Operational determination of multi-qubit entanglement classes via tuning of local operations

0.29 MB·English

by  T. Bastin

Operational determination of multi-qubit entanglement classes via tuning of local
operations
T. Bastin,1 C. Thiel,2 J. von Zanthier,2 L. Lamata,3,4 E. Solano,5 and G. S. Agarwal6
1Institut de Physique Nucl´eaire, Atomique et de Spectroscopie, Universit´e de Li`ege, 4000 Li`ege, Belgium
2Institut fu¨r Optik, Information und Photonik, Max-Planck Forschungsgruppe,
Universit¨at Erlangen-Nu¨rnberg, 91058 Erlangen, Germany
3Max-Planck-Institut fu¨r Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
4Instituto de Matem´aticas y F´ısica Fundamental, CSIC, Serrano 113-bis, 28006 Madrid, Spain
5Departamento deQu´ımicaF´ısica, UniversidaddelPa´ısVasco-Euskal HerrikoUnibertsitatea, Apdo.644, 48080Bilbao,Spain
6Department of Physics, Oklahoma State University, Stillwater, OK 74078-3072, USA
(Dated: January 12, 2009)
9
We present a physical setup with which it is possible to produce arbitrary symmetric long-lived
0
multiqubitentangled states in theinternalground levelsof photon emitters, includingtheparadig-
0
2 maticGHZandWstates. Inthecaseofthreeemitters,whereeachtripartiteentangledstatebelongs
tooneoftwowell-definedentanglementclasses,weproveaone-to-onecorrespondencebetweenwell-
n
defined sets of experimental parameters, i.e., locally tunable polarizer orientations, and multiqubit
a
entanglement classes inside thesymmetric subspace.
J

2 PACSnumbers: 42.50.Ex,03.65.Ud,03.67.Bg,42.50.Dv
1

] Entanglement is a distinctive property of quantum
h
p physics associated with the nonseparable character of
- multipartitequantumsystems. Forthecaseoftwo-qubit
t
n systems,entanglementiswellunderstoodandcanbepre-
a cisely quantified [1]. Apart from the trivial disentangled
u
case,three qubits possess twogenuine tripartite inequiv-
q
alent entanglement classes [2, 3]. Efforts have been done
[
recently towards higher number of qubits [4, 5, 6], in- ……
3
cluding an inductive method [7], though so far no com-
v
prehensiveandscalableclassificationhasbeendeveloped.
0
……
2 In this letter, we introduce a physicalsetup consistingof
7 N emitters, incoherently radiating single photons that
3 may be absorbed remotely by detectors equipped with
.
0 polarizersandproducinglong-livedmultiqubitentangled
1 states among the emitters. We show that it is possible
7 to associate well-defined sets of locally tuned polarizer
0
orientationswithmultiqubitentanglementclasses,allow-
:
v ing their monitoring in an operational manner. Hereby, FIG. 1: (color online). Proposed experimental arrangement.
Xi multipath quantum interferences, associated with qubit N excited emitters are aligned in a row, each of them defin-
permutation symmetry, play a key role in explaining the ing a three-level Λ-system. A long-lived entangled state is
r
obtainedintheN emitterqubitsafterdetectingtheN spon-
a underlying physics.
taneously emitted photons with N detectors equipped with
WeconsiderachainofN equallyseparatedsinglepho- polarizers. The final N-qubit state is tuned and determined
bythe polarizer orientations.
