Minimum Relative Entropy for Quantum Estimation: Feasibility and General Solution

0.19 MB·English

by  Mattia Zorzi

DRAFT 1
Minimum Relative Entropy for Quantum
Estimation: Feasibility and General Solution
Mattia Zorzi, Francesco Ticozzi and Augusto Ferrante
Abstract—We propose a general framework for solving quan- situation in which the system dimension is large and the
tum state estimation problems using the minimum relative availablemeasurementareinsufficienttocompletelydetermine
3
entropy criterion. A convex optimization approach allows us to
1 the state.
decidethefeasibilityoftheproblemgiventhedataand,whenever
0 To address this second issue, a typical approach in both
necessary, to relax the constraints in order to allow for a physi-
2 callyadmissiblesolution.Buildingontheseresults,thevariational the classical and the quantum world is to resort to a MAX-

n analysis can be completed ensuring existence and uniqueness of ENT principle [13], [14], [15], [16], [7], [17], where one
a the optimum. The latter can then be computed by standard, opts for a “maximum ignorance” criterion on the choice of
J efficient standard algorithms for convex optimization, without
parameters that are not uniquely determined by data. The
8 resorting to approximate methods or restrictive assumptions on
MAXENT estimation can indeed be seen as a particular
2 its rank.
case of minimum relative entropy estimation, [18], where the

] Index Terms—Quantum estimation, Kullback-Leibler diver- information-theoretic pseudo-distance of the estimated state
h gence, Convex optimization
with respect to some a priori state is minimized subjected
p
to a set of constraints representing the available data [19],
-
t I. INTRODUCTION [20], [21], [22], [23]. This a priori information introduces a
n
a Quantum devices implementing information processing new ingredient with respect to typical maximum-likelihood
u tasks promise potential advantages with respect to their clas- methods for quantum estimation [7], [9].
q sical counterparts in a remarkably wide spectrum of applica- A quantum minimum relative entropy method for state
[
tions, ranging from secure communications to simulators of estimationhasbeendiscussedin[21],[24],whereapproximate

1 large scale physical systems [1], [2], [3]. solutionsto minimumrelativeentropyproblemsare provided:
v In order to exploit quantum features to the advantage of a the estimates are shown to be good approximation of the
8
desiredtask,tremendouschallengesareposedtoexperimental- optimalsolutionwhenthisiscloseenoughtotheprior.Onthe
5
istsandengineers,andmanyofthesehavestimulatedsubstan- other hand, a way towards the computation of the exact opti-
6
6 tial theoretically-oriented research. Which particular problem mal solution is indicated in [19]: Georgiou has analyzed the
. iscriticaldependsonthephysicalsystemunderconsideration: MAXENT problem for estimating positive definite matrices,
1
from optical integrated circuits to solid-state devices, the providing a generic form for the optimal solution, parametric
0
3 tasks in the device engineering, protection from noise and in the Lagrange multipliers. He has also observed that the
1 control are manifold [3], [2], [4], [5], [6]. However, quantum results can be extendedto the more generalminimumrelative
v: estimation[7]problemsareubiquitousinapplications,beitin entropy problem.
i testingtheoutputofaquantumalgorithm,inreconstructingthe We shall here extend Georgiou’s approach, proving exis-
X
behaviorof a quantumchannelor in retrievinginformationat tence, uniqueness and continuity of the solution with respect
r thereceiverofacommunicationsystem[3],[8],[9],[10],[11], to the measured data, when a generic prior is considered.
a
[12]. In this paper we focus on state estimation problem for The solution can then be computed by standard numerical
finite-dimensionalquantumsystem,namelythereconstruction methods. However,the approachreturnsa meaningfulanswer
of a trace-one, positive semidefinite matrix given from data, only when there is a full-rank admissible solution among the
and in particular on an estimation method that addresses two states compatible with the data. While this appears to be a
criticalproblemsformostreal-worldsituations.Thefirst arises reasonableassumptionasquantumfull-rankstatesaregeneric,
when only a small set of potentially noisy data is available, this is no longerthe case wheneverthe unknownstate is pure
yieldingnophysically-acceptablesolution;thesecondregards or near the boundary of the physical state set. In fact, it is
easy to picture realistic scenarioswhere the effect of noisy or
Work partially supported by the Italian Ministry for Education and biased measurements might actually force the solutions to be
Research (MIUR) under PRIN grant n. 20085FFJ2Z “New Algorithms and
on the bounduary, or even outside of the admissible set [25],
Applications of System Identification and Adaptive Control”, and by the
UniversityofPaduaundertheQUINTETprojectoftheDept.ofInformation [9].Inthelattercasetheconstrainedoptimizationproblem“as
Engineering andtheQFUTUREstrategic Project. is” is not feasible, and one has to relax the constraints.
M. Zorzi and A. Ferrante are with the Dipartimento di Ingegneria
Ourstrategyto solve these issuesis organizedas follows:a
dell’Informazione, Universita` di Padova, via Gradenigo 6/B, 35131 Padova,
Italy ([email protected],[email protected]), generalsettingforposingthefeasibilityproblemandquantum
andF.TicozziiswithwiththeDipartimentodiIngegneriadell’Informazione, minimum relative entropy with data corresponding to linear
Universita` di Padova, via Gradenigo 6/B, 35131 Padova, Italy
constraints is presented in Section II. In Section III, we
([email protected]), and the Deptartment of Physics and
Astronomy,DartmouthCollege, 3127Wilder, Hanover, NH(USA). propose a way to reformulate the feasibility analysis as a
DRAFT 2
convex optimization problem. The solution of this ancillary 3) Outcome frequencies for general measurements: con-
problem, for which we provide a numerical approach in sider repeated measurements of a Positive-Operator Valued
Appendix A, also indicates an optimal way to perturb, or Measure (POVM), that is generalized measurements that can
relax theconstraintsinorderto allowforadmissiblesolutions be used to describe indirect measurements on a system of
to the estimation problem. We also show how the way in interest [3]. A POVM with M outcomes, say k = 1,...M,
which the constraints are relaxed can be tailored to match is associatedto a set of non-negativeoperators Q M such
{ k}k=1
the error distribution or the level of noisy we assume on the that Q =I, playingthe role of resolutionof the identity
k k
measurements. Once the feasibility analysis returns a positive for pProjective measurements. The probability of obtaining the
answer, our approach directly leads to the construction of a k-th outcome can be computed by q = tr(ρQ ), and ex-
k k
reducedproblemforwhichthereexistsapositivedefinitestate perimentallyestimated by qˆ after K repeated measurements.
k
satisfying the givenconstraints.In Section IV, we address the This case in fact includes the first one, and the generalization
corresponding(reduced, if needed) minimum relative entropy to multiple POVM is straightforward.
