Experimental realization of a programmable quantum-state discriminator and a phase-covariant quantum multimeter

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by  Jan Soubusta

Experimental realization of a programmable quantum-state discriminator and a
phase-covariant quantum multimeter
Jan Soubusta,1,2 Anton´ın Cˇernoch,2 Jarom´ır Fiura´ˇsek,3,2 and Miloslav Duˇsek2
1Joint Laboratory of Optics of Palack´y University and Institute of Physics of Academy of Sciences of the Czech Republic,
17. listopadu 50A, 77200 Olomouc, Czech Republic
2Department of Optics, Palack´y University, 17. listopadu 50, 77200 Olomouc, Czech Republic
3QUIC, Ecole Polytechnique, CP 165, Universit´e Libre de Bruxelles, 1050 Bruxelles, Belgium
(Dated: February 1, 2008)
Wepresentanopticalimplementationoftwoprogrammablequantummeasurementdevices. The
firstoneservesforunambiguousdiscrimination oftwo nonorthogonalstates ofaqubit. Thepartic-
ular pair of states to be discriminated is specified by the quantum state of a program qubit. The
second device can perform von Neumann measurements on a single qubit in any basis located on
4
the equator of the Bloch sphere. Again, the basis is selected by the state of a program qubit. In
0
both cases thedata and program qubitsare represented bypolarization states of photons. The ex-
0
perimental apparatusexploits thefact that twoBell states can bedistinguished solely by means of
2
linearoptics. TheoutcomecorrespondingtotheremainingtwoBellstatesrepresentsaninconclusive

n result.
a
J PACSnumbers: 03.65.-w,03.67.-a,42.50.Dv

7
2
I. INTRODUCTION II. PROGRAMMABLE QUANTUM-STATE

1 DISCRIMINATOR
v
6
A. Theory
6
1
1 A general unknown quantum state cannot be deter-
Quantum measurements are inevitable parts of all
0 minedcompletelybyameasurementperformedonasin-
4 quantum devices [1, 2, 3, 4]. In many situations, the
gle copy of the system. But the situation is different if a
0 choice of a particular measurement depends on the task
priori knowledge is available [1, 2, 3] – e.g., if one works
/ to be performed. For instance, in the case of quantum-
h only with states from a certain discrete set. Even quan-
state discrimination the choice of the generalized mea-
p tum states that are mutually non-orthogonalcan be dis-
- surement is given by the specific pair of states that are
t supposedtobediscriminated. Recentlyuniversal(multi- tinguished with a certain probability provided they are
n linearly independent (for a review see Ref. [15]). There
a purpose)quantummeasurementdevices,“quantummul-
are, in fact, two different optimal strategies [16]: First,
u timeters”, were introduced and discussed in several pa-
q pers [5, 6, 7, 8, 9]. Their key property is the possibility thestrategythatdeterminesthestatewiththeminimum
: probability for the error [1, 2] and, second, unambigu-
v to control the choice of the measurement by a quantum
ous or error-free discrimination (the measurement result
i state of the program register, which could be in princi-
X neverwronglyidentifies a state)that allowsthe possibil-
ple unknown. The quantum states of a program register
r corresponding to different measurements are allowed to ity of an inconclusive result (with a minimal probability
a in the optimal case) [17, 18, 19, 20, 21]. We will concen-
be mutually non-orthogonal.