ton emitters, say trapped neutral atoms, trapped ions,
quantum dots, or any other equivalent physical system
with access to similar behaviour. Each emitter defines a
three-level Λ system, where e denotes the excited state gion, each of them being equipped with a polarization
| i
and the two long-lived sublevels, + and , define a filter in front. The far-field detection ensures the era-
| i |−i
qubit. We assume that the transitions between the ex- sure of which-way information of the arriving photons,
cited state and the two lower sublevels have an equal and the polarizers allow the generation of quantum su-
wavenumber and dipole moment, and that they are cir- perpositions of the loweratomic states when considering
cularly polarized, σ and σ , respectively. Figure 1 ex- arbitrary polarizations. As a consequence each photode-
+ −
emplifies the N-emitter case discussed throughout this tection event projects the emitters onto linear combina-
paper. Allemittersareinitiallyexcitedandwewillstudy tionsofthelong-livedstates + and [8]. Thisresults
| i |−i
the cases in which all spontaneously emitted photons atthe endinacoherentsuperpositionbetweenthe qubit
are detected by N detectors located in the far-field re- states ,..., . The indeterminacy of which detector
|± ±i
2
state of the W type [2], the weakly entangled state
|e,e,e(cid:2)
d1clicked α1 β1 W = 1 (1 ,0 ,...,0 +...+ 0 ,...,0 ,1 ),
N φ φ φ φ φ φ
|+,e,e(cid:2) |−,e,e(cid:2) | i √N | i | i
+|e,+,e(cid:2) +|e,−,e(cid:2)
+|e,e,+(cid:2) +|e,e,−(cid:2) (3)
α1 β1
d2clicked α2 β2 α2 β2 with|0φi≡(|+i−eiφ|−i)/√2. Inthe3-qubitcase,where
eachtripartiteentangledstatebelongstooneoftwowell-
+|||+++,,,++e,+,,ee(cid:2)(cid:2)(cid:2) +|||+++,,,−−e,,,−ee(cid:2)(cid:2)(cid:2)+++|||−−−,,,++e,+,,ee(cid:2)(cid:2)(cid:2) +|||−−−,,,−−e,,,−ee(cid:2)(cid:2)(cid:2) defined entanglement classes [2, 3]: GHZ or W family,
+|e,+,+(cid:2) +|e,+,−(cid:2)+|e,−,+(cid:2) +|e,−,−(cid:2)
it turns out that even the rotation of a single polarizer
2α1α2 α1β2+β1α2 2β1β2
d3clicked α3 β3 α3 β3 α3 β3 allows us to switch from one entanglement class prepa-
ration to the other, which are univocally determined by
|+,+,−(cid:2) |−,−,+(cid:2)
|+,+,+(cid:2) ++||+−,,+−,,++(cid:2)(cid:2) ++||−+,,+−,,−−(cid:2)(cid:2) |−,−,−(cid:2) the number of distinct polarizer orientations in the ex-
6α1α2α3 2(α1α2β3+α1β2α3+β1α2α3) 2(α1β2β3+β1α2β3+β1β2α3) 6β1β2β3 perimental setting. Those polarizer manipulations can
be saidto be localwith respecttothe polarizer/detector
FIG. 2: (color online). Pyramid of entanglement paths positions, though they are not local with respect to the
for the case of 3 emitters initialized in the excited state qubit positions: it is well known that one cannot con-
|e,e,ei. The figure illustrates the intermediate states, hor- vert states from different families into each others using
izontally displayed, during three successive photodetection
stochastic local operations and classical communication
events realized by three detectors d1, d2, and d3, equipped (SLOCC) [15]. Remarkably, the above described local
with general elliptical polarizers oriented along ǫ1, ǫ2 and
ǫ3 (ǫi ≡ αiσ++βiσ−), respectively. After each detection, polarization rotations will make possible the transitions
the unnormalized global state of the system is the sum of all
states in the circles weighed by the tagged prefactors. Left- S class W class GHZ class, (4)
↔ ↔
down arrows denote σ+ transitions, and right-down arrows
represent σ− ones. The final state components are shown where “S” stems from “separable”.
inside colored frames and the colored arrows represent the There are several physical systems where the genera-
quantum paths leading to these different components. Only
tionofGHZandWstateswiththreeormorequbitshave
the red circle states are obtained via a single quantum path,
been experimentally achieved: in trapped ions [16, 17],
whiletheblueandgreenonesaretheresultofthreedifferent
in Rydberg atoms crossing microwave cavities [18], and
interfering quantumpaths.
in photonic systems [19, 20]. Furthermore, other mul-
tiqubit entangled states have been realized in different
physical setups [21, 22, 23, 24, 25] with different pur-
has projected which emitter in its ground states implies
poses[26]andpotentialapplicationsinquantuminforma-
theexistenceofmanyquantumpathwaysbetweentheini-
tion. Thoughseveralparadigmaticentangledstateshave
tial fully excited atomic state and each of the final state
beenproducedinthelab,thereisnostudy,toourknowl-
components. This produces a multipath quantum inter-
edge, that associates operationally given experimental
ferenceeffect[9]thatwewilltunebymodifyingthepolar-
configurations with multipartite entanglement classes.
izer orientations. We remark that several experimental
The outlinedbehaviorcanbe understoodfromthe ex-
setups may be in condition to implement the concepts
plicit calculation of the different states of the N-emitter
introduced in this paper [10, 11, 12, 13].
