Robustness of raw quantum tomography

1.1 MB·English

by  M. Asorey

Robustness of raw quantum tomography
M. Asoreya, P. Facchib,c,d, G. Florioe,c,d, V.I. Man’kof,∗, G. Marmog,h,d, S. Pascazioe,c,d, E.C.G. Sudarshani
aDepartamento de F´ısica Te´orica, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain
bDipartimento di Matematica, Universita` di Bari,I-70125 Bari, Italy
cINFN, Sezione di Bari, I-70126 Bari, Italy
dMECENAS, Universita` Federico II di Napoli & Universit`a di Bari, Italy
eDipartimento di Fisica, Universita` di Bari, I-70126 Bari, Italy
fP.N. Lebedev Physical Institute, Leninskii Prospect 53, Moscow 119991, Russia
gDipartimento di Scienze Fisiche, Universita` di Napoli “Federico II”, I-80126 Napoli, Italy
hINFN, Sezione di Napoli, I-80126 Napoli, Italy
iDepartment of Physics, University of Texas, Austin, Texas 78712, USA
1
1Abstract
0
2We scrutinize the effects of non-ideal data acquisition on the tomograms of quantum states. The presence of a weight
n function,schematizingtheeffectsofafinitewindoworequivalentlynoise,onlyaffectsthestatereconstructionprocedure
aby a normalization constant. The results are extended to a discrete mesh and show that quantum tomography is robust
Junder incomplete and approximate knowledge of tomograms.

2
1Keywords: quantum tomography, image reconstruction, integral transforms

]
h
1. Introduction of the detection apparata, and from a theoretical perspec-
p
- tive, the refinement of mathematical inversion techniques
nt Quantum technological applications require extremely for noisy data and the extension of classical tomographic
aaccurateknowledgeofquantumstatesandoftheunderly- techniques to quantum situations [1, 2, 12–16].
u In quantum applications, the most used tomographic
ing quantum dynamical processes. For the application of
q
methods are quantum generalizations of the Radon trans-
[fundamental principles of quantum mechanics and quan-
form [17–19]. This is true both for massive particles and
tumopticstoeffectivelyfosterthereal-lifeimplementation
2 reconstructionmethodsbasedonhomodynephotondetec-
of quantum technologies, accurate quantum state charac-
v
tion. The measure of optical tomograms yields complete
terization is a crucial ingredient at the interface between
4
information on quantum states in terms of tomographic
6theoretical and experimental physics. Applications cover
6wide research areas ranging from nano-science to cosmol- probability distributions [20]. Since, traditionally, one is
1ogy. interested in other equivalent characteristics of quantum
. states, like the Wigner or the Husimi or the diagonal co-
3 One of the most successful quantum state reconstruc-
0tion techniques is quantum tomography [1, 2] with its el- herentstaterepresentationfunction,weshalladdresshere
0 the question of the relation between these quantum state
egant experimental realizations [3–9]. For recent reviews,
1 characteristics and our approach.
see [10, 11]. A large class of quantum states, expressed in
:
v In practice the parameters introduced in the tomogram
termsofWignerfunctions,canbeefficientlyreconstructed
i have many sources of uncertainty and an efficient tomo-
Xbythismethod,includingcoherentstates,Sch¨odingercats,
graphicmeasurementofthequantumstatemustfacethree
kittles,andentangledstates. Inquantumoptics,thestate
r
acan be directly measured by homodyne photon detection major problems: i) a finite window (including the effects
of detectors, entailing coarse graining and/or binning of
[12], providing as output result the optical tomogram.
the data), ii) the presence of random errors (arising both
However, the precise characterization of quantum states
from the sample and the non-ideal precision in controlling
becomes problematic, both for direct measurements and
thequadratures),andiii)adiscretized“mesh”inthedata
reconstruction procedures, when noise, imperfections and
acquisition, that affects the generation of the quadratures
other practical problems deteriorate the quality or limit
of the tomogram.
thesizeofthedataset. Thisentails,fromanexperimental
A realistic approach must define quantum tomograms
perspective, a precise control/manipulation of the quan-
through a convolution with a smooth weight function Ξ
tum system, from the source to the careful optimization
thatspreadsthemarginalsandreducestotheusual“clas-
sical” Radon transform when Ξ becomes a delta function.