trateourattentiontothe unambiguousstatediscrimina-
tion. IthasbeenfirstinvestigatedbyIvanovic[17]forthe
caseoftwoequallyprobablenon-orthogonalstates. Peres
[19]solvedthe problemofdiscriminationoftwostatesin
Inthispaper,wereportontheexperimentalrealization a formulation with POVM measurement. Later Jaeger
ofaprogrammablequantum-statediscriminator(Section and Shimony [20] extended the solution to arbitrary a
II) anda phase-covariantmultimeter (Section III). Both prioriprobabilities. Chefles andBarnett[21]havegener-
programmable detectors involve two qubits, one data alized Peres’ssolution to an arbitrarynumber of equally
qubit and one program qubit. In our optical implemen- probablestateswhicharerelatedbyasymmetrytransfor-
tation, the qubits are encoded into polarizationstates of mation. Unambiguous state discrimination was already
single photons and the required photon pairs are gener- realized experimentally. The first experiment, designed
ated by means of spontaneous parametric downconver- for the discrimination of two linearly polarized states of
sion. TheexperimentexploitsthefactthatapartialBell light, were done by Huttner et al. [22]. The interest in
measurement on polarization states of two photons can thequantumstatediscriminationisnotonly“academic”
be accomplished with linear optics, namely a balanced – unambiguous state discrimination can be used, e.g., as
beamsplitter and two polarizing beamsplitters, followed an efficient attack in quantum cryptography [23].
by photodetectors [10, 11, 12, 13, 14]. Let us now suppose that we want to discriminate un-
2
|V> |+45o> |V> The probability of successful discrimination would be
| +> | +>
d d p=2ab2 =2(a2 a4). (6)
|H> |H> | | | | −| |
| d-> | d-> This probability of success is not optimal in general.
|-45o> As shown in Refs. [17, 18, 19] the optimal probabil-
ity of successful discrimination p = 1 φ+ φ− =
opt − h d| di
FIG.1: Polarization ofthe“data”photon: leftpanel–linear 1 2a2 1 . In fact, the probability (6)(cid:12)correspo(cid:12)nds
polarizations, right panel – elliptical polarizations. to−a(cid:12)“q|u|as−i-cla(cid:12)ssical”discrimination. Byaqu(cid:12)asi-class(cid:12)ical
(cid:12) (cid:12)
approachwemeanaprobabilisticmeasurementwhenone
randomly selects [25] the projective measurement in one
ambiguously between two non-orthogonal states. How-
of two bases that both span the two-dimensional space
ever, we would like to have a possibility to “switch” the
containing both non-orthogonal states of interest (1).
apparatus in order to be able to work with several dif-
One basis consists of the state φ+ and its orthogonal
ferent pairs of states. The switching should be realized | di
complement φ+ . If one finds the result corresponding
by preparing a program register of the programmable | ⊥di
to φ+ he/she can be sure that the state φ+ was not
discriminator in different program states. The program | ⊥di | di
present. Analogously,theotherbasisconsistsofthestate
state thus specifies which pair of quantum states is dis-
φ− and its orthogonal complement.
criminatedby the device. This problemwasinvestigated | di
To get a higher probability of success one would have
theoretically in Ref. [5].
tomakeamoresophisticatedmeasurementthanasimple
Let us have two (non-orthogonal) input states of a
Bell-stateanalysis[5]. However,therearetwoimportant
qubit that should be discriminated:
pointsconcerningtheproposeddiscriminationprocedure:
|φ±di=a|Hdi±b|Vdi, (1) First, a partial measurement in the Bell basis can be
easily realized by linear optics. Namely, states Ψ+ and
where |Hdi and |Vdi represent two “logical” levels of a Ψ− can be distinguished (the rest of the B|ell sitates
qubit, in particular horizontal and vertical linear polar- | i
corresponds to an inconclusive result anyway). Second,
izations of a photon; subscript d stays for “data”. Note
ouraimisnottomaximizethesuccessprobabilitybutto
that these states are supposed to be symmetrically lo-
demonstrateexperimentallythepossibilitytocontrolthe
cated around the state H , as depicted in Fig. 1. In
| di discriminationprocessbythequantumstate ofaprogram
addition let us have a program qubit in a state (index p
qubit. Even to set the bases for the mentioned quasi-
denotes “program”):
classical discrimination correctly, one needs an infinite
φ =a′ H +b′ V . (2) number of bits of classical information. Our procedure
p p p
| i | i | i requires only a single qubit for the same job. This fact
Choosingproperlythe state ofa programqubit andper- reveals the quantum nature of the programming. Even
forming a suitable joint measurement on the data and non-orthogonalstatesofthequantumqubitcarry“useful
programqubitstogetheronecanunambiguouslydiscrim- information” for the discrimination process.
inate any two states of the data qubit that are in agree-
ment with the program.
