Experimental realisation of generalised qubit measurements based on quantum walks

0.75 MB·English

by  Yuan-yuan Zhao

Experimental realisation of generalised qubit measurements based on quantum walks
Yuan-yuan Zhao,1,2 Neng-kun Yu,3,4 Paweł Kurzyński,5,6
Guo-yong Xiang,1,2,∗ Chuan-Feng Li,1,2 and Guang-Can Guo1,2
1Key Laboratory of Quantum Information, University of Science and Technology of China,
CAS, Hefei, 230026, People’s Republic of China
2Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China
3The Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario, Canada
4Department of Mathematics&Statistics, University of Guelph, Guelph, Ontario, Canada
5Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, 117543 Singapore, Singapore
6Faculty of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznań , Poland
5 (Dated: January 22, 2015)
1
0 We report an experimental implementation of a single-qubit generalised measurement scenario
2 (POVM)basedonaquantumwalkmodel. Thequbitisencodedinasingle-photonpolarisation. The
photonperformsaquantumwalkonanarrayofopticalelements,wherethepolarisation-dependent
n
translation is performed via birefringent beam displacers and achange of thepolarisation isimple-
a
J mented with the help of wave-plates. We implement: (i) Trine-POVM, i.e., the POVM elements
uniformly distributed on an equatorial plane of the Bloch sphere; (ii) Symmetric-Informationally-
1
Complete (SIC) POVM;and (iii) UnambiguousDiscrimination of two non-orthogonal qubitstates.
2

PACSnumbers: 03.65.Ta,03.67.Ac,03.67.Lx,42.25.Hz
]
h
p
- I. INTRODUCTION measurement outcome from the pointer. The sharpness
t
n of the measurement comes from the fact that the initial
a spread of the pointer’s state is assumed to be narrow
The basic unit of quantum information is a two-level
u and the translation caused by the interaction with the
q quantum system commonly known as a qubit. Qubits
measured observable is large enough to prevent overlap
[ can be implemented on physical objects such as polari-
between the part of the wave packed that was shifted to
sation of photons or intrinsic angular momentum (spin
1 the right with the part that was shifted to the left.
1/2) of quantum particles. Any quantum computation
v
relies on precise preparations, transformations and mea- Ontheotherhand,therearegeneralisedmeasurement
6
9 surements of such systems. Before actual quantum com- scenarios, the so called Positive-Operator Valued Mea-
0 puterisbuild,onehastomastertheabilitytomanipulate sure(POVM),inwhichoneallowstheprincipalsystemto
5 with single qubits and learn how to readout information interact with an ancillary system, whose state is known,
0 encoded in them. and later performs a von Neumann measurement on the
1. The information readout from a quantum system is joint system. This effectively extends the dimensionality
0 done via a measurement. In most common scenario one of the Hilbert space and one can implement measure-
5 performsthe vonNeumannmeasurementthatprojectsa ments of a quibit with more than two outcomes. As a
1 state of the qubit onto one of two perfectly distinguish- result,onegainsaplethoraofnew possibilities. They al-
:
v able (orthogonal) physical states of the system. Such low one, for example, to perform a tomography of qubit
i measurements are sharp in a sense that once the mea- withasinglemeasurementsetup[1,2],ortodiscriminate
X
surement is done, the outcome of the measurement is between non-orthogonal quantum states [3, 4]. POVMs
ar determined and any repetition of exactly the same mea- were implemented in laboratories using various setups
surement would yield the same outcome value. [5–11].
Physically, von Neumann measurements are realised In [12] it was proposed that they can be implemented
viainteractionofthesystemwiththemeasurementappa- viaadiscrete-timequantumwalkwhichhasbeenrealised
ratus. The pointerof the measurementapparatusis rep- in a laboratory using various physical systems [13–25].
