Statistical estimation of the quality of quantum-tomography protocols
Yu. I. Bogdanov, G. Brida, I. D. Bukeev, M. Genovese, K. S. Kravtsov, S. P. Kulik, E. V. Moreva, A. A. Soloviev, A. P. Shurupov
aa r X i v : . [ qu a n t - ph ] N ov Statistical Estimation of the Quality of Quantum Tomography Protocols
Yu. I. Bogdanov, G. Brida, I. D. Bukeev, M. Genovese, K. S. Kravtsov, S. P. Kulik, E. V. Moreva, A. A. Soloviev, and A. P. Shurupov Institute of Physics and Technology, Russian Academy of Sciences, 117218, Moscow, Russia INRIM, Strada delle Cacce 91 I-10135, Torino, Italy Prokhorov General Physics Institute, Russian Academy of Sciences, Moscow, 119991, Russia Faculty of Physics, Moscow State University, 119992, Moscow, Russia Moscow National Research Nuclear University “MEPHI”, 115409, Moscow, Russia (Dated: July 14, 2018)We present a complete methodology for testing the performances of quantum tomography pro-tocols. The theory is validated by several numerical examples and by the comparison with experi-mental results achieved with various protocols for whole families of polarization states of qubits andququarts including pure, mixed, entangled and separable.
PACS numbers: 42.50.-p, 42.50.Dv, 03.67.-a
I. INTRODUCTION
Quantum information technologies rely on the use ofquantum states in novel data transmission and comput-ing protocols [1–3]. Control is achieved by statisticalmethods via quantum state reconstruction. At presentquantum state and process tomography serves as a prin-cipal instrument for characterization of quantum statespreparation and transformation quality [4–34].In the last few years, a significant effort was made toimprove tomographic methods of quantum state mea-surements. In [35] a method of choosing the best setof measurements from a finite list of possible ones is pre-sented. The basis of consideration for such problems isthe asymptotic theory of statistical estimations of theparameters of the density matrix. Difficulties in solvingoptimization problems are caused by the large number ofparameters to be estimated. In general case, it requiresextensive numerical calculations. The number of possi-ble measurements becomes extremely large if one wantsto cover a multidimensional parameter space of the mea-surement apparatus. Thus, holding a finite number ofsamples in multidimensional parameter space, does notguarantee that the truly optimal set of measurementsis chosen. In other words, even the method is universaland works for qubits as well as for higher dimensional sys-tems, it does not say which particular measurements haveto be performed. It just helps to choose the best mea-surements out of large suboptimal set of possible mea-surements. Another significant problem is the choice ofan adequate parametrization for the quantum states (wewill return to this issue in Sec. VI).The work [36] provides a theoretical approach to op-timal measurements of qutrit systems, where optimal means the use of a minimal number of bases for mea-surements. As expected, the use of mutually unbiasedbases is optimal in that sense, although its experimentalrealization, not discussed in the paper, might be far fromthe optimal one.Another recent work [37] addresses the problem of den- sity matrix reconstruction for large multiqubit states,where a complete state reconstruction is impossible dueto the exponentially growing number of undefined param-eters with the increase of qubit number in the system.The solution given is in finding only the permutation-ally invariant part of the density operator, which gives agood approximation of the true quantum state in manyrelevant cases.In the present paper, we introduce a complete method-ology for statistical reconstruction of quantum stateswhich is based on analysis of completeness, adequacyand accuracy of quantum measurement protocols [38–41]. Completeness of the quantum tomography protocolprovides the possibility of reconstruction of an arbitraryquantum states (both pure and mixed) and it is assessedby means of singular value decomposition of a specialmatrix built upon operators of measurement. Consid-ered SVD allows us to introduce the condition number K , which characterizes the quality of quantum measure-ment protocol. Adequacy employs redundancy of a mea-surement protocol compared to the minimum number ofmeasurements required for reconstruction. It is assessedby consistency of redundant statistical data and mathe-matical model, based on quantum theory. Accuracy ofstatistical reconstruction of quantum states is based ona universal statistical distribution proposed in [40].It is worth mentioning that in real experiments the ac-curacy of the state reconstruction depends on two typesof uncertainties: statistical and instrumental ones. If thetotal number of measurement outcomes (sample size orstatistics) is large enough, the instrumental uncertain-ties dominate over fundamental statistical fluctuationscaused by the probabilistic nature of quantum phenom-ena [17]. Practically, the required statistics, allowing usto exclude statistical fluctuations, depends on the tomo-graphic protocol itself and the total accumulating timeneeded for taking data. From this point of view it wouldbe useful to point out simple and universal algorithms forthe estimation of the chosen protocol on the design stagebefore doing experiments, as well as the sample size fordesirable quality of the state reconstruction.The method considered in the present paper has thefollowing features: (1) It is well suited in the case ofmulti-qubit state tomography. (2) It accepts the recon-struction for mixed states of arbitrary rank as well asfor pure states. (3) It allows to compare various quan-tum measurements protocols with each other and, more-over, with respect to the fundamental fidelity level. (4)It also motivates the experimenter to manage availableresources in the best way as well as to choose an opti-mal quantum measurement protocol. The paper is orga-nized as follows. Section II introduces the mathemati-cal apparatus for general quantum measurement proto-col (for states with discrete variables). In Sec. III wediscuss a generalized statistical distribution for fidelityby introducing specific random value which can be calledthe loss of fidelity. Important operational criterion andcorresponding quantifier (so called conditional number)of the protocols’ quality based on completeness are sug-gested. Section IV relates to analysis of specific exam-ples of tomographic protocols of qubits which are basedon the polyhedrons geometry. Section V collects resultsof numerical and physical experiments for both pure andmixed states of single and pair of qubits. Since it repre-sents a paradigmatic example, here we restrict ourselvesto polarization degrees of freedom only; however, anyother degrees of freedom can be considered. In Sec. VIwe discuss in details the features and advantages of thesuggested approach in context of completeness, adequacyand accuracy. Then we conclude with Sec. VII. II. QUANTUM MEASUREMENT PROTOCOL
An arbitrary s -dimensional quantum state is com-pletely described by a state vector in a s -dimensionalHilbert space when it is a pure state, or by a density ma-trix ρ for a mixed one. To measure the quantum stateone needs to perform a set of projective measurementson a set of identical states.A quantum measurement protocol can be defined by aso-called instrumental matrix X that has m rows and s columns [17–19], where s is the Hilbert space dimensionand m the number of projections in such space. For everyrow, that is, for every projection, there is a correspondingamplitude M j , M j = X jl c l , j = 1 , , . . . , m (1)where we assume a summation by the joint index l , the c l ( l = 1 , , . . . , s ) being the components of the state vec-tor in the Hilbert space of dimension s . The square ofthe absolute value of the amplitude defines the intensityof a process, which is the number of events in 1 s λ j = | M j | . (2)The number of registered events k j is a random variablePoissonially distributed, t j being the time of exposition of the selected row of the protocol and λ j t j the averagevalue, P ( k j ) = ( λ j t j ) k j k j ! exp( − λ j t j ) . (3)It is convenient to introduce special observables Λ j (socalled intensity operators), which are measured by theprotocol during experiment, λ j = tr(Λ j ρ ) (4)Here Λ j = X † j X j is the intensity operator for the quan-tum process X j (the row of the instrumental matrix X ).In this case the intensity operator for quantum processΛ j is a projector, so we haveΛ j = Λ j (5)Formally, in the most general case, Λ j is an arbitrarypositively defined operator [42]. It can be presented as amixture of projection operators described above.Λ j = X k f k X ( k ) † j X ( k ) j (6)Here the index k sums different components of the mix-ture that have weights f k >