Let us suppose that the state of the program qubit B. Experiment
is equal to one of the two states that shall be error-free
discriminated. In particular, let
The scheme of our experimental setup is shown in
φ =a H +b V . (3) Fig. 2. A krypton-ion cw laser (413.1 nm, 95 mW) is
p p p
| i | i | i used to pump a 10-mm-long LiIO nonlinear crystal cut
3
Thenthetotalstateofthedataandprogramqubitreads for degenerate type-I parametric downconversion. We
exploit the fact that the pairs of photons generated by
φ± φ =
| di⊗| pi spontaneous parametric downconversion (SPDC) mani-
a2 b2 a2 b2 fest tight time correlations (i.e., very exact coincidences
√2 ± Φ+ + ∓ Φ− +ab Ψ± , (4)
(cid:20) 2 | i 2 | i | i(cid:21) ofdetectioninstants). Inoursetupthephotonsproduced
by SPDC have horizontal linear polarizations. Different
where
polarization states are prepared by means of half-wave
1 andquarter-waveplates(HWP,QWP).Thepolarization
Ψ± = (H V V H ),
| i √2 | di| pi±| di| pi of the signalphoton representsthe data qubit whilst the
1 polarization of the idler photon serves as the program
Φ± = (Hd Hp Vd Vp ) (5) qubit. The two photons impinge on two input ports of a
| i √2 | i| i±| i| i
50/50 beamsplitter (BS, non-polarizing cube beamsplit-
aretheBellstates. Clearly,ifwewereabletodetectBell ter). A scanning mirror is used in one interferometer
states Ψ+ and Ψ− we could unambiguously discrimi- armin orderto balancethe lengthofboth arms,as indi-
nate st|atesi φ+ |andiφ− . cated by an arrow in Fig. 2. The photons reflected and
| di | di
3
C
C
C
C
FIG. 2: Experimental setup (BS is a beamsplitter, PBS po-
larization beamsplitters, D denotes detectors, and HWP and
QWPhalf- and quarter-waveplates, respectively).
FIG. 3: Detection rates of |Ψ+i (dots, solid line) and |Ψ−i
(diamonds, broken line) for linearly polarized input states.
transmitted by BS pass polarizing beamsplitters (PBS, The upper part corresponds to the situation when the data
polarizing cube beamsplitter) to distinguish horizontal and program states coincide, |φ+di⊗|φpi, the lower part to
andverticalpolarizations. Finally,thebeamsarefiltered thesituation when they are different,|φ−di⊗|φpi.
bycut-offfiltersandcircularapertures,andcoupledinto
multi-mode opticalfibers by lenses. Detectors D ,...,D
1 4
are Perkin-Elmer single-photon counting modules (em- phase of the horizontal components is ϕ+180◦ just for
ploying silicon avalanche photodiodes with quantum ef- geometrical reasons. It means, if one prepares both pho-
ficiency η 50% and dark counts about 100s−1). The tons in the same linear polarizations tilted by 45◦ with
≈
signals from detectors are processed by our home-made respect to the plane of incidence the vectors of electric
4-input coincidence module. With this setup, visibilities field at the interface plane will oscillate in mutually per-
ofHong-Ou-Mandeldip[24]exceeding92%werereached pendicular directions. Consequently,nowitis the triplet
for vertically polarized photons (for other polarizations state Ψ+ that leads to a coincident detection at differ-
| i
visibilities were slightly lower). Higher visibilities could ent outputs.