resented via wave packedand the interaction causes this Quantumwalksmodelanevolutionofaparticleinadis-
wavepackettomoveeithertotheleftorright,depending cretespace. Theparticlemoveseitheronesteptotheleft
onthe value ofthe measuredobservable. Ingeneral,this orright,dependingonastateofatwo-levelsystemknown
value might be undetermined and the pointer goes into asacoin. Quantumwalkswereoriginallyproposedasan
superposition of being to the left and to the right from interference process resulting from a modified version of
its initial position. The actual collapse of wave function a von Neumann measurement in which a pointer state
is usually attributed to the observer who reads out the distribution is much broader than the shift of its posi-
tion[26]. Thepositionofthepointerplaystheroleofthe
quantumwalkerandthequbitthatismeasuredplaysthe
role of the coin. If the interaction between the pointer
∗Electronicaddress: [email protected] and the qubit occurs at the same time as the evolution
2
ofthe qubit, the measuredvalue changesduringthe pro- III. POVM IMPLEMENTATION WITH
cess and the pointer starts to move back andforth. This QUANTUM WALKS
movement leads to an interference and the interference
pattern produced in this process can be interpreted as The probability distribution of a quantum walk, that
a POVM. In fact, in [12] it was shown that any POVM is initially localised at the origin x = 0, depends on the
can be implemented in such a way, provided a necessary initial coin state α +β and on the subsequent
evolution is applied to the qubit. coin operations. A|s→inigle ap|p←liciation of the operator T
In this work we report an experimental implementa- makes the particle to go into superposition α1, +
tion of the above quantum walk POVM scenario. Here, β 1, . In this case the position measuremen|t w→oiuld
we use the optical setup in which the qubit is encoded co|r−resp←onid to a von Neumann measurement of the coin
in a polarization state of a single photon and the posi- in the basis , , where the result corresponds to
tion of the quantum walker is implemented on a pho- finding the p{a←rtic→le}at x = 1 and to→finding it at x =
tonic path [25]. We construct setups realising (i) three 1. However, if one allows quant←um walk to evolve for
POVM elements symmetrically distributed on an equa- −morethanonestepandthecoinoperationtochangefrom
torial plane of the Bloch sphere (Trine-POVM); (ii) one step to another step, then the particle can spread
Symmetric-Informationally-Complete(SIC) POVM; and overmany positionsand the measurementof its location
(iii) Unambiguous Discrimination of two non-orthogonal at x may correspond to a measurement of some POVM
quantum states. element E . Indeed, a proper choice of coin operations
x
can lead to an arbitrary qubit POVM scenario [12].
The POVM elements E (i=1,2,...,n) are nonnega-
i
II. DISCRETE-TIME QUANTUM WALKS tive operators obeying
n
Discrete-time quantum walks are quantum counter-
parts of classicalrandom walks in which a particle takes Ei =11. (4)
Xi
a random step to the left or right. In the classical case
the particle spreads in a diffusive manner (a standard TheydifferfromstandardvonNeumannprojectorsΠ in
i
deviation of its position is proportional to √t) and after thatthey donothavetobe orthonormal(Π Π =δ Π ,
i j i,j i
manystepsaspatialprobabilitydistributionisGaussian. but E E = δ ) and that their number can be greater
i j i,j
6
In quantum case the system is described by two degrees than the dimension of the system n > d. The follow-
of freedom ψ = x,c : the position x=..., 1,0,1,... ing quantum walk algorithm,proposed in [12], generates
| i | i −
and a two-level internal degree of freedom known as a an arbitrary set of rank 1 POVM elements E ,...,E
1 n
{ }
coin c = , . The evolution is unitary and consists of (rank 2 elements can be generated with a modified ver-
← →
two sub-operations U = CT, namely a unitary coin toss sion of this algorithm).
(that is a 2 2 unitary matrix acting only on the coin
× 1. Initiate the quantum walk at position x = 0 with
degree of freedom)
the coin state correspondingto the qubit state one
cosθ e−iβsinθ wants to measure.
C = (1)
(cid:18) eiβsinθ cosθ (cid:19)
− 2. Set i:=1.
and a conditional translation operation 3. While i<n do the following:
T = x+1, x, + x 1, x, . (2) Apply coin operation C(1) at position x = 0
| →ih →| | − ←ih ←| • i
Xx and identity elsewhere and then apply trans-
lation operator T.
Quantum walks with uniform (position-independent)
coin operation spread ballistically (standard deviation • Apply coin operation Ci(2) at position x=1,
proportional to t) and its probability distribution differs
0 1
form the classical Gaussian shape [29, 30]. On the other NOT =
hand, quantum walks with position-dependent coin op- (cid:18)1 0(cid:19)
eration
atpositionx= 1andidentityelsewhereand
−
cosθ e−iβxsinθ then apply translation operator T.