0. Such measurement can beconveniently presented as a reduction of the set of projec-tion measurements where only total statistics is available,while statistical data for individual components are notavailable. The general projection measurement is a par-ticular case of Eq. (6) where f = 1 , f = f = · · · = 0.If the sum of the intensities multiplied by the expositiontime is proportional to a unit matrix, then we say thatthe protocol is brought to a decomposition of unity [42], I = m X j =1 t j Λ j = I E, (7)where I is the constant which defines overall intensityand E is the identity (or unit) matrix. A protocol forwhich the condition (7) holds in the general case can bebrought to the so-called non-orthogonal decompositionof the unity [42]. In this case the protocol analysis issimplified. In mathematics such measurements are con-sidered as the most general extension of traditional vonNeumann measurements, which are based on the orthog-onal decomposition. Even if it is reasonable to require(7) due to the total probability preservation, it is worthmentioning that real experimental protocols often cannotbe brought to decomposition of unity. Indeed, in real ex-periments when event registration scheme is used, theexperimenter often adjusts his device to distinguish onlyone projection of the quantum state (simply loosing datawhich corresponds to the other states). Thereby real ex-periments (due to technical requirements) usually do notprovide registration of the whole statistical ensemble andthus they are not restricted by total probability preser-vation requirement. However, the method suggested inthe present paper is applicable to these cases as well.The normalization condition for the protocol definesthe total expected number of events n summarized by allrows: m X j =1 λ j t j = n, (8)where t j is the acquisition time. Condition (8) substi-tutes the traditional normalization condition for the den-sity matrices, tr( ρ ) = 1In the following we consider the protocols of quan-tum measurements in terms of two important notions–completeness and adequacy. For this purpose, we intro-duce some opportune notation. First, when describingthe whole sequence of quantum protocol measurementseach quantum process intensity matrix Λ j of dimension s × s is pulled into a single string of the length s (toput the second string to the right from the first etc).Then, we assign a weight defined by an exposition time t j to each row in B j and we construct a matrix m × s from these rows, calling it the measurement matrix of thequantum protocol. We assume that m ≥ s .In the case of projection measurements defined by rows X j ( j = 1 , . . . , m ) of instrumental matrix X , the rows B j of the measurement matrix B could be calculatedthrough the use of the tensor product of the row X j andthe complex conjugate row X ∗ j . B j = t j · X ∗ j ⊗ X j . (9)In the following exposition times are assumed to be equalto 1. With this matrix B , the protocol can be compactlywritten in the matrix form: Bρ = T (10)with ρ being the density matrix, given in the form of acolumn (second column lies below the first, etc.). Thevector T of length m records the total number of reg-istered outcomes. The algorithm for solving Eq. (10)is based on the so called singular value decomposition(SVD) [43]. SVD serves as a base for solving inverseproblem by means of pseudo-inverse or Moore-Penroseinverse [43, 44]. In summary, the matrix B can be de-composed as: B = U V S † , (11)where U , ( m × m ), and V , ( s × s ), are unitary matri-ces and S , ( m × s ), is a diagonal, non-negative matrix,whose diagonal elements are “singular values”. Then (10)transforms to a simple diagonal form: Sf = Q (12)with a new variable f , unitary related to ρ via f = V † ρ ,and a new column Q , unitary related to the vector T by the equation Q = U † T . This system is easy to solvebecause S is a diagonal matrix. Its analysis allows classi-fying measurements from the viewpoint of adequacy andcompleteness [41].Let m > s , that is, the number of measurements isgreater than the number of elements in the density ma-trix. The rank of the model q denotes the number of non-zero singular values of the matrix B . By defining q weformulate two important conditions of any tomographyprotocol, namely its completeness and adequacy [41]. Itis obvious that q ≤ s . The last m − q rows in the matrix S are equal to zero. Then it follows that for the systemto be adequate, it is necessary that the last m − q valuesin the characteristic column Q are also equal to zero. Wewill refer to this condition as a measurements adequacycondition. If it does not hold the model is inadequate,that is, statistical data do not correspond to any quan-tum mechanical density matrix. It may mean, for ex-ample, that either the experiment is realized incorrectlyor measurement matrix is wrong. Adequacy means thatthe statistical data directly correspond to the physicaldensity matrix (which has to be normalized, Hermitianand positive). However it is worth noticing that, gener-ally, for mixed state it can be tested only if the protocolconsists of redundant measurements (i.e. if m > s ).Suppose that the model is adequate. The protocolis supposed to be informational complete if the numberof tomographically complementary projection measure-ments is equal to the number of parameters to be es-timated; mathematically completeness means q = s . Ifall singular values are knowingly nonzero, that is, q = s ,then unconditional completeness holds and a solution ex-ists and is unique. Measurement protocol completely re-turns any quantum state (pure and mixed) that could bedefined in the considered Hilbert space.In this case we can determine the factor column divid-ing the elements of characteristic column by the corre-sponding singular values f j = Q j /S j j = 1 , , . . . , s , (13)As a result we obtain the desired density matrix by aunitary transformation: ρ = V f. (14)Due to the unitarity of the matrix V , the factor column f determines the degree of purity of the quantum state:tr( ρ ) = m X j =1 | f j | . (15)Finally, suppose q < s , that is, some singular values areequal to zero. In this case for nonzero values we have: f j = Q j /S j , j = 1 , , . . . , q. (16)Let us call these factors f j , j = 1 , , . . . , q defined factors.At the same time, for nonzero values, we have equa-tions corresponding to uncertainty “zero divided by zero”0 f j = 0 , j = q + 1 , . . . , s . (17)We shall call these factors f j j = q +1 , . . . , s as unde-fined factors. As solutions of the last equations arbitrarycomplex numbers could be used. The considered situa-tion corresponds to the incompleteness of measurementsand this system of equations has an infinite number ofsolutions. However, not all of them correspond to realphysical density matrices. The physical solutions onlycorrespond to that having a Hermitian nonnegative def-inite density matrix. Formally all these solutions couldbe obtained by scanning all possible values of undefinedfactors. It is evident that such a procedure could be car-ried out only when the dimension of undefined factors’space is relatively small.Let us call regularized (normal) a solution correspond-ing to the special choice of all undefined factors: f j = 0for j = q + 1 , . . . , s . Due to the unitarity of relation be-tween the density matrix and the factor column, a regu-larized solution corresponds to the minimum purity levelof the reconstructed state. In this case we have:tr( ρ ) ≥ m X j =1 | f j | . (18)One could see that in the case of incomplete protocols anyadditional measurements could either cause an increaseof the purity or leave it unchanged. If the regularizedsolution already describes a pure state, then new mea-surements will not influence the reconstructed state. Inother words in this case we can obtain complete infor-mation about the considered state despite of incompletemeasurement protocol. We will call such protocols con-ditionally complete, that is, there is completeness undercondition that only specially chosen states are considered;for instance, this is the case of experiments like “whichway” [45].Then we assume that there is unconditional complete-ness ( q = s ). Equations (13) and (14) could be used toapproximate the reconstruction of the density matrix ifwe substitute experimental events (frequencies) into theright side of Eq. (10). However due to statistical fluctua-tions of experimental data the reconstructed matrix willnot always be positive definite (components with smallweights could be reconstructed as negative numbers inthis case). Despite this disadvantage, the method pro-vides a good zero approximation for widely used maxi-mum likelihood method (ML). In this case, the compo-nents with negative weights simply assumed to be zero,then the density matrix is multiplied by a factor that en-sures the correct normalization. Note that ML method it-self is free from the considered disadvantage because pos-itive definiteness lies in the nature of the method. At thesame time the zero approximation, obtained from pseudoinversion method, significantly accelerates the search ofML solution. III. UNIVERSAL STATISTICALDISTRIBUTION FOR FIDELITY LOSSES ANDTHE MAXIMUM POSSIBLE FIDELITY OFQUANTUM STATES RECONSTRUCTION
