PHYSICAL REVIEW A
ATOMIC, MOLECULAR, AND OPTICAL PHYSICS
THIRD SERIES, VOLUME 55, NUMBER 3
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Atominstitut der Österreichischen Universitäten, Schüttelstrasse 115, A-1020 Wien, Austria
~Received 11 September 1996!
An algorithm for quantum-state estimation based on the maximum-likelihood estimation is proposed. Existing techniques for state reconstruction based on the inversion of measured data are shown to be overestimated since they do not guarantee the positive definiteness of the reconstructed density matrix.
PACS number~s!: 03.65.Bz, 02.50.Wp, 42.50.Dv
State reconstruction belongs to the topical problems of
contemporary quantum theory. This sophisticated technique
is trying to determine the maximum amount of information
about the system—its quantum state. Even though the history
of the problem may be traced back to the early days of quantum mechanics, quantum optics opened a new era for state
reconstruction. A theoretical prediction of Vogel and Risken
@1# was closely followed by the experimental realization of
the suggested algorithm by Smithey et al. @2#. Since that
time, many improvements and new techniques have been
proposed @3–12#, to cite without requirements for completeness at least some titles from the existing literature @13#.
Even if the method comes from optics, similar methods
such as quantum endoscopy, are currently being used also in
atomic physics @14#. Homodyne detection of quadrature operators with varying phases of local oscillators (x f , f ) was
used as the measurement technique in the original proposal
@1,2#. The algorithm served to determine the Wigner function
W(x, p) and also other quasiprobabilities representing the
density matrix. Measurement of rotated quadrature operators
may also be used for direct evaluation of the coefficient of a
density matrix in the number-state representation r m,n @4#
and for the analysis of multimode fields @7#. Simultaneous
measurement of the pair of quadrature operators (x,p) using
double homodyne or heterodyne detection directly yields the
Q function Q( a ) @8#. A surprisingly easy technique was sug-
gested by Wallentowitz and Vogel @9# and by Banaszek and
Wodkiewicz @10#. Mixing of the signal and coherent fields
with controlled amplitude on the beam splitter may serve for
reconstruction of the Wigner function and other distribution
functions using the photon counting only. Techniques similar
to the quantum-state reconstruction have been suggested for
indirect observations of particle number; see, for example,
Ref. @15#. Though the techniques are different as far as practical realization is concerned, they all may be comfortably
represented by the formalism of generalized measurement
@16#. As is well known, any measurement may be described
using the probability operator measure ~POM!, P̂( j ) being
any positively defined resolution of the identity operator
P̂( j )>0, * d j P̂( j )51ˆ . The probability distribution of the
outcome predicted by quantum theory is
w r ~ j ! [email protected] r̂ P̂ ~ j !# ,
where r̂ is the density matrix of the state. The measured
variable j represents formally the registered data being in
general a multidimensional vector with the components belonging to both the discrete and continuous spectrum, as
shown in the above-mentioned examples. The key point of
the existing reconstruction techniques—inversion of the relation ~1!—represents a nontrivial problem. The solution
may be formally written as an analytical identity,
*Permanent address: Department of Optics, Palacký University,
17. listopadu 50, 772 07 Olomouc, Czech Republic.
W r~ a ! 5
d j K ~ a , j ! w r~ j ! ,
© 1997 The American Physical Society
W r ( a ) being a representation of a density matrix. In order to
find the representation W( a ) of a density matrix corresponding to an unknown signal, the existing reconstruction techniques apply the relation ~2! on the actually detected statistics w( j ).
Apart from how ingeniously the individual inversions
have been done, some problems are caused by application of
this treatment in quantum theory. In particular, the algorithm
may give a density matrix only for such measured probability distributions, which are given exactly by the relation ~1!.
Deviations between actually detected w( j ) and the true statistics w r ( j ) spoil the positivity of the reconstructed density
matrix. There are at least the following imperfections of the
detected statistics w( j ), which should always be taken into
account: ~i! the sampling error caused by the limited number
of available scanned positions of continuous variable at
which the measurement was done; ~ii! the counting error
caused by the limited set of available data counted at each
position. For example, in Ref. @2# the former one is caused
by the division of the quadrature x f into 64 bins and the
phase into 27 values, whereas the latter one is caused by the
detection of the quadrature x f at each bin. Other errors, such
as imperfections of detectors or external noises, may appear
in practice as well. In the quantum case, the algorithm based
on the inversion provides a result, but does not guarantee the
positive definiteness of the reconstructed density matrix @17#.