∗Correspondingauthor Inthiscaseonegetsakindof“thicktomography,”inanal-
Email address: [email protected](V.I.Man’ko) ogy with signal analysis, where a similar problem arises
Preprint submitted to Physics Letters A January 13, 2011
for signal detection and also requires the use of a window with µ,ν ∈ R. Its complete equivalence with (3) is ex-
function [21, 22]. Notice that in a classical description, pressed by the relation [27, 28]
where tomograms are measured by the attenuation of a (cid:18) (cid:19)
1 X
probing beam scanning a material body, this would cor- W(cid:93)(X,rcosϕ,rsinϕ)= W ,ϕ , (5)
r r
respond to a finite transversal thickness of the scanning
beam. However, this is only an analogy: in the physical valid for any r > 0. Equation (5) is an easy consequence
situation we are describing there is no material body, nor of the fact that the Dirac distribution is positive homoge-
probing beams, and the thickness is in phase space. neous of degree −1. The symplectic tomogram (4) can be
Our aim is to suggest extensions of the quantum tomo- easily inverted by a Fourier transform
graphic techniques by taking into account the finiteness (cid:90) dXdµdν
of the windows, the discreteness of data acquisition and W(p,q)= W(cid:93)(X,µ,ν)ei(X−qµ−pν), (6)
the effects of noise, analogously to signal analysis. Previ- R3 (2π)2
ous work[23, 24]focused onthe definitionof optimization which by (5) yields the inversion formula
strategies given a set of experimental data and nonideal
1 (cid:90) (cid:90) (cid:90)
photodetector efficiency in homodyne detection (see also W(p,q) = dX dr dϕW(X,ϕ)
(2π)2
[25] for a review). R R T
+
This paper is organized as follows. In Sec. 2 we recall ×reir(X−qcosϕ−psinϕ), (7)
the main concepts related to the Radon and symplectic
where T=R/2πZ is the unit circle and R =[0,+∞).
transforms. Section 3 is devoted to the effects of the fi- +
nite window function. In Sec. 4 and 5 we analyze how
thetomogramsaremodifiedbynoiseandadiscretemesh. 3. Finite window
In Sec. 6 we present some numerical results for a partic-
ular example, corroborating our analysis. Finally, in Sec. A finite window function can be accounted for by re-
7 we sketch some conclusions and outline possible future placing the Dirac delta function in the definition of the
research directions. tomogram (4) with a suitable smearing window function
Ξ
(cid:90)
2. Preliminaries: Radon and symplectic trans- W(cid:93)(X,µ,ν)= W(p,q)Ξ(X−qµ−pν)dpdq. (8)
Ξ
forms R2
Now, by Fourier transforming Eq. (8), one gets
We shall start by studying the role of the deformation
associated with the window weight function in the exper- W(p,q) = N (cid:90) dXdµdν W(cid:93)(X,µ,ν)
imental tomogram and its influence on the state recon- Ξ R3 (2π)2 Ξ
struction formula. The following framework is of general
× ei(X−qµ−pν), (9)
validity, and can be applied to massive particles as well as
photons. However,forsimplicity,letρbeagivenquantum where
(cid:98)
ptohrost,oanndstaϕtea,nq(cid:98)daXndtp(cid:98)hethloecpaolsoitsicoilnlaatnordpmhoamseenatnudmquoapderraa-- NΞ = 1 , Ξ(cid:101)(−1)=(cid:90) Ξ(z)eiz dz. (10)