The accuracy of quantum tomography can be definedby a parameter called fidelity [1, 46] F = (cid:20) tr qp ρ (0) ρ p ρ (0) (cid:21) , (19)where ρ (0) is theoretical density matrix and ρ is recon-structed density matrix. The fidelity shows how close thereconstructed state is to the ideal theoretical state: thereconstruction is precise if the fidelity is equal to one.This equation looks quite complex, but it becomes sim-ple if we apply the Uhlmann theorem [46]. According tothis theorem, the Fidelity is simply the maximum possi-ble squared absolute value of the inner product: F = |h c | c i| (20)where c and c are theoretical and reconstructed purifiedstate vectors.We explicitly use the Uhlmann theorem in our al-gorithm of statistical reconstruction of quantum states,based on maximum likelihood method. This fact is veryimportant even if the state is not pure, we have to purifyit by moving into a space of higher dimension [40].It is well known that purified state vectors are definedambiguously. However, this ambiguity does not precludefrom reconstructing a quantum state, which is a veryuseful feature of the suggested algorithm. It is devisedin the way that different purified state vectors producethe same density matrix and therefore the same fidelityduring the reconstruction. This is a key principle forthe proposed procedure and thus reconstruction can beobtained by means of purification. Purification greatlyfacilitates the search of a solution, especially when weneed to estimate a great number of parameters (hundredsor even thousands).The fidelity level (20) has a simple probabilistic inter-pretation. If we choose a known reconstructed vectorand its orthogonal complement as a measurement basisfor an unknown state c then F returns the probabilitythat this unknown state coincides with the reconstructedone.It is equally important that due to the usage of purifi-cation procedure we succeed in formulating a generalizedstatistical distribution for fidelity [40]. Purification pro-cedure will be described below in Sec. VI [formula (58)provides with a transition from the density matrix to apurified state vector]. The value 1 − F can be called theloss of fidelity. It is a random value and its asymptoticaldistribution can be presented in the following form:1 − F = j max X j =1 d j ξ j (21)where d j ≥ ξ j ∼ N (0 , j = 1 , . . . , j max are independent normally distributedrandom values with zero mean and variance equal to one, j max = (2 s − r ) r − s isthe Hilbert space dimension, and r is the rank of mixedstate, which is the number of non-zero eigenvalues of thedensity matrix. In particular j max = 2 s − j max = s − r = s ).This distribution is a natural generalization of the χ distribution. Ordinary χ distribution corresponds to theparticular case when d = d = · · · = d j max = 1 (all com-ponents of vector d are equal to one). In the asymptoticlimit considered by us, the parameters d j are inverselyproportional to the sample size n , that is, d j ∼ n . Thisdependence allows for an easy recalculation when we usedifferent sample sizes.The method of calculating the parameter vector d isbased on Fischer’s information matrix. In this case, theeigenvectors of information matrix define directions ofprincipal fluctuations of the purified state vector whereasthe respective variances of the principal fluctuations areinversely proportional to the eigenvalues of the informa-tion matrix. The method is described in [40] in details.From Eq. (21) one gets the average fidelity loss h − F i = j max X j =1 d j . (22)It is also easy to show that the variance for the fidelityloss is σ = 2 j max X j =1 d j . (23)Moments of higher order for this distribution can be cal-culated analytically. For example the momentum of thirdorder is called skewness and describes, for a random vari-able x , the asymmetry β = M { [ x − M ( x )] } σ (24) M denoting the mathematical expectation. The fourth-order moment is called excess kurtosis, β = M { [ x − M ( x )] } σ − β , β are: β = 8 j max P j =1 d j σ (26) β = 48 j max P j =1 d j σ (27) Let us consider a special case of the quantum state,which is defined by a uniform density matrix. Moreover,this matrix is proportional to the identity matrix. Suchstate can be represented as a “white noise” for whichall weights of principal components are equal. Let us as-sume that the protocol can be brought to projection mea-surements, which form non-orthogonal decomposition ofunity in accordance with (5) and (7). In this case thereis a simple relation between the vector d , of size s − B (theelement of the greatest value should be eliminated fromthis vector of size s ). Denoting the reduced vector ofthe size s − b , one gets the relation between thesevectors: d j = Cnb j , j = 1 , , . . . , s − , (28)where the constant C is given by C = P j b j s − . (29)For multiqubit protocols considered in the next section,which are based on polyhedrons, we have: d j = m l snb j , (30)where m is the polyhedron’s faces number and l is a num-ber of qubits in the register. It is obvious from (30) that d max d min = (cid:18) b max b min (cid:19) . (31)In addition, it can be shown that for such protocols thefollowing equation holds: b max b min = (cid:16) √ (cid:17) l − . (32)In this case the condition number K of the matrix B is K = cond( B ) = (cid:16) √ (cid:17) l . (33)Recall that the condition number of a matrix is the ra-tio of the maximum singular value to the minimum one.Note also that in the definition (33) all singular valuesare taken into account while in the (32) the one is ignoreddue to normalization.Let us introduce a value of fidelity loss, which is inde-pendent on the sample size. L = n h − F i = n j max X j =1 d j . (34)This quantity serves as the main figure of merit of theprecision in the examples analyzed below: the lower thevalue of the loss function (34), the higher is the preci-sion of the protocol. As we can see from the Eq. (34),this value is determined by vector d that defines the gen-eral fidelity distribution. Therefore, the general fidelitydistribution serves as a tool for completely solving theproblem of precision for quantum tomography.As an important example, let us consider the proto-col defined by projection measurements, which form non-orthogonal decomposition of the unity. It can be shownthat the following condition holds in this case14 n ν X j =1 d j = s − ν is the number of parameters to define a state: ν = j max = (2 s − r ) r − d = d = · · · = d ν ,when mean losses approach their minimum level: h − F i min = ν n ( s −
1) (37)Note that the requirement d = d = · · · = d ν not onlydefines a minimum level of mean losses (22), but alsoa minimum of other moments of losses [variance (23),skewness (26) and excess kurtosis (27)].It appears from (37) that the minimum possible loss isgiven by the following equation: L optmin = ν s − . (38)For pure states ν = 2 s −
2, the possible loss is given by: L optmin = s − . (39)For mixed states of full rank ν = s − L optmin = ( s + 1) ( s − . (40)Any protocol for any quantum state cannot have losseslower that those defined by this equation, if the protocolcan be brought to projection measurements, which formnon-orthogonal decomposition of the unity. Note that ifthe protocol cannot be brought to decomposition of unitythen the losses can be lower than defined by this equa-tion. However, this is true only for certain states and animprovement in reconstruction precision for some statesis completely compensated by a significant deteriorationin reconstruction precision for other states. IV. PROTOCOLS BASED ON THEPOLYHEDRONS GEOMETRY
In this section we present analysis of specific examplesof tomographic protocols which are based on the polyhe-drons geometry. The multiqubit protocols described in this section are formed by projective quantum measure-ments on states that are tensor products of single-qubitstates. For example, if a single-qubit measurement pro-tocol is formed by a projection onto a polyhedron with m faces inscribed in the Bloch sphere, then it has m rows(see below). Therefore, the corresponding l -qubit proto-col possess m l rows.Intuitively, projection of the state under considerationonto symmetric solids inscribed in the Bloch sphere leadsto better accuracy of the reconstruction. Shown belowis a study based on the calculation of the loss function(34), which significantly extends the results obtained in[23, 26]. In the term of single-qubit protocols it is worthhighlighting regular polyhedrons and polyhedrons withlesser but still rather high level of symmetry.Regular polyhedrons or Platonic solids are used for themost symmetrical and uniform distribution of quantumstates on the Bloch sphere. Projections are defined bydirections from the center of Bloch sphere to centers ofpolyhedron faces. Therefore, the number of polyhedron’sfaces defines the number of protocol’s rows and is equalto 4 for tetrahedron, 6 for cube, 8 for octahedron, 12 fordodecahedron and 20 for icosahedron.These five bodies form the complete set of regular poly-hedrons. The search of quantum measurements proto-cols with high symmetry on Bloch sphere and number ofrows greater than twenty requires to consider non-regularpolyhedrons, which have high symmetry. As examplesof such polyhedrons we have chosen fullerene (truncatedicosahedron) that defines quantum measurement proto-col with 32 rows (equal to the number of fullerene’s faces)and also a dual to fullerene polyhedron (pentakis dodec-ahedron) which defines quantum measurement protocolwith 60 rows (that is the number of its faces and also thenumber of vertices of fullerene).It is noteworthy that all protocols considered here canbe brought to decomposition of the unity [36].A comparison of the maximal possible fidelity with thefidelity of the protocols considered here shows that as thenumber of polyhedrons’ faces increases fidelity rapidlyconverges to the theoretical limit (in addition, rapidlyincreases a uniformity of fidelity distribution on the Blochsphere). We should mention that the accuracy of thesuggested protocols is much higher in comparison withprotocols exploiting not so highly symmetrical states.As an example we calculated numerically a set of pic-tures that demonstrates the Bloch sphere scanning bymeans of various measurement protocols for the singlequbit pure states. The corresponding colors indicate thevalue of the Fidelity loss function.Figure 1 defines the value of the loss function for theprotocol based on tetrahedron. The minimal losses areequal to 1 as well as for other figures. The maximumlosses for tetrahedron are equal to 3/2. A pair of polyhedra are called dual, if the vertices of one corre-spond to the faces of the other.
