Three-qubit topological phase on entangled photon pairs

0.21 MB·English

by  Markus Johansson

Three-qubit topological phase on entangled photon pairs
Markus Johansson1, Antonio Z. Khoury2, Kuldip Singh1 and Erik Sjo¨qvist1,3
1 Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore
2 Instituto de F´ısica, Universidade Federal Fluminense, 24210-346 Niter´oi - RJ, Brazil and
3 Department of Quantum Chemistry, Uppsala University, Box 518, SE-751 20 Uppsala, Sweden
We propose an experiment to observe the topological phases associated with cyclic evolutions,
generated by local SU(2) operations, on three-qubit entangled states prepared on different degrees
of freedom of entangled photon pairs. The topological phases reveal the nontrivial topological
structureofthelocal SU(2)orbits. Wedescribehowtopreparestatesshowingdifferenttopological
phases, and discuss their relation to entanglement. In particular, the presence of a π/2 phase shift
is a signature of genuine tripartite entanglement in the sense that it does not exist for two-qubit
3 systems.
1
0 PACSnumbers: 03.65.Vf,03.67.Mn,07.60.Ly,42.50.Dv
2

n
a I. INTRODUCTION qubits. All possible values for up to N = 7 have been
J found using a combinatorialalgorithm [9]. Furthermore,

3 a relation between the topological phases and the de-
Topological phases of quantum systems that evolve in
2 gree of nonzero polynomial entanglement invariants has
topologically nontrivial spaces, have attracted consider-
been conjectured [9]. As an example of such a relation,
ableattentioninawide varietyofsubdisciplines inmod-
] the possible topological phases 0,π/2,π, and 3π/2 for
h ern physics. Perhaps the most well-known example of
p suchatopologicalquantityisthe Aharonov-Bohmphase N = 3 can be linked to multipartite entanglement in
- the sense that three-tangle is a polynomial invariant of
t acquired by a charged particle that encircles a shielded
n degree n = 4, namely the hyperdeterminant in the coef-
magnetic flux line [1]. This phase depends only on the
a ficient matrix α [10]. This implies that the allowed
windingnumberoftheparticle’spatharoundtheimpen- klm
u topologicalphases are indeed restricted to integer multi-
q etrable region of magnetic flux, but it is insensitive to
ples of 2π/n=π/2.
[ perturbations of the path.

A topological phase acquired by a pair of entangled In order to realize a multiple qubit system in a pho-
1
v qubitsundergoingcycliclocalunitaryevolutionhasbeen tonic device, one may combine different degrees of free-
8 discovered [2]. The topological interpretation of this dom that can be manipulated independently. Numer-
3 phasereliesontherelationbetweentwo-qubitstatesand ous experiments have employed polarization and orbital
5 therotationgroupSO(3)[3]. Thisisperhapsmostclearly angular momentum (OAM) to implement controlled op-
5 seeninthecaseofthemaximallyentangledstates. These erations [11–15] and spin-orbit Bell inequality [16, 17].
.
1 statesareinone-to-onecorrespondencewiththepointsof Here, we propose an experiment to measure the topo-
0 realprojectivespaceS3/Z2 SO(3)[4,5]. Thetwopos- logical phases for N = 3, in qubits encoded on photon
3 sible topological phases 0 an∼d π can be associated with pairs producedby spontaneousparametric downconver-
1
the two homotopy classes of loops in SO(3). In other sion(SPDC). Eachphotoncarries a polarizationandor-
:
v words the accumulated phase is not affected by contin- bital degreeof freedom. The three qubits areencoded in
i uous deformations of path of the cyclic evolution. The the orbital part of the signal photon and the two polar-
X
topological two-qubit phase has been observed in spin- izations,byprojectingtheorbitalpartoftheidlerphoton
r
a orbittransformationsonalaserbeam[6]andinanuclear on a well-defined Laguerre-Gaussian mode. In this way,
magnetic resonance setting [7]. we demonstrate different three-qubit states that acquire
The notion of topological phase has been extended to thedifferentthree-qubitphasesbyemployinglocalSU(2)
pairs of entangled higher-dimensional quantum systems transformations in Franson loop interferometers on each
[8]. These phases are integer multiples of 2π/d, where d photon. Theobservedphaseswouldbeasignatureofthe
is the Hilbert space dimension of each subsystem. Thus, local orbits and thereby a non-trivial signature of multi-
for such objects, fractional values may occur. The topo- partite entanglement.
logicalphase fora givencyclic localSU(d) evolutionofa
The outline of the paper is as follows. The theory
state |ψi= dk,l=1αkl|kli is restricted by the invariance of topological three-qubit phases arising in local SU(2)
ofthedeterminantdetαkl ofthecoefficientmatrix. Since evolution is described in Sec. II. Sections III-V contain
P
detαkl = 0 for product states, the topological phase is the experimental setup, where the generation of three
only well-defined in the presence of entanglement. different types of three-qubit states are described in Sec.
