Realization of single-qubit positive operator-valued measurement via a one-dimensional photonic quantum walk

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by  Zhihao Bian

Realization of single-qubit positive operator-valued measurement via a
one-dimensional photonic quantum walk
Zhihao Bian,1 Jian Li,1 Hao Qin,1 Xiang Zhan,1 Rong Zhang,1 Barry C. Sanders,2,3,4,5 and Peng Xue∗1,6,7
1Department of Physics, Southeast University, Nanjing 211189, China
2Hefei National Laboratory for Physical Sciences at Microscale and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
3Shanghai Branch, CAS Center for Excellence and Synergetic
Innovation Center in Quantum Information and Quantum Physics,
University of Science and Technology of China, Shanghai 201315, China
4Institute for Quantum Science and Technology, University of Calgary, Alberta, Canada T2N 1N4
5Program in Quantum Information Science, Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
6State Key Laboratory of Precision Spectroscopy,
East China Normal University, Shanghai 200062, China
5 7Beijing Institute of Mechanical and Electrical Space, Beijing 100094, China
1
0 We perform generalized measurements of a qubit by realizing the qubit as a coin in a photonic
2 quantum walk and subjecting the walker to projective measurements. Our experimental technique
n can be used to realize photonically any rank-1 single-qubit positive operator-valued measure via
a constructinganappropriateinterferometricquantum-walknetworkandthenprojectivelymeasuring
J the walker’s position at the final step.

2
2 PACSnumbers: 42.50.Ex,42.50.Dv,03.67.Lx,03.67.Ac

]
h Quantum walks (QWs) exhibit distinct features com- (SIC) POVMs [16, 17].
p pared to classical random walks with applications to Our goal is to realize experimentally a single-qubit
-
t quantum algorithms [1, 2]. The discrete-time QW is a POVM and to discriminate between non-orthogonal ini-
n
process in which the evolution of a quantum particle on tial coin states via executing a properly engineered QW
a
u a lattice depends on a state of a coin, typically a two- whose projective walker measurement is sometimes in-
q level system, or qubit. Controlling the coin degree of conclusive [3]. To achieve a site-specific POVM, we con-
[ freedom indirectly controls the walker, and, through this trol the internal degree of freedom of the measured two-

1 indirect control, the walker’s state can be measured to level coin. Here we report our successful experimental
v infer the coin state. Rigorously speaking, walker-coin realization of POVMs, including unambiguous state dis-
0 entanglement and projective measurement of the walker crimination of two equally probable single-qubit states
4 yields a positive operator-valued measure (POVM) on and a single-qubit SIC-POVM, via a one-dimensional
5
a single qubit [3]. Furthermore any rank-1 and rank- photonic QW.
5
0 2 single-qubit POVM can be generated by a judiciously We focus on rank-1 POVMs, as higher-rank POVMs
. engineeredQW.Herewedemonstrateexperimentallythe canbeconstructedasaconvexcombinationofrank-1ele-
1
capability of performing such generalized measurements ments[3]. Ourexperimentaltechniquecanbeusedtore-
0
5 of a qubit by realizing the walker in the path degree of alizephotonicallyanyrank-1single-qubitPOVMviacon-
1 freedom of a photon and the coin state as polarization structing an interferometric QW and projectively mea-
: and performing optical interferometry with path-based suring the walker’s position at the final step. We char-
v
i photodetector to perform a POVM on the photon’s po- acterize experimental performance by the 1-norm dis-
X
larization state. tance [18] between the walker distribution obtained ex-
ar Realizing a POVM is important as a POVM is needed perimentallyPexp(x)vstheoreticallyPth(x)overinteger-
for generalized acquisition of information thereby asso- valued position x. This distance is
ciated with a multitude of quantum information tasks
such as quantum state estimation and tomography [4], d= 1(cid:88)(cid:12)(cid:12)Pexp(x)−Pth(x)(cid:12)(cid:12), (1)
2
quantum cloning [5], entanglement distillation [6] and
x
generalized quantum cryptography protocols [7]. Single-
and a small distance indicates a successful experimental
qubit POVMs have been performed experimentally [8–
realization.
