Sudden Death of Entanglement induced by Polarization Mode Dispersion

0.37 MB·English

by  Cristian Antonelli

Sudden Death of Entanglement induced by Polarization Mode Dispersion
Cristian Antonelli,1 Mark Shtaif,2 and Misha Brodsky3
1Dipartimento di Ingegneria Elettrica e dell’Informazione and CNISM, Universita` dell’Aquila, 67040 L’Aquila, Italy
2School of Electrical Engineering, Tel-Aviv University, Tel-Aviv 69978, Israel and
3AT&T Labs, 200 Laurel Ave. S., Middletown, NJ 07748 USA
(Dated: January 31, 2011)
Westudythedecoherenceofpolarization-entangledphotonpairssubjecttotheeffectsofpolariza-
tionmodedispersion,thechiefpolarization decoherencemechanisminopticalfibers. Weshowthat
fiberpropagationrevealsanintriguinginterplaybetweentheconceptsofentanglementsuddendeath,
decoherence-freesub-spacesandnon-locality. Wedefinetheboundariesinwhichentanglement-based
quantumcommunications protocols relying on fiberpropagation can be applied.
1
1 PACSnumbers: 03.65.Ud,03.65.Yz,03.67.Hk,42.50.Ex
0
2
Entanglement between particles is a fundamental fea- [13, 15]. Similarly, birefringent crystals have also been
n
tureofquantumphysics. Justasfundamentalisthephe- usedforthecontrolleddemonstrationofdecoherence-free
a
J nomenon of decoherence that takes place when the en- subspaces [16, 17]. Yet, the arbitrary birefringence char-

7 tangledquantumsysteminteractswiththeenvironment. acterizingfiber-optictransmission,producesapreviously
2 One of the most intriguing recent discoveries related to unobserved combination of physical effects.
decoherence is the phenomenon of entanglement sudden Theaccumulationofrandomlyvaryingbirefringencein
]
h death(ESD)[1,2]. Itmanifestsitselfinanabruptdisap- fibers leads to a phenomenon known by the name of po-
p pearance of entanglement once the interaction with the larization mode dispersion (PMD) [18]. Since the anal-
t- environment reaches a certain threshold. [3–5]. Beyond ysis of the general case of PMD is quite cumbersome,
n the interest that it attracts as a fundamental physical we limit ourselves to the simplest regime of operation in
a
phenomenon, decoherence plays a central role in quan- which the optical bandwidth of the photons is small in
u
q tum communications. The security of recent quantum comparisonwith the bandwidth over which PMD decor-
[ key distribution protocols explicitly relies on the nonlo- relates [19]. In this regime, without loosing the essence

calpropertiesofentanglement,quantifiedintermsofthe of the problem, the overalleffect of PMD resembles that
1
v violationofaBell-typeinequality[6–8]. Thereforeestab- of pure birefringence in the sense that it causes an inci-
7 lishing the relation between the violation of non-locality dent pulse to split into two orthogonally polarized com-
1 and ESD, which is very interesting from a standpoint of ponents delayed relative to each other [18]. The polar-
4 basic physics, has a potentially large impact on the new ization states of these two components are known as the
5
area of quantum communications. principal states of polarization (PSP) and the delay be-
.
1 A configuration that provides an excellent platform tweenthem is called the differential groupdelay (DGD).
0
for the controlled study of decoherence is that of polar- In contrast to the controlled environment of [13]-[17],
1
1 ization entangled photon-pairs, distributed over optical both the PSP and the DGD of real fibers vary stochas-
: fibers. In this scheme the main source of decoherence tically in time. Since typical time constants character-
v
i is the residual optical birefringence randomly accumu- izing the decorrelation of PMD in optical fibers are as
X lating along the fiber. While an alternative entangle- long as hours, days and sometimes months [20], PMD
r ment scheme, insensitive to birefringence, has been pro- evolution can be considered adiabatic in the context of
a
posed [9, 10], the ease with which light polarization can quantum communications protocols. Thus the density
be manipulated using standard instrumentation leaves matrix describing the quantum state needs to be evalu-
polarization-entanglement the configuration of choice in atedasafunctionofthe arbitraryvaluesoftheinstanta-
manysituations [11]. Moreover,numeroussourcesofpo- neousPMD.Asaconsequence,theparametersofinterest
larization entangled photons suitable for use with stan- obtained from the so evaluated density matrix are also
dard fibers have recently become available [12]. Hence, PMD dependent. The temporal statistics for those pa-
understandingtherelationbetweennon-localityandESD rameterscouldbe inprincipledeterminedbyapplication
as well as the ultimate limits imposed by fiber birefrin- of the proper PMD statistics. An approach, in which
gence on the distribution of polarization-entangled pho- the randomness of PMD is accounted for in the density
tons in fibers is a problem of utmost importance. matrix itself [21], implicitly assumes ultrafast PMD dy-
The fact that optical birefringence is a major polar- namicsandleadstofundamentallydifferentresults. This
ization decoherence mechanism has been known for a previously unstudied reality, produces important conse-
while. Indeedbirefringentcrystalshavebeenusedexten- quences to the dynamics of decoherence between polar-
sively for the creationand manipulation of specialquan- ization entangled photons.