Recently, the notion of topological phase has been ex- III, the measurement of topological phases is described
tendedtoN-qubitsystems[9]. Thesemulti-qubitphases in Sec. IV, and examples of evolutions that reveal the
may take fractional values for N 3. The number topological phases is given in Sec. V. The paper ends
≥
of possible values increases rapidly with the number of with the conclusions.
2
II. THREE-QUBIT TOPOLOGICAL PHASE qubit. Hence, these two states are entangled in exactly
STRUCTURE the same way.
The states in the GHZ SLOCC-class that do not fall
When considering interconvertibility of three-qubit
inthe X-classhaveonlytwo differenttopologicalphases.
states under stochastic local operations and classical
Sincethe X-classisalowerdimensionalsubset,ageneric
communication(SLOCC),thegenuinelytripartiteentan-
statein the GHZ SLOCC-classis ofthis kind. Anexam-
gled states fall into two classes [18]. These classes are
ple of such a state, with a doubly connected local SU(2)
termed the GHZ-class and the W-class after their repre-
orbit,isabiasedGHZstate ψ =α+++ +β ,
sentatives, the GHZ state and the W state. where α = β and α2+ |βb2gh=zi1 [20|]. Theitwo|−h−om−oi-
By considering interconvertibility under local unitary | |6 | | | | | |
topy classes of cyclic evolutions correspond to the accu-
transformations the two SLOCC-classes can be further
mulated phases 0 and π.
divided into local unitary classes, or in other words, or-
bits of the group of local unitary transformations. The
The X state and a biased GHZ state thus represents
structure of such an orbit constitutes a qualitative de-
the two different topological phase structures present in
scription of the entanglement of the states belonging to
theGHZSLOCC-class. Theremainingthree-qubitstates
theorbit. Thisisthemostdetaileddescriptionoftheen-
with genuine tripartite entanglement belong to the W
tanglement properties that can be given [19]. Since the
SLOCC-class, and have either the topological phases 0
action of the U(1) group is a trivial global phase shift,
and π, or no topological phases at all. We would thus
it is sufficientto considerthe localSU(2)-orbits to study
not see any other sets of topological phases by studying
entanglement properties.
states in the W class.
The structure of the SU(2)-orbits of entangled three-
qubitstateshasbeenstudiedbyCarteretandSudberyin This paper is concerned with three-qubit systems en-
Ref. [20]. InparticularitwasshownthatthelocalSU(2)- coded in the polarization and orbital angular momen-
orbit of a GHZ state |ψghzi = √12(|+++i+|−−−i), tum(OAM)states ofphotons. We willbe describingthe
where + and are orthogonal states, is quadruply polarization states in a basis of right and left circular
| i |−i
connected. The four different homotopy classes of cyclic polarization states + and or alternatively in a ba-
| i |−i
evolutions correspond to the four different accumulated sis of horizontaland vertical polarization states H and
phases, 0,π,π and 3π. Since these are the only phases V . The relation between these basis vectors |is igiven
allowed for2a state w2ith nonzero three-tangle it follows |byi = 1 (H iV ). The OAM states will be de-
|±i √2 | i± | i
that the quadruple connectedness is related to the tri- scribed in terms of a basis of Laguerre-Gaussian modes
partite entanglementmeasuredby the three-tangle. A π2 of first order LG1,0 and LG 1,0, denoted + and
phase shift cannot be generated in a two-qubit system similarly to the circular pola−rization states,| oir in a |b−ai-
andisthereforeameasurablequantitythatindicatesthe sis of the Hermite-Gaussianfirstorder modes HG1,0 and
presence of tri-partite entanglement. HG0,1, denoted h and v similarly to the linear polar-
Fourdifferenttopologicalphasesisinfactnotthemost ization states. T|heirelati|oni between these bases is given
common topological phase structure for local SU(2)- by = 1 (h iv ).