problem, showing it admits a unique full-rank solution. The Inall these scenarios,dataare providedas a set ofrealval-
latter can be computed from the closed-form solution of the uesrepresentingestimatesfˆ ofquantitiesf (thatcanbeeither
i i
primal problem, and a standard numerical algorithm to find p , O or q ), each associated to the state through a linear
i i i
h i
thecorrespondingLagrangemultipliersissuggested.Then,the relation of the form f = tr(ρZ ), where Z have the role of
i i i
solution to the non-reduced, original problem is immediately Π ,O orQ describedabove.Clearlyfˆ f =tr(Z ρ)with
i i i i i i
→
obtained. Some concluding remarks and future directions and probability one as N . This framework is quite general,
→∞
applications are summarized in Section V. andcanbeadaptedtoincludeanycaseifthedataaregivenas
linearconstraints.Anothersignificantsituation that fits in this
framework is when reduced states of a multipartite systems
II. PROBLEM SETTING are available as data [27], [28]. Finally, by the well-known
Choi-Jaimlokowskii isomorphism [29], the same setting, and
A. Quantum States and Measurement Data methods for solution, can be adapted to include estimation of
quantum channels, or quantum process tomography[7], [9].
Consideraquantumn-levelsystem.Itsstateisdescribedby
From a theoretical viewpoint, ρ can be in principle recon-
adensityoperator,namelybyapositivesemidefiniteunit-trace
structed exactly from at least n2 1 averages f = tr(ρZ )
matrix − i i
i = 1...n2 1 when Z ...Z are observables which
1 n2 1
do not carry r−edundantinformation,−namely they form a basis
ρ∈Dn = ρ∈Cn×n |ρ=ρ† ≥0, tr(ρ)=1 , (1) forthe space of traceless Hermitian matrices. In any practical
(cid:8) (cid:9)
application, however, one has to face the following issues:
whichplaystheroleofprobabilitydistributionintheclassical
1) Accurate estimates fˆ of f are only obtained by aver-
probabilityframework.Note that a density matrix dependson i i
n2 1 real parameters. aging overa large quantity of trials; Often only a small
− set of trials is available, and/or the data are subject to
Inthisworkwewillbeconcernedontheproblemofrecon-
significant errors;
structing an unknown ρ from a set of repeated measurement
2) Thenumberofobservablesrequiredfora uniquerecon-
data. This is of course an estimation problemin the statistical
struction of ρ grows quadratically with respect to the
language,whileinthephysicscommunityitisusuallyreferred
dimension of the quantum system, and exponentiallyin
to as state tomography [7].
thenumberofsubsystems.Typicallyonlyasmallsubset
We assume that data are provided in one of the following
of these is available;
forms:
We here analyze the estimation problem when these two
1) Outcome frequenciesfor projective measurements: con-
aspects are taken into account. The first one will lead us
siderrepeatedmeasurementsofa(Hermitian)observable[26],
to consider the feasibility problem, that is, if the problem
O = o Π , where Π is the associated spectral family
k k k { k} admits a physically admissible solution for the given data.
opfososribtPhleogoountaclopmreosjecattioenasc.hTmheeasspuercetmruemnt,{aonkd} rtheperefrseeqnutsenthcye Since errors may affect the fˆi, the reconstructed state may
not be positive semidefinite, or a valid state that satisfies the
of the k-th outcome given a state ρ can be computed as
constraintsmightnoteven exist. The second issue generically
p = tr(ρΠ ). After K measurements of O, we assume we
k k leads to a estimation problem where more than one state
are provided with some experimental estimates of p , i.e. the
k satisfy the constraints, and thus an additional criterion has
experimental relative frequencies of occurences pˆ =#(O =
k to be introduced to arrive at a unique solution. As we said,
o )/K, with #(O = o ) the number of measurements that
k k a typical strategy in this setting is to introduce an entropic
returned outcome o .
k functional,e.g.relativeentropywithrespecttosomereference
2) Observable averages: consider a set of n measured
o state representing a priori information.
observables,representedbyHermitianmatricesO ,wherenow
i
we only have access to the mean values of the outcomes,
B. Statement of the Main Problems
denotedby O (andwithpossibleoutcomeso ),thatcanbe
i i,k
h i
theoreticallycomputedas O :=tr(ρO )andexperimentally Consider the setting described above, where we want to
i i
estimated by Oˆ = oh pˆi . estimate the state of an n-dimensionalquantum systems from
h ii k i,k k
P
DRAFT 3
the real data fˆ p , experimentalestimates of the quantities space generated by the observed operators, span Z ...Z .
{ i}i=2 { 1 p}
f = tr(Z ρ), for the Hermitian matrices Z ...Z , with Thus, by applying the Gram-Schmidt process, starting with
i i 2 p
p n2 1. In addition to these, we introduce an auxiliary X = 1 Z = 1 I, we can compute an orthonormal basis
ob≪servab−le Z = I and the corresponding estimate fˆ = 1. fo1r it: √n 1 √n
1 1
In this way, we include the linear constraint tr(ρ)=1 in the i 1
−
constraints associated with the “data”. We wish now to solve X :=αiZ + αiX , i=2...m p. (5)
i i i l l ≤
the following problem. Xl=1
Problem 1: Given Zi and fˆi , i=1...p, find: By linearity, by associating these basis elements to the esti-
{ } { }
ρ , suchthat ρ 0, fˆ =tr(ρZ ), i=1,...,p. mates
n i i
∈H ≥ (2) f¯ := 1
1
Here, denotesthevectorspaceofHermitianmatricesof √n
n
H
dimensionequalto n. Notice that, if we removethe positivity i 1
constraintρ 0, all otherconstraintsarelinearandidentifya f¯i := αiifˆi+ − αilf¯l, i=2...m, (6)
≥
hyperplanein n. To our aim it is convenientto first address Xl=1
H
a simpler problem: let we obtain a new yet equivalent set of constraints:
:= ρ n ρ 0, fˆi =tr(ρZi) f¯i =tr(ρXi), i=1...m. (7)
S n ∈H | ≥ o
be the set of the density matrices which solve Problem 1. Note that, I span X1...Xm .
