Process reconstruction: From unphysical to physical maps via maximum likelihood

0.33 MB·English

by  Mario Ziman

Process reconstruction: From unphysical to physical maps via maximum likelihood
Ma´rio Ziman1,2, Martin Plesch1, Vladim´ır Buˇzek1,2, and Peter Sˇtelmachoviˇc1
1 Research Center for Quantum Information, Slovak Academy of Sciences, Du´bravsk´a cesta 9, 845 11 Bratislava, Slovakia
2 Faculty of Informatics, Masaryk University, Botanick´a 68a, 602 00 Brno, Czech Republic
(14 January 2005)
We show that the method of maximum likelihood (MML) provides us with an efficient scheme
for reconstruction of quantum channels from incomplete measurement data. By construction this
scheme always results in estimations of channels that are completely positive. Using this property
weusetheMMLforaderivationofphysicalapproximationsofun-physicaloperations. Inparticular,
weanalyzetheoptimalapproximationoftheuniversalNOTgateaswellasaphysicalapproximation
of a quantumnonlinear polarization rotation.
5
Any quantum dynamics [1,2], i.e. the process that tumchannel. Asaresultwecanfindanegativeoperator
0
0 is described by a completely positive (CP) map of a ̺′j, or a channel E, which is not CP.
2 quantum-mechanical system can be probed in two dif- In what follows we will introduce and compare two
n ferent ways. Either we use a single entangled state of schemes how to perform a channel reconstruction with
a a bi-partite system [3], or we use a collection of linearly insufficient measurement data. Firstly, we will consider
J independent single-particle test states [4,1] (that form a rather straightforward “regularization” of the recon-

9 a basis of the vector space of all hermitian operators). structed unphysical map. Secondly we will exploit the
1 Given the fragility of entangled states in this Letter we method of maximum likelihood (MML) to perform an
willfocusourattentionontheprocessreconstructionus- estimation of the channel. We will use these methods
1 ing only single-particle states. to perform a reconstruction of maps based on numerical
v
Thetaskofaprocessreconstructionistodeterminean simulation of the anti-unitary universal NOT operation
2
0 unknownquantumchannel(a“blackbox”)usingcorrela- (U-NOT) [5]. This is a linear, but not a CP map and
1 tionsbetweenknowninputstatesandresultsofmeasure- we will show how our regularization methods will result
1 ments performed on these states that have been trans- in optimal physical approximations of the U-NOT oper-
0 formedbythechannel. Thelinearityofquantumdynam- ation. WewillconcludethattheMMLisatoolthatpro-
5
ics implies that the channel E is exhaustively described vide us with approximations of non-physical operations.
0
by its action ̺ →̺′ =E[̺ ] on a set of basis states, i.e. In order to demonstrate the power of this approach we
/ j j j
h acollectionoflinearlyindependent states̺ , thatplaya will also apply it to obtain an approximation of a non-
j
p roleofteststates. Therefore,toperformareconstruction linear quantum-mechanical map, the so called nonlinear
-
t ofthe channelE wehaveto performacomplete state to- polarization rotation (NPR) [6].
an mography[1] of ̺′j. The number of test states equals d2, Structureofqubit channels. Quantumchannelsarede-
u where d = dimH is the dimension of the Hilbert space scribed by linear trace-preservingCP maps E defined on
q associatedwiththesystem. Consequently,inordertore- asetofdensityoperators[1,2,7]. Thecompletepositivity
v: constructa channelwe haveto determine d2(d2−1)real is guaranteed if the operator ΩE = E ⊗I[P+] is a valid
i parameters,i.e. 12 numbers in the case of qubit (d=2). quantum state [P+ is a projection onto a maximally en-
X In what follows we will assume that test states can be tangled state]. Any qubit channel E can be imagined as
r prepared on demand perfectly. Nevertheless, the recon- an affine transformation of the three-dimensional Bloch
a struction of the channel E can be affected by the lack vector~r(representingaqubitstate),i.e. ~r →~r′ =T~r+~t,
of required information due to the following reasons: i) where T is a real 3x3 matrix and ~t is a translation [7].
