CHSH functions are up
violations compared to the parity measurement scheme. to 2.743 for (a) and (b) and 2.820 for (c) and (d). The
≈
This is consistent with the results of the case with pho- optimizing values of α1 and α2 for (c) and (d) are between
ton loss effects studied in the previous section (see Figs. 0.02 to 1.64 for η1 <0.99 and η2 <0.99.
6(b) and 8(a)).
In detail, for the case of the displaced on-off measure-
ments with the odd ECS (Figs. 11(a) and 11(b)), opti- efficiency η 0.745 is sufficient for the same amount of
≥
mizingvaluesofα andα liebetween0.4to1.4,whichis violation with the asymmetric ECS. When the efficien-
1 2
experimentally feasible [47, 59]. We find that the asym- cies for modes 1 and 2 are η1 = 0.75 and η2 = 1, the
metricECS(Fig11(b))showslargerviolationscompared asymmetric ECS shows the optimized Bell quantity of
to the symmetric ECS(Fig 11(a)). When η =η =η , a 2.305 which is larger than 2.269 for the symmetric ECS.
1 2
detection efficiency of η 0.771 is required for violation In the case of the parity measurements using the
≥
of 2.001 with the symmetric ECS while a smaller even ECSs, the improvement by the asymmetric ECS
O
|B |≥
8
is even larger. For example, comparing η = 0.98 line violation with the asymmetric ECS. This improvement
1
in Fig. 11(c) and (d), we find that there is a violation is even more significant for the case of parity measure-
2.01 for η 0.760 if we use the asymmetric ments particularly when the detection efficiencies of two
Π 2
|B | ≥ ≥
formof ECS. However,much largerefficiency η 0.805 detectors differ much. When the detection efficiency for
2
≥
is needed when we use the symmetric ECS. The opti- mode 1 is η =0.98,the required detection efficiency for
1
mizing values of α and α lie between 0.02 to 1.64 if mode 2 is η &0.81 to show the violation of B 2.01
1 2 2 Π
| |≥
η <0.99 and η <0.99. We also note that if one of the using the symmetric ECS is used, but it is η & 0.76
1 2 2
detectorsisperfect,theoptimizingvalueoftheamplitude when the asymmetric ECS is used. The optimizing am-
of that mode goes to infinity. This behavior can be in- plitudes are found between 0.4 to 1.4 in the case of the
ferredfromtheresultsinSec.IIICwithFig.9(b)forthe on-off measurements and between 0.02 to 1.64 for the
caseofdecoherence,notingthatthedecoherenceandthe parity measurements. It is worth noting that these val-
detection inefficiency give qualitatively the same effects ues of amplitudes for ECSs are within reach of current
(Appendix B). When both the detectors are perfect, of technology [47, 59, 70, 71].
course,thelargeramplitudegivesthelargerviolationfor
Insummary,ourextensivestudyrevealsthattheasym-
each mode as implied in Fig. 4; the maximum violation metric formofECSs andthe asymmetricscheme for dis-
appears when both α and α become infinity.
1 2 tributing entanglement enable one to effectively test the
Bell-CHSH inequality with the same resources.
V. REMARKS
In this paper, we have studied asymmetric ECSs as ACKNOWLEDGMENTS
well as asymmetric lossy environments for Bell-CHSH
inequality tests. We first investigated the violations of
This work was supported by the National Research
Bell-CHSH inequality using perfect on-off detectors and
Foundation of Korea (NRF) grant funded by the Korea
ideal ECSs. We have shown that the asymmetric form
government(MSIP)(No. 2010-0018295)andbytheCen-
of ECS could give larger violations of the Bell-CHSH in-
ter for Theoretical Physics at Seoul National University.
equality in some region of the averagedphoton numbers
The numerical calculations in this work were performed
of the ECSs. On the other hand, in the case of photon
usingChundoongclustersystemintheCenterforMany-
numberparitymeasurements,wecouldnotfindapparent
core Programming at Seoul National University.
improvementofthe Bellviolationsusing the asymmetric
form of ECS with perfect detectors.