11]. POVMs’ wide applications include unambiguous
A standard model of a one-dimensional (1D) discrete-
state discrimination [12–15] and quantum state tomog-
time QW consists of a walker carrying a coin that is
raphy in terms of symmetric informationally complete
flippedbeforeeachstep. Inthecoin-statebasis{|0(cid:105),|1(cid:105)},
the site-dependent coin rotation for the nth step C ∈
x,n
SU(2) is applied to the coin when the walker in the
∗[email protected] position x, followed by a conditional position shift due
2
dependent coin rotations for the first three steps are
(cid:113) 
1−tan2 φ tanφ
C−1,2 = 2 (cid:113) 2 ,
tanφ − 1−tan2 φ
2 2
(cid:18) (cid:19) (cid:18) (cid:19)
0 1 1 1 1
C =σ = ,C = √ (3)
1,2 x 1 0 0,3 2 1 −1
and1elsewhere. Eachstepthesite-dependentcoinrota-
tionsarefollowedbyaconditionalpositionshiftT. Then
the initial walker-coin states (cid:12)(cid:12)ϕ±(cid:11)=|0(cid:105)|φ±(cid:105) evolve into
0
(cid:12)(cid:12)ϕ+(cid:11)=(cid:112)cosφ|3(cid:105)|0(cid:105)+√2sinφ|1(cid:105)|0(cid:105),
3 2
FIG.1: (Coloronline.) Experimentalschematic. (a)Detailed (cid:12)(cid:12)ϕ−(cid:11)=(cid:112)cosφ|3(cid:105)|0(cid:105)−√2sinφ|−1(cid:105)|1(cid:105) (4)
sketch of the setup for realization of unambiguous state dis- 3 2
crimination of two equally probable single-qubit states via
a three-step QW. Single-photons are created via SPDC in respectively. The walker is projectively measured in the
a BBO crystal. One photon in the pair is detected to her-
position basis. If the position measurement outcome is
ald the other photon, which is injected into the optical net-
x = 1 (x = −1), then we ascertain that the initial coin
work. (b) Setup for realization of a qubit SIC-POVM via
state was |φ+(cid:105) (|φ−(cid:105)). If the walker is instead measured
a six-step QW. The initial coin states for realization of SIC-
POVM(cid:12)(cid:12)ψ1,2(cid:11)arepreparedbyaHWP,whereas(cid:12)(cid:12)ψ3,4(cid:11)arepre- in x = 3, we do not know the initial coin state; that is,
pared by a QWP with certain setting angles. Site-dependent x = 3 corresponds to an inconclusive result with prob-
coinflippingisrealizedbyHWPs(andQWPs)withdifferent ability η = |(cid:104)φ |φ (cid:105)| = cosφ. The probability of
err + −
setting angles placed in different optical modes. (c) Detailed the inclusive result depends on the similarity of the two
interferometric setup formed by the third and fourth BDs,
states,whichagreeswiththeIvanovic-Dieks-Peresbound
whichareusedintheexperimentalrealizationofaqubitSIC-
obtained from the optimum strategy of this kind for un-
POVM.
ambiguous state discrimination [12–14]. For the extreme
case of φ = π/2, the measurement is projective and the
probability η is 0.