eachtest state is representedby a finite ensemble, corre- This form guarantees that the transformation E is her-
spondingly, measurements performed at the output can mitian and trace preserving. The CP condition defines
result in an approximate estimation of transformed test (nontrivial)constraintsonpossiblevaluesofinvolvedpa-
states; ii) the set of test states is not complete; and iii) rameters. Infact,thesetofallCPtrace-preservingmaps
incomplete measurements on transformed test states are forms a specific convex subset of all affine transforma-
performed. In these cases some of the parameters that tions.
determine the map E cannot be deduced perfectly from The matrix T can be written in the so-called
the measured data. In order to accomplish the channel singular-value decomposition, i.e. T = R DR with
U V
reconstruction additional criteria have to be considered. R ,R corresponding to orthogonal rotations and D =
U V
Inthis Letterwewillpayattentionto the casei), that diag{λ1,λ2,λ3} being diagonal where λk are the singu-
istypicalforexperiments-onecannotprepareaninfinite larvaluesofT. This meansthatanymapE is amember
ensemble, so frequencies of the measured outcomes are of less-parametricfamily of maps of the “diagonalform”
only approximationsof probabilitydistributions. Conse- Φ , i.e. E[̺] = UΦ [V̺V†]U† where U,V are unitary
E E
quently,thereconstructionofoutputstates̺′ canleadus operators. The reduction of parameters is very helpful,
j
to unphysical conclusions about the action of the quan- and most of the properties (including complete positiv-
1
ity) of E is reflected by the properties of Φ . The map equally distributed events (clicks) to outcome statistics,
E
E is CP only if Φ is. Let us note that Φ is determined i.e. addition of random noise.
E E
not only by the matrix D, but also by a new translation As we have seen from above, an important role in the
vector~τ =R ~t, i.e. under the action of the map Φ the state reconstruction is played by the total mixture 1I,
U E 2
Bloch sphere transforms as follows r →r′ =λ r +τ . which is an average over all possible states. An average
j j j j j
A special class of CP maps are the unital maps, that overallCPmapsisthemapAthattransformsthewhole
transform the total mixture into itself. In this case state space into the total mixture, i.e. A[̺]= 1I [8].
2
~t = ~τ = ~0, and the corresponding map Φ is uniquely The reconstruction of qubit channels consists of the
E
specified by just three realparameters. The positivity of knownpreparationof(atleast)fourlinearlyindependent
the transformation Φ results into conditions |λ | ≤ 1. test states ̺ and a state reconstruction of correspond-
E k j
On the other hand, in order to fulfill the CP condition ing four output states ̺′. The process estimation based
j
we need that the four inequalities |λ1 ±λ2| ≤ |1±λ3| on the correlations ̺j → ̺′j is certainly trace-preserving
are satisfied. These conditions specify a tetrahedron ly- (if all ̺′ are positive). Though the complete positivity
j
ing inside a cube of all positive unital maps. In this is problematic. The average channel A can be used to
case the extreme points represent four unitary transfor- correct (“regularize”) improper estimations E to obtain
mations I,σx,σy,σz (see Fig.1). a CP qubit channel Ec
I 1 ~0
E =kE +(1−k)A= . (2)
c (cid:18) k~t kT (cid:19)
This method of channel regularization uses the same
principle as the method for states, i.e. it is associated
Z with an addition of random noise into data.
Let us try to estimate what is the critical value of k,
X optimal i.e. the amountofnoise that surely correctsany positive
NOT Y
map. Trivially, it is enough to set k =0. In this case we
completely ignore the measured data and the corrected
U−NOT
map is A. However,we are interested in some nontrivial
FIG.1. UnitalCP mapsareembededin theset of allpos-
lower bound, i.e. in the largest possible value of k that
itive unital maps (cube). The CP maps form a tetrahedron
guarantees the complete positivity. Let us consider, for
with four unitary transformations in its corners (extremal
simplicity, that the map under consideration is unital.
points) I,x,y,z corresponding to the Pauli σ-matrices. The
un-physical U-NOT operation (λ1 = λ2 = λ3 = −1) and its Then the worst case of a positive map that is not CP, is
represented by the universal NOT operation.