Ξ(cid:101)(−1) R
ture in a homodyne experimental setup. The homodyne
tomogram is given by In operator terms the state reconstruction is achieved by
(cid:90) dXdµdν
W(X,ϕ)=Tr{ρ(cid:98)δ(X−q(cid:98)cosϕ−p(cid:98)sinϕ)}, (1) ρ(cid:98)=NΞ R3 (2π)2 WΞ(cid:93)(X,µ,ν)ei(X−q(cid:98)µ−p(cid:98)ν). (11)
where Tr{A} denotes the trace of the operator A. Ex-
This is one of our central results: the inverse reconstruc-
pressed in terms of the Wigner function
tion map is independent of the window function, the only
1 (cid:90)
W(p,q)= (cid:104)q−ξ|ρ|q+ξ(cid:105)e2ipξ dξ, (2) relic of the latter being the normalization constant NΞ,
π R (cid:98) thatcanbefixedbythenormalizationofonesingletomo-
gram. Thus, no matter how involved is the shape of the
the tomogram (1) reads
window function, an exact reconstruction of the state can
(cid:90)
bealwaysachievedbymeansof(9)or(11). Thisproperty
W(X,ϕ) = dpdq W(p,q)
R2 is very interesting because in practical cases the experi-
×δ(X−qcosϕ−psinϕ). (3) mentalwindowfunctionΞisunknown,buttheresulttells
us that the exact state reconstruction is possible without
TheRadontransformoftheWignerfunction(3)hasbeen
any precise information about Ξ. This result is also inde-
generalized to the following symplectic, or M2, transform
pendent of the features of the initial state, that can even
[13, 26]
benonstationary. Interestingly,thisresultcanbeextended
(cid:90) to situations were noise is present. This will be done in
W(cid:93)(X,µ,ν)= W(p,q)δ(X−qµ−pν) dpdq, (4)
the next section.
R2
2
4. Noise yields the symplectic tomograms
2(X2/r2+σ2(2σ2+1))
We start from the following important observation: the W(cid:93)(X,µ,ν) = √
effects of a spread function in the tomogram (8) are de Ξ πr(2σ2+1)5/2
facto equivalent to the presence of noise, due to quan- ×e−X2/r2(2σ2+1), (18)
tum efficiency and/or thermalization, in a homodyne to-
(cid:112)
mogram(e.g.,ifthequadratureisdeterminedwithafinite with r = µ2+ν2. Note that
precision). The origin of such noise is due to the fact that
2σ2
any counting homodyne statististics is just a sampling of W(cid:93)(0,µ,ν)= . (19)
probabilities. Namely, any statistical event can be only Ξ (cid:112)πr2(2σ2+1)3
approximatelysampledandinthecourseofrepeatedmea-
Therefore, the presence of a window function/noise pro-
surements,theoutputsarealwaysfluctuating. Allthiscan
vokes a reduction of “visibility” of the tomogram (that
be seen as a jitter in the detected statistics.
would vanish for σ = 0). However, a perfect state recon-
The equivalence between such a jitter and a finite win-
struction is obtained by Eqs. (9) or (11).
dow can be seen as follows: if there is some jitter X +ξ
This proves that an exact reconstruction can be ob-
on X, where ξ is a random variable with a finite variance,
tainedeveninthepresenceofnoise,providedonehasvery
one obtains
(in the limit, infinitely) accurate control over the position
W(ξ)(X,µ,ν)=Tr{ρ δ(X+ξ−qµ−pν)}. (12) of the quadrature. This extends the central result of the
(cid:98) (cid:98) (cid:98)
preceding section.
