Experimental Investigation of Nonideal Two-qubit Quantum-state Filter by Quantum Process Tomography

0.26 MB·English

by  Yoshihiro Nambu

Experimental Investigation of Nonideal Two-qubit Quantum-state Filter by Quantum
Process Tomography
Yoshihiro Nambu and Kazuo Nakamura
Fundamental and Environmental Research Laboratories,
NEC, 34 Miyukigaoka, Tsukuba, Ibaraki 305-8501, Japan
We used quantum process tomography to investigate and identify the function of a nonideal
two-qubit quantum-state filters subject to various degree of decoherence. We present a simple
decoherencemodelthatexplainstheexperimentalresultsandpointoutthatabeamsplitterfollowed
byapost-selection processisnot,ascommonlybelieved,asinglet-statefilter. Intheidealcaseitis
a triplet-state filter.
PACSnumbers: 03.65.Wj,42.50.Dv
5
0
0 Characterizationofquantumdevicesisimportant,and standard operator basis {Xˆk}3k=0 that spans two-qubit
2 the characterization of single- and two-qubit devices is operator space[16], where Xˆ = i j (i,j = 0,1) and
hiji
n particularly important because a single-qubit rotation ij :=2i+j. Then can be expa|nidhed| as
a gate and a two-qubit controlled-NOT (C-NOT) gate are h i E
J two building blocks of the quantum computer[1]. The 3
5 quantumsystemsinactualdevicesmayundergononuni- E(ρˆ)= X (cid:0)χS(cid:1)[kl][mn]Xˆk⊗XˆlρˆXˆm† ⊗Xˆn†, (1)
tary state evolution because they interact with their en- k,l,m,n=0
4 vironments, so identifying and understanding the func-
v tion of such nonideal devices are of practical impor- where [kl] := 4k + l[15]. A 16 16 positive Hermitian
3 tance. Nonideal devices can be characterized by quan- matrix χS fully characterizes th×e function of quantum
7
tumprocesstomography(QPT)[2,3,4],andthismethod device and is called the process matrix in the standard
0
5 has been used to characterize single-qubit transmission operator basis[10].
0 channels[5, 6] and rotation gates for photons[7], gates We can transform χS into the process matrix in any
04 ftowro-eqnusbemitbqluesanotfumtw-sot-aqtuebfiitltesyr[s9t]emansdina CN-MNOR[T8],gaatnedfoar oerpaetroartosprabcaes.isNsoette{tAhˆαa}t1αA5ˆ=0mthaaytbspeadnesfitnheedtewitoh-qerubloitcaolply-
/ α
h photons[10]. or nonlocally. For example, the Kronecker products of
p thePaulioperatorbasisBˆ =Eˆ Eˆ (k,l =0, ,3),
ThisletterreportstheQPTofaquantum-statefilter— [kl] k l
t- the beamsplitter (BS) in a Hong-Ou-Mandel (HOM) where Eˆk = σˆ0/√2,σˆ1/√2,σˆ2/√⊗2,σˆ3/√2 and·σ·ˆ·0 de-
n { } { }
interferometer[11,12]—asademonstrationhowthefunc- notes an identity operator, is a local operator basis,
ua tion of two-qubit devices can be identified experimen- whereas the outer product Cˆ[kl] = Φk Φl , where Φk
| ih | | i
q tally. The result of an earlier investigation of the nearly are Bell states, is a nonlocal one. The operator basis Aˆ
α
: idealfilter[9]wasinconsistentwitha commonbelief that is, in general, unitarily related to the standard operator
v
Xi athneiodueatcloBmSesfowllhoewreedthbeytawopoinspt-ustelpechtoiotonnpsrcoocmeses—ouotnilny bwahseirseXˆUk ⊗isXaˆl1.6T1h6atuinsi,taAˆrαy =maPtri3kx,l.=T0(hUen)[kl]αcaXˆnkb⊗eXeˆxl-,
r two output ports are taken into account—behaves as a panded using Aˆ×as follows: E
a α
singlet-state filter [12, 13], but the reason for this in-
consistency was not fully understood. To resolve this 15
problem, we have investigated the function of the non- (ρˆ)= χA Aˆ ρˆAˆ†, (2)
E X (cid:0) (cid:1)αβ α β
ideal filters subject to various degree of decoherence
α,β=0
and have identified two leading contributions to the
operator-sum representation of the quantum operation where χA = U†χSU. Therefore, χA is given by the uni-
forthem[1,2,14]. Herewepresentamodelthatexplains tary transform of χS.