In the example of Ref. @2#, the positive definiteness of the
reconstructed matrix has not been checked explicitly, but can
be judged according to the papers @4,5,18#. Here the negative
part of the photocount distribution indicates the spoiling of
positive-definiteness. Even if there were a connection between the dimension of Hilbert space where this happens and
the number of phases @18#, a rigorous way to treat the positive definiteness within reconstruction has not yet been suggested. The goal of the existing techniques is the estimation
of the error bars of the coefficients parametrizing an unknown density matrix @19# involving also the possible nonpositive parts. Ordinary reconstruction techniques, therefore,
determine the coefficients of a particular representation of
the density matrix as a fluctuating variable rather than a
positive-definite density matrix itself. As pointed out by
Jones @20# in his Ref. @12#, the failure of similar methods is
the rule rather than the exception in the case where the measured data underdetermine the solution. Instead of inversion
of the detected data, a technique motivated by quantum information theory @16,21# and by phase-shift estimation @22#
will be suggested in this paper. Previous reconstruction techniques will be embedded into the common scheme based on
the maximum-likelihood estimation.
Many parameters characterizing the quantum state should
be estimated in state reconstruction. As pointed out by Helstrom @16#, this may be done by restricting the dimension of
Hilbert space, and accepting some residual uncertainty. Similarly, Jones @20# investigated the fundamental limitations of
quantum-state measurement using Bayesian methodology.
On the contrary, realistic measurements, such as those in the
existing techniques, will be anticipated here. Assuming the
repeated ~or multiple! measurement performed on the n copies of the system, the output of the observation may be parametrized by the set of states ~projectors! formally denoted
as u y 1 & , . . . , u y n & , repetition of a particular outcome being
allowed. Pure states represent here the case of sharp measurement, whereas unsharp measurement involving the finite
resolution should be represented by an appropriate POM.
Since formal considerations are valid for both these cases,
the notation of sharp measurement will be kept in the following for the sake of simplicity. Maximum-likelihood estimation ascribes to such a measurement the state r̂ maximizing
the likelihood functional
L~ r̂ ! 5
)i ^ y iu r̂ u y i & .
The aim of this contribution is to find this state and to clarify
the fluctuations of such a prediction. As the mathematical
tool, the inequality between the geometric and arithmetic averages of non-negative numbers q i will be used,
( ) ni q i ) 1/n < (1/n) ( i n q i . The equality is achieved if and
only if all the numbers q i are equal. The variables will be
formally replaced by q i 5x i /a i , where x i >0 are positive
and a i .0 are auxiliary positive nonzero numbers. In the
following the n-dimensional vectors will be denoted, in
boldface, by a, x, y, etc. Assume now that the numbers q i are
chosen from the given set of values so that the value q i
appears k i times in the collection of n data. Hence k i represents the frequency, f i 5k i /n being the relative frequency
( 8i f i 51. Parametrization explicitly revealing the frequency
will be denoted by an upper prime in sums and products,
indicating that the index runs over a spectrum of different
values. Without loss of generality the variable x may be interpreted as probability ( 8i x i 51, since the normalization
may always be involved in auxiliary variables a. The relation, known as Jensen’s inequality @23#, then reads
( 8i f i a ii .
In this form it represents a remarkably powerful relation
since the equality sign may be achieved for an arbitrary probability x5a. For example, the Gibbs inequality @21# follows
as a special case choosing the parameters a i 5 f i , since the
inequality ~4! may be rewritten as 2 ( 8i f i ln( f i / x i )<0. These
formal manipulations are tightly connected to the maximization of the likelihood function. Using the definition
x i 5 ^ y i u r̂ u y i & ,
a i being a subject of further consideration, the likelihood
functional may be simply estimated as
„L~ r̂ ! …1/n 5
) 8i ~ ^ y iu r̂ u y i & ! f < ) 8j a fj Tr$ r̂ R̂ ~ y,a! % .
The operator R̂ is given, in general, by nonorthogonal decomposition as
R̂ ~ y,a! 5
( 8i aii u y i &^ y iu .