By setting
E[δ(X+ξ)]=Ξ(X), (13) 5. Discreteness of data acquisition
where E denotes the expectation value over the random Sofarwehaveassumedthatthewindowfunctioncanbe
finite(thicktomograms),butonehasaccesstoallpossible
variable ξ, one gets
tomograms in a continuous range of parameters. We have
E[W(ξ)(X,µ,ν)]=W(cid:93)(X,µ,ν) (14) notdiscussedtherobustnessoftomographywithrespectto
Ξ
thediscretenessofdataacquisition. Letusassumethatthe
and the reconstruction can be done exactly as above. experimental tomograms are gathered only on a sequence
Thejitterprovidesextranoiseanddeformstheoutputof of discrete values of X and ϕ, namely
homodyne experiments by affecting the basic inequalities
(cid:18) (cid:19)
2π
of quantum mechanics expressed in terms of tomograms. W =W kT,m , k ∈Z, m∈Z (20)
For this reason, it is of practical interest to take it into k,m N N
account when extracting information on the uncertainty
where T > 0, Z = Z/NZ and, for convenience, N is
relations from the measurements of tomograms. In prac- N
an odd positive integer. If the tomograms have a limited
tice this can be done, for example, by using the output of
bandwidth,thenforsufficientlysmallvaluesofT andN−1
photon homodyne measurements [3, 11], given as optical
onecanexactlyreconstructthewholefamilyoftomograms
tomograms(1). Thequadrature(co)variancesaredirectly
from the knowledge of the experimental ones. This is the
expressed in terms of simple integrals containing optical
contentoftheNyquist-Shannonsamplingtheorem[30,31].
tomograms [20, 29] and can be used to yield a noise limit
More precisely, consider the Fourier transform of W
contribution to the experimental accuracy of uncertainty
relations in homodyne photon detection. (cid:90) (cid:90) dϕ
W(cid:102)(ω,(cid:96))= dX e−iωX e−i(cid:96)ϕW(X,ϕ). (21)
Let us look at a simple but significant example of ex- 2π
R T
perimentalrelevance. ConsidertheWignerfunctionofthe
If its support is compact and satisfies
firstexcitedstateofaharmonicoscillator(orasinglepho-
ton state) (cid:16) π π(cid:17) (cid:16) N N(cid:17)
suppW(cid:102)⊂D, D = − , × − , , (22)
T T 2 2
2(p2+q2)−1
W(p,q)= e−(p2+q2). (15) one has
π
T (cid:88) (cid:88)
Its Radon transform reads W(cid:102)(ω,(cid:96)) = N Wk,me−iXkωe−iϕm(cid:96)
2X2 k∈Zm∈ZN
W(X,ϕ)= √ e−X2. (16) ×χ (ω,(cid:96)), (23)
π D
where X =kT, ϕ =2πm/N, and
The introduction of a Gaussian window function k m
(cid:26)
1 1 if (ω,l)∈D
Ξ(X)= √ e−X2/2σ2, (17) χ (ω,(cid:96))= (24)
2πσ2 D 0 if (ω,l)∈/ D.
3
Letusbrieflyshowhow(23)isobtained, byconsidering with X =kT and sincx=x−1sinx.
k
the less common situation of a function on the torus T: Therefore, the tomograms are given, for N odd, by the
(cid:90) dϕ (cid:88) (generalized) Shannon-Whittaker interpolation formula
f(cid:101)((cid:96))= e−i(cid:96)ϕf(ϕ), f(ϕ)= ei(cid:96)ϕf(cid:101)((cid:96)). (25)
2π
T (cid:96)∈Z W(X,ϕ) = (cid:88) (cid:88) W sinc(cid:18)πX−Xk(cid:19)
Suppose f(cid:101)((cid:96)) = 0 for (cid:96) ∈/ J, with J = (−N/2,N/2)∩N. k∈Zm∈ZN k,m T
For definiteness let us assume N odd. Then J ={−(N − (cid:18) (cid:19)
ϕ−ϕ
1)/2,...,(N −1)/2}, and one can consider the periodic ×SN 2 m , ∀X ∈R,∀ϕ∈T.
extension of f(cid:101)
(35)
(cid:88)
f(cid:101)N((cid:96))= f(cid:101)((cid:96)−kN), (26)
For N even the formula is the same, by replacing N by
k∈Z
N −1 (since in that case the angular sampling points are
whose restriction to J coincides with f(cid:101), namely, N −1 instead of N). Remarkably, for sufficiently small T
and N−1 the reconstruction of limited-bandwidth tomo-
f(cid:101)((cid:96))=f(cid:101)N((cid:96))χJ((cid:96)). (27)
grams is faithful and there is no information loss. It is in-
Thispropertywouldnotbetrueiff(cid:101)didnotvanishoutside teresting to observe that the application of signal-analysis
J, and the phenomenon called aliasing would occur. One techniques to quantum tomography can be quite straight-
gets forward. For example, from the mathematical point of
(cid:90) dϕ (cid:88) view, the extension of Eqs. (32) and (35) of quantum to-
f(cid:101)N((cid:96))= 2π e−i(cid:96)ϕf(ϕ) eikNϕ. (28) mograms to a torus is very natural.