C = x x (3)
x (cid:18) eiβxsinθx cosθx (cid:19) i:=i+1.
− •
canbeusedtoobservelocalisation[31,32],ortosimulate Since in this workwe areonly interestedinthe measure-
physicalsystems with non-homogenousinteractions [33]. ment statistics, and not in the post-measurement states,
In [12] it was shown that quantum walks with position we simplified the algorithm in [12] and omitted the last
andtime-dependentcoinoperationsC canbealsoused step. The POVM elements that are generated depend
x,t
to implement POVMs. solely on the form of operators C(1) and C(2).
i i
3
IV. EXPERIMENT A. Trine POVM
In our experiment, frequency-doubled femtosecond
The experimental setup in Fig. 1(a) is used to con-
pulse (390 nm, 76 MHz repetition rate, 80 mw aver-
struct the trine POVMs, 2 ψi ψi (i=1,2,3),
agepower)fromamode-lockedTi:sapphirelaserpumpa 3| 3ih 3|
type-I beta-barium-borate(BBO,6.0 6.0 2.0mm3, θ=
× ×
29.9) crystal to produces the degenerate photon pairs.
After being redirected by the mirrors (M1 and M2, as
ψ1 = H
in the Fig. 1(a)) and the interference filters (IF, λ=3 | 3i | i
△ 1
nm,λ=780nm), thephotonpairsgeneratedinthe spon- ψ2 = (H √3V )
taneousparametricdown-conversion(SPDC)processare | 3i −2 | i− | i (5)
1
coupledintosingle-modefibersseparately. Singlephoton ψ3 = (H +√3V ).
state is prepared by triggering on one of these two pho- | 3i −2 | i | i
tons, and the coincidence counting rate collected by the
avalanche photo-diodes (APD) are about 4 104 in one
×
minute.
Accordingto the settings of the coinoperators,the opti-
calaxesofBD1andBD2mustbealigned,inotherwords,
they form an interferometer. When rotating HWP1 and
HWP3by22.5◦,weobservedthattheinterferencevisibil-
IIFF HHWWPP QQWWPP BBBBOO PPBBSS BBDD AAPPDD
ity of the interferometer was about 99.8 and the system
((aa)) 44
ttrriiggggeerr HHWWPP11 HHWWPP22 0022 wwaesbsetgainblteoosveetrth2.e5chororuersspoonfdtiinmgecscoainle.opAerfatteorrsaliingneiancgh,
NNOOTT HHWWPP33 NNOOTT step. For the Trine POVMs, we have
MM22 BBDD11 BBDD22 BBDD33 BBDD44
ppuummpp MM11
CCCoooiiinnnccciiidddeeennnccceee
DDDeeecccttteeeccctttiiiooonnn
((bb)) 1 √2 1
C(1) =1,C(2) = ,
1 1 r3(cid:18) 1 √2(cid:19)
66 − (6)
44
HHWWPP22 HHWWPP44 HHWWPP55 0022 C(1) = 1 1 1 ,C(2) =1,
HHWWPP11 2 r2(cid:18)1 1(cid:19) 2
QQWWPP22 NNOOTT −
HHWWPP33 NNOOTT
QQWWPP11 NNOOTT BBDD55 BBDD66
BBDD11 BBDD22 BBDD33 BBDD44
where C(2) and C(1) are realized by rotating HWP2 and
1 2
FIG.1: Experimentalsetup. (a)Experimentalsetupforcon- ◦ ◦
HWP3by17.32 and22.5 respectively. Theinitialtrine
structing the trine POVMs corresponding to |ψ3ii and realiz- coin states ψi are constructed by rotating HWP1 by
ing the unambiguous state discrimination. (b) Optical net- 0◦, 30◦ an|d3i30◦, while the anti-trine states ψ¯i are
work for constructing SIC-POVMs. Initial coin states are cons−tructedbyrotatingitby45◦,15◦ and 15◦.|A3tilast,
prepared by passing the single photons through a polarizing −
every output port’s detection efficiency are calibrated so
beam splitter (PBS), a half-wave plate (HWP) HWP1 and a
quarter-waveplate(QWP)QWP1inaspecificconfiguration. that the differences among them are below 5%.