tum states, such as the MEMS [13, 14] or Werner states In this letter we formulate a quantitative approach to
2
studying PMD-induced disentanglement. We consider with f(τ) being the inverse Fourier transform of f˜(ω).
the evolution of an arbitrary two-photon state maxi- As can be deduced from the form of Eq. (2), f(τ)2
| |
mally entangled in polarization as each photon propa- is proportional to the probability density function that
gates through a fiber with PMD. Our studies demon- Bob’s photon precedes Alice’s photon (or vice versa)
strate that the unequalandincreasingdifferentialdelays by τ. For brevity, we will denote the polarization
in both arms always lead to entanglement sudden death dependent part in the tensor product (1) by ψ =
p
[1, 2] for all but two special PSPs orientations. That is, u ,u +eiα u′ ,u′ /√2, whereas the expres|sioin in
| A Bi | A Bi
ourchannelcausesanabruptdroptozeroinconcurrence (cid:2)Eq. (2)willbeshortlyd(cid:3)enotedas f(t t ) . Theover-
A B
| − i
whileasinglephotonsubjectedtothesameenvironment all state is then expressed as ψ = f(t t ) ψ .
A B p
| i | − i⊗| i
depolarizes asymptotically. Contrary to that, when both Letusnowrepresentthestate ψ intermsoftheprin-
| i
delays are sufficiently close in value there exists a range cipal states of the PMD in the two arms. We denote by
of PSP orientations for which concurrence does not van- s ,s′ and s ,s′ thepairsofJonesvectorsthatcor-
{ A A} { B B}
ish. This is related to the existence of decoherence free respondtothe PSPalongthe pathsofphotonsAandB,
subspaces and offers an opportunity for non-local PMD respectively. We now represent ψ in the basis of the
p
| i
compensation. Finally, when only one photon experi- PSP modes as follows
ences PMD, the concurrence decays gradually for every
PSP orientation. Besides concurrence, we calculate the ψ = η s ,s +eiα˜1 s′ ,s′ /√2
S parameter of the Clauser-Horne-Shimony-Holt Bell’s | pi + η1(cid:0)|sA,sB′ i eiα˜2|sA′ ,sBi(cid:1)/√2, (3)
inequality. Comparing the loss of entanglement (concur- 2 | A Bi− | A Bi
(cid:0) (cid:1)
renceC =0)andviolationoflocality(S >2)wediscover
an intriguing empirical relation between them. where the coefficients η1 and η2 are given by
We assume a source in which a pair of polarization
entangled photons is generated either via spontaneous η1 =(sA·uA)(sB ·uB)+eiα(sA·u′A)(sB ·u′B) (4)
parametric down conversion [22], or by using four–wave η =(s u )(s′ u )+eiα(s u′ )(s′ u′ ) (5)
mixing [23, 24]. In either case, the quantum state of the 2 A· A B · B A· A B · B
generated photon-pair can be expressed as
and where α˜ is defined through the relation η =
i i
|ψi=Z d2πωf˜(ω)|ω,−ωi⊗ |uA,uBi+√e2iα|u′A,u′Bi, (1) s|ηtia|teexnpo(cid:0)rim(αal−izaα˜tii)o/n2,(cid:1)|.η1A|2ls+o,|ηn2o|2te=th1.atT,haesqisuaimntpitlyied|η1b|y2
isrelatedtothealignmentbetweentheinputtwo-photon
wherethe left-hand sideofthe tensorproductrepresents state (1) and the PSP. Thus, for example, in the case of
thefrequencywaveform,whereastheright-handsiderep- uA =sA and uB =sB, the value of |η1|2 is unity. In the
resents polarization modes. The frequency variable ω presence of PMD the arrivaltime of the A-photon is de-
denotes the offset from the central frequency, which is layedbyτA/2inthesA polarizationandadvancedbythe
′
equal to the pump-frequency in sources relying on four– same amount in the sA polarization, and the B-photon
wave mixing and to half the pump frequency in sources undergoesasimilarprocess. Therefore,theoutputstate,
based on spontaneous parametric down conversion. The i.e. the two-photon state after propagating through me-
function f˜(ω) represents the effect of phase-matching as dia with PMD, can be expressed as
well as the possible effects of filters, as we shall see be-
η τ τ
tlohwat. Ndoωticf˜e(ωth)a2t=no2rπmTa−li1zawtiiotnh oTfbtheiengstathtee |iψntiegimraptliioens |ψouti = √12|f(tA−tB − A−2 B)i⊗|sA,sBi