FIG. 1. (Color online) Tetrahedron and distributions fidelityloss over the Poincar´e-Bloch sphere for the protocol basedon tetrahedron geometry. Color bar shows level of averagefidelity loss L . Figure 2 presents a cube and an octahedron. Thesepolyhedrons are dual to each other. The maximum lossesare equal to 9/8 in both cases.Figures 3 and 4 present a dodecahedron and an icosa-hedron, as well as fullerene and a polyhedron that is dualto the latter: one can clearly see that when the numberof projections grows the maximum losses converge to theminimum possible losses. In the limit of infinite numberof points on the Bloch sphere we get an optimal pro-tocol for which the precision of reconstruction does notdepend on the reconstructed state at all. The price forthat would be an infinite growth of the performed mea-surements, so in a real experiment one should sacrificethe fidelity losses (accuracy) to use a realistic (limited)number of measurements. However, this point is quitecommon for any quantum tomography protocols exploit-ing redundant (with respect to dimension of the recon-structed state) number of measurements.For the sake of completeness, fullerene and its dualprotocols are presented on the last picture, while Table I
FIG. 2. (Color online) Shapes of solids and distributions ofaverage fidelity loss over the Poincar´e-Bloch sphere for theprotocol based on cube and octahedron geometry. Color barshows level of average fidelity loss L . FIG. 3. (Color online) Shapes of solids and distributions ofaverage fidelity loss over the Poincar´e-Bloch sphere for theprotocol based on dodecahedron and icosahedron. Color barshows level of average fidelity loss L . presents the results of numerical experiments for purequantum states with the number of qubits from 1 to 3.It is worth noting that the algorithm of numerical opti-mization does not garantee the finding of the global op-tima. Using the numerical procedure we have only foundhypothetical maximum loss values.The precision of each protocol can be characterized bythe following bounds L min ≤ L ≤ L max . This inequal-ity defines a rather narrow range, where the precision ofquantum state reconstruction is localized definitely. FIG. 4. (Color online) Shapes of solids and distributions ofaverage fidelity loss over the Poincar´e-Bloch sphere for theprotocol based on fullerene and its dual polyhedron. Colorbar shows level of average fidelity loss L . Numerical calculations demonstrate that the minimumpossible losses L min are defined by the theoretically de-rived optimal limit L min = L optmin = s − L max , shown in the Ta-ble I, as the result of numerical experiments, one canevince that for single-qubit protocols as the number ofprojections grows the maximum losses converge to theminimum possible losses. In the limit of an infinite num-ber of points on the Bloch sphere one gets an optimal pro-tocol for which the precision of the reconstruction doesnot depend on the reconstructed state.However, this is not true for multiqubit protocols. Fortwo-qubit protocols, the maximum possible losses ap-proach the level L max ≈ .
38 while the minimum pos-sible level is equal to L optmin = 3. Similarly for three-qubitstates the values are L max ≈ . L optmin = 7. Theseresults are due to the fact that these protocols are basedon projections only onto non-entangled states.Here we should stress again that from the theoreticalpoint of view in the multiqubit case the discussed proto-cols are not the best possible ones because they do not in-volve projections onto entangled states. In that case theprecision will be somewhat smaller than the minimumpossible limit. ( L max ≈ .
38 compared to L optmin = 3 fortwo-qubit states and L max ≈ . L optmin = 7for three-qubit states).It is also worth noting that, though the protocols basedon polyhedrons with small number of faces (tetrahedron,cube) are somewhat less precise, they are much easier inpractical implementation.When considering tomography of mixed states it isworth noting that there is no finite upper limit for preci-sion losses (losses can be infinitely large L → ∞ ). Suchlarge losses are inherent to mixed states, which are closeto pure ones. In fact the number of real parameters thatdefine a mixed state of full rank in Hilbert space of di-mension s is equal to s −
1, which is significantly greaterfor large s than for a pure state that takes only 2 s − d of dimen-sion s −
1, that defines distribution of precision loss, andthe vector of singular values of the measurement matrix B . Corresponding estimate for L min for the protocolsconsidered here is given by the following equation: L min = n X j d j min = X j m l sb j = 10 l − . (41) This value depends on the number of qubits, but doesnot depend on the type of polyhedron. It defines theminimum possible losses for the considered protocols thatdo not use projections on entangled states.Recall that in the general case for any protocols includ-ing those ones that involve projections onto entangledstates minimum (optimal) losses are described by: L optmin = ν s −
1) = (2 l + 1) (2 l − . (42)By comparing the Eqs. (41) and (42) one can seethat these protocols provide minimum possible (optimal)losses for reconstruction of mixed states of full rank onlyfor single-qubit states. Therefore, for multi-qubit casesprotocols that provide minimum possible losses duringquantum states reconstruction should necessarily includeprojections onto entangled states.Let us consider few examples demonstrating the fea-tures of the developed approach. Figure 5 presents theresults of numerical experiments testing the universalstatistical distribution for fidelity; 200 experiments wereconducted with sample size 1 million each.The measurement protocol is based on tetrahedron.We considered a four-qubit state that represents a mix-ture of GHZ state and uniform density matrix (whitenoise): ρ = f E
16 + (1 − f ) | GHZ ih GHZ | , (43)where E is the unit matrix of size 16 × | GHZ i is the state of Greenberger-Horne-Zeilinger: | GHZ i = √ ( | i + | i ), and f is the weight of the uniformdensity matrix (white noise). In our case f = 0 . χ criterion.Then we consider the dependence of reconstructionprecision on the weight of the “white noise” component.The value of fidelity can belong to a wide interval, thusit is convenient to use a new variable z = − log(1 − F ).Here and below log denotes the common logarithm. Thenew variable z defines the number of nines in numericalrepresentation of fidelity, for example, z = 3 means that F = 0 . n is equal to one million as well. It is evident that thehigher the white noise weight the higher the precision ofreconstruction. It is not difficult to explain this fact inthe context of previous discussion. Namely the “whitenoise” is the best one for mixed state reconstruction. TABLE I. Results of numerical experiments that define maximum precision losses L max for protocols based on the pohyhedrongeometry. 1 qubit 2 qubits 3 qubits( s = 2, L min = 1) ( s = 4, L min = 3) ( s = 8, L min = 7)Tetrahedron ( m = 4) 3/2=1.5 4.442971458 ≈ . m = 6) 9/8 = 1.125 ≈ . ≈ . m = 8) 9/8=1.125 3.4708(3) ≈ . m = 12) 36/35 ≈ . ≈ . m = 20) 45/44 ≈ . ≈ . m = 32) ≈ / ≈ . ≈ . m = 60) 1.0041037488 ≈ . ≈ . z for three qubit state with different value of mixtureGHZ and “white noise”states. The measurement protocol isbased on dodecahedron. Sample size n is equal to one million. And finally, Fig. 7 presents the behavior of the recon-struction precision for Bell and GHZ states when thenumber of qubits grows from two to eight and the samplesize is one million. The measurement protocol is basedon tetrahedron. It is evident that the precision of recon-struction falls as rapidly as the width of the distribution
FIG. 7. (Color online) Density distribution of the scaled fi-delity z for Bell and GHZ states. The measurement protocolis based on tetrahedron. Sample size n is equal to one million. with increasing the number of qubits. V. EXPERIMENT AND ANALYSIS
For testing the theoretic approach, described in theprevious sections, we have prepared a whole family ofpolarization states for qubits and ququarts. In particularwe considered both pure and mixed qubit states and pure,mixed, entangled and separable states of ququarts.