optimal completely positive approximation quantum univer-
sal NOTgate (λ1 =λ2 =λ3 =−1/3) are shown. Universal NOT. The logical NOT operation can be
generalizedinto the quantumdomainasa unitarytrans-
formation |0i→|1i,|1i→|0i. However, this map is ba-
Qubit channel estimation and its regularization. Re- sisdependentanddoesnottransformallqubitstates|ψi
constructions of states and processes share many fea- into their (unique) orthogonal complements |ψ⊥i. Such
tures. Therefore we briefly remind us basic concept of universal NOT(ENOT :|ψi→|ψ⊥i)isassociatedwiththe
the state reconstruction using finite ensembles of identi- inversion of the Bloch sphere, i.e. ~r → −~r, which is not
cally prepared states. In this case one can obtain from a CP map. It represents an unphysical transformation
estimated mean values of observable a negative density specifiedbyλ1 =λ2 =λ3 =−1. Thedistance(seeFig.1)
operator of a qubit ̺ = 1(I +~r·~σ) with |~r| > 1. The betweenthismapandthetetrahedronofcompletelypos-
2
reconstructedoperatorhasalwaysunittrace,buttheas- itive maps is extremal, i.e. it is most un-physical map
sociatedvector~r can point out of the Blochsphere. One amonglineartransformationsandcanbeperformedonly
canarguethattheproperphysicalstateistheclosestone approximatively. A quantum “machine” that optimally
to the reconstructed operator, i.e. a pure state with ~r implements an approximation of the universal NOT has
c
pointingintothesamedirection. Formallyitcorresponds been introduced in Ref. [5]. The machine is represented
to a multiplication ~r by some constant k, i.e. ~rc = k~r. by a map E˜NOT = diag{1,−1/3,−1/3,−1/3}. The dis-
The correction by k can be expressed as tance [8] between the U-NOT and its optimal physical
approximationreads
1 1
̺ =k̺+(1−k) I = (I +k~r·~σ) (1)
c
2 2 d(E˜NOT,ENOT)= d̺Tr|(ENOT−E˜NOT)[̺]|=1/3. (3)
Z
states
and it can be understood as a convex addition of the to-
tal mixture 21I represented by the center of the Bloch The CP conditions imply that the minimal amount
sphere, i.e. ~0 = (0,0,0). In other words, the correc- of noise necessary for a regularization of the universal
tion consists of the addition of completely random and NOT gate corresponds to the value of k = 1/3, i.e.
2
λ1 = λ2 = λ3 = −1/3 (see Fig.1). The channel repre- FIG.2. Thedistanced(ENOT,Eest)asafunctionofthenum-
sentingthispointcorrespondstothebestCPapproxima- ber of measured outcomes N in the logarithmic scale. We
tion of the universal NOT operation, i.e. to the optimal used 6 input states (eigenvectors of σx,σy,σz) and we “mea-
universal NOT machine originallyintroducedinRef.[5]. sured” operators σx,σy,σz. The distance converges to the
theoretical value1/3 that correspondstotheoptimaluniver-
One wayhow to interpretthe “regularization”noise is
sal NOT.
to assume that the qubit channel is influenced by other
quantumsystems(the physicsbehind adilationtheorem
[5]).
Thereasonwhywehavetoconsideranoiseinarecon-
struction of quantum maps is that we deal with incom-
plete measurement statistics (e.g., test states are rep-
resented by finite ensembles). As a result, the recon-
structed assignment ̺ → ̺′ = E[̺ ] is determined not
j j j
only by the properties of the map E itself but also by
the character of the estimation procedure. In this situa-
tion, the map itself can be unphysical, but if we request
that the estimation procedure is such that the complete
positivity of the estimated map is guaranteed then the
resultoftheestimationisaphysicalapproximationofan
unphysicaloperation. Inordertoproceedweassumethe FIG.3. We present analytical as well as numerical results
method of maximum likelihood. of an approximation of a non-linear map Eθ given by Eq. (6)
for different values of the parameter θ. The numerical (“ex-
The maximum likelihood method is a general estima-
perimental”) results shown in the graph in terms of a set of
tion scheme [9,10] that has already been considered for
discretepointswitherrorbarsareobtainedviatheMML.For
a reconstructionof quantum operations fromincomplete
every point (θ), the nonlinear operation was applied to 1800
data. It has been studied by Hradil and Fiura´ˇsek [11],
input states that have been chosen randomly (via a Monte
and by Sachci [12] (criticized in Ref. [13]). The task of
Carlo method). These input states have been transformed
the maximum likelihood in the process reconstruction is
according to the nonlinear transformation (6). Subsequently
to find out a map E, for which the likelihood is maximal.