timeRof th|e det|ectors used in the set-up. In what fol- η2 τA+τB ′
+ f(t t ) s ,s
lows, the state (1) will be referred to as the input state √2| A− B − 2 i⊗| A Bi
of the system, which consists of the two optical paths η∗eiα˜2 τ +τ
2 f(t t + A B) s′ ,s
that lead the entangled photons from the source of en- − √2 | A− B 2 i⊗| A Bi
taasnAglleicmeenatndtowBaorbd.s iTtsheusteerrsm, scounventiaonndalluy′ refearrreedthtoe + η1∗eiα˜1 f(t t + τA−τB) s′ ,s′ . (6)
A,B A,B √2 | A− B 2 i⊗| A Bi
Jonesvectorsthatcorrespondtotheexcitedpolarization
statesofAlice’s andBob’sphotons,respectively. We use
The density matrix that characterizes the detected field
primes to denote orthogonality in polarization space, so
′ ′ ′ ′ ′ ′
that uA,B ·u′A,B = 0. The phase factor α is introduced itsragcivinegnboyveρr=thRedttiAmdetBmhotAd,etsBi|sψopuetrifhoψromute|dtAi,ntBoir,dwehretroe
for consistency with the experimental generation of po-
account for the fact that the photo-detection process
larization entangled photon pairs [23]. Notice that the
is not sensitive to the photon’s time of arrival (within
frequency dependent part in (1) can be re-expressed as
the detector’s integration window). The elements of
dω the resulting density matrix, establishing the correspon-
Z 2πf˜(ω)|ω,−ωi=ZZ dtAdtBf(tA−tB)|tA,tBi (2) dence (sA,sB) ↔ 1, (s′A,sB) ↔ 2, (sA,s′B) ↔ 3, and
3
sity matrix can be reduced to a Bell-diagonal form by a
proper change of basis [29]. This enables a fully analyt-
e ical evaluation of C and S. Note that a Bell-diagonal
c
n matrix is defined by three real parameters only. In the
e
urr caseofPMDtheyareτA,τB and|η1|2 andthefunctional
c dependence of C and S on them is given in [29].
n
o In the simplest case of PMD present in only one of
C
1 2 0 1 2 0 the two fibers, as described, for example, by τB = 0,
(cid:75)120.5 0 6 4(cid:87)A(cid:117)B (cid:75)120.5 0 6 4(cid:87)A(cid:117)B the concurrence is given by C = |Rf(τA)|; in this case
the concurrence is independent of the PSP orientation
FIG. 1: Concurrence versus τA and |η1|2. In (a) τB = τA, andcanonlydecayasymptoticallywithτA. Correspond-
whereas in (b) τB ≃ 1.7B−1. Light-green: S > 2. Dark- ingly,theS parameteracquiresthemaximumvaluecom-
green: S <2. Red: C =0 and S <2. patible with such concurrence, that is S = 2√1+C2
[30], indicating unconditional violation of Bell’s inequal-
ity. Note that the two cases η = 1 and η = 0,
1 1
′ ′ | | | |
(s ,s ) 4 are then given by where concurrence simplifies to C = R (τ τ ) and
A B ↔ | f A − B |
C = R (τ +τ ) respectively, are equivalent to that
ρ = ρ = η 2/2 | f A B |
11 44 1 of single-arm PMD, with corresponding nonzero DGDs
| |
ρ22 = ρ33 = η2 2/2 equal to τA τB and τA+τB. Remarkably, for τA =τB
| | −
ρρ31 == −ρρ42 ==η1∗ηη∗2ηR∗fei(ατRB)/(τ2 )/2 DanGdD|ηm1|a=gn1it,utdhee. cTonhcisurrreesnuclteiissquuniittey,inretegraersdtliensgsaonfdthiet
21 − 43 − 1 2 f A isdirectlyrelatedtotheconceptofdecoherence-freesub-
ρ41 = (η1∗)2eiαRf(τA−τB)/2 spaces [16, 17]. In this situation the output state, when
ρ = (η∗)2eiαR (τ +τ )/2, (7) expressedin the basis of the PSP,is a superposition of a
23 − 2 f A B state in which both photons are delayed, with a state in
∗
where Rf(τ) = T dtf (t)f(t + τ) is the autocorrela- which they are both advanced (see Eq. (3)), such that
tion function of f(Rt), normalized such that Rf(0) = 1. they reach the detectors simultaneously. Since in this
The function f˜(ω) which defines the frequency con- state,knowledgeofthephoton’stimesofarrivaldiscloses
tents of the two generated photons accounts for the no information on their polarization states, tracing out
phase matching spectrum, as well as for filtering ap- time involves no loss of information. Decoherence-free
plied to the two generated photons. In this case f˜(ω) = subspaces would not be allowed if PMD dynamics were
f˜ (ω)H (ω)H ( ω), where H (ω) and H (ω) denote fast on the scale of measurements, as assumed in [21].