T k∈Z By plugging (35) into (7) one can obtain a reconstruc-
By recalling Poisson’s formula tion formula of the Wigner function in terms of the tomo-
graphic samples (20). By writing
(cid:88) 2π (cid:88)
eikNϕ = δ(ϕ−ϕ ), (29)
k∈Z N k∈Z k sinc(cid:18)πX−Xk(cid:19)= T (cid:90) +π/T eiω(X−Xk) dω (36)
with ϕk = 2kπ/N, and setting k = jN +m with k,j ∈ Z T 2π −π/T
and m∈Z one gets
N
andbyperformingtheintegrationoverX onegetsaDirac
(cid:88) 2π (cid:88) (cid:88)
eikNϕ = δ(ϕ−ϕ −2jπ). (30) delta function δ(ω+r) whose integral yields
N m
Therekf∈oZre, m∈ZN j∈Z W(p,q) = (cid:88) (cid:88) Wk,m(cid:90) dϕSN(cid:18)ϕ−2ϕm(cid:19)
1 (cid:88) k∈Zm∈ZN T
f(cid:101)N((cid:96))= N m∈ZNe−i(cid:96)ϕmf(ϕm). (31) ×(2Tπ)2 (cid:90) π/T drreir(Xk−qcosϕ−psinϕ).
0
Byplugging(31)into(27)onegetstheangulardependence (37)
of (23). The linear dependence is obtained analogously by
replacing T with R and Z with Z. We get
N
By Fourier inverting (27) one gets
T (cid:90) π/T αsinα+cosα−1
f(ϕ) = (cid:88)ei(cid:96)ϕf(cid:101)N((cid:96)) (2π)2 dr reirβ = 4Tα2
0
(cid:96)∈J
αcosα−sinα
(cid:88) 1 (cid:88) +i , (38)
= f(ϕ ) ei(cid:96)(ϕ−ϕm) 4Tα2
m N
m∈ZN (cid:96)∈J with α = πβ/T. Since the imaginary part is odd with
(cid:18) (cid:19)
= (cid:88) f(ϕ )S ϕ−ϕm , (32) respect to the rotation ϕ → ϕ+π, its integral over the
m N 2
m∈ZN torus vanishes and we finally obtain
with W(p,q) = (cid:88) (cid:88) W (cid:90) dϕS (cid:18)ϕ−ϕm(cid:19)
sin(Nx) k,m N 2
SN(x)= Nsinx , (33) k∈Zm∈ZN T
(cid:20)
cos(α (ϕ;q,p))−1
whichistheextensionofthesamplingtheoremtofunctions × k
on the torus T and their discrete spectra. For a function 4Tαk(ϕ;q,p)2
(cid:21)
g(X) on the line, the analogous, well-know formula reads +αk(ϕ;q,p)sin(αk(ϕ;q,p)) , (39)
g(X)=(cid:88)g(X ) sinc(cid:18)πX−Xk(cid:19), (34) 4Tαk(ϕ;q,p)2
k T withα (ϕ;q,p)=π(X −qcosϕ−psinϕ)/T (andN odd).