orbits belonging to the GHZ SLOCC-class. Using the |±i √2 | i± | i
canonical form of three-qubit states of Carteret et al. It is useful to note that the X state prepared in a ba-
[21],itcanbeseenthatthe setoflocalSU(2)orbitsthat sis of circular polarization states and Laguerre-Gaussian
exhibit four topological phases forms a subset of the lo- modes is the GHZ state, up to a relative phase factor
calSU(2)orbitsofthe GHZSLOCC-classparameterized i of the two terms, in a basis of horizontal and verti-
−
by four real parameters, while the full set of local SU(2) calpolarizationstatesandHermite-Gaussianmodes. For
orbits is parameterizedby four realandone complex pa- example, if the X state in the + , basis has been
{| i |−i}
rameter. The statesofthis subsetcan,up to localSU(2) encodedin the polarizationandOAMstates of aphoton
operations, be written on the form pair, such that the first and last qubit are encoded in
polarization states and the middle in the OAM state of
Xa,b,c,d = a +++ +b + one of the photons, the same state in the H , V and
| i +|c +i +|d −−+i , (1) h , v basis would be 1 (HhH iV{v|V i).| i}
|− −i |−− i {| i | i} √2 | i− | i
wherea,b,c,d C 0 suchthat a2+ b2+ c2+ d2 = We will consider the X state, the GHZ state, and a
∈ \{ } | | | | | | | |
1. We will refer to this class of states as the X-class. biasedGHZstateinthe + , basissincethisallows
{| i |−i}
A distinguished member of this class is the three-qubit us to implement cyclic local SU(2) evolutions that re-
state ψ for which a = b = c = d = 1 termed the vealthe topologicalphasesandlie completely within the
| Xi 2
three-qubit X state in Ref. [22]. This state is maximally set of operators that diagonalize in the + , basis.
{| i |−i}
entangledinthe sense that allreduceddensity operators Considering the X state there are evolutions in each ho-
forthe individualqubits areproportionaltothe identity. motopyclassthatdiagonalizeinthe + , basis,and
{| i |−i}
Note that the X state can be brought to the GHZ state thusallowsallpossibletopologicalphasestobeobserved.
by application of a Hadamard transformation on each This is true also for the biased GHZ state.
3
III. QUANTUM STATE PREPARATION where we have grouped together the signal degrees of
freedom. Now we shall discuss separately the two entan-
Ourexperimentalproposalisbasedonthespontaneous gled three-qubit states of interest. We further show how
parametricdownconversion(SPDC) sourceofentangled to prepare certain product states that are used to inves-
photonsfirstdemonstratedinRef. [23],andlaterusedin tigate the role of entanglement in the topological phase
other experiments [24, 25]. There, two adjacent nonlin- measurements.
ear crystals cut for type I phase match are spatially ori-
entedwiththeir opticalaxismutuallyorthogonal. Start-
A. X State
ing from a linearly polarized laser, a quarter waveplate
(QWP-p) can be used to produce a circularly polarized
pump, and generate pairs of polarization entangled pho- First,wewillseehowtoproducethethree-qubitquan-
tons of the kind tum state showing the π/2 topological phase. The pro-
posed setup is sketched in Fig. 1. In order to simply
HH iVV
understand the setup, it is useful to recall that the X
ψ = | i− | i , (2)
pol
| i √2 state in the + , basis corresponds to a GHZ state
{| i |−i}
in the H , V basis. Therefore, following the setup,
where the first term on the right hand side comes from {| i | i}
we shall be seeking for this state. First, an astigmatic
theV componentofthepumpwhilethesecondonecomes
mode converter can be used to transform the signal LG
from the H component.
mode to a horizontal first order HG mode [32], giving
Inordertorealizethe three-qubitsystem,wemayadd
the orbitalangularmomentum(OAM) quantumstateof HhH iVhV
the photon pair [26]. As already demonstrated [27–29], ψ1 ψ2 = | i− | i . (6)
| i→| i √2
thespatialcorrelationsimposedbythephasematchcon-
dition in parametric down conversion are manifested in This state could also be produced by pumping the crys-
the OAMtransferfromthe pump to the downconverted tals with the first order Hermite-Gaussian mode h, still
photons, giving rise to an OAM entangled state of the filteringtheidlerwiththesinglemodefiber. Inthiscase,
form the signal mode with optimal spatial overlapwith pump
and idler is also h. This would exempt the use of the
ψ = C m, l m , (3)
oam m mode converter, making the system alignment consider-
| i | − i
Xm ably easier.
Then, a spin-orbit controlled NOT (CNOT) gate is
where m and l are the topological charges of signal and
used to flip the signal HG mode conditioned to its po-
pump photons, respectively. Then, OAM conservation
larization. The CNOT gate is a Mach-Zehnder interfer-
imposes that the added topological charge of signal and
ometer with input and output polarizing beam splitters
idler equals that of the pump, leading to a superposi-
(PBS). A Dove prism (DP) oriented at 45o and inserted
tion of all components compatible with this condition.