pm A B A B
−
the transferfunctions ofAlice’sandBob’sfilters,respec- The dependence of the two-photon state decoherence
tively, and where f˜pm(ω) represents the phase-matching onthePMDparametersinthegeneralcaseismorecum-
spectrum. In most applications, the filters are much bersome,asitisgovernedbythetwoDGDvaluesτ and
A
narrower than the phase matching spectrum, in which τ andbythePSPorientation,accountedforby η . For
B 1
case f˜pm(ω) can be replaced by a constant, such that illustrative purposes, we plot in Fig. 1 the conc|ur|rence
f˜(ω) H (ω)H ( ω). Notice that the effect of PMD asafunctionofτ (normalizedtoB−1)and η 2 fortwo
A B A 1
∝ − | |
scales with the width of the autocorrelation function differentsettings ofτ , soastodescribe PMDeffects for
B
R (τ), whichis in turn determined by the overlapband- the most relevant realizations of PMD parameters. In
f
width of Alice’s and Bob’s filters, namely by the width Fig. 1a we plot the concurrence for the case of identical
of H (ω)H ( ω)2. For illustrative purposes,in all the DGDs, τ = τ , whereas in Fig. 1b the value of τ is
A B B A B
num| erical exam−ple|s considered in what follows, we will fixedandequalto1.7B−1. Therangeofvaluesforwhich
assume that H (ω)2 and H (ω)2 are Gaussian func- entanglement disappears entirely (i.e. C = 0) is empha-
A B
| | | |
tions of root mean square bandwidth B and central fre- sized in red color, whereas the green colored regions of
quencies ω and ω = ω respectively. thesurfacecorrespondtosettingsinwhichentanglement
A B A
−
We now turn to the characterization of the degree of exists(i.e. C >0). InthelightgreenareaS >2,whereas
entanglement of the PMD-affected two-photon state. In inthedarkgreenareaS 2. InthecasewheretheDGD
≤
thepresenceofPMD,tracingoverthetimeofarrivalputs values are equal (Fig. 1a), the concurrence approaches
the system in a partially mixed state. The extent of this unity when η 2 1. That is because in this situation
1
| | →
process can be quantified with the help of several pro- theinputstateisgivenbythefirstterminEq. (3),andit
posedentanglementmetrics[25–27]. Wechoosetocalcu- is not affected by the loss of time of arrival information.
latepurity,concurrence[27]andBell’sS parameter,thus In the opposite limit, when η = 0, the concurrence is
1
| |
assessing the largest possible violation of Bell’s inequal- givenbyC = R (2τ ) andreducestowardszeroasymp-
f A
| |
ity in the CHSH definition [28]. Because the individual totically for largeDGD values. Figure 1b also illustrates
photonstatesaremaximallymixed,thetwo-photonden- thatconcurrencecanbeunityonlywhentheDGDvalues
4
Fig. 2also marksthe S =2 boundary. Belowandto the
6
left of this line S >2 for all η 2 values, but to its right
1
| |
S may be smaller than 2 for some relative PSP orien-
tations. Intriguingly, this non-locality boundary (thick
4
B gray line) nearly perfectly reproduces the boundary to
× B 0.5 the ESD-free region(the white area)if the scale on both
τ
axes is stretched by a factor of 1.5.
2
Toconclude,wehavecarriedoutwhatwebelievetobe
thefirstquantitativeanalysisofthedecoherenceofpolar-
0 0 ization entangled photons propagating in optical fibers.
0 2 4 6
τ × B Our study shows that PMD leads to entanglement sud-
A
den death in all but a well defined restricted set of re-
alizations of fiber birefringence. In addition, we demon-
FIG.2: ESDprobability(seetext)versusτA andτB normal-
ized to B−1. The grey line is the boundary S = 2. Below strated how decoherence-free subspaces can be reached
and to the left of the grey line S > 2 for all values of |η1|2, via non-local PMD compensation. The ultimate limits
whereas to its right S <2 for some range of |η1|2. imposed by fiber birefringence to applications based on
non-local properties of polarization entanglement were
showntobeintriguinglyrelatedwiththephenomenonof
in the two arms are equal. entanglement sudden death.
The most interesting feature in Fig. 1 is the abrupt
transition of the concurrence to zero when either τ ,
A
or η 2 are varied continuously. The abrupt decay of
1
| |
concurrence is contrasted with the asymptotic decoher-
ence that would be experienced by a single photon if it [1] J.H. Eberly and T. Yu,Science 316, 555 (2007)
[2] T. Yuand J. H. Eberly,Science 323, 598 (2009)
evolved in the same environment. Indeed, the purity of
[3] M.P. Almeida et al.,Science 316, 579 (2007)
a polarized single-photon pulse characterized by a Jones [4] J. Laurat et al.,Phys. Rev.Lett. 99, 180504 (2007)
vector u and a time-mode distribution g(t), transmit- [5] K.AnnandG.Jaeger, Found.ofPhysics39,790 (2009)
ted through a fiber with DGD τ and PSP s, is given by [6] A.K. Ekert, Phys.Rev.Lett. 67, 661 (1991)
p = 1 2u s2 1 u s2 1 Rg(τ)2 , with Rg(τ) [7] A. Acin, S. Massar, and S. Pironio, New J. Phys. 8, 126
− | · | −| · | −| |
the autocorrelati(cid:0)on function(cid:1)o(cid:2)f g(t). This(cid:3)is a manifes- (2006)
tationoftheentanglementsudden-death(ESD)thathas [8] A. Acin et al., Phys.Rev.Lett. 98, 230501 (2007)
[9] R.T. Thew et al., Phys.Rev.A 66, 062304 (2002)
beenpreviouslyreportedinother physicalsystems [1–5].