∈ { }
Problem 2 (Feasibility): Determine if is not empty. Let Y1...Yn2 m be an orthonormal completion of
When the problem is feasible, in geneSral contains more X1...Xm to a b−asis of n. Accordingly, all the Hermitian
than one solution, and in principle any solutSion in fits the matrices,andin particularHdensityoperators,canbeexpressed
data. We focus on choosing a solution that has mSinimum as
m n2 m
distance with respect to an a priori state. In the same spirit −
ρ= α X + β Y , (8)
of MAXENT problem, this corresponds to give maximum i i i i
Xi=1 Xi=1
priority to fitting the data, and then choosing the admissible
with tr(ρX ) = α . In particular, all the Hermitian matrices
solution that is the closest (in the relative-entropy pseudo- i i
satisfying the linear constraints in (7) depend on n2 m
distance) to our a priori knowledge on the systems. To
−
parameters β =[β ...β ]:
this aim, consider the (Umegaki’s) quantum relative entropy 1 n2 m
−
between ρ n, and τ int( n) [30]: n2 m
∈D ∈ D −
ρ=ρ˜ + β Y (9)
S(ρ τ)=tr(ρlogρ ρlogτ). (3) 0 i i
k − Xi=1
Assuming τ in the interiorand with the usual conventionthat where we have defined the (not necessarily positive) pseudo-
0log(0)=0,wedonothavetoworryaboutunboundedvalues state associated to the constraints:
of S(ρ τ). m
k
Problem 3 (Minimum relative entropy estimation): Given ρ˜ = f¯X . (10)
0 i i
the observables Z1...Zp and the corresponding estimates Xi=1
fˆ1...fˆp, solve In the light of this observation, the feasibility problem
minρi≥m0ize S(ρkτ) subject to tr(ρZi)=fˆi, i=1...p. (4) Rconn2s−ismts sinucchhetchkaitngρ if≥the0r.eTeoxistthsisatailmea,stwoeneinvtreocdtourceβa∈n
Here, τ represents the a priori information on the considered auxiliary problem. Intuitively, the idea is the following: given
quantum system. We set σ =I if no information is available. any Hermitian matrix ρ˜0, there always exists a real µ such
Inthis situation,(3) isthe oppositeofthe quantumentropyof that ρ˜0 + µI is positive definite. More precisely, if ρ˜0 is
ρ,andweobtainaMAXENTproblem.Notethat,thesolution not positive definite already, it is easy to see that such a µ
to the Problem above may be singular. will need to be positive. On the other hand, if ρ˜0 is already
positive, the perturbed matrix remains positive semi-definite
III. FEASIBILITYANALYSIS for some small, negative µ. Studying the minimal µ that
correspond to a positive semidefinite matrix offers us a way
A. An auxiliary problem
to understand whether our constraints allow for physically
We start by addressing the feasibility problem, i.e. to admissible solutions.
determine when is not empty. In addition, whenever the Let us formalize these idea: we define c :=
S
problem is not feasible, we show how to determine a suitable 0 ... 0 1 T Rn2 m+1, and
−
perturbation of the fˆ that makes our problem feasible. We ∈
i (cid:2) (cid:3)
will show that the{co}rresponding only contains singular n2 m
−
S H(v):=ρ˜ + v Y +v X
density matrices when the constraints are relaxed. 0 i i n2 m+1 1
Note that, the constraints are linear in fˆ and Z , and can Xi=1 −
i i
be linearly combined: αfˆi + βfˆk = tr[ρ(αZi + βZk)] for withv = v1 ... vn2 m+1 andweconsiderthefollow-
each α,β R 0 and i,k = 1...p. Consider the vector ing minim(cid:2)um eigenvalue−problem(cid:3).
∈ \{ }
DRAFT 4
Problem 4: Given ρ˜0 as in (10) and Y1...,Yn2 m an Proof:NotethatG(v1,...,vn2 m):=ρ˜0+ in=2−1mviYi
orthonormal completion of span X1...Xm , solve − represents the parametric family of−Hermitian mPatrices (not
{ }
necessary positive semidefinite) satisfying constraints in (7).
minimizecTv subjectto v := v H(v) 0 .
∈I { | ≥ } Define ε := vn2 m+1, thus Problem 4 can be rewritten in
(11) the following−way:−
Lemma 3.1: Problem 4 always admits solution.
ε
Proof: First of all, notice that Problem 4 is a convex maximizeε subject to G(v1,...,vn2 m) I. (12)
optimizationproblem,andthe objectivefunctioncTv islinear − ≥ √n
and continuous over the set . Then, the proof is divided in Let ε◦ = µ be the solution of the above problem. If
three steps. I ε◦ < 0, the−parametric family G(v1,..., vn2 m) does not
Step1:We showthatcTv =v is boundedfrom below contains positive semidefinite matrices, accordi−ngly Problem
n2 m+1
boansIis:sainndceIX1,..s.p,aXnmX,Y11,,......,,X−Ymn2−,mthfeormmastarinceosrthoYniormarael G1(dvo1e,s..n.o,tvna2dmmit)scoolunttiaoinn.sIaftεle◦a≥st o0n,ethpeospiatirvaemseetrmicidfeafimniiltye
traceless. Thu∈s, { } { } matrixandPro−blem1admitssolution.Moreoverif ε◦ >0,the
parametric family contains at least one matrix ρ ε◦ I > 0
n2 m ≥ √n
tr[H(v)] = tr[ρ˜ + − v Y +v X ] which is positive definite. On the contrary, for ε◦ = 0 there
0 i i n2 m+1 1 only exist positive semidefinite matrices which are singular.
Xi=1 −
=tr(ρ˜ )+√nv =1+√nv
0 n2 m+1 n2 m+1
− −
and tr[H(v)] 0 for each v . Hence, An effective numerical approach for the solution to the
≥ ∈ I
cTv =v 1 for each v . problem is described in Appendix A.
Step n22−:m+1 ≥Le−t √n us con∈siIder v = In the light of the previous result, if µ < 0 Problem 1 is
0
0 ... 0 √n( λ (ρ˜ )+1) where λ (ρ˜ ) feasible and contains at least one positive definite solution.
min 0 min 0
− ∈ I S
(cid:2)denotes the minimum eigenvalue(cid:3) of ρ˜0. Accordingly, As we will see in Section IV, this condition ensures that a
cTv = √n( λ (ρ˜ )+1) and Problem 4 is equivalent to minimum relative entropy criterion will lead to an admissible
0 − min 0
minimize cTv overthe closed sublevelset = v H(v) solution in . The remaining cases need to be studied more
0
I { | ≥ S
0, 1 v √n( λ (ρ˜ ) + 1) . We carefully.WestartbyshowinghowtomakeProblem1feasible
want−to√nsho≤w tnh2a−tm+1is≤bounde−d amnidn a0ccording}ly⊂coImpact when it is notbe so forthe given constraints. It turns outthat
0
I
(recall that we are working in a finite dimensional space). a minimally relaxed problem is feasible and only contains
S
This can done by proving that a sequence vk such that singulardensitymatrices.Next,wedealwiththecaseinwhich
k 0
vk cannot belong to . It is the(cid:8)refo(cid:9)re≥sufficient to Problem1isfeasibleandallitssolutionsaresingular,showing
0
kshowk t→hat∞theminimumeigenvaIlueoftheassociatedHermitian how they all share a minimal kernel and how to construct
matrix H(vk) tends to as vk with v a reduced problem with a full-rank solution for which the
m2 n+1
bounded. Note that the −af∞fine mkap vk → ∞H(v) is inje−ctive, minimum relative entropy methods work.