in the (V) arm makes the transverse mode conversion
The probabilityamplitudes C associatedwithapartic-
m
h v on this arm. After the CNOT gate the three-
ular OAM partition is proportional to the spatial over-
| i → | i
qubit quantum state becomes the desired X state:
lap between signal, idler, and pump transverse modes
[30]. Now, the three-qubit realization can be achieved
HhH iVvV
by pumping the SPDC source with a Laguerre-Gaussian ψ = | i− | i
X
| i √2
mode with l =+1 and detecting the idler photon with a
single mode fiber (SMF) that admits only the l m=0 +++ + + + + + +
− = | i | −−i |− −i |−− i .(7)
component. Then, coincidence measurements should be 2
obtained only for signal photons with m = +1. There-
fore, the postselected spin-orbit quantum state is
B. Biased GHZ state
ψ = ψ +,0 . (4)
SO pol
| i | i⊗| i
Sincethesubspaceoffirstorderparaxialmodeshavea InordertoproducethebiasedGHZstateshowingonly
qubitstructure[31],wecannowencodetwoqubitsonthe topological phase π, the setup shown in Fig. 2 can be
signalphotons,namelytheirpolarizationandOAM,and used. First, a half waveplate (HWP-p) with a suitable
a single qubit on the idler polarization. From now on, orientation is placed on the pump laser to set its polar-
we shall omit the idler OAM since no operations other ization to produce the partially entangled state
than detection filtering will be performed in this degree
of freedom. Therefore, the initial three-qubit state gen- |ψp′oli=α|HHi+β|VVi, (8)
erated is
so that the initial three qubit state will be
H +H iV +V
|ψ1i= | i√−2| i , (5) ψ1′ =αH +H +β V +V . (9)
| i | i | i
4
For example, the product state
X STATE DP H + V h + v H + V
ψ = | i | i | i | i | i | i
prod
Signal PBS | i √2 ⊗ √2 ⊗ √2
LG+1 MC CNOT e−iπ4 + +eiπ4 e−iπ4 + +eiπ4
QWP−p NLC = | i |−i | i |−i
PBS √2 ⊗ √2
HHVV e−iπ4 + +eiπ4
| i |−i (12)
⊗ √2
Idler
ψ has the same probability distribution as the X state for
X
eachindividual degree of freedom in both the H , V
{| i | i}
basis and the + , basis.
{| i |−i}
FIG. 1: X state preparation setup. This state is readily prepared by the setup shown in
Fig. 3 when the pump polarization is set to V and
the down converted photons are created at the prod-
uct state H + H . In the signal arm, the mode con-
| i
verteristhenorientedtomakethetransformation +
BIASED GHZ STATE DP QWP−s | i→
(h + v )/√2, and the H polarizationpasses unaffected
LG Signal CNOT PBS th|rioug|hithe CNOT gate. Then, two half-waveplates can
+1 be used to set signal (HWP-s) and idler (HWP-i) polar-
HWP−p NLC
PBS izations to (H + V )/√2, thus producing ψ .
prod
| i | i | i
A product state with the same probability distribu-
HHVV
tions as the biased GHZ state in the + , basis
Idler QWP−i could be {| i |−i}
ψ
bghz ψ = αH +β V αh +β v
| p′rodi | i | i ⊗ | i | i
(cid:16) (cid:17) (cid:16) (cid:17)
FIG. 2: Biased GHZ preparation setup. ⊗ αe|Hi+βe|Vi e e
(cid:16)α+ +β (cid:17)α+ +β
= e| i |e−i | i |−i
√2 ⊗ √2
With the astigmatic mode converterremoved,the trans- α+ +β
| i |−i, (13)
formation + is performed in the (V) arm of the ⊗ √2
| i→|−i
CNOT gate, giving
where α= α√+2β and β =iα√−2β. |ψp′rodi can be produced
in the same way as ψ , but with suitable settings of
|ψ2′i=α|H +Hi+β|V −Vi. (10) the moede converter|aenprdodtihe HWPs in order to provide
the coefficients α and β. Both ψ and ψ could
Now,twoquarterwaveplatesinsertedonsignal(QWP-s) alsobeproducedbytailoringth|eppruomdipmod|ep′irnodoirderto
andidler(QWP-i)paths,makethepolarizationtransfor- optimizethespaetialoveerlapbetweenpump,idler,andthe
mations H + and V neededtoproducethe desiredsignalmode,withouttheuseofamodeconverter
| i→| i | i→|−i
desired biased GHZ state on the signal arm.
The role played by entanglement in the topological
phase evolution can be investigated with two-photon in-
ψ =α+++ +β . (11)
bghz
| i | i |−−−i terferometry, as we shall see in section V. The interfer-
ence patterns produced by entangled states are clearly
distinguished from those expected for product states.
IV. TOPOLOGICAL PHASE MEASUREMENT
C. Product states
Under local unitary operations, the quantum state of
the three-qubit photon pairs evolve keeping their entan-
In order to investigate the role of entanglement in the glement unaltered. The topological nature of the phase
topologicalphase measurements,it is important to com- evolutionisstronglydependentonentanglement,sothat
pare the quantum states discussed above with product it is important to identify entanglement signatures on
states that are equivalent to the X state and the biased the state evolution. As in Ref. [6], signatures of entan-
GHZstateinwhatregardsthe singlequbitprobabilities. glement can be found on interference patterns between
5
+
A
PRODUCT STATE DP HWP−s
PBS
LG Signal CNOT
+1
MC
HWP−p NLC Ω
PBS
B D
HHVV
Idler HWP−i
ψ
C
prod
FIG. 3: Product state preparation setup.