We then studied Bell-CHSH violations under the ef-
fectsofdecoherenceontheECSs. Weconsideredtwodif-
ferent schemes for distributing entanglement under pho- Appendix A: Correlation functions for on-off and
tonloss,i.e.,symmetricandasymmetricschemes. Inthe parity measurements
symmetricscheme,thephotonlossesoccurinbothmodes
oftheECSs. Ontheotherhands,photonlossoccursonly
Here, we present the explicit forms of the correlation
in one of the two modes in the case of the asymmetric
functions defined in Eqs. (4) and (7) under lossy effects.
scheme. We showed that the asymmetric form of ECS
Instead of computing the correlation function for every
can increase the amount of violations significantly under
case,wecalculatetheresultsforanECSunderthebeam
theasymmetricschemecomparedtothecaseofthesym-
splitters with transmissivities η and η for mode 1 and
1 2
metric scheme. For example, when the normalized time
mode 2, respectively,andshowthat these results areap-
is r = 0.2, the Bell-CHSH function using photon on-off
plicable to all the cases discussed in this paper.
detectorsfortheasymmetriclossschemeshowsthemax-
The beam splitter operator with transmissivity √η
imum value of 2.145, which is much larger than that for
the symmetric case, 2.054. A similar improvement can between modes a and b can be represented by Bˆab =
be made in the case of photon number parity measure- exp[(cos−1√η)(aˆ†aaˆb − aˆaaˆ†b)] [72]. If a coherent-state
ments; when r = 0.1 with the ECS of average photon dyadic α β for mode C is mixed with the vacuum for
| ih |
number 4.19, the optimized Bell-CHSH function un- mode v, the initial coherent-state dyadic becomes
≈
der the symmetric loss scheme is about 2.014 but it is
2.1W31euhnadveeratlhseoainsyvmesmtigeatrtiecdscehffeemctes.of inefficient detec- Trv[BˆCv(|αihβ|)C ⊗(|0ih0|)vBˆC†v]
1
tors. We show that the asymmetric form of ECSs lowers =exp (1 η)(α2+ β 2 2αβ∗) √ηα √ηβ .
the detectionefficiency requiredfor violationofthe Bell- −2 − | | | | − | ih |
(A1)
CHSH inequality. For example, a detection efficiency (cid:2) (cid:3)
of η 0.771 is required for a Bell-CHSH violation of
≥
2.001 with the symmetric ECS, but a smaller ef- We now apply this result to a pure ECS ECS mixed
O ±
|B |≥ | i
ficiency η 0.745 is sufficient for the same amount of withthe vacuummodesthroughtwobeamsplitterswith
≥
9
transmissivities η and η , and the result is It is straightforward to check that if we let η = η = t
1 2 1 2
then the state will be exactly the same to Eq. (9). In
ρ±[α1,α2,η1,η2] addition, η1 =1 with η2 =t2 will give Eq. (10).
= 2 √η α √η α √η α √η α
1 1 1 1 2 2 2 2
N± | ih |⊗| ih |
e−2[(1(cid:8)−η1)α21+(1−η2)α22]
±
√η α √η α √η α √η α
1 1 1 1 2 2 2 2
| ih− |⊗| ih− |
(cid:2)+ √η1α1 √η1α1 √η2α2 √η2α2
|− ih |⊗|− ih | Using the result, the correlation functions for on-off
+ √η1α1 √η1α1 √η2α2 √η2(cid:3)α2 . measurement and parity measurement are obtained as
|− ih− |⊗|− ih− |
(cid:9)
EO(ξ,χ)=Tr[ρ±Oˆ1(ξ) Oˆ2(χ)]
⊗
= 2 2e−22α21−2α22 1∓2e−2α21−2α22−χ2i−χ2r−ξi2−ξr2 eα21η1+χ2i+χ2rcos 2α1√η1ξi
±