Qubits.
The experimental setup for generation andmeasurement of qubit states is shown in Fig. 8. Set-ting up the experiment we pursuit the following goals:(1) The setup should allow performing the transition be-tween pure and mixed polarization states of qubits. (2)Three different measurement protocols (R4, K4, B36)should be realized. (3) The setup should allow perform-ing measurements with different sample size (statistics).To satisfy the first request we used a wide-bandlight source (incandescence lamp) passing through amonochromator and thick birefringent plates. This pro-vides a variable spectral range around the chosen centralwavelength 1.55 µ m and a controllable phase delay be-tween basic polarization modes. Therefore, by changingthe spectral range of the light (within 1–23 nm) we wereable to vary the purity of the output polarization state. Aparallel light beam was formed by a SM F
28 single-mode0fiber with F F C -1550 micro-objectives placed at itsinput and output. A Glan-Thompson prism was used forpreparation of the horizontally polarized state | H i serv-ing as initial state for the following transformations. Asa result, the original pure state was transformed into amixed state with a degree of purity, depending on thespectral width of the detected radiation. A smooth tran-sition from pure states to total mixtures was achievedwith increasing spectral width of radiation. Since the“measurement part” of the setup had a finite spectralband, the polarization states at different spectral com-ponents within this band were integrated, which corre-sponded to the registration of mixed-polarization state. FIG. 8. (Color online) Experimental setup for different to-mographic reconstructions of qubits with variable degree ofmixture. Here H, V are Glan-Thompson prisms, h is a thickquartz plate, h is a thick tilt plate, h d is a driving phaseplate, h , are tomographic phase plates, and D is a detector. For the preparation of pure polarization states ofqubits we selected a spectral range about 1 nm. The ini-tial state | H i passed through a phase plate h d (436 µ m)oriented at 0 ◦ , ◦ . So the states under consideration hadthe following form respectively: | Ψ i = | H i , | Ψ i = 0 . | H i − . i | V i . (44)The second requirement is achieved with a standardquantum tomography method that used two achromaticquartz plates h =441 µ m, h =313 µ m served for recon-struction of polarization states. These plates were ori-ented at particular angles α, β (with respect to the verti-cal axis), so light passing through the plates and follow-ing vertical polarizer was projected onto the necessaryset of states required by protocols J4 and R4. In the firstprotocol J4, suggested in [5], projective measurementsupon some components of Stokes vector were performed: | H i , | V i , | ◦ i = √ ( | H i + | V i ) , | R i = √ ( | H i + i | V i ).Basically the projective measurements can be chosen ar-bitrarily, in particular if the measured qubits were pro-jected on the states possessing tetrahedral symmetrythen the protocol transforms to R4. There are severalworks showing that due to the high symmetry such pro-tocol provides a better quality of reconstruction [23–25].Another protocol B9 exploits a single plate and fixed po-larizer. The corresponding measurements have been per-formed for each of the nine orientations of the plate witha step of 20 ◦ . We have chosen the “optimal” thickness ofthe plate h =313 µ m and achieved the better conditionnumber ( K = 2 .
7) for reconstruction of mixed states.Also for testing theoretical predictions with this protocolwe used other plates h =824 µ m and h =358 µ m and corresponding condition numbers were K = 14 . K = 183 . δ = πh ∆ n/λ , andthe orientation angle β , where ∆ n is the birefringenceof the plate material at a given wavelength λ and h isits geometric thickness. It is worth to mention that theparameters of this protocol can be chosen by doing theexperiment, depending on available resources: in somesense this choice can be done in an optimal way, but us-ing the plate with optimal condition number. Figure 9shows the calculated condition numbers K and maxi-mum losses L on the Poincar´e-Bloch sphere as functionsof the optical thickness of the phase plate for the B9protocol. The dependencies are periodic with a periodof π . Poor conditionality occurs at δ = 0 , π , π . Bothquantities K and L tend to infinity at these particu-lar points. For this protocol the best (lowest) achiev-able parameter K takes the value 1.85. For example, toreach such a value one might choose plates with opticalthickness δ = 0 . π or δ = 0 . π . The correspondingcondition numbers for protocols J4, R4 are K J4 ≈ . K R4 = √ ≈ .
73. Particularly for our experiment wehave chosen a set of following three plates: h =313 µ m(optimal), h =824 µ m (medium), h =358 µ m (nonop-timal) with condition numbers K = 2 . K = 14 . K = 183 .
9. The optimal value of maximum losses L opt = 1 .
47 occurs at the points δ = 0 . π, . π . Thecorresponding losses are somewhat lower than those forthe R4 protocol ( L = 1 . L = 4 . δ . However, thesingular values of K and L correspond to the same δ value. Thus, both criteria allow separating the regions ofthe protocol parameters for which the results would beunsatisfactory. Unfortunately, for a large dimension ofthe Hilbert space for reconstructed states, the scanningof L values is a complicated computational problem. Forthis reason, to optimize such protocols, we suggest to usethe parameter K .Figure 10 demonstrates the crucial difference betweenthe optimal and nonoptimal approach for the exam-1 FIG. 9. (Color online) Calculated condition numbers K andmaximum losses max( L ) on the Poincar´e-Bloch sphere versusthe optical thickness δ of a phase plate for the B9 protocol.The dashed and dash-dotted lines are for the R4 and J4 pro-tocols, respectively. ple of considered protocol B9. Here the optimal pro-tocol [Figs. 10(a) and 10(b)] corresponds to the platewith thickness h =293.8 µ m and condition number K =2 . h =358 µ m and condition number K = 183 . . Figures 10(a) and 10(b), whichcorrespond to the optimal protocol, describe the case ofguaranteed number of nines in Fidelity being not lessthan z min = 3 .
83 (so that the mean fidelity (22) forall states on Bloch sphere can not be lower). At thesame time, Figs. 10(c) and 10(d), which correspond tothe nonoptimal protocol, stand for a very low level offidelity ( z min = 0 . z min , z max ] barely changes. A comparison ofFig. 10(c) with Fig. 10(d) shows that in the nonoptimalcase increasing the number of projections little affects thedistribution of Fidelity on Bloch sphere.For protocol B9 the statistical reconstruction of theprepared states (44) has been performed at given sam-ple sizes. One of the considered states | Ψ i belongs toan area of small losses for all three plates, and the sec-ond state | Ψ i was changed out of this area. As anexample, we present Fig. 11 that shows the calculatedwidths of fidelity distributions at 1% and 99% quantilesas well as the experimentally reconstructed values for thestates (44). Two polarization states (44) were measuredand reconstructed using three phase plates chosen above.Experiments were performed with different sample sizes; FIG. 10. (Color online) Difference between the (a) and (b)optimal and (c) and (d) nonoptimal approach for protocol B9.(a) and (c) and (b) and (d) correspond to 20 ◦ and 1 ◦ steps ofthe plate orientation, respectively. that is, the total number of the pulses coming from single-photon detector in a fixed time was varied and served asa parameter of the problem.The approach described above provides the ideal ac-curacy level for quantum state reconstruction. It meansthat the fluctuations of the estimated quantum statescannot lead to uncertainties smaller than this limit. Thepresence of instrumental errors and uncertainties makesthis level to be exceeded. Indeed, Fig. 11 shows that,above some sample size, the experimental value of fidelityfalls out the theoretical uncertainty boundary shownas dotted lines (corresponding to 1% significance levelwhich characterizes a given protocol). This happenssince instrumental uncertainties prevail over the statis-tical ones and indicates that either state preparationstage or measurement procedure were not performed ac-curately enough. It is clearly seen that both theoreticaldistributions and experimental points for protocol withnonoptimal choice of plates strongly depend on an initialstate. The quality of reconstruction (fidelity) is varied inthe range 0.6030–0.9999 for both reconstructed states forthe protocol with the plate h =358 µ m (nonoptimal) andin the range 0.9950–0.9997 for the plate h =313 µ m (op-timal). It means that if the set of initial states is known(as it happens in quantum process tomography) one canchoose the parameters of the protocol that are not opti-mal for an arbitrary state, but provide the highest accu-racy for the selected set of states. In both cases, whenthe input states are unknown, like in quantum state to-mography, it is better to use optimal choice of the plateswith minimal condition number, that will ensure highaccuracy of the reconstruction of arbitrary states. Mixed States: modeling and experiment.