FIG. 5: Paths on thePoincar´e sphere.
D
1
by the DHWPs and DDP in the single qubit Poincar´e
φs φo representation. Starting from a + state, a HWP with
| i
itsfastaxisorientedatangleθ makesthetransformation
DHWP DDP θs C + θ+π/4 , where θ represents a linear po-
| i→| i→|−i | i
larization state along a direction rotated by the angle θ
with respect to the horizontal. Therefore, a sequence of
BS BS
SMF two HWPs oriented at θ and θ+φ, respectively, makes
φi D2 the cycle |+i→|θ+π/4i→|−i→|θ+φ+π/4i→|+i,
which corresponds to path A B C D A in
→ → → →
θ Fig. 5. Since this path is composed of two geodesic seg-
DHWP i
ments, enclosing a solid angle Ω=φ, a purely geometric
ψ
phase φ/2 is acquired by single qubits initially prepared
at + . Of course, the same solid would be enclosed by
| i
aninitialstate ,butinoppositedirection,givingage-
FIG. 4: Franson interferometry. |−i
ometric phase φ/2. Therefore, each degree of freedom
−
follows the local SU(2) operation:
the evolved and the initial state. Two-photon interfer- eiφ/2 0
U(φ)= , (14)
encecanbeachievedwiththewellknownFransonsetup, 0 e iφ/2
−
whereeachphotonfromaquantumcorrelatedpairissent (cid:20) (cid:21)
throughtwoalternativepaths,alongandashortone[33– given in the + , basis, so that the overall three-
{| i |−i}
36]. Whenthedelaytimebetweentheshortandthelong qubit state will be transformed according to U(φs)
⊗
paths is largerthan the detection time window, the two- U(φo) U(φi),whereφs,φo,andφicorrespondtosignal
⊗
photon coincidence count exhibits interference patterns. polarization, OAM, and idler polarization, respectively.
Each coincidence count may result from both photons Under these local operations, product and entangled
followingeither the shortor the longpaths. Photons go- states evolve differently in what regards the overlap be-
ing through different paths do not coincide. Moreover, tween the initial and the evolved states: ψ U(φs)
h | ⊗
each arm of an SPDC source has a considerably short U(φo) U(φi)ψ . Inordertoaccessthesedifferencesand
⊗ | i
coherence length, so that no single photon interference investigate the role played by entanglement and its rela-
can occur. Then, the overlap between the evolved and tionshiptothethree-qubittopologicalstructure,wemust
the initial state appears as the fringe visibility when the perform interferometric measurements where a proper
threequbitsareindividuallyoperatedinonearmandleft background for comparison between product and entan-
unchanged in the other, as sketched in Fig. 4. gled states can be established. A possible strategy is
Inordertosimplifytheexperimentalproposal,stillbe- suggested in Fig. 4, where dynamical phases θs and θi
ing able to cover the topological structure of the three- are deliberately added in one arm of the interferometer.
qubit local SU(2) orbits, we shall be dealing with diago- As these dynamical phases are continuously varied, the
nal unitary operations in the + , basis. For each coincidence count exhibits an interference pattern that
polarization transformation, a{|seiqu|−enic}e of two HWPs should evolve as the single qubit unitary operations are
(DHWP) oriented at 0 and φ, respectively, can be used. applied. Infact, the coincidence countis proportionalto
The same kindofdiagonaltransformationforOAM(LG
C =C0[1+ cos(θ+Φ)] , (15)
basis)canbeperformedbyasequenceoftwoDPs(DDP) V
at different orientations. These schemes are sketched in where C0 is the coincidence offset, ψ is one of the se-
| i
Fig. 4. Itisinstructivetoviewtheoperationsperformed lected states discussed above, θ = θ + θ is the total
s i
6
dynamical phase added and
1 3πt 3 πt
Φ ≡ arg[hψ|U(φs)⊗U(φo)⊗U(φi)|ψi], C(t,θ)=C0 1+ 4cos θ− 2T + 4cos θ+ 2T .
ψ U(φs) U(φo) U(φi)ψ . (16) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
V ≡ |h | ⊗ ⊗ | i| (18)
Therefore, the absolute value of the overlapbetween the
Wecanseethatthereisareappearanceofmaximalfringe
initial and the evolved states is related to the fringe
V visibility for t = 1 with the expected fringe shift π.
visibility, while the overlapphase Φ, i.e., the Pancharat- T 2
nam relative phase [37, 38], translates to a fringe dis- Moreover, there are no values of t for which the inter-
T
placement. For a cyclic evolution, the fringes should ference fringes disappear. This illustrates that, in con-
recover maximal visibility and exhibit the accumulated trasttothecaseofmaximallyentangledtwo-qubitstates
phase shift, which is of topological nature for entangled [3], nontrivial topological phases can be obtained with-
states. However, the role of entanglement must be cap- out going through a state orthogonal to the initial one.
turedfromsignaturesontheevolutionoftheinterference The coincidence intensity in Eq. (18) for selected values
pattern,astheindividualunitaryoperationsareapplied. of t is shown in the left panel of Fig. 6.