err
to the outcome of the coin flipping for each step T = NowweestablishtheuniversalityoftheQWprocedure
(cid:80) |x+1(cid:105)(cid:104)x|⊗|0(cid:105)(cid:104)0|+|x−1(cid:105)(cid:104)x|⊗|1(cid:105)(cid:104)1|. Theunitary for generation of an arbitrary rank-1 POVM via an ex-
x
operation for the nth step is U =T (cid:80) |x(cid:105)(cid:104)x|⊗C . ample of a properly engineered six-step QW generating
n x x,n
a single-qubit SIC-POVM. For example we choose
We commence with the simplest non-trivial case,
namambieglyuoausthstraeete-sdteispcrQimWinaaptipornooafchtwtoosiimngpllee-mquebnittastnatuens-. (cid:12)(cid:12)ξ1(cid:11)=|0(cid:105),(cid:12)(cid:12)ξ2(cid:11)= √1 (cid:16)|0(cid:105)+√2|1(cid:105)(cid:17), (5)
3
Twononorthogonalpurestatescanalwaysbeencodedas
(cid:12)(cid:12)ξ3(cid:11)= √1 (cid:16)|0(cid:105)+λ√2|1(cid:105)(cid:17),(cid:12)(cid:12)ξ4(cid:11)= √1 (cid:16)|0(cid:105)+λ∗√2|1(cid:105)(cid:17)
3 3
(cid:12)(cid:12)φ±(cid:11)=cosφ2 |0(cid:105)±sinφ2 |1(cid:105). (2) satisfying |(cid:104)ξi|ξj(cid:105)|=3−1/2 for i(cid:54)=j and 1(cid:80)4(cid:12)(cid:12)ξi(cid:11)(cid:10)ξi(cid:12)(cid:12)=
2 i
1forλ=ei2π/3. Nowweconstructfourstatesorthogonal
totheabovestates(5)andpreparethecoinstateinoneof
Our objective is to discriminate these two states with the four states initially. Thus, the QW procedure starts
equal prior probability for three outcomes: conclusively with four initial coin states
measuring one or the other state (2) or obtaining an in-
conclusive measurement result. We prepare an initial (cid:12)(cid:12)ψ1(cid:11)=|1(cid:105),(cid:12)(cid:12)ψ2(cid:11)= √1 (cid:16)√2|0(cid:105)−|1(cid:105)(cid:17),
coin state in either of the two states (2). For a properly 3
einnigtiinaleecroeidnQstWatepsroacreridvuerse,atthdeiffwearleknetr pwoitshititohneddiisffterribenut- (cid:12)(cid:12)ψ3(cid:11)= √1 (cid:16)√2|0(cid:105)−λ|1(cid:105)(cid:17), (6)
3
tions. By projective measurement onto the walker’s po-
sition, initial coin states can be discriminated. (cid:12)(cid:12)ψ4(cid:11)= √1 (cid:16)√2|0(cid:105)−λ∗|1(cid:105)(cid:17).
3
Forourrealization,thecoinisinitiallypreparedin|φ±(cid:105)
and the walker starts from the origin |x=0(cid:105). The site- The site-dependent coin rotations for the first six steps
3
are
(cid:18) (cid:19) (cid:18) (cid:19)
1 1 −1 1 −1 1
C = √ ,C = √ ,
1,2 2 −1 −1 0,3 2 1 1
C =C =C =σ , (7)
−1,2 −1,4 −1,6 x
√
C = √1 (cid:18) 2 √1 (cid:19),C = √1 (cid:18)e−iπ3 eiπ6 (cid:19),
1,4 3 1 − 2 0,5 2 eiπ3 e−iπ6
and 1 elsewhere. The coinoperatorschosenhere depend
only on the states we aim to discriminate. Following the
six-stepQWprocedureincludingspecificsite-dependence
coinrotations,theinitialstatesofthewalker-coinsystem
(cid:12)(cid:12)ϕi(cid:11)=|0(cid:105)(cid:12)(cid:12)ψi(cid:11) (i=1,2,3,4) evolve to
0
(cid:12)(cid:12)ϕ1(cid:11)= √1 (−|4(cid:105)−i|2(cid:105)+i|0(cid:105))|0(cid:105),
6
3
FIG. 2: (Color online.) Experimental data for unambigu-
(cid:12)(cid:12)ϕ26(cid:11)= √1 (cid:0)|6(cid:105)−e−iπ3 |2(cid:105)−eiπ3 |0(cid:105)(cid:1)|0(cid:105), ous state discrimination via a photonic QW. Measured posi-
3 tion distributions for the three-step QW with site-dependent
(cid:12)(cid:12)ϕ36(cid:11)= √13(cid:0)|6(cid:105)−e−iπ6 |4(cid:105)−|2(cid:105)(cid:1)|0(cid:105), ccooienfficainednts(aφ) oinfit(cid:12)(cid:12)iφa±l(cid:11)cofoinr ustnaatmeb(cid:12)(cid:12)iφg+u(cid:11)ouasnsdta(tbe)d(cid:12)(cid:12)iφsc−ri(cid:11)m; ivnaartiioouns.