7→
since Y ...Y ,X are linearly independent.Accordingly
1 n2 m 1
H(vk) −as vk . Since H(vk) is an Hermitian B. Forcing the feasibility condition (case µ>0)
k k → ∞ k k → ∞
matrix, H(vk) has an eigenvalue η such that η as
k k The parameter µ given by the auxiliary problem described
| | → ∞
vk . By construction tr[H(vk)] = 1+√nvk
k k → ∞ n2 m+1 above reveals if the original problem is feasible, but also
andvnk2 m+1 isboundedinI0.Thustr[H(vk)]<∞,na−mely suggests an “optimal” way to relax unfeasible constraints so
the sum−of its eigenvalues is always bounded. Thus, there
thattheymakeProblem1feasible.Infact, fromthe definition
exists an eigenvalue of H(vk) which approaches −∞ as of µ, we know that there exist v1...vn2 m R such that
k . So, is bounded. − ∈
0
Ste→p 3∞: SinceIcTv is continuous over the compact set 0, by n2−m
Weiestrass’ theoremwe concludethat cTv admits a minIimum ρ˜µ :=ρ˜0+ viYi+µX1 ≥0,
point over . Xi=1
0
I and:
tr(ρ˜ )=1+√nµ.
We need to take into account that the vector which mini- µ
mizes cTv over may not be in general unique. However, to From this positive operator, in order to obtain a density
I
our aim we are more interested in the sign of the minimum. operator, we only need to normalize the trace by defining:
Proposition 3.1: Let µ = mincTv. Then, the following
1
v ρ := ρ˜ (13)
facts hold: ∈I 1+√nµ µ
12)) IIff µµ><00,, tthheennPPrroobblleemm11iissnfeoatsfibealesibanled there exists at = 1 X + m f¯i X +n2−m vi Y .
1 i i
√n 1+√nµ 1+√nµ
least one positive definite matrix satisfying constraints Xi=2 Xi=1
in (7) Thisimpliesthattheoriginalproblemcanbemadefeasibleby
3) Ifµ=0,thenProblem1 isfeasibleandallthematrices uniformly,“isotropically”contractingthedata f¯ ofafactor
i
satisfying constraints in (7) are singular. 1/(1+√nµ) and, in light of the fact that µ i{s a}solution to
DRAFT 5
Problem4,thatthisistheminimumamountofcontractionthat the perturbed constraints
makes Problem 1 feasible. Moreover,the correspondingset
1
only contains singular solutions. S f¯1 =
√n
However, the entries in the data set f¯ may differ in
i 1
their reliability, and one would like to b{e a}ble to relax the f¯i = 1+√nµ fˆm′ , i=2...m. (17)
correspondingconstraintsaccordingly.This is complicatedby ′
the fact that the original Z may notbe orthogonal,and the Thus, the corresponding Problem 1 is feasible and only
data we are contracting{arei}in fact the linearly transformed contains singular solutions. S
output of the Gram-Schmidt orthonormalization described
above. C. Case µ=0
Thisweighedrelaxationcanberealizedasfollows:consider In the limit case µ=0, not only all solutions are singular,
the initial setting of Section II, where we have p observables but they share a key property.
Z1...Zp (not necessarily orthonormal), with Z1 = I and Proposition 3.2: Assume that, with the definition above,
fˆ1 = 1. Define the reliability indexes 0 < d2...dp 1 µ = 0. Then there exists a kernel which is common for
associated to each observable Z2...Zp. More precisely,≤the all ρ . K
more fˆi is reliable, the closer to one di is. This information P∈roSof: Let us assume µ = 0, accordingly Problem 1
can be extracted, for example, from an error analysis on the does only admit singular solutions, with dimker(ρ) > 0
measurementprocedures,with di associatedtothenormalized ρ . Pick a solution ρ◦ with kernel of min-
reciprocal of the variances. ∀imal d∈imSension. Suppose by con∈tradSiction that there exists
WhenweobtaintheorthonormalgeneratorsX1...Xm,the ρ¯ such that ker(ρ◦) * ker(ρ¯). Taking into account
∈ S
Gram-Schmidt process induces a linear transformation on the p (0,1), we define ρ := pρ +(1 p)ρ¯ . Accordingly
◦
original estimates fˆ,...,fˆ: dim∈ ker(ρ) < dimker(ρ ) which is a−contr∈adSiction, since ρ
1 p ◦ ◦
has kernel with minimal dimension on . We conclude that
S
f¯1 fˆ1 K=ker(ρ◦)⊆ker(ρ) ∀ρ∈S.
. .
 .. =T  ..  (14) This directly implies the following block-form for all the
 f¯m   fˆp  solCuotiroonlslatroyP3r.o1b:leLmet1ρ. be a solution with minimal
◦
∈ S
kernel and consider its block-diagonalform
1 0 K
where T = √n Rm×p. ρ 0
Inorderto(cid:20)mTo1difyTth2e(cid:21)da∈ta fˆ accordingtotheirreliability ρ◦ =U(cid:20) 0◦1 0 (cid:21)U†,
i
{ }
indexes, we define the new set of data: where U is a unitary change of basis consistent with the
Hilbert space decomposition = so that ρ > 0.
H K⊥ ⊕K ◦1
fˆ Then, the set of all the solutions of Problem 1 is
fˆ 1
 ...1′ =T kd2...fˆ2 . (15) S =(cid:26)ρ=U(cid:20) ρ01 00 (cid:21)U† |ρ1 ≥0, f¯i =tr(ρXi)(cid:27). (18)
 fˆ   
 m′   kd fˆ  AsconsequenceofCorollary3.1,wecanfocusonareduced
 p p 
version of Problem 1, by considering optimization only on
owfhearefakcto>r kmdaix>{d−i11a}c.cIonrdtihnigs twhaeyirfrˆe2l,i.a.b.il,iftˆyp ianrdeeaxmesp.lTifiheids athpeplsiucapbploertsionfceρ◦ρ,◦1foisr pwohsiictihvethdeemfininitiem,usemerSeelacttiivoeneInVt.ropy is
will allow for the maximum contraction to be applied to the
most noisy estimates. IV. STATEESTIMATIONWITHMINIMUMRELATIVE
In orderto computethe minimum µ thatmakes the original ENTROPYCRITERION
problem feasible perturbing the data consistently with their A. Reduced Problem
reliability indexes, we can solve Problem 4 with respect to
In the previous part of the paper we showed how to check
the new pseudo-state:
the feasibility of Problem 1 given the constraints associated
to the data and, if needed, how to relax the constraints in
m
ρ˜ = fˆX . (16) such a way that the corresponding Problem 1 is feasible. In
′0 i′ i
general, however, the set of solutions is not constituted by
Xi=1 S
onlyoneelement.Inthissection,weshowhowtochoose,and
It is easy to see that if the original problem was unfeasible, thencompute,asolutionin accordingtheminimumquantum
S
thismodifiedproblemisunfeasibleaswellforklargeenough, relative entropy criterion.
since all the fˆ corresponding to traceless operators are mul- Given the results of the previous sections, we can assume
i
tiplied for a factor Kd >> 1. Let µ > 0 be the parameter that either Problem 1 admits at least one (strictly) positive
i ′
givenby the auxiliaryproblemwhen ρ˜ is considered.By the definite solution, or we can resort to a reduced problem for
′0
resultsoftheprevioussubsectionand(13)above,weconsider which a full rank solution exists. In fact, if only contains
S
DRAFT 6
singular matrices (case µ = 0, or after relaxation of the is the adjoint operator of L. Define f¯= f¯ ... f¯ T.