T
We shall investigate these signatures numerically in the Another cyclic evolution in the same homotopy class
next section. can be generated by the unitary operator UX2(t) given
by
V. NUMERICAL RESULTS
3t 1 t
φ (t) = π π 1 H
s
− − T − 3 − T
To demonstrate the presence of the topologicalphases (cid:20) (cid:21) (cid:20) (cid:21)
3t t 1 2 t
ofthe twodifferenttopologicalstructuresrepresentedby φ (t) = π π 1 H 1 H
o
the X state and the biased GHZ state we give examples − − T − T − 3 − 3 − T
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21) (cid:20) (cid:21)
of cyclic unitary evolutions in each homotopy class for 3t t 2
φ (t) = π π 2 H 1 ,
bothstates. Theevolutionoftheinterferencepatternfor i − − T − T − 3 −
these entangledstates,asthe unitaryevolutionis gradu- (cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
(19)
allyimplemented,iscomparedtotheevolutionofthein-
terference patterns ofproductstates with the same local for 0 t T, where H is the Heaviside step function
statistics in the + , basis. Since we consider uni- define≤d by≤H(x) = 0 for x < 0 and H(x) = 1 for x > 0.
{| i |−i}
tary evolutions that are diagonal in the + , basis, The coincidence intensity as a function of θ and t in this
{| i |−i}
no difference in the interference patterns between an en- case is
tangledstateandsuchaproductstate canbe attributed
to the local degrees of freedom. Any difference is thus
due to entanglement. 3t 3πt
C(t,θ) = C0 1+H 1 cosθcos
− T 2T
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)
3t 3πt
A. X state +H 2 sinθsin .
T − 2T
(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(20)
A set of cyclic evolutions of the X state resulting in
a π2 phase shift is generated by the unitary operators Again we can see the expected fringe shift π2 with max-
U(φs(t)) U(φo(t)) U(φi(t)), where t [0,T], such imal fringe visibility for t = 1. For the cyclic evolution
that φ (0⊗) = φ (0) =⊗φ (0) = 0 and φ (T∈) = φ (T) = T
s o i s o generated by UX2(t), as opposed to that generated by
cφiid(eTn)ce=in−teπn.sitFyorCsuacshaafnunecvtoilountioonf θo,fφ|ψ,Xφi,, athned cφoinis- UX1(t),theinterferencefringesdisappearfor 31 ≤ Tt ≤ 32,
s o i meaning that the evolution takes the system through
given by the expression
states orthogonal to the initial state during the evolu-
tion. With respect to the evolution of the fringe visibil-
itytherearethusqualitativelydifferentevolutionsinthe
1 φ φ +φ
s o i
C = C0 1+ cos θ+ cos same homotopy class. The coincidence intensity in Eq.
2 2 2
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (20) for selected values of t is shown in the left panel of
1 φ φ φ T
s o i Fig. 7.
+ cos θ cos − .
2 − 2 2 Toverifythatthefringeshiftsareduetoentanglement
(cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
(17) weconsidertheproductstate ψprod ,definedinEq. (12),
| i
whichhasthesameprobabilitydistributionsforthelocal
One unitary operator of this kind is UX1(t) given by degrees of freedom as the X state in both the H , V
{| i | i}
φ (t)=φ (t)=φ (t)= πt/T. If the X state is evolved and the + , bases. The coincidence intensity as a
s o i
− {| i |−i}
byUX1(t)thecoincidenceintensityasafunctionoftand function of θ,φs,φo, and φi for ψprod , given that it is
| i
θ is given by subjected to a unitary U(φ ) U(φ ) U(φ ), is
s o i
⊗ ⊗
7
1 1
0.8 0.8
0 0
C 0.6 C 0.6
2 2
θ)/ θ)/
C( 0.4 C( 0.4
0.2 0.2
0 0
0 −π/2 −π −3π/2 −2π −5π/2 −3π 0 −π/2 −π −3π/2 −2π −5π/2 −3π
θ θ
FIG. 6: (Color online) The coincidence intensity C(θ) as a function of θ for selected valuesof t of theevolution generated by
T
UX1(t) for |ψXi (left) and |ψprodi (right). The different interference intensities correspond to the values Tt = 0 (solid black),
t = 1 (dash-dottedblack), t = 1 (dashed blue), t = 3 (dotted blue),and t =1 (solid blue), respectively.