(c) Measured probability η for inconclusive results vs pa-
(cid:12)(cid:12)ϕ46(cid:11)= √1 (cid:0)|6(cid:105)−eiπ6 |4(cid:105)−|0(cid:105)(cid:1)|0(cid:105). (8) rametersφ,whicharerelateedrrtothestatetobediscriminated;
3 compared to theoretical predictions. Error bars are smaller
than portrayed by the symbols. (d) Position distribution for
Evidently the final walker-coin states have differing sup-
the three-step QW with initial coin state |H(cid:105), which is an
port over position x so by measuring the position of the equally-weighted superposition of (cid:12)(cid:12)φ±(cid:11) with φ = 45◦. The
walker, the initial coin state can be determined with blueandredbarsshowtheexperimentaldataandtheoretical
(cid:12) (cid:69)
some degree of certainty. Specifically, (cid:12)ϕ1,2,3,4 cannot predictions, respectively. Error bars indicate the statistical
(cid:12) 6 uncertainty.
be found at x=6, x=4, x=0 and x=2, respectively.
Thus, we realize all elements (cid:12)(cid:12)ξi(cid:11)(cid:10)ξi(cid:12)(cid:12)/2 (i=1,2,3,4) of
a qubit SIC-POVM through a QW procedure.
move up a 2.7mm lateral displacement into a neighbor-
Therealizationofunambiguousstatediscriminationof
ing mode, which interfere with the vertical photons in
two equally probable single-qubit states via a three-step
the same mode. Certain pairs of BDs form an interfer-
QW is shown in Fig. 1(a). The coin qubit is encoded
ometer, which are placed in sequence and need to have
in the horizontal |H(cid:105) = |0(cid:105) and vertical polarization
their optical axes mutually aligned.
|V(cid:105) = |1(cid:105) of photons. The walker’s positions are rep-
We attain interference visibility of 0.992 for each step.
resentedbylongitudinalspatialmodes. Thepolarization
Output photons are detected using avalanche photo-
degenerate photon pairs are generated via type-I spon-
diodes (APDs, 7ns time window) with dark-count rate
taneous parametric down-conversion (SPDC) in 0.5mm-
of less than 100s−1 whose coincidence signals, monitored
thicknonlinear-β-barium-borate(BBO)crystal,whichis
using commercially available counting logic, are used to
pumped by a CW diode laser with 90mW of power. For
post-selecttwosingle-photonevents. Thewalkerposition
1D QWs, triggering on one photon prepares the other
probabilitiesareobtainedbynormalizingthecoincidence
beam at wavelength 801.6nm into a single-photon state.
counts on each mode with respect to the total count for
Theinitialcoinstatecanbepreparedbythehalf-wave
each respective step.
plate (HWP) or quarter-wave plate (QWP) right after
thepolarizingbeamsplitter(PBS)showninFig.1. After Forsite-dependentcoinflipping,theopticaldelayusu-
passingthroughthePBSandwaveplate(WP)thedown- ally needs to be considered. Fortunately, in our experi-
converted photons are steered into the optical modes of mentonunambiguousstatediscrimination,onlythefirst
the linear-optical network formed by a series of birefrin- and second BDs form an interferometer. For the third
gent calcite beam displacers (BDs) and WPs. The site- step of the QW, the photons in mode x = 2, which are
dependent coin rotations C for the nth step can be all in |H(cid:105), move up to mode x = 3 after the last BD
x,n
realized by HWP and QWP with specific setting angles and thus do not interfere with other photons. Thus, no
placed in mode x. optical compensate is needed and the difficulty of the
TheconditionalpositionshiftisimplementedbyaBD realization of the experiment is decreased.