1 m
constraints), by Corollary 3.1 we have that the set of solution SinceProblem5isaconstrainedconvexop(cid:2)timizationproble(cid:3)m,
is we take into account its Lagrangian
= ρ=U ρ1 0 U ρ 0, f¯ =tr(ρX ) (19) (ρ,λ) = tr(ρlogρ ρlogτ) λ,f¯ L(ρ)
S (cid:26) (cid:20) 0 0 (cid:21) † | 1 ≥ i i (cid:27) L − − −
= tr(ρlogρ ρlogτ)+(cid:10)L (λ),ρ (cid:11)λ,f¯
∗
− h i−
forsomeunitarychangeofbasisU consistentwiththeHilbert = tr[ρ(logρ logτ +L (λ))] λ,f¯(cid:10) ((cid:11)27)
∗
space partition = . Accordingly for each ρ , − −
H K⊥ ⊕K ∈ S (cid:10) (cid:11)
constraints in (7) can be rewritten in the following way where λ Rm is the Lagrange multiplier. Note that (,λ)
∈ L ·
is strictly convex over where denotes the cone
f¯i =tr(ρXi)=tr(cid:18)U(cid:20) ρ01 00 (cid:21)U†Xi(cid:19)=tr(ρ1X¯i) (20) of the positive definite mHant,r+ices. ThusH,nit,s+minimum point is
given by annihilating its first variation
I
where X¯ := I 0 U X U with n < n. δ (ρ,λ;δρ) = tr[(logρ+I logτ +L (λ))δρ](28)
i † 1 (cid:20) 0 (cid:21) ∈ Hn1 1 L − ∗
Accordingly P(cid:2)roblem (cid:3)1 is equivalent to the corresponding
foreachdirectionδρ .Accordingly,theuniqueminimum
reduced problem with X¯ ...X¯ and f¯ ...f¯ . The corre- ∈Hn
1 m 1 m point for (,λ) is
sponding set of solutions is L ·
= ρ ρ 0, f¯ =tr(ρ X¯ ) (21) ρ(λ)=elogτ−I−L∗(λ) (29)
S1 1 ∈Hn1 | 1 ≥ i 1 i
(cid:8) (cid:9) and
which contains the positive definite solution ρ . Once chosen
◦1 (ρ(λ),λ) (ρ¯,λ), ρ¯ . (30)
a solution ρˆ1 1, the corresponding original solution is L ≤L ∀ ∈Hn,+
∈S
If there exists λ such that ρ(λ ) , i.e. f¯ = L(ρ(λ )),
ρˆ=U(cid:20) ρˆ01 00 (cid:21)U†. (22) then (30) implies◦ ◦ ∈ S ◦
In the effort of keeping a simple notation, we will not S(ρ(λ◦) τ) S(ρ¯ τ), ρ¯ . (31)
k ≤ k ∀ ∈S
distinguish between the reduced and the full problem in the
Thus,ifweareabletofindλ Rmsuchthatf¯ L(ρ(λ ))=0,
followingdiscussion,thereforeusingρforeitherthefullorthe ◦ ∈ − ◦
then ρ(λ ) is the unique solution to Problem 5. This issue is
reduced state, X for either the full or reduced observable, ◦
andnforthedi{mein}sionofthefullHilbertspaceorthereduced solved by considering the dual problem wherein λ◦ (if there
exists) maximizes the following functional over Rm
oneasneeded.Wecanconsiderthefollowingsimplerproblem,
restricted to strictly positive matrices: inf (ρ,λ) = (ρ(λ),λ)
resPproonbdlienmg5es:tiGmiavteens tf¯h1e..o.bfs¯mer,vasbollevseX1...Xm and the cor- ρ∈Hn,+L = L−tr(elogτ−I−L∗(λ))− λ,f¯ .(32)
(cid:10) (cid:11)
minimize S(ρ τ) subject to tr(ρX¯i)=f¯i, i=1...m. The existence of such a λ◦ is proved in Section IV-C.
ρ>0 k Moreover, we suggest how to efficiently compute it.
(23)
Remark 4.1: When µ=0, only contains singular matri-
S
ces.Insteadofconsideringthereducedproblemaswedid,one
B. Lagrangian and Form of the Full-Rank Solution couldtrytoconsiderProblem3withrelaxedconstraint ρ 0.
≥
Now we are ready to derive a solution method for problem In this situation the Slater’s condition [31, 5.2.3], however,
the entropic criterion. Consider the linear operator associated does not hold because does not contain positive definite
S
to the above constraints: matrices.Hence,we cannotconcludethatρ(λ ) isthedesired
◦
solution of the primal problem. Moreover, for any λ, would
L : n Rm be ρ(λ ) > 0. This means that ρ(λ ) is not the solution to
H → ◦ ◦
tr(ρX1) Problem 3.
.
ρ  .. . (24)
7→
 tr(ρXm)  C. Dual Problem: Existence and Uniqueness of the Solution
 
Given λ= λ1 ... λm T Rm and ρ n, The dual problem consists in maximizing (32) over Rm
∈ ∈H which is equivalent to minimize
(cid:2) (cid:3)
m m
L(ρ),λ = λitr(ρXi)=tr(ρ λiXi)= ρ,L∗(λ) J(λ)=tr(elogτ−I−L∗(λ))+ λ,f¯ . (33)
h i h i
Xi=1 Xi=1 (cid:10) (cid:11)
(25) Thisfunctionalwillbereferredtoas dualfunctionthroughout
where this Section. Before to prove the existence of λ which
◦
minimizes J we need to introduce the following technical
L∗ : Rm →Hn results.