notbe reacheddue tothe factthatthesplitting ratiosof So,ifdetectorsD1 andD3 orD2 andD4 clicktogether
thecouplerwerenotexactly50/50andduetotheimper- in our setup the state Ψ+ is detected (at the input of
| i
fectionsofthewaveplatesandpolarizationdistortionson the beamsplitter) — this corresponds to the recognition
miArrsorasl.ready mentioned our experiment is based on the oinf sctoainteci|dφe+dnic.eItfhdeestteactteorsΨD−1 awnads “Dp2reosreDnt3”ainndthDe4incpliuckt
pHoilsbsiebritlistpyatcoedoeftetchtetphoelatwriozaptiaornticsutalatresBoeflltswtoatpeshointotnhse. —detseoct,o|rφ−deiithiserdeΦte+cteodr. |ΦIf−biowthasp“hportoesnesnte”ntienrtthheeisnapmuet
| i | i
This task can be done by means of passive linear optical — this represents the inconclusive result of the discrimi-
elements,namelybyabeamsplitterandtwopolarization nation.
beamsplitters as shown in Fig. 2 [10, 11, 12, 13, 14]. Each data point at presented plots has been derived
from 10 one-second measurement periods. The accuracy
The simplest theoretical model of the beamsplitter
of polarization-angle settings was better than 1◦.
leads to the conclusion that if one fetches Bell states at ±
the inputs the only one of them that results in a coin-
cident detection at two different outputs of the beam-
splitter is the singletstate Ψ− . However,in case of the C. Results
| i
“real” beam-splitting cube one must take into account
that the two photons strike upon a beamsplitter in op- We tried to discriminate a broad variety of pairs of
posite directions. So, if the mutual phase of the vertical elliptically polarized states
components of the electric-field vectors at the interface
plane (we mean the vectors corresponding to the fields φ± =(xcosϑ+iysinϑ) H (xsinϑ iycosϑ) V ,
| di | di± − | di
coming from the two opposite inputs) is ϕ the mutual (7)
4
φ+ φ ,C−+ denotesthedetectionrateof Ψ− when
C t|hdeii⊗np|utpiwas in the state φ+ φ , C+− d|enoties the
C detection rate of Ψ+ whe|ndtih⊗e i|nppuit was in the state
φ− φ ,andC|−−idenotesthedetectionratesof Ψ−
| di⊗| pi | i
when the input was in the state φ− φ .
| di⊗| pi
The relative(with respectto allconclusive results)er-
ror rates – i.e., fractions of events when we get Ψ+
instead of Ψ− or viceversa – are about 5%. This v|aluei
| i
isapproximatelythesameforallmeasuredpolarizations.
Ideally the error rate should be zero but due to techni-
C cal imperfections of the setup it gets this small nonzero
C value.
The main parameter – the probability of success – is
plotted in Fig. 5 for four different ellipticities. Symbols
denoteexperimentaldata,linesrepresenttheoreticalpre-
dictions (8). The first curve corresponds to linearly po-
larizedphotons(actingasdataandprogramqubits), the
others,insequence,toellipticallypolarizedphotonswith
ε = 12◦,24◦, and 36◦. As can be seen the data are in a
good agreement with the theory.
The probability of success is calculated as an average
FIG. 4: Detection rates of |Ψ+i (dots, solid line) and |Ψ−i of normalized detection rates corresponding to success-
(diamonds,brokenline)forellipticallypolarizedinputstates, ful events (i.e., correct conclusive measurement results)
ε = 24◦. The upper part corresponds to the situation when for two possible input states (that are in correspondence
the data and program states coincide, |φ+di⊗|φpi, the lower with the program state). The averaging takes into ac-
part tothe situation when theyare different,|φ−di⊗|φpi. count that both the inputs |φ+di⊗|φpi and |φ−di⊗|φpi
appear with the same probability 1/2. The “normaliza-
tion”meansthatthedetectionrateisdividedbythepair-
where x and y are the half-axes of the polarization el- generation rate, i.e., by the number of all measurement
lipse and ϑ is the angle between horizontal axis and events per time unit. This quantity is obtained from the
the direction of x-half-axis (see Fig. 1). The elliptic- coincidence measurement on the shoulder 150µm out of
ity tan(ε) = y/x. We started with linear polarizations thedip(seeFig.6). Inthis situationthe twoone-photon
(ε = 0) and then proceeded with elliptical polarizations packets generated by downconversion no longer overlap
with three different ellipticities (namely for ε=12◦,24◦, atthebeamsplitterandeachofthemrandomly“decides”
and36◦). Theangleϑwas,inallthecases,scannedfrom whether to go through or to be reflected.