The goal ofthe second experiment with qubits was the generalizationof the developed approach to the family of mixed states ofpolarization qubits [48]. As an example we have testedthree protocols again, namely J4, R4, and B36. Theprotocol B36 has been considered at optimal parameters(phase plate h =313 µ m). This protocol is similar tothe one described above (B9), but measurements were2 FIG. 11. (Color online) Reconstruction of pure qubit states by protocol B9 at different thicknesses of the plate. Vertical barsshow 1% and 99% quantiles for fidelity distributions. Dotted lines connecting lower bar ends point out critical significant levels. performed at 36 consistent orientations of the phase plate(0 ◦ –360 ◦ , step 10 ◦ ) instead of 9.As we have pointed out before, the condition num-bers K for R4 and J4 protocols take the following values: K R4 = √ ≈ . K J4 ≈ .
23 while for B36 it becomes K B36 ≈ .
7. Thus, we expect that the symmetrical proto-col R | Ψ i = | H i which passed through one or twothick birefringent plates, oriented at 45 ◦ , where partialdecoherence between vertical and horizontal basic polar-izations took place. For that purpose we used a quartzplate with a thickness of h =10 mm and additional tiffplate with a thickness of h =4 mm. Changing the widthof the spectrum it is possible to pass from completelycoherent (pure) case to a very narrow spectral line to acompletely incoherent (i.e. mixed in polarization), whenthe difference between the optical lengths for two orthog-onal polarizations exceeds the coherence length of theradiation under consideration: h ( n ⊥ − n k ) quartz + h ( n ⊥ − n k ) tiff ) ≫ l coh , (45)where l coh ≈ λ △ λ ≈ µ m . Thus, at the output of plates h , h components withorthogonal polarizations evolve with a random phase,which leads to a mixed state.For determining the accuracy of quantum tomographyand fidelity, we need to calculate the density matrix ofa prepared mixed state. For this purpose we divide thefrequency spectrum of the radiation in small parts and represent the polarization state of a qubit as a superpo-sition of states corresponding to different frequencies inthe spectrum: | Ψ i = P k a k | Φ( ω k ) i , | Φ( ω k ) i = c ( ω k ) | H i + c ( ω k ) | V i , (46)where amplitudes a k are defined by the spectrum shape.The density matrix of the state before transformation(46) has the form: ρ in = | Ψ ih Ψ | = X k,j a k a ∗ j | Φ( ω k ) ih Φ( ω j ) | (47)The next step of modeling includes the calculation ofphase plates action on the qubit state. The unitary trans-formation on state (47) is given by matrix G ( ω k ) = t k r k − r ∗ k t ∗ k ! (48)where t k = cos δ k + i sin δ k cos 2 α,r k = i sin δ k sin 2 α, δ k = π ( n ko − n ke ) h/λ k . (49)Here t k and r k are the amplitude transmission and re-flection coefficients of the wave plate at fixed frequency, δ k is its optical thickness, h is the geometrical thickness, α is the orientation angle between the optical axis ofthe phase plate and vertical direction. The measurementpart of the experimental setup does not distinguish fre-quency modes, so the theoretical polarization mixed statewill be described by a reduced density matrix: ρ = X k | a k | G ( ω k ) | Φ( ω k ) ih Φ( ω k ) | G + ( ω k ) . (50)3 FIG. 12. (Color online) Theoretical distribution of the real and imaginary parts of the density matrix for states with varyingdegrees of purity. The spectral widths of the spectrum are (a) 1.6 nm, (b) 7 nm, (c) 13 nm. Distribution (d), for 22 nm,corresponds almost to a completely mixed state.
The corresponding density matrixes are obtained byintegrating formula (50) with the distribution of weights | a k | as a function sinc ( x ). Figure 12 shows the graph-ical representations of the real and imaginary parts ofthe theoretical qubit density matrices at different widthsof the spectrum: 1.6 nm, 7 nm, 13 nm, 22 nm. Whenincreasing the width of the spectrum the non-diagonalcomponents, responsible for the correlation, decay andthe state becomes completely mixed.In the experiment we have prepared four polarizationstates of qubit with following degrees of mixture: 3%(1.6 nm), 30% (7 nm), 66% (13 nm), 100% (22 nm),where the state purity is analyzed by calculating the stateentropy defined as S = − P n =1 λ n log λ n , λ n being theeigenvalues of the density matrix ρ . For each protocolvarious measurements at different sample sizes were per-formed. The results are shown in Fig. 13.Figure 13 presents the theoretical distributions of fi-delity at 1% and 99% quantiles for each considered pro-tocol. Experimental values are indicated by points. Thedashed lines connecting the 1% quantiles show a theoreti-cal lower boundary of the fidelity distribution and dependon the type of protocol: the lower line, the greater con-dition number and the function of losses. Fig. 13 demon-strates that for a small sample size statistical uncertain-ties dominate over the instrumental ones. Starting fromsample size about (2 − × the experimental points fallout the theoretical uncertainty boundary and the proto-col achieves a coherent sample size. This means that in-strumental uncertainties (the accuracy of setting angles,thicknesses of phase plates, etc.) prevail over the levelof statistical errors. From this point a further increaseof sample size does not improve the quality of the recon-struction of the quantum state. However, a comparisonof experimental results with the theoretical distributionserves as effective method for setup adjusting, stabilityof the preparation/measurement systems, etc. Furthermore, from Fig. 13 one can evince that the ac-curacy of the reconstruction increases with the degree ofmixture: for the states close to a pure ones (3% mixture)the fidelity achieves an average value 0.93–0.94, whilethe best accuracy is achieved for totaly mixed state cor-responding to the center of the Bloch sphere, F > . Ququarts: pure and mixed states, preparation and mea-surement.