[10] S. Tanzilli et al., Eur. Phys.J. D 18, 155 (2002)
In general, PSP orientation in optical fibers varies [11] A. Poppe et al, Optics Express, 12, 3865 (2004); Hu¨bel
faster than the DGD, resulting in a uniform distribution et al,Optics Express, 15, 7853 (2007)
of the parameter η1 2 between 0 and 1. For each given [12] S.X.WangandG.S.Kanter,IEEEJ.SelectedTopicsin
combination of τ| a|nd τ , we evaluate the fraction of Quantum Electronics 15, 1733 (2009)
A B
the interval between 0 and 1 in which ESD occurs. This [13] M. Barbieri et al.,Phys. Rev.Lett. 92, 177901 (2004)
[14] N.A. Peters et al., Phys.Rev.Lett. 92, 133601 (2004)
quantity, which can be interpreted as the probability of
[15] A.G. Whiteet al., Phys.Rev.A 65, 012301 (2001)
ESD (conditioned on the DGD values) is illustrated in
[16] P.G. Kwiatet al.,Science 290, 498 (2000)
Fig. 2. Note the existence of a completely white re- [17] J.B. Altepeteret al.,Phys.Rev.Lett.92,147901 (2004)
gion that shows the range of DGD values in which en- [18] J.P.GordonandH.Kogelnik,Proc.Natl.Acad.Sci.USA
tanglement does not disappear for any value of η1 2. In 97, 4541 (2000)
contrast to that, the darker tones mark areas fo|r w|hich [19] M. Shtaif and A.Mecozzi, Opt.Lett. 25, 707 (2000)
ESD occurs for some range of η 2. The color progres- [20] M. Brodskyet al.,J.LightwaveTechnol.24, 4584 (2006)
1
| | [21] P.S.Y.Poon,C.K.Law,Phys.Rev.A77,032330(2008)
sivelyturnsdarkforhighlydifferingDGDvalues. Onthe
[22] D.C.BurnhamandD.L.Weinberg,Phys.Rev.Lett.25,
other hand, when increasing DGDs remain nearly equal,
84 (1970)
the probability of ESD is about 0.5 [29]. [23] X. Li et al.,Optics Express 12, 3737 (2004)
Manyapplicationsinvolvingentanglementrelydirectly [24] H.TakesueandK.Inoue,Phys.Rev.A70,031802(2004)
on the violation of Bell’s inequality [7, 8] and, there- [25] G.S. Jaeger et al.,Phys. Rev.A 67, 032307 (2003)
fore, we compute the maximum value of the CHSH S– [26] R. Horodecky et al.,Rev. Mod. Phys. 81, 865 (2009)
parameter[28]forthedensitymatrixEq. (7)[31]. InFig. [27] W.K. Wootters, Phys.Rev. Lett.80, 2245 (1998)
[28] J.F. Clauser, M.A. Horne, A. Shimony, and R. A. Holt,
1, the range of parameters in which S > 2 —meaning
Phys. Rev.Lett. 23, 880 (1969)
thattheCHSHinequalityisviolated—iscoloredbylight
[29] Supplement to thisLetter.
green, whereas the dark green part of the surfaces rep- [30] F. Verstraete and M. M. Wolf, Phys. Rev. Lett. 89,
resents states in which C > 0, but there is no violation 170401 (2002)
of the CHSH inequality (S 2). The thick gray line in [31] R., I.and M. Horodecki, Phys.Lett. A200, 340 (1995)
≤
Supplementary material for the manuscript entitled
“Sudden Death of Entanglement induced by Polarization Mode Dispersion”
∗
Cristian Antonelli
Dipartimento di Ingegneria Elettrica e dell’Informazione and CNISM,
Universita` dell’Aquila, 67040 L’Aquila, Italy
Mark Shtaif
School of Electrical Engineering, Physical Electronics,
Tel-Aviv University, Tel-Aviv 69978, Israel
1
1
0 Misha Brodsky
2
AT&T Labs, 200 Laurel Ave. S., Middletown, NJ 07748 USA

n (Dated: January 31, 2011)
a
J In this supplement we derive analytical expressions for the purity, concurrence and S pa-

7 rameterofthe Clauser-Horne-Shimony-Holt-typeBell’sinequality, whichcharacterizeapair
2 of polarization entangled photons transmitted through two optical fibers with polarization

mode dispersion.