Relation ~6! simply follows from the definition ~3! and from
the inequality ~4!. Further treatment is distinguished by the
following specifications of auxiliary parameters a.
Reconstructions of the wave function. Condition a i 5 f i
tends to considerable simplifications. Since the measurement
need not be complete, R̂(y,a5f)<1̂, the right-hand side of
the relation ~6! reads
~ 6 !5
) 8j f jf Tr r̂ ( 8i u y i &^ y iu < ) 8j f fj
This represents a state-independent upper bound. The necessary condition for the equality sign in Eq. ~6! is given by the
conditions ^ y i u r̂ u y i & /a i 5const for any i, whereas the equality sign appears in relation ~8! for complete measurements.
These relations, together with the normalization of relative
frequencies, tend to the necessary condition for searched
state r̂ ,
^ y i u r̂ u y i & 5 f i .
( 8i f iu y i &^ y iu .
Unfortunately, such measurements do not reveal information
about the full density matrix since the nondiagonal elements
are lost, as, for example, in the case of particle number measurement. Techniques dealing with orthogonal measurements
are therefore not suitable for full state reconstruction, which
should be based on the usage of nonorthogonal states. On the
other hand, in these cases the completeness and the existence
of a solution of Eq. ~9! cannot be guaranteed. Quantum analogy of the Gibbs inequality corresponds to an overestimated
upper bound and tends to the conditions imposed by reconstruction techniques.
Maximum-likelihood estimation. The problems with the
existence of a state achieving the upper bound descend obviously from the fixing of the auxiliary parameters a. The
remedy is to keep them free as a subject of further optimization. For any positively defined operator R̂5 ( i l i u r i &^ r i u ,
l i >0, and any density operator r̂ , the simple lemma holds
Tr~ r̂ R̂ ! <maxi l i .
The inequality sign is achieved for the density matrix corresponding to the spectral projector of operator R̂ with maximal eigenvalue. Using this lemma, the estimation of the
right-hand side of the inequality ~6! then reads
~ 6 ! <l ~ y,a!
independently on the index i. Finally, the maximumlikelihood estimation determines the desired state as
u c (y,a) & , where vector a is given by the solution of the set
of nonlinear equations ~13!. The uncertainty of such a
quantum-state estimation may be, according to the Bayesian
formulation @20#, characterized by the likelihood functional
~3!. Since the interpretation of the probability distribution on
the space of states is rather complicated, the uncertainty of
the prediction may be involved in an alternative way. The
measured data are fluctuating according to the distribution
function P(y) depending on the true state of the system.
Fluctuations of quantum-state estimates may be represented
by the sum of independent contributions
r̂ MLE5 ^ u c ~ y! &^ c ~ y! u & y5
This is merely the experimental counterpart of the relation
~1! and hence the starting point of reconstruction based on
inversion. The relation ~9! may be simply inverted in the
case of orthogonal measurements, which may be considered
as complete on the given subspace, tending to the solution
r̂ f 5
) 8i a if ,
where l(y,a) denotes formally the maximal eigenvalue of
the operator R̂(y,a) with the corresponding eigenvector
u c (y,a) & . Equality signs in the chain of inequalities are
achieved simultaneously if and only if
z^ y i u c ~ y,a! & z2
dyP ~ y! u c ~ y! &^ c ~ y! u .
This density matrix shows how closely the maximum likelihood method allows us to estimate an unknown state hidden
in the measured statistics P(y). Unfortunately, the proposed
method is rather complicated and examples of reconstructions specified above should be solved separately, case by
case. Considerable technical difficulties may be caused, for
example, by possible degeneracy of operator R̂ reflecting the
structure of performed quantum measurement. This particular question is beyond the scope of this contribution and
represents an advanced program for further reinterpretation
of existing reconstruction techniques.
A developed technique may be illustrated on simple but
theoretically worthwhile examples. Quantum-state reconstruction after the measurement of a Hermitian operator with
an orthogonal spectrum is the simplest problem. The solution
corresponds to the application of the Gibbs inequality, since
the relation ~9! may be solved in this case. The quantum state
is then reconstructed after each measurement by the density
matrix ~10!. This is a consequence of the possible degeneracy of the operator R̂ mentioned above. The treatment
based on the Gibbs inequality is overestimated in general.