T 4 T 2 T 4 T
1 1
0.8 0.8
0 0
C 0.6 C 0.6
2 2
θ)/ θ)/
C( 0.4 C( 0.4
0.2 0.2
0 0
0 −π/2 −π −3π/2 −2π −5π/2 −3π 0 −π/2 −π −3π/2 −2π −5π/2 −3π
θ θ
FIG. 7: (Color online) The coincidence intensity C(θ) as a function of θ for selected valuesof t of theevolution generated by
T
UX2(t) for |ψXi (left) and |ψprodi (right). The different interference intensities correspond to the values Tt = 0 (solid black),
t = 1 (dash-dottedblack), t = 1 (dashed blue), t = 3 (dotted blue),and t =1 (solid blue), respectively.
T 4 T 2 T 4 T
basis are the evolutions U(φ (t)) U(φ (t)) U(φ (t))
s o i
⊗ ⊗
t [0,T] such that φ (0) = φ (0) = φ (0) = 0 and
C =C0 1+cosθcos φs cos φo cos φi . φs∈(T) = φo(T) = φi(Ts) = 2πo. Howevier since these
2 2 2 −
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) arealsocyclicevolutionsofthe productstate ψprod the
| i
(21) reappearance of maximal fringe visibility and π phase
shift cannot be attributed to the entanglement of the X
We can see that fringe visibility for φ (T) = φ (T) =
s o state.
φ (T)= π iszero. For ψ the onlyvaluesofφ ,φ ,
i − | prodi s o To observe a π phase shift that cannot be attributed
and φ that gives maximal fringe visibility are 0 and 2π.
i to local degrees of freedom we must implement a cyclic
Thus, the reappearance of maximal fringe visibility for
evolutiongeneratedby unitariesthatarenotdiagonalin
the value π of φ ,φ and φ with a π fringe shift, is
− s o i 2 the + , basis. We recall however that the X state
due to the entanglementof the X state. The coincidence {| i |−i}
is identical to the GHZ state in a different basis. For
intensity in Eq. (21) for the evolutions of ψ gener-
ated by UX1(t) and by UX2(t) at selected |vapluroedsiof Tt is tuhneitGarHieZslsetaadteintghetoreaaπrepehvaosleutsihoinftstgheantercaatnedbebyatdtriaibguotneadl
shown in the right panel of Figs. 6 and 7 respectively.
to entanglement.
Note that the cyclic unitary evolutions that give max-
imal fringe visibility for ψ are alsocyclic evolutions
prod
| i
of the X state. This however holds only for the diagonal
unitary operators we are considering. A more general B. GHZ and biased GHZ state
cyclic evolution of ψ is not typically a cyclic evolu-
prod
| i
tion of ψ .
| Xi We consider the GHZ state ψ = 1 + ++ +
The set of cyclic evolutions generating the π-phase | ghzi √2| i
shift of the X state that are diagonal in the {|+i,|−i} √12| − −−i and the biased GHZ state |ψbghzi = 12| +
8
1 1
0.8 0.8
0 0
C 0.6 C 0.6
2 2
θ)/ θ)/
C( 0.4 C( 0.4
0.2 0.2
0 0
0 −π/2 −π −3π/2 −2π −5π/2 −3π 0 −π/2 −π −3π/2 −2π −5π/2 −3π
θ θ
1 1
0.8 0.8
0 0
C 0.6 C 0.6
2 2
θ)/ θ)/
C( 0.4 C( 0.4
0.2 0.2
0 0
0 −π/2 −π −3π/2 −2π −5π/2 −3π 0 −π/2 −π −3π/2 −2π −5π/2 −3π
θ θ
FIG. 8: (Color online) The coincidence intensity C(θ) as a function of θ for selected valuesof t of theevolution generated by
′ T
U (t) for |ψ i (upperleft), |ψ i (upperright), |ψ i (lower left), and |ψ i (lower right). The different interference
bghz bghz prod ghz prod
intensities correspond to the values t = 0 (solid black), t = 1 (dash-dotted black), t = 1 (dashed blue), t = 3 (dotted
T T 4 T 2 T 4
blue), and t =1 (solid blue), respectively.
T
++ + √3 . The GHZ state and the biased GHZ of t is shown in the lower left panel of Fig. 8. The co-
i 2 |−−−i T
state share a set of cyclic evolutions that give rise to a π incidence intensity for the evolution of ψ generated
bghz
| i
phase shift and are diagonal in the + , basis. We by U (t) on the other hand will include an additional
bghz
{| i |−i}
recall however that the GHZ state and the biased GHZ term and is given as a function of t and θ by
statearerepresentativesofdifferentclassesoftopological
phase structures of the local SU(2) orbits since for the
πt 1 πt
GHZ state there are cyclic evolutions resulting in a π2 C(t,θ)=C0 1+cosθcos sinθsin .