]
h
p PACS numbers: 03.65.Ud; 03.65.Yz; 03.67.Hk; 42.50.Ex
-
t
n
a 1. INTRODUCTION
u
q
[ In our Letter [1] we derive the density matrix describing the polarization state of two

polarization-entangled photons transmitted over optical fibers with polarization mode dispersion
1
v (PMD). The density matrix of Eq. (7) in [1] (we will further down refer to this equation as Eq.
7
(M7)) is expressed in the frame of reference of the principal states of polarization (PSP), in which
1
4 the derivation is greatly simplified and the final result appears quite compact. However, the an-
5 alytical evaluation of some of the parameters characterizing the entanglement of the two-photon
.
1 state is more convenient in a different frame of reference, in which the density matrix appears in
0
the so-called X-shape [2, 3].
1
In the first section of this supplement we study the purity of the two-photon state. In the
1
: following sections we provide analytical formulae for concurrence and CHSH S-parameter.
v
i
X
r 2. PURITY OF THE TWO-PHOTON STATE
a
The purity of the two-photon state can be evaluated through straightforward algebra from
P
Eq. (M7)
4 4
η1 + η2 2 2 2 2
= | | | | + η1 η2 Rf(τA) + Rf(τB)
P 2 | | | | (cid:2)| | | | (cid:3)
1
4 2 4 2
+ η1 Rf(τA τB) + η2 Rf(τA +τB) . (1)
2(cid:2)| | | − | | | | | (cid:3)
Obviously, the purity is equal to 1 when τ = τ = 0, that is when there is no PMD at all.
A B
2
Yet interestingly, it remains equal to 1 when η1 = 1 and the DGD values are equal, re-
| |
gardless of their magnitude. This property is a manifestation of the decoherence-free subspace
concept. On the other hand, as the DGD values become larger than the inverse overlap band-
−1
width of Alice’s and Bob’s filters B (see [1] for the definition of B), the purity approaches
4 2 4
η1 1+ Rf(∆τ) /2+ η2 /2, with ∆τ = τA τB. Then, the minimum purity with respect to
| | (cid:2) | | (cid:3) | | −
2
2 2
η1 and η2 is 1+ Rf(∆τ) / 2+ Rf(∆τ) /2 and it ranges between 1/4, for ∆τ = 0, and 1/3,
−(cid:2)1 | | (cid:3) (cid:2) | | (cid:3)
for ∆τ B .
≫
3. EXPRESSION FOR CONCURRENCE AND S-PARAMETER
In this section we derive the expressions for the concurrence and the S parameter of the CHSH
version of Bell’s inequality in terms of the parameters characterizing the PMD of the two fiber
arms. This goal is most conveniently achieved by first expressing the density matrix in a basis
where only the terms on its two diagonals are not zero, a form known as X-shape representation
[2, 3]. Subsequently the procedure requires defining the matrix S whose components are given by
s = Trace(ρσ σ ), where σ (n = 1, 2, 3) are the Pauli matrices. Then the concurrence and
nm n m n
⊗
the S parameter are expressed in terms of the eigenvalues of STS, where T denotes transposition.
Eventually, the eigenvalues of STS are related to the PMD parameters.
3.1 The X-shape representation of the two-photon density matrix
Since the individual photons are completely depolarized, the two-photon state belongs to the
class of the Bell-diagonal states [4] and hence to that of theso-called X-states [2, 3]. Namely, there
exists a representation in which the density matrix takes the form of
ρ11 0 0 ρ14
 
0 ρ22 ρ23 0
ρ= , (2)
 0 ρ32 ρ33 0 
 
ρ41 0 0 ρ44 
where all entries are real with ρ11 = ρ44, ρ22 = ρ33 = 1/2 ρ11, ρ14 = ρ41 and ρ32 = ρ23. In this
−
representation concurrence is given by the formula [3]:
C = 2max 0, ρ23 ρ11, ρ14 ρ22 . (3)
{ | |− | |− }
The representation in which the density matrix is in the form of Eq. (2) is constructed as follows.
Using the coefficients s = Trace(ρσ σ ) which form a 3 3 real matrix S, we find the
nm n m
⊗ ×
eigenvectors ofSTSandofSST,whichcouldbeviewedasvectorsrepresentingstatesofpolarization
in the Stokes space [5]. One then chooses the two orthogonal Jones vectors that correspond to one
of the eigenvectors of STS to span the subspace corresponding to the first photon. Similarly an
orthogonal pair of Jones vectors corresponding to one of the eigenvectors of SST is chosen to span
the subspace corresponding to the second photon. The overall space, which is the tensor product
of the two subspaces, is then spanned by the two pairs of Jones vectors that were selected [4]. In
this process, the eigenvector of STS and the eigenvector of SST that the desired representation
is based upon, are selected arbitrarily. Moreover, it is irrelevant which of the two pairs of Jones
vectors is assumed to represent a specific photon’s subspace, namely the two pairs can always be
interchanged without compromising the process.