TheconditionalpositionshiftsareimplementedbyBeamDis-
placers(BDs)andthecoinoperatorsindifferentpositionsare Fig. 2 shows that the results in our experiment agree
realized by wave plates with different angles (Table 1). The with theoretical predictions. The ratios 2/3 : 1/6 : 1/6
indices in thefiguredenote theposition of walker. (0:1/2:1/2)forthecasesof ψi (ψ¯i ,where ψi ψ¯i =
| 3i | 3i h 3| 3i
0) are given in theory and the detailed numerical results
of the probability distributions can be found in Table II
One-dimensional discrete time quantum walk system and III. To visualize that the setup has constructed the
has two degrees of freedom, x and c, x is the position Trine POVMs, it is important to demonstrate that we
of the particle and c is the state of the coin. In our cannot find states ψ¯1 in position 4, ψ¯2 in position 0
experiment they are encoded in the longitudinal spa- and ψ¯3 inposition|23.iFig. 2(b)andT|ab3lei IIIshowthat
| 3i
tial modes and polarizations H , V of the single pho- the probabilities of these events are indeed very close to
| i | i
tons respectively. In this case, the conditional transla- zero, with an average value of 0.0085. In addition, the
tion operator as given by Eq.(2) is realized by the de- results for states ψi also indicate the coefficients of the
signed BD, that does not displace the vertical polarized POVM we constr|uc3tied is 2, see Fig. 2(a). The errors
3
photons (x,V x 1,V ) but makes the horizon- in our experiment mainly stem from the imperfect wave
| i → | − i
tal polarized ones undergo a 4 mm lateral displacement platesandtheinterferometersandthecountingstatistics
(x,H x+1,H ). of the photons.
| i→| i
4
C2 C1 C2 C1 C2
1 2 2 3 3
◦ ′ ◦ ′ ◦ ′ ◦ ′
HWP 67 30 67 30 17 38 142 30 −−
◦
QWP −− −− −− 150 −−
TABLE I: The configurations of the QWPs and HWPs to
realize thecoin operators for constructing theSIC-POVM.
For these settings, BD1 and BD2, BD3 and BD4 form
twointerferometerswhoseinterferencevisibilityareboth
above0.993. TheHWP1andQWP1withdifferentangles
in front of a polarizing beam splitter (PBS) are used to
produce ψi and the corresponding orthogonal states ψ¯i
4 4
(TableVIandTableVII).AsshowninFig. 3,theresults
arealsoingoodaccordancewiththeoreticalratios 1 : 1 :
2 6
1 : 1 for ψi and 1 : 1 : 1 :0 for ψ¯i.
6 6 4 3 3 3 4
(cid:3)
FIG.2: ResultsfortrinePOVMs. Histogramshowstheprob-
abilities of counting rates in position 0, 2 and 4 with input
states ψi(a) and ψ¯i(b), respectively. All results are normal-
3 3
ized so that the sum of the counts in these three positions is
1. The theoretical values are shown as the blue lines, which
are 2/3, 1/6, 1/6 for ψi and 1/2, 1/2, 0 for ψ¯i(i=1, 2, 3);
3 3
error bars are too small toidentify.
B. SIC POVM
The optical network in Fig. 1(b) construct the SIC
POVMs, 1 ψi ψi (i=1,2,3,4),
2| 4ih 4|
ψ1 = H
| 4i | i
1 2
ψ2 = H + V
| 4i −√3| i r3| i (cid:3)
ψ3 = 1 H +ei23π 2 V (7)
| 4i −√3| i r3| i FIG. 3: Results for SIC POVMs. Histogram showing the
normalized probability of counting rate in position 0, 2, 4
|ψ44i=−√13|Hi+e−i23πr23|Vi. athnedor6erteicsaplevcatilvueelsyawreitshhtohweninapsuthtestbaltueeψli4n(eas),awnhdicψh¯4ia(rbe).giTvehne
by 1/2, 1/6, 1/6, 1/6 for states ψi and 1/3, 1/3, 1/3, 0 for
The coin operators states ψ¯ ; error bars are too small4to identify.