For a further verification of our approach weprepared also a family of quqarts biphoton polarizationstates [49], which can be easily converted into either en-tangled (in polarization) or factorized states, both pureand mixed. ρ = (1 − p ) | Ψ pure ih Ψ pure | + p ( | H H ih H H | + | V V ih V V | ) , (51)where | Ψ pure i = c | H H i + c e iϕ | V V i (52)with real amplitudes c and c and relative phase shift ϕ . The coefficient p defines the degree of mixture.The set-up is schematically depicted in Fig. 14. Forgeneration of pure biphoton-based polarization ququarts( p = 0) we used a set of two orthogonally orientedtype-I BBO crystals (1 mm), cut for collinear, frequencynon-degenerate phase-matching around the central wave-length of 702 nm. The crystals were pumped by a600 mW cw-argon laser operating at 351 nm. The Glan-Thompson prism (V) at vertical polarization and thehalf-wave plate ( λ p /
2) placed in front of crystals allowedrotating the polarization of the pump by the angle φ ,which controlled the real amplitudes c and c in (52).A set of quartz plates QP introduced the relative phaseshift ϕ between the horizontal and the vertical compo-nents of the UV pump. If φ = 0 we prepared the state4 FIG. 13. (Color online)Reconstruction of qubits states with various degree of mixture. Vertical bars show 1% and 99% quantilesfor fidelity distributions. Dotted lines connecting lower bar ends point out critical significant levels. (a) mixed 3%, (b) mixed30%, (c) mixed 66%, and (d) mixed 100%. | Ψ i = | V V i , if φ = 22 . ◦ and ϕ = 3 π/ | Ψ i = √ ( | H H i − | V V i ) ≡ | Φ − i . Tomaintain stable phase-matching conditions, BBO crys-tals and QP were placed in a closed box heated at fixedtemperature. The lens L coupled SPDC light into themonochromator M (with 4 nm resolution), set to trans-mit “idler” photons at 710 nm. The conjugate “signal”wavelength 694 nm was selected automatically by meansof the coincidence scheme.In order to prepare a mixed state it is necessary tointroduce quantum distinguishability among the bipho-ton ququart basis states in a controllable manner. Forthe preparation of states with various degree of mix-ture the double-crystal scheme was complemented by aquartz plate with thickness h , which was putted in thereflected arm of Brown-Twiss scheme. A thick quartzplate with vertically oriented optical axis introduced a de- lay between vertically and horizontally polarized photonsthat led to their temporal distinguishability, and hence,to the emergence of a mixture. Changing the width ofthe frequency spectrum, which, in our case, was deter-mined by the width of the monochromator slit, or theplate thickness it was possible to change the visibility ofpolarization interference. Our goal was the preparationand reconstruction (by several protocol, including non-optimal ones) of a set of states with different degree ofentanglement and set of mixed two-qubit states.The reconstruction part for the ququart is shown inFig. 14. First, the photon pair that forms the biphoton-ququart was split into two spatial modes by using abeam splitter BS. A photon in each arm, then, under-went a polarization state transformation with the use ofa couple of zero-order wave plates ( λ/ λ/
4) for proto-cols. Two Si-APD’s linked to a coincidence scheme with5
FIG. 14. (Color online) Experimental setup for different to-mographic reconstructions of photon pairs with variable po-larization entanglement and degree of mixture. Ar laser: ar-gon laser with wavelength 351 nm, M: mirror, V: verticaloriented Glan-Thompson prism, BBO: nonlinear Barium Bo-rate crystals, UVF: ultraviolet filter, λ p / λ p /
4: half-wave,quarter-wave plates, L: lens with focus 20 cm, BS: beamsplit-ter, IF: interfilter, D: avalanche photodetectors, CC: coinci-dence circuit. K in common logarithmic scale for the set of thicknesses ofthe first and second quartz retardant plates. Specificallythe first plate Wp1 was varied within the limits from1.100 to 1.350 mm with a step 0.001 mm and a secondplate, Wp2, was varied within the limits from 0.300 to0.550 mm with the same step. Here the thicknesses of thefirst and second quarts retardant plates are shown alongthe axis x, y , correspondingly. Blue color indicates areaswith low value of condition number K , while the red oneindicates areas with high value of K . It is obvious, that FIG. 15. (Color online) Dependence of log( K ) on plates thick-nesses for protocol B144. the arbitrary choice of plates most likely will be wrongfrom the point of view of completeness of matrix B andprotocol optimality. In other words this figure serves assome sort of a “navigation map” for the measurementprotocol. So to achieve good quality of the measurementone should select the plates thicknesses in blue areas andavoid red areas. For the present protocol the minimumachievable K is 11.35. For example, to reach such a valueone can choose plates with thickness Wp1=1.207 mm andWp2=0.520 mm. In our specific experimental configu-ration we have selected the plates Wp1=1.303 mm andWp2=0.440 mm. Such thicknesses of the plates for theprotocol B144 were chosen to show the difference in thequality of reconstruction.For all three protocols used in the experiment we havecalculated the condition numbers K , which assume thefollowing values: K R16 = 3, K J16 ≈ K B144 ≈ | Φ − i = √ ( | H H i − | V V i ).Figure 16 shows average fidelities, calculated accordingto (21) as functions of a sample size for each protocol. Itturns out that the difference between curves disappearsat sufficiently large sample size (10 )[50], but for the samequality of state reconstruction the correct choice of pro-tocol allows using a smaller set of statistical data, i.e. fi-nally reduces total acquisition time. Figure 16 shows thatprotocols are ranged in accuracy as following: R16, J16,and B144 in complete agreement with the range given bycondition number K . Figure 17 presents accuracy distri-butions calculated according to (21) for the sample size6 FIG. 16. (Color online) Dependence of the average fidelity onnumber of registered events forming the sample for Bell statereconstructed by protocols J16, K16 and B144. × .It is clearly seen that the density distribution for theR16 protocol is narrower than the one for J16 and B144and localizes in the region of lower losses or higher fideli-ties. The distribution for B144 is broader and lower incomparison with R16 and J16. Obviously, the narrowerdistribution of fidelity indicates a better reconstructionquality in the sense that the reconstruction procedurereturns a better-defined state. Thus, Figure 17 confirmsour expectation based on estimation of condition number K : R16 achieves the best results.Let us consider now the experimental results. As an FIG. 17. (Color online) Density distribution of the scaledfidelity z (lower abscissa) at n = 3 × for Bell state re-constructed by protocols J16, K16 and B144. Upper abscissapresents regular fidelity. example, Fig. 18 shows both experimental points andtheoretical distributions for the set of pure ququart stateswith various degree of entanglement: C ϕ = 0, C ϕ =0 . C ϕ = 0 . C ϕ = 1 correspondingly. Here C ϕ i isquantity concurrence, which is defined as C = 2 | c c − c c | and C = 0 for a separable state and C = 1 for amaximally entangled state.Figure 19 presents experimental points and theoreticaldistribution for the set of mixed ququart states with vari-ous degree of mixture: p = 1 , . , . p = 1 correspondsto totaly mixed state, p = 0 . a priori calculated expectations is note-worthy.Figure 19 shows that the accuracy of reconstruction in-creases with the degree of mixture. However from a sta-tistical point of view there is a problem. A small weight ofthe mixed states, adding to the original pure state, leadsto a small change in the statistical data while the numberof parameters for ququart state increases from 6 to 15.As a result the amount of new information that arisesin quantum measurements is not sufficient to achieve anadequate estimation of all the parameters. VI. DISCUSSION
Many remarkable works in quantum state tomogra-phy developed different theoretical approaches. In thepresent section we shall briefly discuss the features andadvantages of our approach in this framework.One of the main issues in quantum tomography is thechoice of adequate parameterization for quantum states.Bloch sphere representation is one option. In this repre-sentation density matrix ρ is defined in Hilbert space ofdimension s by the following equation [5, 35, 51] ρ = 1 s E + ν j σ j , (53)where E is the s × s identity matrix, ν j is a vector inreal Euclid space of size s −
1, and j changes from 1 to s −
1. Respective Hermitian basis matrices σ j meet thefollowing conditions:tr( σ j ) = 0 , tr( σ j σ k ) = δ jk . (54)Note, that for single-qubit states when s = 2, matrices σ j coincide with ordinary Pauli matrices (multiplied by √ ).The radius of 3D Bloch sphere in this representation isalso equal to √ .One may easily calculate Bloch vectors ν j from theknown density matrix ρ : ν j = tr( ρσ j ) . (55)However, Bloch parameterization has a significant draw-back. For every real vector ν Eq. (53) guarantees that7
FIG. 18. (Color online) Reconstruction of ququart states with various degree of entanglement. Vertical bars show 1% and 99%quantiles for fidelity distributions. Dotted lines connecting lower bar ends point out critical significant levels. the density matrix ρ is Hermitian, but does not guaranteethat it is positively defined. As a result, we can obtain anon-physical density matrix using Eq. (53). Separationof admissible density matrices from those that can notphysically exist is a non-trivial task for all dimensionshigher than two. In particular, for s > s −
1. This drawback of Bloch parameterizationcomplicates the use of Fisher information matrix and cal-culation of statistical estimates precision in general.An adequate and convenient procedure of parameteri-zation is important for numerical procedures of calculat-ing density matrix from experimental data because forevery iteration the approximate solution must lie in therange of physical density matrices only. For instance,such issues may arise for problems of estimation of quan-tum state by maximum likelihood method.The problem of estimating a density matrix by max- imum likelihood method was considered in [8, 14–16].The respective likelihood equation has the following form[8, 51, 52]: Rρ = ρ, (56)where R is some operator.The iteration procedure from j th to j +1th step for thisequation has the following form: ρ j +1 = Rρ j . Such pro-cedure, however, does not even provide Hermitian prop-erty for density matrix.Another procedure that is widely used [51] ρ j +1 = ( Rρ j + ρ j R ) provides the Hermitian property but doesnot guarantee positive definiteness in general. Fortu-nately there are known robust, well-converging proce-dures providing with both Hermiticity and positive defi-niteness simultaneously [54].In this paper we consider a natural approach for quan-tum theory that is based on representing the quantumstate as a mix of pure states [17, 18, 40]. Consider trans-8 FIG. 19. (Color online) Reconstruction of ququart states with various degree of mixture. Vertical bars show 1% and 99%quantiles for fidelity distributions. Dotted lines connecting lower bar ends point out critical significant levels. (a) p = 0 .