simulations of random projective measurements have been
By the definition we assume that the estimated map has
performed. With these “experimental” data a linear opera-
to be CP. Let us now briefly describe the principal idea
tionwasnumericallysearchedfor,whichmaximizestheprob-
in more details.
ability to obtain the same results. The resulting approxima-
Giventhemeasureddatarepresentedbycouples̺k,Fk tion specified by a value of λ (error bars shown in the graph
(̺ is one ofthe test states andF is a positive operator representthevarianceinoutcomesforsubsequentruns)trans-
k k
correspondingtotheoutcomeofthemeasurementusedin forms the original Bloch sphere as it is shown in an inset for
the kth run of the experiment) the likelihood functional aparticularvalueθ=3. Thefigure(a) corresponds toresult
is defined by the formula obtained by MML, and the figure (b) has been obtained via
analyticcalculations. WeseethattheoriginalBlochsphereis
transformedintoanellipsoid,oneaxisofwhichissignificantly
N N
longer than the remaining two axes, that are of a compara-
L(E)=−log p(k|k)=− logTrE[̺ ]F , (4)
k k
ble length. The mean of these two lengths corresponds to
kY=1 Xk=1
the parameter λ that specifies the map. The analytical ap-
proximationE˜ ofthenonlinearNPRmapischaracterizedby
θ
where N is the total number of “clicks” and we used the parameter λ that is plotted (solid line) in the figure as a
p(j|k)=TrE[̺k]Fj for a conditional probability of using function of theparameter θ.
test state ̺ and observe the outcome F . The aim is
k j
tofinda physicalmapE thatmaximizesthis function,
est
i.e. L(E )=max L(E). Thisvariationaltaskisusually Numerical results. Our approach is different from
est E
performed numerically. those described in Refs. [11–13] in the way, how we find
the maximum of the functional defined in Eq.(4). The
parametrization of E itself, as defined by Eq.(2), guar-
) antees the trace-preservingcondition. The CP condition
est 0.375 isintroducedasanexternalboundaryforaNelder-Mead
E
;T simplex scheme. Therefore, there is no need to intro-
ENO 0.350 duce Lagrange multipliers. We use the eigenstates of
(
d 1/3 σ ,σ ,σ as the collection of test states. The data are
x y z
generated as (random) results of three projective mea-
3 5 7 9 11
surementsσ ,σ ,σ appliedinordertoperformthe out-
x y z
log2 (N/18) put state reconstruction. In order to analyze the con-
3
vergence of the method we have performed the recon- λy,z →pz. In the process of minimalization the param-
struction for different number of clicks and compare the eter p behaves trivially and equals to one. It means that
distancebetweentheoriginalmapENOT andtheestimated E˜θ is of the form Eλ = diag{1,λ,λ,1}. Our task is to
map E . The result is plotted in Fig.2, where we can minimize the distance d(E ,E )= d̺|E [̺]−E [̺]| and
est θ λ θ λ
see that the distance converges to 1/3 as calculated in tofindthe physicalapproximationRE˜ ,i.e. the functional
θ
Eq.(3). For N = 100×18 clicks, i.e. each measurement dependence of λ on θ.
is performed 100 times per particular input state, the We plot the parameter λ that specifies the best physi-
algorithm leads us to the map calapproximationoftheNPRmapinFig.3. Inthesame
figurewealsopresentaresultofthemaximumlikelihood
1 0 0 0
estimation of the NPR map based on a finite number of
−0.0002 −0.3316 −0.0074 0.0203
E =  (5) “measurements”(for technicaldetails see the figure cap-
est 0.0138 −0.0031 −0.3334 0.0488
tions). We conclude that the MML is in an excellent
−0.0137 0.0298 −0.0117 −0.3336
  agreement with our analytical calculations.