m
λ λ X (26) First of all, note that λ⊥,f¯ = 0 for each λ⊥
i i ∈
7→Xi=1 [RangeL]⊥. In fact, if λ⊥(cid:10)∈ [R(cid:11)angeL]⊥ = kerL∗, then
DRAFT 7
L (λ ) = 0. Since = , there exists ρ ofallnotethatthe minimumsingularvalue α ofL restricted
∗ ⊥ n,+ f n,+ ∗
such that f¯=L(ρ ).ST∩huHs, 6 ∅ ∈ H to RangeL=[kerL ] is strictly positive, accordingly
f ∗ ⊥
λ⊥,f¯ = λ⊥,L(ρf) = L∗(λ⊥),ρf =tr(L∗(λ⊥)ρf)=0. kL∗(λi)k≥αkλik→∞. (40)
(cid:10) (cid:11) (cid:10) (cid:11) (cid:10) (cid:11) (34) This means that L (λ ), which is an Hermitian matrix, has
∗ i
We conclude that λ⊥ does not affect J, i.e. at least one eigenvalue βi such that βi approach infinity.
| |
If β , then the first term of J tends to infinity and
J(λ+λ⊥)=J(λ), ∀λ⊥ ∈[RangeL]⊥. (35) domiin→ates−t∞he second one, accordingly J(λi) . In the
→ ∞
Wemaythereforerestrictthesearchoftheminimumpointfor remaining possible case no eigenvalue of L(λi) approaches
J over RangeL. and βi . Thus, L∗(λi) MI where M R is
−∞ → ∞ ≥ ∈
Proposition 4.1: J is strictly convex over RangeL. a finite constant and the first term of J takes a finite value.
Since is not empty, there exists ρ such
Proof:SinceJ istheoppositeof (ρ(λ),λ), itisconvex n,+ f n,+
over Rm. The first and the secondLvariation of J(λ) in that f¯S=∩L(Hρf) and ∈ H
direction δλ∈Rm are: λi,f¯ =hL∗(λi),ρfi=tr(ρf21L∗(λi)ρf12)≥M, (41)
δJ(λ;δλ) = tr 1(e(1−t)(logτ−I−L∗(λ))L∗(δλ) (36) wher(cid:10)e we (cid:11)exploited the fact that tr(ρf)=1. This means that
− Z0 λ ,f¯ cannot approach . Finally, ρ21L (λ )ρ12 ,
·et(logτ−I−L∗(λ)))dt+ δλ,f¯ (cid:10)beciaus(cid:11)e ρ > 0. It fol−lo∞ws that ρ21Lk f(λ ∗)ρ21i, afnkd →hen∞ce
1 (cid:10) (cid:11) f f ∗ i f
= tr elogτ−I−L∗(λ)dtL∗(δλ)+ δλ,f¯ L∗(Λi), have at least one eigenvalue tending to ∞. Accord-
− Z0 (cid:10) (cid:11) inglyJ(λi) asi .Weconcludethat isbounded
= tr(elogτ I L∗(λ)L (δλ))+ δλ,f¯ (37) and accordin→gly∞compa→ct.∞By Weierstrass’ theoMrem, J admits
− − ∗
−
(cid:10) (cid:11) minimum point over .
M
δ2J(λ;δλ) = tr[ 1e(1−t)(logτ−I−L∗(λ))L∗(δλ) Falignoalrliyth,mλ◦wmithaybabcektcraocmkpinugte,dsebeySee.cgt.ioenmVplIoyining[3a2],Nwewhtiocnh
Z
0 globally converges.
et(logτ I L∗(λ))L (δλ)dt].(38)
− − ∗
·
V. CONCLUSIONS
Here, we exploited the expression for the differential of the
Theproposedsetofanalyticresultsandalgorithmsprovides
matrix exponential (see [19, Appendix IA]). Define Q =
t
et(logτ I L∗(λ)) which is positive definite for each t R. a general method to find the exact minimum relative entropy
− −
∈ estimate of a quantum state under general assumptions, that
Thus,
is, without requiring the constraints to include a full-rank
1 solution to begin with. A generalsolution to the problem was
δ2J(λ;δλ) = tr(Q L (δλ)Q L (δλ))dt (39)
Z0 1−t ∗ t ∗ missing for quantum estimation, and our feasibility analysis
1 is of interest for the classical case as well. Summarizing, we
1 1
= Z0 tr(Qt2L∗(δλ)Q1−tL∗(δλ)Qt2)dt≥0. hoafvPerporbolepmose1d,:w(i)hiAchnuismoefricinatlemreestthoondthoisdeocwidne,tahnedfecaosmibpiluittye
Assume now that δλ RangeL. If δ2J(λ;δλ) = 0, then theminimumnecessaryrelaxationoftheconstraintswhenever
tr(Q12L (δλ)Q L (δ∈λ)Q12) = 0. Since Q > 0 for each necessarytoobtainatleastonesolution;(ii)Asasideproduct,
t ∗ 1 t ∗ t t
t R, it follow−s that L (δλ) = 0. Since δλ RangeL, we are able to determine whether there is at least a full-rank
∗
we∈get δλ = 0. We conclude that δ2J(λ;δλ) >∈0, for each solution. When this is not the case, we proved there exists,
δλ=0, i.e. the statement holds. and devised a way to determine, the maximalcommonkernel
6 of all the solution; (iii) We extended Georgiou’s approach to
In the light of the previous result, the dual problem admits at
maximum entropy estimation to our setting, and analyzed in
most one solution, say λ , over RangeL. If such a λ does
◦ ◦ depthboththeprimalandthedualoptimizationproblems.The
exist,thenδJ(λ;δλ)=0 δλ RangeLwhichisequivalent
to L(elogτ I L∗(λ◦))+∀f¯=∈0. It means that ρ(λ ) satisfies general form of a full-rank solution of the reduced problem,
− − − ◦ namely the one we obtain by removing the common kernel,
constraints in (7) and it is therefore the unique solution to
is given explicitly and depends on the solution of the dual
Problem 5. It remains to show that such a λ does exist.
◦ problem. The latter is proven to be a convex optimization
Proposition 4.2: J admitsaminimumpointoverRangeL.
problem with a unique solution, λ , which can be obtained
Proof: We have to show that the continuous function J ◦
by standard numerical algorithms.
takes minimum value over RangeL. Observing that J(0) =
It is also possible to show,λ is continuouswith respectto
1etr(τ), we can restrict our search over the closed set thedatasetf¯(see AppendixB)◦.Since,inviewof(29),ρ(λ )
◦
:= λ Rm J(λ) J(0) RangeL. is continuous with respect to λ◦, we can conclude that the
M { ∈ | ≤ }∩ solution to Problem 5 is continuous with respect to the data
We shall show that is bounded. To this aim, consider a f¯ ...f¯ . This is of course a desirable property, and ensures
1 m
M
sequence λ , λ RangeL such that λ . It is that for a increasing accuracy of the estimates, the computed
therefores{uffii}cii∈eNnt tois∈how that J(λ ) ,kasiki → ∞. First solution will converge to the actual state.
i
→∞ →∞
DRAFT 8
Possible extensions of the present framework include ap- Accordingly,theNewtonalgorithmwithbacktrackingglobally
plications to state reconstruction from local marginals and converges, [31, 9.5.3]. Moreover, the rate of convergence is
its connection to entanglement generation [27], [28], as well quadraticinthelaststage.Finally,notethatthefoundsolution
as comparison with the existing approximate results [21] in vˆq satisfies the following inequalities, [31, p. 566],
physically meaningful situation. Lastly, the advantage offered n
cTv cTvq cTv + (47)
by the introduction of an a priori state in the estimation ◦ ◦
≤ ≤ q
problem can be exploited to devise recursive algorithms, that
where v is a solution to Problem 4 and n the accuracy of
update existing estimate in an optimal way relying only on ◦ q
cTvq with respect to the optimal value cTv .
partial data. ◦
This method works well only setting a moderate accuracy.