0◦to90◦withthestepof4◦. Ellipticalpolarizations(cor-
As the splitting ratio of the used beamsplitter is
respondingtostates φ± )wereproducedfromhorizontal
| di slightly different for vertical and horizontal linear polar-
ones by means of a quarter-wave plate rotated by angle
izationsthenormalizationmeasurementsweredonewith
α = ±ε with respect to the horizontal axis. Then the 45◦ linear polarization states at the inputs. For the set
directionofthex-half-axisofthepolarizationellipsewas
of mainmeasurements (i.e., Bell measurementswith bal-
tilted by a half-wave plate rotated by β = (ε+ϑ)/2.
± anced armsofHong-Ou-Mandelinterferometer)withthe
Letusrecallthatthetheoreticalprobabilityofsuccess- states φ+ φ the input polarization of the normal-
ful discrimination of states (7) is ization|mdeia⊗sur|empient were 45◦, 45◦, for the set of main
p=2(|a|2−|a|4), where |a|2 =x2cos2ϑ+y2sin2ϑ. (8) m45e◦a.surementswiththestates|φ−di⊗|φpitheywere−45◦,
Figure 3 shows the coincidence rates measured for Because our setup and detection electronics are tai-
linear polarization states. The upper part shows the lored to the measurement of the Bell states Ψ+ and
measurement of Bell states Ψ+ and Ψ− when the Ψ− we can measure only the events when det|ectoirs D
1
| i | i | i
data and programstates coincide (i.e., the input state is andD orD andD clicktogetherorwhendetectorsD
3 2 4 1
φ+ φ ),thelowerpartshowsthesamemeasurement and D or D and D click in coincidence. For the given
|indcia⊗se| wphien they are different (i.e., the input state is 45◦ po2larizat3ions bo4th these two rates outside the dip
φ− φ ). Statisticalerrorsare smallerthan the sym- shouldbe equalto 1/4 of the pair-generationrate. How-
| di⊗| pi
bols of points. Graphs in Fig. 4 illustrate the same mea- ever, there could be deviations from this 1/4 due to the
surement but with pairs of elliptically polarized states non-ideal splitting ratio of the beamsplitter. Neverthe-
(ε=24◦, i.e., tan(ε)=0.45). less, the sum of these two quantities is immune against
Throughout the whole paper, C++ denotes the de- this imperfection and proportional to 1/2 of the pair-
tection rate of Ψ+ when the input was in the state generation rate.
| i
5
C
C
C
C
ucc
Ps
FIG. 5: Probability of successful operation as a function of FIG. 6: Detection rates as functions of the mirror position.
angle ϑ. Symbols denote experimental data, lines represent Here C++ is a coincidence rate at D1 and D3 or D2 and
theoreticalpredictions(8). Thefirstcurvecorrespondstolin- D4 when the input polarizations were 45◦,45◦. The quan-
earlypolarizedphotons,theothers,insequence,toelliptically tityC+− isacoincidencerateatthesamedetectorsforinput
polarized photonswith ε=12◦,24◦, and 36◦. polarizations −45◦,45◦. Analogously, C−+ and C−− are co-
incidence rates at D1 and D2 or D3 and D4.