In summary, we have shown that theoretical methods
which is very close [d(E ,E ) = 0.0065] to the best
est app
that are designed for a reconstruction of quantum maps
approximation of the NOT operation, i.e. E =
app
fromincompletedatecanbemodified(seethemethodof
diag{1,−1/3,−1/3,−1/3}.
maximumlikelihoodwithanexplicitlyincorporatedcon-
WeconcludethatforlargeN theMMLreconstruction
dition of the complete positivity) for an efficient deriva-
gives us the same result as a theoretical prediction de-
tion of optimal (given the figure of merit) approxima-
rived in Ref. [5]. ¿From here it follows that the MML
tions of un-physical maps. The derivation of physical
helps us not only to estimate the map when just in-
approximationsofun-physicalmapsbasedonreconstruc-
complete data are available, but also serves as a tool
tion schemes as discussed in the Letter is very natural.
to derive physical approximations of unphysical maps.
Thisviewissupportedbythefactthatnumerical“recon-
The reconstruction procedure guarantees that the esti-
structions”arecompatible withthe analyticalresults. It
mation/approximation is physical. In order to illustrate
shouldbepointedoutthatwehavechosentworelatively
the power of this approach we will find an approxima-
simple examples (the universal NOT gate that is a non-
tion of a non-linear quantum mechanical transformation
CP linear map and the nonlinear polarization rotation)
that is even “more” unphysical than the linear though
where analytical solutions can be found due to intrinsic
antiunitary U-NOT operation.
symmetriesoftheconsideredmaps. Assoonasthemaps
Nonlinear polarization rotation Letusconsideranon-
donotpossessuchsymmetriesanalyticalcalculationsare
linear transformation of a qubit defined by the relation
essentially impossible and the power of our method be-
[6]:
comes obvious.
Eθ[̺]=eiθ2hσzi̺σz̺e−iθ2hσzi̺σz. (6) ThiswasworksupportedinpartbytheEuropeanUnion
projects QUPRODIS and CONQUEST, and by the Slo-
Unlike the universal NOT this map is nonlinear. Four vak Academy of Sciences via the project CE-PI.
teststatesarenotsufficienttogetthewholeinformation
about the action of nonlinear maps. Consequently, the
fabricated data must use all possible input states (that
cover the whole Bloch sphere) as test states, but still we
useonlythree differentmeasurementsperformedonout-
comesthataresufficientforthestatereconstruction. We
note that a straightforward regularization via the addi- [1] M.A.Nielsen and I.L.Chuang, Quantum Computation
tion of noise cannot result in a CP map unless the orig- and Quantum Information(CUP, Cambridge, 2000)
inal map is not completely suppressed by the noise, i.e. [2] J.Preskill, Lecture Notes on Physics 229: Quantum The-
the regularization leads to a trivial result E = A. How- ory of Informationand Computation(1998),availableat
http://www.theory.caltech.edu/people/preskill/
ever, as we shall see, the maximum likelihood approach
[3] G.M.D’Ariano et al., Phys.Rev.Lett. 86, 4195 (2001)
givesusareasonableandnontrivialapproximationofthe
[4] J.F.Poyatos et al., Phys.Rev.Lett. 78, 390 (1997)
transformation (6).
[5] V.Buˇzek et al., Phys.Rev.A 60, R2626 (1999)
Firstly, we presentan analytic derivationof a physical
[6] C.Vinegoni et al., J.Opt.A 2, 314-318 (2000)
approximation of E . This approximation is the closest
θ [7] M.B.Ruskai et al., Lin.Alg. Appl. 347, 159 (2002)
physical map E˜ , i.e. d(E˜ ,E ) = min. The map E ex-
θ θ θ θ [8] M.Ziman et al., quant-ph/0406088
hibits two symmetries: the continuous U(1) symmetry [9] R.A.Fisher, Proc. Cambridge Phil. Soc. 22, 700 (1925)
(rotations around the z-axis) and the discrete σ sym-
x [10] Z.Hradiletal.,inQuantumEstimations: TheoryandEx-
metry (rotation around the x-axis by π). The physical periment (Springer-Verlag,Berlin, 2004)
approximationE˜θ shouldpossessthesepropertiesaswell. [11] J.Fiur´aˇsek et al., Phys. Rev.A 63, 020101(R) (2001)
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ofthe Blochvectorarerestrictedasfollowsx→λx,y → [13] J.Fiur´aˇsek and Z.Hradil, quant-ph/0101048
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