To improve the accuracy, we can iterate the above Newton
APPENDIX
algorithm and in each iteration we gradually increase q in
A. Numerical solution for the feasibility problem order to find a solution vξ with a specified accuracy ξ > 0
[31, p. 569]:
We propose a Newton-type algorithm with logarithmic
barrier for numerically find one solution v◦ to Problem 4. 1) Set the initial conditions q0 > 0 and vq0 =
T
Using the same approach of [9, Section 4], we resort to the 0 ... 0 √n( λ (ρ˜ )+1) int( ).
min 0
− ∈ I
approximate problem 2) (cid:2)At the k-th iteration compute vqk i(cid:3)nt( ) by minimiz-
∈ I
min gq(v) (42) imngethgoqkd pwrietvhiosutasrltyinpgrepsoeinnttedv.qk−1 by using the Newton
v int( )
∈ I 3) Set q =βq with β >1 closer to one.
k+1 k
where q >0, and 4) Repeat steps 2 and 3 until the condition n < ξ is
qk
satisfied.
g (v):=qcTv logdet(H(v)). (43)
q
− Finally, we deal with the problem to compute a solution
Recall that we defined to Problem 1 with kernel when µ = 0. In this situation,
K
consider the non-empty convex set
= v H(v) 0 ,
I { | ≥ }
n2 m
and notice that g is continuous and strictly convex over −
q =w ρ˜0+ wiYi 0, (48)
cthoempseotneinntt(vI). Morecoavnerbelimrevs→tri∂cItegdq(tvo)be=lon+g∞to aancdlostehde I∗  | Xi=1 ≥ 
and boundedni2n−temrv+a1l. Accordingly the approximatedproblem where w = w1 ... wn2 m T Rn2−m. Thus, the set
admitsauniquesolution,denotedbyvˆq,whichisnumerically of solutions(cid:2)to Problem 1 is − (cid:3) ∈
computed by employing the Newton algorithm with back-
n2 m
tracking, [31, 9.5]. Here, we can choose as initial guess the −
vector vˆq0 = 0 ... 0 √n(−λmin(ρ˜0)+1 T ∈ int(I). S =ρ˜0+ Xi=1 wiYi |w∈I∗. (49)
Concerning th(cid:2)e l-th Newton step, we have to sol(cid:3)ve
We wish to compute a matrix ρ having kernel cor-
◦
∆vˆql =−Hv−ˆql1∇Gvˆql (44) responding to the minimum commo∈n kSernel K. To this aim
consider the following problem.
where Problem 6: Pick u Rn2 m at random and solve:
−
tr(H(vˆq) 1Y ) ∈
l. − 1 wu =arg min(w u)T(w u). (50)
∇GHvvˆˆqlq == qc− trt(Hr(H(vˆ(qlvˆ)ql−..)1−Y1nX2−1)m)  (45) No•teI(tw∗haits:uc)oTm(pwactu()i.eis.wccl∈oonIs∗teinduao−nudsbanodunsdtr−eidct)lyconvexon .
l • − − I∗
tr(H(vˆq) 1Y H(vˆq) 1Y ) tr(H(vˆq) 1Y H(vˆq) 1Y ) ... By Weiestrass’ theorem, the above problem admits a unique
l − 1 l − 1 l − 1 l − 2
 tr(H(vˆql)−1Y...2H(vˆql)−1Y1) tr(H(vˆql)−1Y2H(vˆql)−1Y2) ...(46) ρs˜o0lLu+etitounsnkw=2d−u1em.finueiYρi◦,∈u S:=. Iρ˜t0is+ePasykn=2t−o1mseweiutYhiat∈PrSobalnemd ρ6uc:a=n
be rewPritten in terms of the matrix ρ◦,u as follows:
are the gradientand the Hessian computedat ˆvq, respectively.
l ρ ,u =argmin ρu ρ 2, (51)
Note that, it is not difficult to prove that ◦ ρ k − kF
∈S
1) aTlhgeorithmsequeisncecon{taviqln}eld≥0 ingentheeratedcompbayct tsheet twhheecrelokse·sktFmdaetnrioxteisnthe(FwriotbherneisupsemctattorixthneorFmro.bTehnuius,sρn◦o,urmis)
= v int( ) 1 cTv cTvq ; to the matrix ρu beloSnging to the hyperplane characterized
2) Ng is tnwic∈e diffeIren|ti−abl√enan≤d stron≤gly co0novex on ; by the constraints (7). Hence, randomly generating a finite
q
3) The Hessian of gq is Lipschitz continuous on N. N sequence{u1...ul}ofelementsinRn2−m weobtainasubset
DRAFT 9
:= ρ◦,u1...ρ◦,ul contained in . Construct the convex [8] D.Petz,QuantumInformationTheoryandQuantumStatistics. Springer
Ucombin{ation ρ¯:= 1l } lj=1ρ◦,uj. TheSn ρ¯ has minimal kernel [9] VMe.rlZago,rz2i,00F8..Ticozzi,andA.Ferrante,“Onquantumchannelestimation
if eitherone of the ρ◦P,uj belongsto the interiorof , or ρ◦,uj withminimalresources,”on-linepreprint,vol.arxiv.org/abs/1106.2105,
S
belongto differentfacesof , whichis a compactconvexset. 2011.
By the randomized construcStion, the probability of remaining [10] M. Mohseni, A. T. Rezakhani, and D. A. Lidar, “Quantum-process
tomography: Resource analysis of different strategies,” Phys. Rev. A,
on the boundary of becomes small as l grows.
vol.77,p.032322,2008.
S
Concerningthecomputationofwu,wecanresortaNewton- [11] I. Bongioanni, L. Sansoni, F. Sciarrino, G. Vallone, and P. Mataloni,
type algorithm with logarithmic barrier named exterior-point “Experimental quantum process tomography of non-trace-preserving
maps,”Phys.Rev.A,vol.82,no.4,p.042307,2010.
method, [33, Chapter 4].
[12] P.Villoresi,T.Jennewein,F.Tamburini,C.B.M.Aspelmeyer,R.Ursin,
C. Pernechele, V. Luceri, G. Bianco, A. Zeilinger, and C. Barbieri,
B. Continuity of λ with respect to f¯ “Experimental verification of the feasibility of a quantum channel
◦ between spaceandearth,” NewJournalofPhysics,vol.10,p.033038,
We show that the solution λ is continuous with respect to 2008.