system after the successive photon detection events. A
As will be shown explicitly below, it is always possi-
properly located detector in the far-field with a gen-
ble to find suitable polarizerorientationsto produce any
eral elliptical polarizer oriented in the xy plane of the
desiredstate totallysymmetric withrespectto permuta-
circularly polarized light along the polarization vector
tions of the emitters. Hereby, linear polarizers allow the ǫ ασ +βσ with arbitrary complex coefficients α
+ −
generationofastateoftheGHZtype[14],themaximally an≡d β (α2 + β 2 = 1) can implement the operator ac-
entangled state | | | |
tion [27]
GHZ 1 +,...,+ +eiφ ,..., , (1) N N
| Ni≡ √2(cid:0)| i |− −i(cid:1) Dˆ(ǫ)=αX|+ijhe|+βX|−ijhe|, (5)
j=1 j=1
with arbitraryrelative phaseφ, or a separable (product)
up to an insignificant prefactor. Here, the sum over j
state
runs over all emitters and e is the projection oper-
j
|±i h |
ator from state e to state for emitter j. Starting
SN 1φ,...,1φ , (2) with N emitters|ini their exci|t±eid state e,...,e , the de-
| i≡| i | i
tection of the N emitted photons by N detectors with
with 1 (+ +eiφ )/√2. Anintermediatepolarizer polarizer configuration ǫ ,...,ǫ (ǫ α σ +β σ )
φ 1 N i i + i −
| i≡ | i |−i ≡
configuration permits the generation of the multiqubit projects the emitter system onto the final state ψ =
f
| i
3
(e−iθkσ +eiθkσ )/√2 oriented along angles
ǫ1 y ǫ2 ǫ1 y y + −
polarizer x x x
orientation: π φ π
ǫ3 ǫ2=ǫ3 ǫ1=ǫ2=ǫ3 θ = + +(k 1) , k =1,...,N, (8)
ǫ1(cid:1)=ǫ2(cid:1)=ǫ3 ǫ1(cid:1)=ǫ2=ǫ3 k h2Ni 2N − N
entacnlgalsesm:ent GHZ W Separable
or any configuration resulting from permuting the in-
dices. Here, the term inside the square brackets is only
FIG. 3: (color online). Operational monitoring of tripartite present for the case of an even number of emitters. The
entanglement classes by changing the polarization configura-
state GHZ appears naturally when using all distinct
N
tions. If the three polarizer orientations are all different, the | i
polarizer orientations uniformly shared out over 2π.
finalstate belongs tothe GHZ class; if two polarizer orienta-
In contrast, we are left with the separable state S
tions are different, the final state belongs to the W class; if | Ni
all polarizers are identically oriented, the final state belongs of Eq. (2) with linear polarizers all identically oriented
to theS class. along the angle φ/2 :
φ
ˆ(ǫN)... ˆ(ǫ1)e,...,e . Due to the symmetry proper- θ1 =...=θN = 2. (9)
D D | i
ties ofthose operators,the finalstate aswellasallinter-
Thestate W isalsogeneratedwithlinearpolarizers,
mediatestatesaretotallysymmetricwithrespecttoper- | Ni
allexceptoneorientedidenticallyalongφ/2withthelast
mutationsoftheemitters. Ifentanglementisproducedat
orthogonalto the N 1 first,
theendofthedetectionprocess,weareensuredthatonly −
genuine multipartite entangled states belonging to the φ φ π
accessible symmetric entanglement classes will be gen- θ1 =...=θN−1 = , θN = , (10)
2 2 ± 2
erated. The intermediate states produced in the course
of the successive photon detection events can be ordered or any configuration obtained with permutation of the
in a pyramidal manner, displaying the various possible indices.
quantum paths towards the generation of the desired fi- Forthreeemitters,thoseparticularresults,associating
nal state (see Fig. 2 which exemplifies the case N = 3). the states S3 , W3 , and GHZ3 withcertainchoicesof
| i | i | i
At the end of the N photon detection process, the sys- polarization angles, suggest a wider physical picture. In
tem is found in a coherent superposition of all possible fact, we can associate in an operational manner specific
product states of the N-qubit system in the basis. polarizer configurations of our proposed physical setup
All probability amplitudes related to the differ|e±nit quan- with the three paradigmatic entanglement classes ap-
tum paths add up and yield interference terms (tagged pearing in the 3-qubit case : the number of distinct po-
prefactors on Fig. 2). The final state ψ is found to be larizerorientationsidentifiesunivocallytheentanglement
f
a linear combination of all symmetric| Diicke states with class, be S, W, or GHZ, as shown in Fig. 3.