Thus the probabilityofsuccess is calculatedasfollows
space[5,8]. Nevertheless,itispossibletoencodePOVMs
1 C++ C−− thatrepresent,inacertainsense,thebestapproximation
Psucc = 2(cid:20)2(C+++C−+) + 2(C−−+C+−)(cid:21), (9) of the required projective measurements. A specific way
sh sh sh sh ofapproximationofaprojectivemeasurementisa“prob-
abilistic” measurementthat allows for some inconclusive
where C are the above mentioned detection rates mea-
sh
sured outside the dip; in particular, C++ denotes the results. In this case, instead of a two-component pro-
sh
coincidence rate of D and D or D and D with 45◦, jective measurement one has a three-component POVM
1 3 2 4
45◦ polarizations at the inputs, etc. These quantities andthethirdoutcomecorrespondstoaninconclusivere-
sult. The optimal multimeter should minimize the error
werealwaysmeasuredjustbefore,orjustafterthecorre-
rate at the first two outcomes for a fixed fraction of in-
spondingsetofmainmeasurementsinorderto minimize
conclusive results. As a limit case it is possible to get
the influence of the long-term fluctuations (like a laser-
an error-free operation. Such a multimeter performs the
power drift etc.). The errors derived from the statistical
exact projective measurements but with the probability
fluctuationsofindividualdetectionratesaresmallerthan
of success lower than one.
thesymbolsofpointsinFig.5. Thesystematicerror(the
The optimal phase-covariant multimeters for the pro-
shift of all the data values) is due to non-unit visibility.
gramstates Ψ consistingofN copiesofthebasisstate
p
ψ , Ψ =| iψ ⊗N [26], were determined in [8]. For
+ p +
| i | i | i
III. PHASE-COVARIANT QUANTUM the simplest case N =1 that corresponds to a one-qubit
MULTIMETER program the optimal (fixed) POVM acting on data and
programqubits together reads:
A. Theory
1 η
Π = Ψ± Ψ± + − (Φ+ Φ+ + Φ− Φ− ),
±
| ih | 2 | ih | | ih |
Nowwewillconsideradifferentkindofquantummulti- Π = η(Φ+ Φ+ + Φ− Φ− ), (11)
?
meter. NamelysuchonethatcanperformvonNeumann | ih | | ih |
measurementsonasinglequbitinanybasis ψ , ψ where η [0,1], and
+ −
{| i | i} ∈
located on the equator of the Bloch sphere,
1
Ψ± = (0 1 1 0 ),
ψ (φ) = 1 (0 eiφ 1 ), (10) | i √2 | di| pi±| di| pi
±
| i √2 | i± | i 1
Φ± = (0 0 1 1 ). (12)
d p d p
| i √2 | i| i±| i| i
where φ [0,2π) is arbitrary. The particular measure-
∈
ment basis is selected by a quantum state of a program The fidelity of the multimeter can be defined as the
register. probabilityofacorrectmeasurementoutcomeinthecase
It is impossible to perfectly encode such projective of a conclusive result, assuming the data register is pre-
measurements into states in finite-dimensional Hilbert paredinthebasisstate ψ or ψ withprobability1/2
+ −
| i | i
6
each. Themeanfidelityisthenobtainedbyaveragingthe
fidelity over all measurement bases, i.e. over the phase
φ [0,2π). The average fidelity of the phase-covariant
∈
multimeterdependsontheprobabilityofinconclusivere-
sult in the following way,
3 2P
I
F = − .
4(1 PI) PI
−
The effective POVM on the data qubit only (this
POVM depends on the program state) is given by
π = (1 P )[F ψ (φ) ψ (φ)
± I ± ±
− | ih |
+(1 F) ψ (φ) ψ (φ)],
∓ ∓
− | ih |
π? = PI11.
For η = 0 we have an ambiguous (error-prone) op-
eration with no inconclusive results (P = 0,F = 3/4)
I
whilst for η = 1 we get an unambiguous (error-free) but FIG. 7: Probability of inconclusive results for unambiguous
probabilistic measurement device (from time to time we operation as a function of phase φ. Symbols denote exper-
imental data, solid line represents the theoretical prediction
obtain an inconclusive result; P =1/2,F =1) [27].