4
(1) (2) 1 1 1
C =1,C = − ,
1 1 √2(cid:18) 1 1(cid:19)
C(1) = 1 −1 1 ,C(2) = 1 √2 1 , (8)
2 √2(cid:18) 1 1(cid:19) 2 √3(cid:18) 1 √2(cid:19)
−
C3(1) = √12(cid:18)ee−iiπ3π3 ee−iiπ6π6(cid:19),C3(2) =1 C. Unambiguous state discrimination
arerealizedby waveplates invariousconfigurations(de- For the unambiguous state discrimination of states,
tails in Table I). ψ± = cos(θ/2)H sin(θ/2)V , we can use the same
| i | i± | i
5
1 on a quantum walk model presented in [12]. Our results
ψ− match the theoretical predictions. We believe that these
0.9 ψ+
Theory kind of experimental setups can be used in the future to
0.8
implement other types of generalised measurement sce-
0.7
narios with multiple outcomes and rank 2 POVM ele-
ess0.6 ments, and to study quantum walks with position and
c
uc0.5 time-dependent coin operations. Finally, we would like
s
P−0.4 to mention that similar results had been reported while
0.3 we were working on this experiment [27, 28].
0.2
0.1
0
0 1 2 3 4 5 6 7 8 9 10
θ(π/20)
VI. ACKNOWLEDGEMENTS
FIG. 4: Theoretical and experimental successful probability
v.s. θ whichisrelatedtothestatetobediscrimination; error The authorswould like to thank Yongsheng Zhang for
bars are too small to identify. helpful discussion. The work in USTC is supported by
National Fundamental Research Program (Grants No.
2011CBA00200andNo. 2011CB9211200),NationalNat-
setup as in Fig. 1(a), with ural Science Foundation of China (Grants No. 61108009
and No. 61222504). P.K. is supported by the National
C(1) =1,
1 Research Foundation and Ministry of Education in Sin-
gapore.
1 (tanθ)2 tanθ
C(2) =q − 2 2 ,
1 tanθ 1 (tanθ)2 (9)
 2 −q − 2 
C(1) = 1 1 1 ,C(2) =1,
2 r2(cid:18)1 1(cid:19) 2
− VII. APPENDIX
The state after four quantum walk steps becomes
θ Thedetailedresultsaboutψi,ψ¯i,ψi andψ¯i areshown
ψ =√cosθ 4,H +√2sin 2,H , (10) 3 3 4 4
+ in Table II, Table III, Table IV and Table V. Table VI
| i | i 2| i
andTableVIIaretheanglesoftheHWP1andQWP1for
or
preparingstates ψi andψ¯i. The mainparametersofun-
4 4
ψ− =√cosθ 4,H √2sinθ 0,H . (11) ambiguous state discrimination and the detailed results
| i | i− 2| i can be found in Table VIII and Table IX.
Therefor, if the photon is detected at position x=0 one
knows that the coin was definitely in the state ψ− . If
it was detected at position x = 2 one knows t|hatithe state P0 P2 P4
ψ1 0.1684(20) 0.1711(19) 0.6604(25)
coin was definitely in the state ψ . Finally, if it was 3
detected at position x=4 one ga|in+sino information and ψ32 0.6540(26) 0.1731(20) 0.1729(21)
ψ3 0.1753(21) 0.6466(25) 0.1782(21)
the discrimination was unsuccessful. 3
preInpaoreudr ienxpπersimteepnstf,rotmhe0inpuθt stπa.teTshψe+staatneds fψo−rvaarre- TABLEII:P0,P2andP4arethenomarizedprobabilitiesofcount-
20 ≤ ≤ 2 ingrateinposition0,2and4.
ious θ are prepared by rotating the HWP, placed before
the polarizing beam splitter (PBS), and the coin oper-
ator C(2), C(1)and the NOT operator are realized by a
1 2
HWP rotating in the angle of 1arcsin(tanθ), π and π
2 2 8 4
respectively. FromFig. 4wecanseethatthe probability
state P P P
of the successful discrimination is increasing with θ. For 0 2 4
more details see Tables VIII and IX. ψ¯31 0.5018(27) 0.4977(27) 0.0005(01)
ψ¯2 0.0151(06) 0.4897(27) 0.4952(26)
3
V. CONCLUSIONS ψ¯33 0.4933(25) 0.0081(05) 0.4987(25)
We experimentally realisedthree generalisedmeasure- TABLE III: P0, P2 and P4 are the nomarized probabilities of
ment scenarios for a qubit. These scenarios are based countingrateinposition0,2and4.