k excitations [28, 29], D (k) : AccordingtoDu¨ret al.[2],theGHZclassisformedby
N
|−i | i statescharacterizedbyanonvanishing3-tangle[31],the
N Wclassbystateswithazero3-tangleandnon-zerosingle
|ψfi=N Xck|DN(k)i, (6) qubit von Neumann entropies, while the separable state
k=0 classischaracterizedbyzerovaluesoftheseentanglement
measures. In our proposed scheme, the 3-tangle τ of the
where is a normalization prefactor, and
N final state ψ reads
f
| i
c =(Ck)1/2 β ...β α ...α , (7)
k N X i1 ik ik+1 iN 4
16i16=...6=iN6N τ = 4 α1β2 α2β1 2 α1β3 α3β1 2 α2β3 α3β2 2
27N | − | | − | | − |
(11)
with Ck the binomial coefficient of N and k. Thereby,
N andvanishesonlywhen2polarizersareequallyoriented.
wecangenerateanysymmetricstatewithrespecttoper-
In this case, the local entropies vanish only when the
mutations of the emitters using suitable elliptical polar-
thirdpolarizercoincideswiththetwofirst. Whenswitch-
izer orientations. This is a direct application of Vieta’s
ingfromaconfigurationwiththreedistinctpolarizerori-
formulas[30]: anarbitrarysymmetricalstatecanbe ex-
entationstoaconfigurationwiththreeidenticalones,via
panded in the Dicke state basis as Nd D (k) and is
Pk k| N i the intermediate case where two of them are equal, we
produced in our setup using K polarizers oriented along
transit successively from the GHZ class, to the W class,
vectors ǫ with α /β identifying to the K roots of the
i i i andendintheSclass,asseeninFig.3. Thisremarkably
polynomial P(z) = PNk (−1)K−kqCNk/CNKdkzk with K showsthepotentialityoftheproposedsetupforassociat-
the degree of this polynomial, the remaining polarizers ing operationally a physical setting with 3-qubit entan-
oriented along σ . glementSLOCCclassesofstatessymmetricwithrespect
+
For example, the maximally entangled state GHZ topermutationsoftheemitters. Theseresultsencourage
N
of Eq. (1) is generated with linear polarizers|π i a possible generalization to the arbitrary N-qubit state.
θk ≡
4
Finally,wegiveanestimationoftheexpectedfidelityof (2006).
ourscheme,e.g.,forgeneratingthe GHZ stateusing 4 [13] D.L.Moehring,P.Maunz,S.Olmschenk,K.C.Younge,
4
adjacentionslocalizedinalineartra|p. Assiuminga5µm D.N.Matsukevich,L.-M.Duan,andC.Monroe, Nature
separation between ions with a 5 nm confinement in the 449, 68 (2007).
[14] D. M. Greenberger, M. Horne, A. Shimony, and A.
transverse direction and an azimuthal detection window
Zeilinger, Am. J. Phys. 58, 1131 (1990).
of1◦,we estimate afidelity ofabout90%for the genera-
[15] Only approximated conversions can be realized. See e.
tion of the 4-qubit state, largely above the 50% minimal
g. P. Walther, K. J. Resch, and A. Zeilinger, Phys. Rev.
value for provinggenuine four-qubitentanglementofthe Lett. 94, 240501 (2005).
state [32]. With usual ellipsometry techniques, polarizer [16] D. Leibfried et al.,Nature 438, 639 (2005).
orientation uncertainties can be made insignificant com- [17] H. H¨affneret al., Nature 438, 643 (2005).
pared to the finite size effect of the detection window. [18] A.Rauschenbeutel,G.Nogues,S.Osnaghi,P.Bertet,M.
Brune, J.-M. Raymond, and S. Haroche, Science 288,
With an excitation rate of several tens of MHz [8], the
2024 (2000).
counting rate of the needed fourfold coincident events
[19] J.-W. Pan, D. Bouwmeester, M. Daniell, H. Weinfurter,
can be of the order of several tenths of Hz using CCD and A.Zeilinger, Nature 403, 515 (2000).
cameras covering a fair area in the detection plane. In [20] B. P. Lanyon and N. K. Langford, quant-ph/0802.3161
general, the counting rate decreases with the number of (2008).
qubits. This might limit the scalability of the scheme as [21] J.-W. Pan, M. Daniell, S. Gasparoni, G. Weihs, and A.
in similar experiments related to entanglement produc- Zeilinger, Phys. Rev. Lett. 86, 4435 (2001).
[22] Z. Zhao, Y.-A. Chen, A.-N. Zhang, T. Yang, H. J.
tion [8, 13, 19, 25].