+eα22η2+ξi2+ξr2cos (cid:2)2α χ √η 2eα21η1+α22η2cos(cid:0)2 α √η ξ +α(cid:0)√η χ (cid:1)
2 i 2 1 1 i 2 2 i
−
e−2 α21+α22 +2(cid:0)e− ξr−α1√η(cid:1)1(cid:1)2− χr−α2√η2 2−χ2i(cid:0)−ξ(cid:0)i2 +2e− α1√η1+ξr 2−(cid:1)α(cid:1)2√η2+χr 2−χ2i−ξi2
±
e− (cid:0)ξr−α1√(cid:1)η1 2−ξi2(cid:0) e− α1√(cid:1)η1+ξ(cid:0)r 2−ξi2 e(cid:1)− α2√η2+χr 2−(cid:0)χ2i e− χ(cid:1)r−α(cid:0)2√η2 2−χ2i(cid:1),
− − − −
EΠ(ξ,χ)=Tr(cid:0)[ρ±Πˆ1(ξ)(cid:1) Πˆ2(χ)] (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
⊗
1
= exp 2 α2(η +1)+α2(η +1)+χ2+χ2+ξ2+ξ2
2 2e−2α21−2α22 − 1 1 2 2 i r i r
±
2e4(α21η1+α22η2)cos(4(cid:0)α1√(cid:0)η1ξi+4α2χi√η2)+e2(α21−2α1ξr√η1+α22−2α2χr(cid:1)√(cid:1)η2) e8α1√η1ξr+8α2χr√η2 +1 ,
±
(cid:16) (cid:16) (cid:17)(cid:17)
where ξ = ξr + iξi and χ = χr + iχi, and Oi(ξ) and transmissivity √η would be
Π (ξ) were defined in Eqs. (3) and (6). By substituting
i
η = η = 1, we can get a correlation functions for the γ γ √ηγ √ηγ
1 2 | ih |−→| ih |
case of perfect detectors. Likewise, η1 =η2 =t gives the γ γ exp[ 2(1 η) γ 2] √ηγ √ηγ .
correlationfunctionsfordistributionstrategyAandη = | ih− |−→ − − | | | ih− |
1
1, η2 = t2 gives the correlation function for strategy B. Now we apply the displacement operator Dˆ(ξ) =
Aswewilldiscussbelow,wealsocanusetheseresultsfor
exp[ξaˆ ξ aˆ] to photon lost dyadics. Then the states
† ∗
the cases of imperfect detectors even though they differ −
would be
in the order ofthe displacement operatorsandthe beam
splitters. γ γ √ηγ ξ √ηγ ξ
| ih |−→| − ih − |
γ γ exp[ 2(1 η) γ 2+√η(ξ∗γ ξγ∗)]
| ih− |−→ − − | | −
√ηγ ξ √ηγ ξ . (B1)
Appendix B: The order of photon loss and | − ih− − |
displacement operator in correlation
Next,letus considerthe caseweemploythe operators
tooppositeorder,displacementfirstandphotonlosssec-
In this section, we show that ECSs give the same op- ond. After the displacement, the dyadics would be
timized value of the Bell-CHSH function independent to
the order of the displacement and the photon loss by γ γ γ ξ γ ξ
| ih |−→| − ih − |
a beam splitter. Note that it is sufficient to only con-
γ γ exp[ξ γ ξγ ] γ ξ γ ξ .
∗ ∗
sider two coherent-state dyadics of the form γ γ and | ih− |−→ − | − ih− − |
| ih |
γ γ . This is because ECSs only contain this kind ApplyingphotonlosstothesedyadicsusingEq.(A1),we
| ih− |
of dyadics and a photon loss by a beam splitter is su- finally obtain
perlinear(linear in the space of density matrices) and a
displacement operator is linear. γ γ √η(γ ξ) √η(γ ξ)
| ih |−→| − ih − |
First, suppose that the states undergo a beam split- γ γ exp[ 2(1 η) γ 2+η(ξ γ ξγ )]
∗ ∗
ter first and displacement second. From the above, the | ih− |−→ − − | | −
√η(γ ξ) √η( γ ξ) . (B2)
dyadics after the transmission to the beam splitter with
| − ih − − |
10
These results show that if we replace ξ with √ηξ in means that the optimized values of the Bell-CHSH func-
Eq. (B1), it will be exactly the same to Eq. (B2). This tionsfor alldisplacementvariablesdonotdependonthe
order of loss and displacement.
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