Inthisrepresentation onecanverify, usingEq. (2)alongwiththedefinitionofthes elements,
nm
that S is diagonal and therefore
2
(ρ14 +ρ23) 0 0
STS = 1  0 (ρ14 ρ23)2 0  (4)
4 − 2
 0 0 (ρ11 ρ22) 
−
This form will be later used for relating the concurrence and the S parameter to the quantities
characterizing PMD through the eigenvalues of STS.
3
3.2 Expressing concurrence in terms of the eigenvalues of STS
The aim of this section is to demonstrate the following expression for concurrence in terms of
the eigenvalues of STS:
1 1/2 1/2 1/2
C = max 0,λ +λ +λ 1 , (5)
n 1 2 3 o
2 −
wherewedenotetheseeigenvaluesasλ1,λ2 andλ3. Forthecalculationthatfollows, itisconvenient
to assume that the eigenvalues are sorted such that λ1 λ2 λ3.
≥ ≥
Our first goal is to express the Bell-diagonal density matrix in terms of λ1, λ2 and λ3. Let us
consider a basis for subspaceA such that the correspondingStokes unit vector ˆb is an eigenvector
A
of STS belonging to the largest eigenvalue, namely STSˆbA = λ1ˆbA. Consider also a basis for
subspace B such that the corresponding Stokes unit vector ˆb is an eigenvector of SST defined
B
though ˆb = Sˆb / Sˆb , where denotes vector length.
B A A
|| || ||·||
We denote the correlation between the measurements of coincident photons along the po-
larizations defined by the Stokes vectors ˆb and ˆb as E(ˆb ,ˆb ) = P (1 P ), where
A B A B c c
− −
Pc = ρ11+ρ44 = 2ρ11 istheprobabilityofacoincidentdetectionofphotonsAandB passingthrough
polarizer beam splitters aligned to ˆb and ˆb . From Eq. (7) of [4], E(ˆb ,ˆb ) =ˆb Sˆb = λ1/2, it
A B A B A B 1
1/2 ·
follows that ρ11 = ρ44 = (1+λ1 )/4. Also from the condition of unit trace of the density matrix,
ρ22 = ρ33 = (1 λ11/2)/4. This yields λ1 as the third diagonal element of STS.
−
The remaining off-diagonal terms of the density matrix follow from equating the first and the
second diagonal elements of STS to λ2 and λ3 (in some arbitrarily assumed order). This yields two
1/2 1/2 1/2 1/2
possiblesolutions: either ρ23 = ρ32 = λ2 +λ3 and ρ14 = ρ41 = λ2 λ3 ,or ρ23 = ρ32 =
1/2 1/2 | | 1|/2 | 1/2 | | | | − | | | |
λ2 λ3 and ρ14 = ρ41 = λ2 +λ3 . However, only one of them is compatible with nonzero
− | | | | 1/2 1/2
values of concurrence. In fact, using Eq. (3) for the choice ρ23 = ρ32 = λ2 +λ3 and ρ14 =
| | | | | |
1/2 1/2 1/2 1/2 1/2 1/2 1/2 1/2
ρ41 = λ2 λ3 yields C = maxn0, λ1 +λ2 +λ3 1,λ1 +λ2 λ3 1o/2 = 0.
| | − − − − −
Indeed,sincealleigenvaluesarelessthanunityandλ1 isthelargest,thefollowinginequalityistrue,
1/2 1/2 1/2 1/2 1/2 1/2
λ +λ +λ 1 0. Ontheotherhand,thepositivityofρimpliesthatλ λ 1 λ ,
1 2 3 2 3 1
− 1/2 − 1≤/2 1/2 − ≤1/2− 1/2
which yields λ1 +λ2 λ3 1 0. Hence the correct solution is ρ23 = ρ32 = λ2 λ3
1/2 − 1/2 − ≤ | | | | −
and ρ14 = ρ41 = λ2 +λ3 and the density matrix reads as:
| | | |
1/2 1/2 1/2
1+λ 0 0 ζ(λ +λ )
1 2 3
 
1 0 1 λ1/2 ξ(λ1/2 λ1/2) 0
1 2 3
ρ= 4 0 ξ(λ12/−2 λ13/2) 1 −λ11/2 0 . (6)
 ζ(λ1/2+λ1/2) 0− −0 1+λ1/2 
2 3 1
where ζ and ξ can arbitrarily be set to 1. This indetermination reflects the phase ambiguity in
±
the conversion from Stokes to Jones representation of a polarization state [5].
Finally, applying the simplified concurrence formula Eq. (3) to the density matrix in Eq. (6)
yields the expression for concurrence given in Eq. (5).