I
Clearly, when η =1 then POVM (11) is just a projec- PI =1/2.
tive measurement on the Bell states Ψ+ , Ψ− , and on
| i | i
therestofthefourdimensionalHilbertspace(thatcorre-
8◦. Theinconclusive-resultrateisa“complement”tothe
spondstoinconclusiveresults). Ifη <1wejust“reinter-
rate of “conclusive” results – i.e., both the correct and
pret”someinconclusiveresultsas“conclusive”ones. I.e.,
erroneous ones. Thus
we will treat randomly selected (with probability 1 η)
−
inconclusive results as results “+” or “ ” (at random). 1 C+++C−+ C−−+C+−
− P =1 + . (13)
Therefore, it is quite enough to test experimentally only I − 2(cid:20)2(C+++C−+) 2(C−−+C+−)(cid:21)
sh sh sh sh
the unambiguous version (η = 1) of our phase-covariant
quantummultimeterasalltheothervariants(0 η <1) The notation is the same as above.
≤
can be obtained manipulating the measured data only. GraphinFig.7showsthe dependence ofthe probabil-
ityofinconclusiveresults(foranunambiguousoperation)
onphase φ for input data states ψ and ψ (that co-
+ −
B. Experiment incide with or are orthogonal to|theiprogr|amistates, re-
spectively). Symbolsdenoteexperimentaldata,solidline
The simplest (N =1) phase-covariantmultimeter can represents the theoretical prediction. The derived errors
be experimentally implemented using the same setup as are smaller than the symbols of points. The experimen-
for the programmable quantum-state discriminator (see tal data fit well with the calculated value of PI = 1/2.
Fig. 2). The “logical values” 0 and 1 of qubits will be As expected, the probability of inconclusive results does
represented by horizontal, H , and vertical, V , linear- notdependonthe measurementbasisdeterminedbythe
| i | i
polarization states of photons, respectively. phase φ.
The states ψ (φ) (10) are prepared from states H The relative error rate (with respect to all conclusive
+
bytwowavep|lates. Tihe firstone isaquarter-wavepl|atei results)— i.e., a fractionof events when we get Ψ+ in-
rotated by α = φ/2 (with respect to the horizontal steadof Ψ− orviceversa—isabout3%. Ideall|y,inithe
− | i
axis) and the second one is a half-wave plate rotated by considered case of the unambiguous operation, the error
β = (90◦ φ)/4. The states ψ (φ) are prepared with rate should be zero, but due to technical imperfections
−
− | i
the wave plates rotatedby angles α and β. This cor- of the setup it gets non-zero values.
responds to the shift of φ by 180−◦. Dete−ction of Ψ+
| i
corresponds to the measurement result connected with
the basisstate ψ (φ) , detectionof Ψ− correspondsto IV. CONCLUSIONS
+
| i | i
the measurement result connected with the basis state
ψ−(φ) . Everything else means an inconclusive result. Our experimental results clearly illustrate that the
| i
measurement on the data qubit can be quite efficiently
controlled by the quantum state of the program qubit
C. Results (program register). In particular, we emphasize that a
classicalsettingoftheanglebetweenthestatesthatshall
We have measured coincidence rates corresponding to be unambiguously discriminated and a classical descrip-
detections of Ψ± and calculated probabilities of incon- tionofthemeasurementbasisincaseofprojectivephase-
clusive results|foriphases φ from 90◦ to +90◦ with step covariantmeasurementwouldrequireinfinitelymanybits
−
7
of classicalinformation, while only one quantum bit suf- preparatory stage of the experiment. This research was
ficesinthepresentcasetoobtainanerror-free (although supportedundertheprojectLN00A015oftheMinistryof
probabilistic) operation. Education of the Czech Republic. JF also acknowledges
supportfromtheEUundertheprojectCHIC(IST-2001-
32150).
Acknowledgments
The authors would like to thank to Martin Hendrych
and Miroslav Jeˇzek for their help and advises in the
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