Provided that Eq. ~9! is fulfilled in some special cases, then
the solution should coincide with the prediction of the
The cases of strongly underdetermined data are also
simple, when the state is estimated after single detection
n51. Assume for concreteness the standard ‘‘measurement
of Q function’’ corresponding to the detection of coherent
states u y & 5e yâ 2y * â u 0 & . If the value y is detected, the system is with the highest likelihood just in the state u y & . Provided that system was in a coherent state u a & , the output
fluctuates as z^ a u y & z2 / p . Estimation after single detection
then yields the density matrix of superposition of coherent
signal a and the thermal noise @24# with the mean number of
particles equal to 1,
d 2 ye 2 u y2 a u u y &^ y u .
The difference between the true state and its estimation is
negligible in the case of classical fields, but considerable in
the quantum domain.
Estimating the quantum state after multiple detection of
coherent states, the matrix R̂ should be diagonalized. Using
the assumption for eigenstates as u w & 5 ( 8i V i u y i & , linear
equations for desired coefficients V i and eigenvalues l follow as
8 V i C ki 5lV k ,
where C ki 5C *
ik 5 ^ y k u y i & ,C ii 51. This solution determines
the coefficients a according to the relation ~13! as
u ( i8 V i C ki u 2 /a k 5const for any index k. Let us illustrate this
strategy in the case of double detection n52 yielding the
values y 1 and y 2 . Parameters are given as f 1 5 f 2 51/2 and
without loss of generality a 1 51, a 1 /a 2 5x. The secular
equation for the maximal eigenvalue l reads
l 2 2(11x)l1x2x u C 12u 2 50, yielding easily solutions for
the maximal eigenvalue and its eigenvector. Equations ~13!
impose the single condition as u C 12u l5 Ax(l211 u C 12u 2 ).
This nonlinear system of equations may be easily solved
yielding an expected solution, such as l511 u C 12u , x51.
The projector is given by the normalized Schrödinger-catlike state,
A2 ~ 11 u C 12u !
~ e iargC 12u y 1 & 1 u y 2 & ).
The density matrix ‘‘reconstructing’’ the coherent state is
then given as
r̂ MLE5 2
d 2 y 1 d 2 y 2 e 2 u y 1 2 a u 2 u y 2 2 a u u w &^ w u .
The proposed method describes easily the cases in which the
data seem to be underdetermined. There is also a strong ef-
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@13# Besides these, there are still other methods based on the principle of maximum entropy. An unknown state is approximated
on the class of density matrices specified by moments of a
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fort to apply the developed technique to the case of large
data sets estimating properly the quantum state in the cases
of realistic measurements.
Even if the problem of positive definiteness used for motivation may seem trivial, it has far-reaching consequences.
Since a quantum state comprises the maximum possible information about the system, its proper description is of fundamental interest. The method based on maximum likelihood
addresses the state reconstruction in close analogy to the ordinary methods, completing rather than denying them in the
following way: For measured data, whenever the ordinary
reconstruction schemes provide a positive matrix, the solution coincides with the maximum-likelihood estimation. The
predictions will differ only if the ordinary scheme yields a
matrix that is not positively defined.
Unfortunately, the proposed technique is nonlinear and
the evaluation of realistic data for existing state reconstructions represents a nontrivial problem. Since the algorithm
never admits the existence of ‘‘negative probabilities,’’ the
conjecture concerning accuracy may be formulated as the
following statement: Enforcing positiveness will enhance the
uncertainty of state estimation, in comparison to the ordinary
techniques. These and other questions deserve further attention, since an information-theoretic approach allows us to
analyze the state reconstruction free of any additional assumptions.
I am grateful to Professor H. Rauch for the hospitality of
Atominstitut der Österreichischen Universitäten and to T.
Opatrný for discussions concerning state reconstruction. This
work was supported by the East-West Program of the Austrian Academy of Sciences and by a grant from the Czech
Ministry of Education, No. VS 96028.
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@16# C. W. Helstrom, Quantum Detection and Estimation Theory
~Academic Press, New York, 1976!.
@17# Positive definiteness of r̂ , i.e., the property ^ x u r̂ u x & >0 for any
state u x & , is very complicated for testing. It should not be confused with the positivity of the Wigner function, which need
not be positive.
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