3.3. Expressions for the eigenvalues of STS in terms of the PMD parameters
Theanalytical derivation of theeigenvalues λ requirestosolve a thirdorderalgebraic equation.
i
This task is in general quite cumbersome. However, for the density matrix in the shape of Eq.
(M7) this is possible under two conditions that make the matrix real. First, we make a realistic
assumption on the symmetry of Alice’s and Bob’s filters, H (ω)H ( ω) = H ( ω)H (ω).
A B A B
| − | | − |
4
Under this assumption R (τ) is real. Secondly, we perform a change of basis. In particular,
f
we use a basis (s˜ , s˜′ ) for the subspace A where s˜ = eiϕA/2 s and s˜′ = e−iϕA/2 s′
| Ai | Ai | Ai ′ | Ai | Ai | Ai
|ws˜iBthi =ϕAei=ϕB/α2|s+BIima[nlodg(|ηs˜1′Bηi2)=]. eS−iimϕBil/a2r|sly′B,iwweituhseϕaB b=asIism[(l|os˜gB(iη,1|ηs˜B2)i]). fIonr tthhies sruepbrsepsaecnetaBtiownhtehree
density matrix of Eq. (M7) can be simplified by setting α = 0 and by replacing η1 and η2 with
their absolute values η1 and η2 , thus making the matrix real. In this representation the matrix
| | | |
STS appears as:
m11 0 m13
 
STS = 0 m22 0 (7)
m13 0 m33 
with
2 2 2 2 2 2
m11 = (cid:2)|η1| Rf(τA −τB)−|η2| Rf(τA +τB)(cid:3) +4|η1| |η2| Rf(τB) (8)
2 2 2
m22 = η1 Rf(τA τB)+ η2 Rf(τA +τB) (9)
(cid:2)| | − | | (cid:3)
2 2 2 2 2
m33 = 4|η1| |η2| Rf(τA)+(2|η1| −1) (10)
2 2
m13 = 2 η1 Rf(τA τB) η2 Rf(τA+τB) η1 η2 Rf(τA)
− (cid:2)| | − −| | (cid:3)| || |
2
+ 2η1η2 Rf(τB)(2η1 1) (11)
| | | | −
The eigenvalues of STS can be evaluated with the following result
2 2 1/2
ℓ1 = (m11+m33)/2+ (m11+m33) /4 (m11m33 m13) (12)
(cid:2) − − (cid:3)
ℓ2 = m22 (13)
2 2 1/2
ℓ3 = (m11+m33)/2 (m11+m33) /4 (m11m33 m13) , (14)
−(cid:2) − − (cid:3)
where ℓ1, ℓ2 and ℓ3 are the unsorted eigenvalues of STS. The concurrence is then given by
1 1/2 1/2 1/2
C = max 0,ℓ +ℓ +ℓ 1 . (15)
n 1 2 3 o
2 −
The largest violation of CHSH inequality is given by S = 2√λ1+λ2, where λ1 and λ2 are the two
largest eigenvalues of STS [6]. From Eqs. (12) and (14) it follows that ℓ1 > ℓ3, and therefore
S = 2 ℓ1+max ℓ2,ℓ3 . (16)
p { }
Equations (15) and (16) are the main result of this supplement. They give the expressions for
largest violation of Bell’s inequality and concurrence as a function of the instantaneous PMD
values.
3.4 Entanglement sudden death threshold for large differential group delays
We conclude this supplement by evaluating the concurrence of the output two-photon state in
a particular case of interest, that is when the two DGDs are much larger than the inverse photon
bandwidth. In this regime one can derive the following expression
2
C = maxn0, 1+ Rf(∆τ) η1 1o, (17)
(cid:2) | |(cid:3)| | −
where ∆τ is the difference between the two DGDs. This results can be used to characterize a
threshold seen in Fig. 1a of the Letter. In that figureit appears that for two equal DGDs thereis a
5
2
threshold value of η1 = 0.5, above which ESD does not occur for increasing DGDs. In fact, such
| |
athresholdexists as longas thedifferencebetween two DGDs isfinite. Usingtheabove asymptotic
2 −1
expression for concurrence this threshold can be estimated as η1,th = [1+ Rf(∆τ) .
| | | |(cid:3)
∗
Electronic address: [email protected]
[1] C.Antonelli,M.Brodsky,andM.Shtaif,“SuddenDeathofEntanglementinducedbyPolarizationMode
Dispersion,” Phys. Rev. Lett. (2011)
[2] T. Yu and J. H. Eberly, Optics Communications 264, 393 (2006)
[3] T. Yu and J. H. Eberly, Qantum Information & Computation 7, 459 (2007)
[4] M. Horodecki, R. Horodecki, Phys. Rev. A 54, 1838 (1996)
[5] J.P. Gordon and H. Kogelnik, Proc. Natl. Acad. Sci. USA 97, 4541 (2000).
[6] R. Horodecki, I. Horodecki, and M. Horodecki, Phys. Lett. A 200, 340 (1995)