k∈Z k k
4
Let us now consider an experimental situation in which of a finite window (or equivalently the presence of noise)
there is some uncertainty on the linear and angular posi- and those that are a consequence of partial data acquisi-
tion of the quadrature, that is tion. The problems that arise in non-ideal situations are
under control in two limiting cases: when the window size
(cid:16) (cid:17)
Wk(ξ,m) =W Xk(ξ),ϕ(mξ) , σ and the sampling steps s(=T,1/N) are well separated,
2π i.e. σ (cid:28) s or s (cid:28) σ. The first case can be analyzed
Xk(ξ) =T(k+ξk(1)), ϕ(mξ) = N (m+ξm(2)), (40) throughEq.(39),thesecondonethroughEqs.(9)-(10). It
would be interesting to understand whether a single gen-
with k ∈ Z and m ∈ Z , where {ξ(1)} and {ξ(2)} are eral formula exists, that accounts at the same time for
N k m
two sequences of independent identically distributed ran- the consequences of all these effects and from which both
domvariableswithfinitestandarddeviationsσ(1)andσ(2). above-mentionedcasesariseassuitablelimits. Suchafor-
ξ ξ
Under hypothesis (22), one gets mula would also enable us to elucidate whether the sim-
ple expression we obtained in terms of the normalization
W(cid:102)(ω,(cid:96)) = T E(cid:104)(cid:88) (cid:88)W(ξ) e−iXk(ξ)ωe−iϕm(ξ)(cid:96) constant (10) can be generalized to the case of a finite
N k,m
k∈Zm∈ZN number of measurements, or when noise is combined with
(cid:105) othersourcesofuncertainty. Thisinterestingaspectisleft
×χ (ω,(cid:96)) . (41)
D for future research. The following section is devoted to a
partial numerical investigation of this problem.
A Fourier transform, followed by a Radon inversion (11)
yields on average a perfect reconstruction of the state.
Therefore,quadratureuncertaintiesandunbiasednoisedo
6. Robustness of tomograms
not affect the result.
Let us prove Eq. (41). Consider a function g(X) on the
So far, our analysis has taken into account the effects
line, with limited bandwidth, namely, g(ω)=0 for ω ∈/ I,
(cid:101) of noise and discreteness of data. We have seen that for
where I =(−π/T,+π/T). One can write sufficientlysmallT andN−1 thereconstructionoflimited-
h(ξ)(ω) = (cid:88)g(Xk(ξ))e−iXk(ξ)ω bmaantdiownidlothsst[osemeoEgrqa.m(3s5i)safnadithfofulllowanindgtchoemremiesntnso]. iAnflosor-,
k∈Z quadrature uncertainties and unbiased noise do not affect
= (cid:88)(cid:90) d2πν g(cid:101)(ν)e−iXk(ξ)(ω−ν) tBhyecroemcobnisntirnugctailolnth[seeseeEreqs.u(lt4s1)waenndowfoldloiswciunsgsctohme mroebnutsst]-.
k∈Z
= (cid:90) d2πν g(cid:101)(ω−ν) (cid:88)e−iXk(ξ)ν (42) nsoeussrcoesf oqfuuannctuermtaitnotmyoagnrdamlims iutnatdieornsth(deisacfroertee-nmeesnstoiofntehde
k∈Z sampling and finite precision in the determination of the
By taking the expectation value, and by using Poisson’s quadrature,sincewehavealreadyarguedthatafinitewin-
formula (29) one gets dow is equivalent to the presence of noise). Let us look
again at the tomogram (16) and numerically investigate
E(cid:104)(cid:88)e−iXk(ξ)ν(cid:105) = E(cid:2)e−iξν(cid:3)(cid:88)e−iXkν the effects that arise due to a discrete mesh and the pres-
k∈Z k∈Z enceofnoiseontheWignerfunction(15). Weshallrecon-
= 2πE(cid:2)e−iξν(cid:3)(cid:88)δ(cid:18)ν− 2πk(cid:19). sNtru=ct5,thaendW(cid:80)ignerfruenpcltaicoendbbyyu(cid:80)singEq.,(w3i9t)hwKith=T40=. 0.1,
T T k∈Z |k|≤K
k∈Z Figure 1 shows the tomographic reconstructions of the
(43) Wigner function (15), starting from its tomograms (16).