◦
the data set f¯. To this aim we take into accountthe following [13] E.T.Jaynes,“Information TheoryandStatistical Mechanics,” Physical
ReviewSeries,vol.106,no.4,pp.620–630,1957.
result, see [34, Theorem 3.1]. [14] ——, “Information Theory and Statistical Mechanics. II,” Physical
Theorem A.1: Let A be an open and convex subset of a ReviewSeries,vol.108,no.2,pp.171–190,1957.
[15] J. Burg, Maximum entropy spectral analysis, ser. Stanford Exploration
finite-dimensional euclidean space V. Let h : A R be a
→ project. StanfordUniversity, 1975.
strictly convexfunction,and suppose that a minimumpoint x¯ [16] D. Petz, “Entropy, von Neumann and the von Neumann entropy,” in
of h exists. Then, for all ε >0, there exists δ > 0 such that, JohnvonNeumannandtheFoundationsofQuantumPhysics,M.Redei
andM.Stoltzner, Eds. Kluwer,2001.
for p∈Rn, kpk<δ, the function hp :A→R defined as [17] M. Ziman, “Incomplete quantum process tomography and principle of
maximalentropy,” Phys.Rev.A,vol.78,p.032118,2008.
h (x):=h(x) p,x (52)
p [18] J. Shore and R. Johnson, “Axiomatic derivation of the principle of
−h i
maximum entropy and the principle ofminimum cross-entropy,” IEEE
admits a unique minimum point x¯ , and moreover
p Trans.Inf.Theor.,vol.26,no.1,pp.26–37,Jan.1980.
[19] T. Georgiou, “Relative entropy and the multivariable multidimensional
x¯ x¯ <ε. (53)
p momentproblem,” Information Theory,IEEETransactions on,vol.52,
k − k
no.3,pp.1052–1066,2006.
Consider [20] V.Vedral,“Theroleofrelativeentropyinquantuminformationtheory,”
J(λ,f¯)=tr(elogτ−I−L∗(λ))+ λ,f¯ (54) [21] SR.evO.lMivaorde.sPahnyds.M,v.oGl..7A4.,Pnoar.is1,,“pQpu.a1n9t7u–m23e4s,tim20a0ti2o.nviatheminimum
wherewemakethedependenceofJ uponf¯(cid:10).The(cid:11)n,theunique [22] Mku.llPbaavckonenantrdopFy.Tpriicnoczizpil,e,“”DPishcyrse.teR-teivm.eA,clvaosls.ic7a6l,apn.d0q4u2a1n2t0u,m20m0a7r.ko-
minimum point is vian evolutions: Maximum entropy problems on path space,” J. Math.
Phys.,vol.51,p.042104,2010.
λ(f¯)=arg min J(λ,f¯). (55) [23] F. Ticozzi and M. Pavon, “On time-reversal and space-time harmonic
λ Rm processes for markovian quantum channels,” Quantum Information
∈ Processing,vol.9,no.5,pp.551–574,Oct.2010.
Letδf Rm be a perturbationof f¯. We haveJ(λ,f¯+δf)= [24] S.L.Braunstein, “Geometryofquantum inference,” Physics Letters A,
J(λ,f¯)∈+λTδf. Applying the previous theorem, where δf vol.219,no.3-4,pp.169–174,1996.
[25] A. Aiello, G. Puentes, D. Voigt, and J. P. Woerdman, “Maximum-
is p, we have: ε > 0 δ > 0 s.t. if δf < δ then
J(λ−,f¯+δf) adm∀its a uniqu∃e minimum pointk k lpipk.el8ih1o7o–d81e9s,tiMmaartio2n00o6f. mueller matrices,” Opt. Lett., vol. 31, no. 6,
[26] J.J.Sakurai,ModernQuantumMechanics. Addison-Wesley,NewYork,
λ(f¯+δf)=arg min J(λ,f¯+δf) (56) 1994.
λ∈Rm [27] W.Hall,“Compatibilityofsubsystemstatesandconvexgeometry,”Phys.
and Rev.A,vol.75,p.032102,Mar2007.
λ(f¯+δf) λ(f¯) <ε. (57) [28] F. Ticozzi and L. Viola, “Stabilizing entangled states with quasi-local
k − k quantum dynamical semigroups,” Phil. Trans. R. Soc. A, vol. 370, no.
Accordingly, the map f¯ λ(f¯) is continuous. 1979,pp.5259–5269, 2012.
7→ [29] M. D. Choi, “Completely positive linear maps on complex matrices,”
Linearalgebra anditsapplications, vol.10,pp.285–290,1975.
REFERENCES [30] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quan-
tum Information (Cambridge Series on Information and the Natural
[1] R. P. Feynman, “Simulating physics with computers,” International Sciences). CambridgeUniversity Press,2004.
JournalofTheoretical Physics,vol.21,1982. [31] S.BoydandL.Vandenberghe, Convex Optimization. New York,NY,
[2] D. Bouwmeester, A. Ekert, and A. Zeilinger, Eds., The Physics of USA:Cambridge University Press,2004.
QuantumInformation:QuantumCryptography,QuantumTeleportation, [32] A.Ferrante,M.Pavon,andM.Zorzi,“Amaximumentropyenhancement
Quantum Computation. Springer-Verlag, 2000. forafamilyofhigh-resolutionspectralestimators,” IEEETrans.Autom.
[3] M.A.NielsenandI.L.Chuang,QuantumComputationandInformation. Control,vol.57,no.2,pp.318–329,Feb.2012.
Cambridge University Press,Cambridge, 2002. [33] A. Fiacco and G. McCormick, Nonlinear Programming: Sequential
[4] D.D’Alessandro, Introduction toQuantumControlandDynamics, ser. Unconstrained Minimization Techniques. Society for Industrial and
Applied Mathematics & Nonlinear Science. Chapman & Hall/CRC, AppliedMathematics, 1987.
2007. [34] F. Ramponi, A. Ferrante, and M. Pavon, “On the well-posedness of
[5] H.M.WisemanandG.J.Milburn,QuantumMeasurementandControl. multivariatespectrumapproximationandconvergenceofhigh-resolution
Cambridge University Press,2009. spectralestimators,”System&ControlLetters,vol.59,no.3-4,pp.167–
[6] C.AltafiniandF.Ticozzi, “Modelingandcontrol ofquantumsystems: 172,2010.
Anintroduction,”IEEETrans.Aut.Cont,,vol.57,no.8,pp.1898–1917,
2012.
[7] M.ParisandJ.Rehacek,Eds.,QuantumStateEstimation(LectureNotes
inPhysics,vol.649),ser.LectureNotesinPhysics. Springer,2004,vol.
649.