withlength28mmandclearaperture10mm×10mm. The The measured probability distributions of a three-step
opticalaxisofeachBDiscutsothatverticallypolarized QW for unambiguous state discrimination are shown in
photons are directly transmitted and horizontal photons Fig. 2. We choose a different coefficient φ and prepare
4
(a ) 0.4 |y 1> E xperim ent (b ) 0.4 |y 2> θH =17.63◦andθH =45◦respectivelytorealizethesite-
T heory dependentcoinrotationsC ,andthephotonsinthose
±1,4
bability00..23 bility00..23 tcwauosme oodfetshientsemrfaelrlesienpamraotdieonxs=be0twateetnhethfoeurntehigBhDbo.rBineg-
Pro0.1 roba0.1 modes, it is difficult to inset a HWP in the middle mode
P x = 1 and avoid the photons in the neighboring modes
0.0 0.0 passing through it.
0 2 4 6 0 2 4 6
x x In our experiment, we place a HWP with θ =17.63◦
H
(c ) 0.4 |y 3> (d ) 0.4 |y 4> in both modes x=1 and x=3 followed by a HWP with
the same angle in mode x = 3 and a HWP with 45◦ in
0.3 0.3
ility0.2 ility0.2 mode x = −1. Thus, the photons in modes x = ±1 do
b b not suffer an optical delay and interfere with each other
a a
b b
ro0.1 ro0.1 with a high visibility. The polarizations of photons in
P P
mode x = 3 are not changed after two HWPs with the
0.0 0.0
0 2 4 6 0 2 4 6 same angle. The photons in mode x=3 do not interfere
x x
with those in the other modes, though there is optical
FIG. 3: (Color online.) Experimental data of a qubit SIC- delay between them. Hence optical compensation is not
POVM via a photonic QW. Measured probability distribu- required.
tions of the six-step QW with the site-dependent coin ro-
tations and four different initial coin states (cid:12)(cid:12)ψi(cid:11) with i = Themeasuredprobabilitydistributionsofsix-stepQW
for a qubit SIC-POVM are shown in Fig. 3, which agree
1,2,3,4 in (a)-(d) respectively.
well with the theoretical predictions. Using the experi-
mental distribution of the QW with the initial coin state
theinitialcoinstatetothecorrespondingstate|φ±(cid:105). For (cid:12)(cid:12)ψ1(cid:11)asanexample,aftersixstepstheprobabilityP(6)is
measured as 0.0149±0.0007 and thus is very small com-
eitherofthetwostates, thephotonsundergoingtheQW
paredtotheprobabilitiesofthephotonsbeingmeasured
networkaremeasuredatthemodex=3forinclusivere-
in the other modes, i.e., P(0)=0.3246±0.0037,P(2)=
sult and x=±1 for conclusive results. Two pronounced
0.3277±0.0038,P(4) = 0.3327±0.0038, which ensures
peaks for each φ shown in the probability distribution in
thatoneoftheelementsofaqubitSIC-POVMisrealized
Figs. 2(a,b) validate the demonstration of unambiguous
successfully. The small distance d<0.043 demonstrates
state discrimination. With φ increasing from 45◦ to 90◦
strong agreement between theoretical and measured dis-
theprobabilityofinconclusiveresultsη ofthediscrimi-
err tributionaftersixsteps. Dominantsourcesofexperimen-
nationofthestate|φ+(cid:105)decreasesfrom0.7139±0.0030to
talerrorsareinaccuracyofanglescontrolledbytheWPs
0.0070±0.0060(from0.7125±0.0031to0.0080±0.0071to
and imperfect non-unit visibility.