The Wigner function in the noiseless case is shown in the
Therefore,
upper panel and is practically indistinguishable from the
(cid:104) (cid:105) 1 (cid:88) (cid:18) 2πk(cid:19) (cid:104) (cid:105) original (15). Notice that, even though the tomogram
E h(ξ)(ω) = g ω− E e−i2πkξ/T ,
T (cid:101) T is not band limited and the number of points K is fi-
k∈Z nite, the reconstruction by means of (39) is practically
(44)
alias free. The reconstructions in the central and lower
panel are affected by noise, as in (40) with zero means
that, when multiplied by χ (ω), yields
I
and σ(1) = σ(2) = 0.5: these are large values, of the same
(cid:104) (cid:105) 1 ξ ξ
E h(ξ)(ω)χ (ω) = g(ω)E[1], (45) order of the sampling periods T and 2π/N (one should
I T (cid:101)
notice, however, that in this particular case the noise on
because of the condition on the bandwidth of g. Since the angular position of the quadrature does not affect the
E[1] = 1, we get the linear dependence of (41), the proof procedure due to the symmetry of the state considered).
of the angular dependence being analogous. Thecentralpanelreferstoasinglerealizationofthenoise,
A comment seems in order. In our analysis we sepa- while the lower panel to an average over 10 realizations of
rately addressed the effects that arise from the existence the noise.
5
The reconstruction error is measured by the distance
1
ε = ||W −W ||
2 rec B
1(cid:90)
= |W(p,q)−W (p,q)|dpdq, (46)
2 rec
B
where B is the box considered (in our case B =
[−1.5,1.5]2). We obtain ε = 1.4 × 10−4,0.11,0.028 for
the upper, central and lower panel, respectively. This
should be compared with (cid:107)W(cid:107) = 1.01. It is apparent
B
both from Fig. 1 and the above numerical results that the
reconstructionisrobustatthesametime againstnoiseand
discretization effects.
7. Conclusions
In conclusion, we have discussed the practical problems
that arise in quantum (homodyne) tomography. First of
all, the presence of a weight function has been shown
to introduce only a normalization constant, which entails
no loss of information and permits the exact reconstruc-
tion without any precise knowledge of the window func-
tion of the experimental setup: on average, quadrature
uncertainties and unbiased noise do not affect the recon-
struction. Second, the discretization procedure affects the
global reconstruction in the same way as it does in the
classicalNyquist-Shannonsetting: ifthebandwidthofthe
tomogram is limited, there is no information loss for a
sufficiently dense sampling. We have also discussed the
most general case, in which both problems arise simulta-
neously. Although we were not able to derive a general
formula, from which both sources of non-ideal data ac-
quisitions are present at the same time, and from which
both situations investigated here are derived as limiting
cases, we proved by numerical methods the robustness of
the overall reconstruction against the different sources of
imperfections. Thevalidityoftheapproachproposedhere
should be tested in conjunction with other refined theo-
retical tools based on the maximum likelihood estimation
[14, 15, 23].
Since nowadays quantum states reconstruction is based
onthemeasurementoftomograms,therobustnessagainst
Figure 1: Reconstruction starting from the tomograms (16) of the non-idealdataacquisitionprovidessolidfoundationtothe
WignerfunctionW(p,q)inEq.(15). WesetN =5,T =0.1andK=
compatibility of tomographic experiments performed in
40inEqs.(39)and(41). Upperpanel: nonoise. Thereconstructed
andtheoriginalfunctionareindistinguishable. Centralpanel: single different laboratories by different methods. Our results
noise realization, σ(1) = σ(2) = 0.5. Lower panel: average over 10 provide therefore a solid basis for the theoretical and phe-
ξ ξ
realizationsofthenoise. nomenological analysis of real tomograms.
8. Acknowledgments
V.I.M. was partially supported by the Russian Founda-
tionforBasicResearchunderProjectNo. 09-02-00142and
Italian INFN and thanks the Physics Department of the
UniversityofNaplesforthekindhospitality. P.F.andG.F.
acknowledgesupportthroughtheprojectIDEAofUniver-
sity of Bari. The work of M.A. and G.M. was partially
6
supported by a cooperation grant INFN-CICYT. M.A.
was also partially supported by the Spanish CICYT grant
FPA2009-09638 and DGIID-DGA (grant 2009-E24/2).
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