6
θ θ C(2) P P
1/2 1 theory e
state P P P P
0 2 4 6 ◦ ′ ◦ ′
π/20 2 15 2 15 0.0123 0.013(06)
ψ1 0.1662(19) 0.1654(19) 0.1934(20) 0.4749(26)
4 π/10 4◦30′ 4◦34′ 0.0489 0.050(10)
ψ2 0.1585(18) 0.1571(18) 0.5220(27) 0.1625(20)
4 3π/20 6◦45′ 6◦57′ 0.0109 0.103(15)
ψ3 0.5015(26) 0.1676(19) 0.1695(19) 0.1614(19)
4 π/5 9◦ 9◦29′ 0.191 0.181(19)
ψ4 0.1885(20) 0.4843(25) 0.1623(19) 0.1649(19)
4 ◦ ′ ◦ ′
π/4 11 15 12 14 0.293 0.285(22)
◦ ′ ◦ ′
3π/10 13 30 15 19 0.412 0.402(23)
TABLEIV:Thenomarizedprobabilitiesofcountingratesforstate
ψi inposition0,2,4and6. 7π/20 15◦45′ 18◦54′ 0.546 0.531(24)
4
◦ ◦ ′
2π/5 18 23 18 0.691 0.673(22)
◦ ′ ◦ ′
9π/20 20 15 29 21 0.844 0.832(18)
◦ ′ ◦
state P P P P π/2 22 30 45 1.000 0.996(03)
0 2 4 6
ψ¯1 0.3341(24) 0.3367(25) 0.3289(24) 0.0003(01)
4
ψ¯42 0.3182(23) 0.3283(24) 0.0051(04) 0.3485(24) TisApBreLpEareVdIbIyI:HθW, tPh1e raontgalteedretlaoteθd1/2to; Cth12e, tihnepuatngstleatoefψH+WwPh2ictho
ψ¯3 0.0040(03) 0.3209(24) 0.3671(25) 0.3080(23) realize the operator C12. Ptheory and Pe represent the theoretical
4 andtheexperimental successfulprobability.
ψ¯4 0.3152(24) 0.0005(01) 0.3647(24) 0.3196(24)
4
TABLEV:Thenomarizedprobabilitiesofcountingratesforstate θ θ C(2) P P
ψ¯4i inposition0,2,4and6. −π/20 −21◦/125′ 2◦115′ 0t.0h1eo2r3y 0.0127e(05)
◦ ′ ◦ ′
−π/10 −4 30 4 34 0.0489 0.0511(11)
ψ1 ψ2 ψ3 ψ4 −3π/20 −6◦45′ 6◦57′ 0.0109 0.1096(15)
4 4 4 4
HWP1 0◦ −27◦22′ 17◦38′ 45◦ −π/5 −9◦ 9◦29′ 0.191 0.1889(19)
QWP1 0◦ 35◦16′ −27◦22′ −27◦22′ −π/4 −11◦15′ 12◦14′ 0.293 0.2883(23)
◦ ′ ◦ ′
−3π/10 −13 30 15 19 0.412 0.4092(24)
TABLEVI:TheanglesofHWP1andQWP1usedtoprepare ◦ ′ ◦ ′
−7π/20 −15 45 18 54 0.546 0.5340(24)
thestates ψi.
4 ◦ ◦ ′
−2π/5 −18 23 18 0.691 0.6812(22)
◦ ′ ◦ ′
−9π/20 −20 15 29 21 0.844 0.8367(17)
ψ¯1 ψ¯2 ψ¯3 ψ¯4 −π/2 −22◦30′ 45◦ 1.000 0.9951(03)
4 4 4 4
◦ ◦ ′ ◦ ′ ◦
HWP1 45 17 38 −27 22 0
◦ ◦ ′ ◦ ′ ◦ ′ TABLE IX: θ, the angle related to the input states ψ− which is
QWP1 0 35 16 −27 22 −27 22
prepared by HWP1 rotated to θ ; C2, the angle of HWP2 to
1/2 1
realise the operator C12. Ptheory and Pe represent the theoretical
TABLEVII:TheanglesofHWP1andQWP1usedtoprepare andtheexperimental successfulprobability.
thestates ψ¯i.
4
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