discriminate the state |φ−(cid:105)) shown in Fig. 2(c), agreeing
InsummaryweexperimentallyshowthatQWsareca-
with the Ivanovic-Dieks-Peres bound.
pableofperforminggeneralizedmeasurementsonasingle
Taking a superposition of |φ±(cid:105) as an initial coin state
qubit. OurdemonstrationemploysanovelphotonicQW
a|φ+(cid:105)+b|φ−(cid:105) (non-normalized), with a,b ∈ R, the ra-
with site-dependent coin rotation for realizing a general-
tio of the probabilities for the two conclusive results is
ized measurement [3]. The key experimental advance to
a2/b2, which is also demonstrated in our experiment.
realize a QW-based generlalized measurement is the ap-
In Fig. 2(d) we show with a = b the probabilities of
plication of site-dependent coin rotations to control the
x = 1 and x = −1 are measured approximately equal,
coin’s internaldynamics andtherebyeffect theevolution
i.e.,P(1)=0.0854±0.0015andP(−1)=0.0850±0.0015.
of the walker. We have thus demonstrated a new and
We characterize the quality of the experimental QW
versatileapproachtogeneralizedqubitmeasurementsvia
by its 1-norm distance. The unambiguous state discrim-
photonic quantum walks.
ination is confirmed by direct measurement and found
We would like to thank P. Kurzyn´ski, C. F. Li and
to be consistent with the ideal theoretical values at the
Y. S. Zhang for stimulating discussions. This work has
level of the small average distance d < 0.02 and the fi-
been supported by NSFC under grants 11174052 and
delityofthecoinstatemeasuredintheposition(x=±1)
11474049, the Open Fund from SKLPS of ECNU, the
F >0.9911.
CAST Innovation fund, NSERC, AITF, and the China
The realization of a qubit SIC-POVM via a properly
1000 Talents program.
engineered six-step QW is shown in Fig. 1(b). For site-
dependent coin flipping, the challenge is placing the WP
into a given optical mode without influencing the pho-
tons in the other modes. For example, in Fig. 1(c) for
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1015 (2012), URL http://dx.doi.org/10.1007/ where |H(cid:105)=(1,0)T and |V(cid:105)=(0,1)T represent the hor-
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vancesInAtomic,Molecular,andOpticalPhysics,edited
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vol. 52 of Advances In Atomic, Molecular, and Optical
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TABLE I: The coefficients of the states to be discriminated
10.1103/PhysRevLett.104.153602.
(column1),theinitialcoinstates(column2),thecorrespond-
ing parameters for the HWP settings (columns 3-5), the ex-
perimentaldatafortheprobabilityofinconclusiveresults(col-
umn 6) and 1-norm distance from the theoretical predictions
SUPPLEMENTARY MATERIAL (column 7). Error bars indicate the statistical uncertainty.
METHODS AND EXPERIMENTAL ANALYSIS
FOR REALIZATION UNAMBIGUOUS STATE
DISCRIMINATION
METHODS AND EXPERIMENTAL ANALYSIS
FOR REALIZATION OF SIC-POVM
For a photonic quantum-walk approach to realize an
unambiguous state discrimination of two equally proba-
ble single-qubit states, the initial coin states are This single qubit SIC-POVM can be realized by using
a six-step photonic quantum walk. The initial coin state
(cid:12)(cid:12)φ±(cid:11)=cosφ|H(cid:105)±sinφ|V(cid:105), (9) criagnhtbeafptererptahreedPbByS.aFHoWr ePxaomrqpulea,ttrhe-ewianviteiapllactoein(QstWatPes)
2 2
6
FIG. 4: (Color online.) Real and imaginary parts of the density matrices ρi measured for four different initial coin states (cid:12)(cid:12)ψi(cid:11)
with i=1,2,3,4.