our experimental results and resolve the above inconsis- To obtain χS, it is sufficient to determine the as-
tency. sociated four-qubit state[17]. Let us consider a four-
Let us start by introducing the process matrix[10], qubit system composed of the two two-qubit systems
whichoffersus the mostgeneraldescriptionof the quan- 1-2 and 3-4 and introduce the double-ket notation
tum device allowed in quantum mechanics, and showing I := (Iˆ Iˆ) Φ to denote the bipartite state, where
| ii ⊗ | i
how to obtain it experimentally. Consider a two-qubit Φ = 1 i,i is the unnormalized maximally en-
| i Pi=0| i
device whose function is to map a two-qubit input den- tangled state[4, 16]. Let us introduce the unnormal-
sity operator ρˆ to a two-qubit output density operator ized four-qubit density operator ˆ associated with
in E
ρˆ = (ρˆ ), where is a completely positive linear by ˆ = (13) (24)(I I D I I ), where E
out in E 12 34
E E D E ⊗I | ii hh |⊗ | ii hh | I
map[14]. The map is considered a superoperator act- is an identity map and the superscript and subscript re-
E
ing on positive operators[15]. Let us introduce the set of spectively denote the space on which each map acts and
2
the space to which the operator belongs[17]. Using Eq. put density matrices because the state filtering is trace-
1 and X :=(Xˆ Iˆ)Φ , we can rewrite ˆ as decreasingprocesswhoseoutputintensitydependsonthe
k k E
| ii ⊗ | i D input state.
3
DˆE = X (cid:0)χS(cid:1)[kl][mn]|Xkii|XliihhXm|hhXn|. (3) bˆ cˆ
k,l,m,n=0
aˆ dˆ
If we note that X i,j , Eq. 3 indicates that χS BS
| hijiii ≡ | i 266nm
agrees exactly with the density matrix of the associated
pulse
state ˆ giveninthe standardstatebasis. Evaluationof (cid:1) H H coincidence
E
D
the process matrix is thus reduced to evaluation of the NL crystal QWP
assToociaotbetdaisntateˆEDˆ,E.it is sufficient to evaluate all the (BBO) HWPPOLIF
maps (Xˆ DXˆ ) for all the elements of the stan- delay PMT
hiji hkli
E ⊗
dard operator basis Xˆ Xˆ , since we can ob-
hiji ⊗ hkli FIG. 1: Experimental setup to investigate the beamsplitter.
tain the permutation equivalent of ˆ —that is, ˜ =
E E
Pˆ Pˆ Pˆ ˆ Pˆ Pˆ Pˆ = (12) D (34)(I DI
23 12 34DE 34 12 23 I ⊗ E | ii13hh | ⊗ We used optical trombones to equalize the two path
I I )—by using the following identity:
| ii24hh | lengths to within the coherence length l of the down-
c
3 converted photons ( 200 µm). Two two-photon am-
≈
˜ Xˆ Xˆ (Xˆ Xˆ ), (4) plitudes contributing to coincidence detection—i.e.,that
DE ≡ X k⊗ l⊗E k⊗ l both photons are reflected (r-r) or that both are trans-
k,l=0
mitted (t-t)— were initially (τ = 0) aligned to overlap
where Pˆ is a permutation operator between the sys- bothspatially andtemporally,giving a HOMdip visibil-
ij
tems i and j. To evaluate these maps, we need ity of 85% as shown in Fig. 2. In this case, the BS is
only to use state tomography[18] to evaluate the 16 commonlythoughttoworkasasinglet-statefilter[12,13].
two-qubit output states (ρˆ(1) ρˆ(2)) 3 associated To investigate the nonideal filter, we introduced a delay
{E i ⊗ j }i,j=0 τ in one arm by moving one of the trombone prisms.
with the set of 16 separable input states, where ρˆ
i
0 0, 1 1, + +, (cid:9) (cid:9) , + =1/√2(0 + 1 ),an∈d The larger the τ is, the smaller the overlap of the two
{| ih | | ih | | ih | | ih |} | i | i | i amplitudes is and the more the temporal and polariza-
(cid:9) =1/√2(0 +i1 ) [2, 3, 4]. We therefore can obtain
| i | i | i tioninformationofthe photonpairisentangled. Wewill
the process matrix directly from the experimental data
show later that this results in decoherence if we observe
without using a matrix inversionprocedure[1, 2].
only the polarization information.
We used the above procedure to obtain the pro-
cess matrices for nonideal two-qubit quantum-state fil-
ters. The experimental setup is shown in Fig. 1.