T − 2 T
phase shift while for the biased GHZ state there are no (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
such evolutions. (23)
The cyclic evolutions of these states that result in
The coincidence intensity in Eq. (23) for selected values
a π phase shift are generated by unitaries U(φ (t))
s ⊗ of t is shownin the upper left panel ofFig. 8. For both
U(φo(t)) U(φi(t)), where t [0,T], such that φs(0) = T
⊗ ∈ theGHZ stateandthebiasedGHZstatethe coincidence
φo(0)=φi(0)=0,and φ1(T)+φ2(T)+φ3(T)= 2π.
± intensity depends only on the sum φ (t)+φ (t)+φ (t)
There are no evolutions generated by unitaries of this s o i
and not on the φ (t) individually. For both states there
kind that takes the biased GHZ state through a state i
is a reappearance of maximal fringe visibility at t = 1
orthogonal to the initial state. Thus, the interference T
and the interference fringes are shifted by a π phase.
fringes never disappears. The GHZ state on the other
To see the signature of the entanglement in the inter-
hand is evolved through an orthogonal state.
ference pattern of ψ , we also consider the product
One unitary operator of this kind is U (t) given by bghz
bghz | i
the choice φ (t) = φ (t) = φ (t) = 2πt. The coincidence state
s o i 3T
intensityfortheevolutionof ψ generatedbyU (t)
ghz bghz
| i
will be a function of t and θ given by
1 √3 1 √3
ψ = + + + +
πt | p′rodi 2| i 2 |−i!⊗ 2| i 2 |−i!
C(t,θ)=C0 1+cosθcos . (22)
T 1 √3
(cid:20) (cid:18) (cid:19)(cid:21) + + , (24)
The coincidence intensity in Eq. (22) for selected values ⊗ 2| i 2 |−i!
9
whichhasthesameprobabilitydistributionsforthelocal VI. CONCLUSIONS
degreesoffreedominthe + , basisas ψ . The
bghz
{| i |−i} | i
interferenceintensityasafunctionofθandtwhen ψ
| p′rodi We propose an experimental scheme to observe the
is evolved by U (t) is
bghz topological phases acquired by special classes of three-
qubit states. These phases revealthe nontrivial topolog-
ical structure of the local SU(2) orbits. In particular,
πt 7 πt
C(t,θ) = C0 1+cosθcos 1 sin2 observation of the π/2 topological phase shift would be
3T − 4 3T
(cid:26) (cid:20) (cid:21)(cid:20) (cid:18) (cid:19)(cid:21) a signature of multiqubit entanglement as this phase ex-
+3sinθsin πt 1 13sin2 πt . ists only for more than two qubits. The experimental
2 3T − 12 3T proposal is within the technological resources available
(cid:20) (cid:21)(cid:20) (cid:18) (cid:19)(cid:21)(cid:27)
(25) in most quantum optics laboratories, and can be imple-
mented in a short term. Furthermore, the insensitivity
Comparing Eqs. (23) and (25) we see that the reap- to continuous path deviations of the unitary evolutionis
pearance of maximal fringe visibility for t =1 is absent a robust feature of the topological phases with potential
T
for ψ . Moreover the fringe shift is not equal to π. applications to quantum information processing.
| p′rodi
Thus, the reappearance of fringe visibility and π phase
shiftisthusduetoentanglement. Thecoincidenceinten-
sity for ψ , evolved by U , is shown for selected
values of| tp′roindiupper right panbgehlzof Fig. 8 alongside the Acknowledgments
T
coincidence intensity of ψ .
bghz
| i
To see the signature of entanglement in the interfer- M.J., E.S., and K.S. acknowledge support from the
ence pattern of ψ in Eq. (22) we compare with the National Research Foundation and the Ministry of Ed-
ghz
| i
interference patterns of ψ . The coincidence inten- ucation (Singapore). A. Z. Khoury acknowledges fund-
prod
| i
sity for ψ is given by Eq. (21) and we can see that ing from the Brazilian agencies Coordenac¸a˜o de Aper-
prod
| i
there is no reappearance of maximal fringe visibility for fei¸coamento de Pessoal de N´ıvel Superior (CAPES),
φ (t) = φ (t) = φ (t) = 2π. Thus, the reappearance for Funda¸ca˜o de Amparo `a Pesquisa do Estado do Rio de
s o i 3
ψ canbe attributed to entanglementandthe phase Janeiro (FAPERJ-BR), and Conselho Nacional de De-
bghz
| i
shift is of topological nature. The coincidence intensity senvolvimento Cient´ıfico e Tecnol´ogico (CNPq). A. Z.
for ψ , evolved by U , is shown for selected val- Khoury is deeply grateful for the hospitality during the
prod bghz
ues|of t iin the lower right panel of Fig. 8 alongside the visittoCentreforQuantumTechnologiesattheNational
T
coincidence intensity for ψ . University of Singapore, Singapore.
ghz
| i
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