(cid:12)(cid:12)ψ1(cid:11)and(cid:12)(cid:12)ψ2(cid:11)canbepreparedbyaHWPwithanglesset 1,2,3,4) with the measured probability distribution the
to θH =45o and −17.63o, and the other two initial coin initial coin state can be reconstructed. The real and
states (cid:12)(cid:12)ψ3(cid:11) and (cid:12)(cid:12)ψ4(cid:11) can be prepared by setting angle of P(0) P(2) P(4) P(6) d
QWP to θQ = −152.63o and 117.37o, respectively. The (cid:12)(cid:12)ψ1(cid:11) 0.3246(37) 0.3277(38) 0.3327(38) 0.0149(07) 0.0149(33)
single-qubit rotation realized by QWP is (cid:12)(cid:12)ψ2(cid:11) 0.3398(38) 0.3135(36) 0.0345(11) 0.3123(36) 0.0401(32)
(cid:12)(cid:12)ψ3(cid:11) 0.0335(10) 0.3137(36) 0.3432(38) 0.3104(36) 0.0425(32)
R (θQ)=(cid:18)cos2θQ+isin2θQ (1−i)sinθQcosθQ(cid:19), (cid:12)(cid:12)ψ4(cid:11) 0.3158(36) 0.0329(10) 0.3419(38) 0.3094(35) 0.0415(32)
QWP (1−i)sinθQcosθQ sin2θQ+icos2θQ
TABLE III: The measured probability distribution of the
(12)
walker P(x) and 1-norm distance d from the theoretical pre-
where θQ is the angle between the optic axes of QWP
dictionsareshowncorrespondingtofourdifferentinitialcoin
and horizontal direction.
states. Error bars indicate the statistical uncertainty.
The site-dependent coin rotations can be realized by
HWP and QWP with specific angles show in Table II
placed in certain modes. imaginary parts of the density matrices ρ =ρRe+iρIm
corresponding to the states (cid:12)(cid:12)ψi(cid:11) are showin as i i
θH θQ θH θQ θH θQ
0 0 1 1 −1 −1
(cid:18) (cid:19) (cid:18) (cid:19)
step 1 − − − − − − 0.0299 0.0093 0 0.0038
ρRe = ,ρIm =
step 2 − − −22.5o − 45o − 1 0.0093 0.9701 1 −0.0038 0
step 3 67.5o − − − − − (cid:18) (cid:19) (cid:18) (cid:19)
0.6247 −0.4132 0 −0.0322
step 4 − − 17.63o − 45o − ρRe = ,ρIm =
step 5 52.5o 45o − − − − 2 −0.4132 0.3753 2 0.0322 0
step 6 − − − − 45o − (cid:18)0.6209 0.2386(cid:19) (cid:18) 0 0.3432(cid:19)
ρRe = ,ρIm =
3 0.2386 0.3791 3 −0.3432 0
TABLE II: The setting parameters of the HWPs and QWPs
(cid:18) (cid:19) (cid:18) (cid:19)
whichareusedtorealizesite-dependentcoinrotationsforthe ρRe = 0.6188 0.2370 ,ρIm = 0 −0.3465
six-step quantum walk. The subscripts denote the optical 4 0.2370 0.3812 4 0.3465 0
modes where the wave plates are placed and “−” means no (13)
corresponding wave plate is used.
Figure 4 shows the histograms of the real and imaginary
The measured probability distribution of the walker parts of the density matrices obtained. The fidelities of
P(x) and 1-norm distance d from the theoretical pre- the measured density matrices with respect to the the-
dictions are shown corresponding to four different initial oretically predicted states (i.e., initial coin states) are
coin states are shown in Table III. 0.9701±0.0108, 0.9311±0.0131, 0.9330±0.0086, and
Moreover given a SIC-POVM {(cid:12)(cid:12)ξi(cid:11)(cid:10)ξi(cid:12)(cid:12)/2} (i = 0.9342±0.0102, respectively.