3500
A 0.3-mm-thick type-I crystal (BBO) was pumped by )
c
frequency-tripled pulses generated by 100-fs pulses from se3000 B
aMHmzo)d.e-PloacirksedofThio:sraizpopnhtiarlelylapsoelrar(izreepde5ti3t2io-nnmraptheot≈o8n2s n 10 2500 C
were generated and those traveling in two directions 9◦ e (i2000
from the pump beam were selected by 5-mm irises (1 m nc1500 A
e
from the crystal) and identical 3-nm-width filters (IF) d1000
placed before the detectors. They were injected into a ci
n 500
HOM interferometer[9, 11, 12] having a broadband non- oi
c
polarizing cube BS (Suruga Seiki model S322-20-550). 0
The horizontally(H) and vertically(V) polarizedstates -400 -200 0 200 400
(respectively 0 and 1 ) and the other two states re- delay (fs)
| i | i
quired for the QPT were prepared by half- and quarter-
waveplates(HWPandQWP)placedimmediatelybefore
FIG.2: HOMdipobservedinthecoincidencerecord. Arrows
theBS.Down-convertedphotonswereincidentontheBS
indicatedelaysτ of(A)0,(B)100,and(C)350fsintroduced
at45◦. After theBS,polarizationanalyzersconsistingof
in one of thetwo arms.
aQWPandaHWPfollowedbyaGlan-Thompsonprism
(POL)andasingle-photoncountingmodule(PMT)per-
We obtained the process matrices χF associated with
formed projective measurements onto the set of the 16
three-delayconditions (τ =0, 100,and350fs). To make
product states ρˆ(1) ρˆ(2) 3 . Only signals coincident
{ i ⊗ j }i,j=0 the results clear, we represent them here in the nonlocal
at the two PMTs were recorded, and the associated 16 operator basis Fˆ = (Eˆ Eˆ )Uˆ , where Uˆ is a swap
[ij] i j S S
outputstates (ρˆ(1) ρˆ(2)) 3 wereevaluatedbystate operator. Theyareshown⊗inFig. 3,wherespaceissaved
{E i ⊗ j }i,j=0
tomography[18]. Note that we used unnormalized out- by showing only the real parts. What is most interest-
3
ing is that the contribution of (χF) increases with sum of the identity operator Uˆ associated with trans-
1515 I
increasing delay τ. mission and the operator Uˆ (θ ,θ )Uˆ associated with
3 1 2 S
This result is consistent not with the common belief reflection, where Uˆ (θ ,θ ) accounts for the helicity re-
3 1 2
but with what was pointed out by Mitchell et al.[9]. In- versal and the extra p-s phase shift upon reflection.
deed, we found that our results are well reproduced if Although the above description may suffice when τ =
weassumethat (ρˆ)hasanoperator-sumrepresentation 0,thetemporal(T)spaceofthetwo-photonstateshould
E
with two Kraus operators ˆ− and ˆ+[1, 14]: be taken into account when τ =0. Concerning the tem-
P P 6
poral state space, it would be reasonable to assume that
(ρˆ)=(1 p)ˆ−ρˆˆ−†+pˆ+ρˆˆ+†, (5)
the BS leaves the state untouched on transmission and
E − P P P P
ˆ± = U Uˆ (θ ,θ )Uˆ , (6) swaps the state on reflection. We thus assume that the
I 3 1 2 S
P T ±R function ofthe BSis describedby the followingoperator
where p (0 p 1/2) is a parameter that can
entangled with respect to the polarization and temporal
≤ ≤
be interpreted as the degree of decoherence and where
state spaces:
Uˆ (θ ,θ ) = eiθ1σˆ3/2σˆ e−iθ2σˆ3/2σˆ , which explains
3 1 2 3 3
tahsewloaurlgde bceonetxrpibeucttieodnfoorf⊗t(hχeF)n1o5n1id5eainlsstienagdleto-fst(aχtFe)fi0l0- PˆPT =TUˆIP ⊗UˆIT−RUˆ3P(θ1,θ2)UˆSP ⊗UˆST, (8)
ter. The Kraus operators ˆ− and ˆ+ correspond re- where the superscripts indicate the space on which each
P P
spectively to the operator for the ideal state-filter and operatoracts. We further assume that input state is un-
the noise operator. Note that Kraus operators are not entangled, i.e., that it is a product state of polarization
unique and are not necessarily orthogonal[1]. Figure 4 state ρˆ and temporal state ωˆ. Then, by partially trac-
showstheprocessmatricescalculatedusingthemeasured ing over the temporal degree of freedom, we obtain the
value / =0.76 and appropriate values of the param- following output polarization state (ρˆ) consistent with
R T E