Note added.–After submissionofthiswork,webecame Briegel, and J.-W. Pan, Nature 430, 54 (2004).
[23] C. A.Sackett et al.,Nature 404, 256 (2000).
awareofaniceexperimentalobservationofanentirefam-
[24] D. Leibfried, M. D. Barrett, T. Sch¨atz, J. Britton, J.
ily of four-photon entangled states [33]. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D.
L.L. thanks Alexander von Humboldt Foundation for J. Wineland, Science 304, 1476 (2004).
the funding through a Humboldt Research Fellowship. [25] N. Kiesel, C. Schmid, G. T´oth, E. Solano, and H. Wein-
G.S.A. thanks NSF grant NSF-CCF-0524673 for sup- furter, Phys. Rev. Lett. 98, 063604 (2007).
[26] P. Kok, H. Lee, and J. P. Dowling, Phys. Rev. A 65,
porting this collaboration. C.T.andJ.v.Z.gratefullyac-
052104 (2002).
knowledgefinancialsupportbytheStaedtlerfoundation.
[27] G. S. Agarwal, J. von Zanthier, C. Skornia, and H.
E.S. thanks to Ikerbasque Foundation, EU EuroSQIP Walther, Phys. Rev. A65, 053826 (2002).
project, and UPV-EHU Grant GIU07/40. T.B. thanks [28] InanN spin-1 system,theDickestates,usuallydenoted
2
F. Bastin, P. Mathonet, P. Lecomte and M. Rigo from by |S,mi, are defined as the simultaneous eigenstates of
University of Li`ege for helpful discussions. both the square of the total spin operator Sˆ2 and its
z-component Sˆz, where S(S+1)~2 and m~are thecor-
responding eigenvalues. The N +1states with thehigh-
est value S = N/2 form a special subset of Dicke states
that are symmetric under qubit permutation, and can
[1] W. K.Wootters, Phys. Rev. Lett. 80, 2245 (1998). be characterized by the numberof excitations k and de-
[2] W. Du¨r, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, noted |DN(k)i. For details see R. Dicke, Phys. Rev. 93,
062314 (2000). 99 (1954).
[3] A.Ac´ın,D.Bruß,M.Lewenstein,andA.Sanpera,Phys. [29] C. Thiel, J. von Zanthier, T. Bastin, E. Solano, and G.
Rev. Lett. 87, 040401 (2001). S. Agarwal, Phys. Rev. Lett. 99, 193602 (2007).
[4] F. Verstraete, J. Dehaene, B. De Moor, and H. Ver- [30] Vieta’s formulas are sum-of-roots and product-
schelde, Phys. Rev. A 65, 052112 (2002). of-roots identities for polynomial. See e. g.
[5] L. Lamata, J. Le´on, D. Salgado, and E. Solano, Phys. E. W. Weisstein, “Vieta’s Formulas.” From
Rev. A 75, 022318 (2007). MathWorld - A Wolfram Web Resource.
[6] L.ChenandY.-X.Chen,Phys.Rev.A74,062310(2006). http://mathworld.wolfram.com/VietasFormulas.html.
[7] L. Lamata, J. Le´on, D. Salgado, and E. Solano, Phys. [31] The3-tangle isameasureofgenuinetripartiteentangle-
Rev. A 74, 052336 (2006). ment that is invariant underpermutations. For a formal
[8] B.B.Blinov,D.L.Moehring, L.-M.Duan,andC.Mon- definition see V. Coffman, J. Kundu, and W. K. Woot-
roe, Nature 428, 153 (2004). ters, Phys. Rev. A 61, 052306 (2000).
[9] C. Cabrillo et al., Phys. Rev. A 59, 1025 (1999); X.-L. [32] M. Seevinck and J. Uffink, Phys. Rev. A 65, 012107
Fenget al.,Phys. Rev. Lett. 90,217902 (2003); C.-S.Yu (2001).
et al.,Phys. Rev. A75, 044301 (2007). [33] W. Wieczorek, C. Schmid, N. Kiesel, R. Pohlner, O.
[10] U.Eichmann et al.,Phys. Rev. Lett. 70, 2359 (1993). Gu¨hne,andH.Weinfurter,Phys. Rev. Lett. 101,010503
[11] Y.Miroshnychenkoet al., Nature 442, 151 (2006). (2008).
[12] J. Beugnon, M. P. A. Jones, J. Dingjan, B. Darqui´e, G.
Messin, A. Browaeys, and P. Grangier, Nature 440, 779