3, (b) p = 0 .
7, and (c) p = 1. formation of the density matrix to diagonal form. ρ = U DU † . (57)Here U is an unitary matrix (its columns define eigen-vectors of density matrix), D - is a diagonal non-negativematrix (its diagonal is formed by eigenvalues of densitymatrix that we shall put in decreasing order). Let therank of density matrix be r (1 ≤ r ≤ s ). We shall elim-inate all non-zero rows and columns of matrix D andshall leave only the first r columns in matrix U . Thenthe density matrix could be represented in the followingcompact form. ρ = LL † , (58)where L = U D .Here L is a complex matrix of dimension s × r thatdefines the actual parameterization of density matrix.It represents a purified state vector (probability ampli-tudes). For representation as a column vector, one shouldsimply place the second column under the first one and soon. Note that the purified state is defined ambiguously,because the density matrix does not change during trans-formation L → L ′ = LV, (59)where V - is an arbitrary unitary matrix of size r × r .It is crucial that the ambiguity (59) does not affect theprocedure of statistical reconstruction of density matrix,because all possible purified state vectors define the samedensity matrix.We can choose the unitary transformation V such thatin matrix L ′ all elements on the principal diagonal willhave real strictly positive values, while elements higherthe principal diagonal shall be equal to zero.For a density matrix of full rank, such matrix L ′ shallform a lower triangular matrix. In mathematics suchdecomposition is usually called Cholesky decomposition.Representation of density matrix in such form was origi-nally studied in [10]. Note that from the point of view of physics, Cholesky representation is just one of the possi-ble ways of recording a purified quantum state.A purified state vector represents the most simple andmost natural in view of physics form of parameterizationof density matrix. It is important that parameterizationbased on purification of quantum state radically simpli-fies the theory of statistical estimates.For example, the precision of estimates by maximumlikelihood method is defined by matrix of full informationthat is analogous to Fishers information matrix in rela-tion to estimation of quantum state vector [17, 18, 40].The matrix of full information is defined by the follow-ing equation: H = 2 X j t j (Λ j c )(Λ j c ) + λ j . (60)The object is defined in real Euclidian space of dimension2 rs . To obtain state vector c in this representation, oneshould place the imaginary part of purified state vectorunder its real part. Intensity matrices Λ j represent ex-pansion of matrices (6) to the considered Euclid space.The sum in Eq. (60) is taken for all measurement proto-col rows. Matrix H becomes a real symmetric matrix ofdimension 2 rs × rs .If the quantum measurement protocol is complete,then for the information matrix H ν H = (2 s − r ) r outof 2 rs eigenvalues are strictly positive, while the other r are equal to zero. One may formulate a universal sta-tistical distribution (21) based on information matrix H that will describe precision of statistical reconstructionof quantum states.Eigenvectors of matrix H corresponding to non-zeroeigenvalues define directions of fluctuations of a quantumstate and its norm, while eigenvectors that correspond tozero eigenvalues define directions of insignificant fluctua-tions that are due to the ambiguity (59) in definition ofthe purified state vector.While information matrix H defines detailed charac-teristics of precision of reconstruction of an arbitraryquantum state, the measurement matrix B (9) defines9the quality of the protocol as a whole. The conditionnumber K separates complete and well defined quantummeasurement protocols from incomplete and ill-definedones.The approach developed above is based on analysis ofcompleteness, adequacy and precision of quantum mea-surements and it could be applied to analysis of qualityand efficiency of arbitrary protocols of quantum measure-ments.As a simple example, let us consider a single-qubit pro-tocol proposed in works [35, 53] (see Fig. 1 in [35] andFig. 3 in [53]). The considered approach is an optimizedmeasurement of state c = 1 √ ! . Measurements are conducted by means of a set of half-and quarter- wave plates that lead to projections to thefollowing basis states ψ = 1 √ sin(2 h ) + i sin(2( h − q ))cos(2 h ) − i cos(2( h − q )) ! .ψ = 1 √ cos(2 h ) + i cos(2( h − q )) − sin(2 h ) + i sin(2( h − q )) ! . where h and q are orientation angles for half-wave andquarter-wave plates respectively. This measurement de-fines two strings of instrumental matrix X . The authorsof the protocol proposed a set of four measurements ofsuch type (8 rows in total). Measurements are defined bythe following orientation angles for the plates: h = 0 ◦ , q = 0 ◦ ; h = 20 ◦ , q = 45 ◦ ; h = 25 ◦ , q = 45 ◦ ; h = 45 ◦ , q = 0 ◦ .A complete analysis of precision losses (34) during re-construction of single-qubit pure quantum states usingthe protocol is given in Fig. 20 (little precision loss statesare marked blue, large precision losses a marked brown). FIG. 20. (Color online) Analysis of precision losses for proto-col proposed in [35, 53].
We see that for the state c = 1 √ ! the protocol is indeed the optimal one because the lossesare at minimum L min = 1. However for a number of otherstates the protocol is rather not optimal (maximum lossesare quite large L max ≈ . L min = 1 to L max = 9 / K for the protocol isapproximately 4 .
07 times higher than for protocols basedon regular polyhedrons, which also certifies its lower qual-ity.
VII. CONCLUSIONS
In this paper, we proposed a methodology to assess thequality and efficiency of quantum measurement proto-cols. It was shown that the proposed approach, based onthe analysis of completeness, adequacy and accuracy, canbe successfully applied to arbitrary quantum states andmeasurement protocols. Analysis of the completeness ofthe protocol allows to answer the question: Is the setof operators considered sufficient to assess an arbitrarypure or mixed quantum state? The operational crite-rion of the protocols’ quality follows from the analysisof completeness. The criterion is based on the conditionnumber of the measurements matrix K . An analysis ofthe adequacy allows one to answer the question about thecorrespondence between the statistical experimental dataand the quantum-mechanical mathematical model. Theanalysis of the adequacy guarantees the correctness of thefollowing procedures of statistical quantum state recon-struction by using the maximum likelihood method. Thedeveloped method of statistical reconstruction is basedon on the procedure of quantum state parameterizationusing the procedure of purification. Such parametriza-tion allows us to introduce a universal distribution for Fi-delity, which provides all information about the accuracyof the the quantum state reconstruction. Various numer-ical examples and results of physical experiments havebeen analyzed for testing theoretical predictions both forpure and mixed states and demonstrating approach va-lidity. ACKNOWLEDGMENTS
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