eters θ , θ , and p. As shown in Fig. 4, Eqs. 5 and 6 Eqs. 5 and 6:
1 2
well reproduced the experimental results. It follows that
theoperator ˆ− describingidealoperationdeviatesfrom
the singlet-stPate because of the factor Uˆ3(θ1,θ2), which (ρˆ)=TrT ˆPT (ρˆ ωˆ)(cid:16)ˆPT(cid:17)†
is consistent with Ref. [9]. E P ⊗ P
To understand these results, we need a decoherence =(1 p)ˆ−ρˆˆ−†+pˆ+ρˆˆ+†, (9)
− P P P P
model. Let us first call our attention to the function of
the BS in the polarization space. Let aˆ andˆb (cˆand dˆ) wheretheparameterp dependsontheoverlapofthetwo-
photonamplitudes contributing to coincidence detection
denote input (output) field operators of the BS, and the
polarization be indicated by subscripts (e.g., by aˆ and (r-r or t-t) and thus on the delay τ. Ideally, p is zero if
H
aˆ fortheH-andV-polarizedfieldoperators). Ifweuse τ =0,increasesasτ increases,andapproaches1/2when
V
τ . Figure 4 shows that this model largely explains
aright-handedcoordinatesystemdefinedbythepositive
→∞
directionsofH-,V-polarizedfield(pandscomponents) the experimental results, except that θ1 is not actually
equal to θ as in the model. The reason is uncertain,
andpropagationvector(Fig. 1)todescribetheinputand 2
butthisinequalitymaybeduetopolarization-dependent
output fields, their operator equations are given by:
/ ignored in the model.
cˆ =√ aˆ +iei(Γ+π)√ ˆb , RWTe conclude that the function of the ideal BS is de-
H H H
dˆH =ie−Ti(Γ+π)√RaˆH +√RTˆbH, swcerlilbaesdtbhyecPˆo−effi, wciheincthsdepaenndds .onThtheesepp-sarpahmaesteersshiofbtvais-
cˆV =√ aˆV +iei∆√ ˆbV, ously depend on the deRtailedTdesign of the BS. Properly
T R
dˆV =ie−i∆√RaˆV +√TˆbV, (7) mdeuslitganneedouBsSly,waotulledasstatfiosrfyphRo/tTons=th1aatnhdav∆ea−pΓar=tic0ulsair-
where and arerespectivelythe transmissionandre- wavelengthandareincidentat45◦. Inthisidealizedcase,
flectionTcoefficRientsofthe BS.We haveassumedthatthe ˆ− agreeswithoneofthefourBellstates: atripletstate.
P
pandscomponentsofthefieldrespectivelyacquirephase This shows that the common belief that an ideal BS be-
shiftsΓ+πand∆uponreflection. Thefactorπ accounts haves as a singlet-state filter is a misconception due to
for abrupt change in p-s phase shift due to nonadiabatic overlooking the helicity reversal of the photon upon re-
changes in the k vector upon reflection, which has been flection at the BS.
frequentlynotedinthepapersconcerningthetopological In summary, we used QPT to investigate the nonideal
phase effect in optics [20, 21, 22]. Note that the helicity two-qubit quantum-state filter and identify its function.
of a photon in a state (cid:9) changes sign upon reflection We showed that our results are satisfactorily explained
| i
fromtheBSbutnotupontransmissionthroughtheBSif by a simple model with several reasonable assumptions
∆ Γ=0. Ifwe considerthe outputfields that cancon- thatacommonbeliefaboutthefunctionofaBSfollowed
−
tribute to coincidence measurement, we conclude from by post-selection process is mistaken.
Eqs. 7 that the BS projects the input state onto ˆ− We thank S. Ishizaka and A. Tomita for their helpful
P
with θ = θ = ∆ Γ, which may be finite for a non- discussions. This work was supported by the CREST
1 2
polarizingBS.Note−thattheprojector − isthecoherent programof the Japan Science and Technology Agency.
P
4
A B C
1155 1155 1155
00 00 00
115500 115500 115500
decoherence
FIG.3: Theprocess matrices ofthestate filterobtained for delaysτ of (A)0, (B) 100, and (C) 350 fs. Onlythereal partsof
thematrices are shown (in arbitrary units). Vertical arrows indicate thediagonal elements (χF)1515.
A B C
1155 1155 1155
00 00 00
115500 115500 115500
decoherence
FIG. 4: Calculated process matrices corresponding to Fig. 3 (see text). The following parameters were used: (A) p = 0.14,
(B) p=0.325, (C) p=0.5 and the common parameters R/T =0.76, θ1 ≈0.41π, and θ2 ≈0.076π.
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