Exact and approximate solutions for the quantum minimum-Kullback-entropy estimation problem
Carlo Sparaciari, Stefano Olivares, Francesco Ticozzi, Matteo G. A. Paris
EExact and approximate solutions for the quantum minimum-Kullback-entropyestimation problem
Carlo Sparaciari ∗ Dipartimento di Fisica dell’Universit`a degli Studi di Milano, I-20133 Milano, Italia.
Stefano Olivares † Dipartimento di Fisica dell’Universit`a degli Studi di Milano, I-20133 Milano, Italia andCNISM UdR Milano Statale, I-20133 Milano, Italy
Francesco Ticozzi ‡ Dipartimento di Ingegneria dell’Informazione, Universit`a di Padova, I-35131 Padova, Italia andDepartment of Physics and Astronomy, Dartmouth College, 6127 Wilder, Hanover, NH 03755 (USA).
Matteo G. A. Paris § Dipartimento di Fisica dell’Universit`a degli Studi di Milano, I-20133 Milano, Italia andCNISM UdR Milano Statale, I-20133 Milano, Italy (Dated: November 8, 2018)The minimum Kullback entropy principle (mKE) is a useful tool to estimate quantum states andoperations from incomplete data and prior information. In general, the solution of a mKE problemis analytically challenging and an approximate solution has been proposed and employed in differentcontext. Recently, the form and a way to compute the exact solution for finite dimensional systemshas been found, and a question naturally arises on whether the approximate solution could be aneffective substitute for the exact solution, and in which regimes this substitution can be performed.Here, we provide a systematic comparison between the exact and the approximate mKE solutionsfor a qubit system when average data from a single observable are available. We address bothmKE estimation of states and weak Hamiltonians, and compare the two solutions in terms of statefidelity and operator distance. We find that the approximate solution is generally close to the exactone unless the initial state is near an eigenstate of the measured observable. Our results provide arigorous justification for the use of the approximate solution whenever the above condition does notoccur, and extend its range of application beyond those situations satisfying the assumptions usedfor its derivation.
PACS numbers: 03.65.Wj, 42.50.Dv
I. INTRODUCTION
Let us consider a situation where a quantum system isprepared in a known state and , after some time and un-known evolution, some measurements are performed inorder to gather information on the final state. The exactsolution to this state estimation problem is provided byquantum tomography, which however requires the mea-surement of a complete set (i.e. a quorum ) of observables[1].Measuring a quorum of observables may be experi-mentally challenging, or require too many resources, andtherefore it is worth exploring the case where the set ofobservables that can be measured on the system is in-complete [2]. In this case, we cannot obtain completeinformation about the state of the system from the out-comes of these measurements, i.e. the measurements are ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: matteo.paris@fisica.unimi.it not fully determining the state of the system. We thusneed some additional ingredient to fill the informationgap and single out a quantum state that is compatiblewith the data, and with the information that is possiblyavailable prior to the measurements [3, 4].When we have no a priori information, e.g. becausethe initial state of the system is unknown or the interac-tion with the environment is strong enough to wash outany initial information, the problem may be attacked us-ing the maximum entropy principle (ME) [5, 6]. Withthe ME we take all the states (density matrices) com-patible with the evidences, i.e. reproducing the correctprobabilities of the observed data, and pick up the onemaximizing the Von Neumann entropy. In this way, theonly knowledge about the state is that coming from themeasurements made on the system, without the additionof any unwanted piece of extra information which is notavailable from the experimental evidences [7, 8].On the other hand, there are several situation of inter-est where some a priori information is indeed available,in the form of a a priori state. This may be due to someconstraints imposed to the physical preparation of thesystem, or to the fact that the coupling of the systemwith the rest of the universe is weak, so that the state a r X i v : . [ qu a n t - ph ] J u l remains close to the initial preparation. In these cases,the minimum Kullback entropy (mKE) principle [9, 10]provides an effective tool to include this new ingredientin the solution and to complement the experimental data,thus allowing to obtain a unique estimated state.The mKE principle has received attention in the recentyears and has been applied to both finite and infinite-dimensional systems [11–14]. In particular, applicationsto qubit and harmonic oscillator systems have been ini-tially put forward upon exploiting an approximate solu-tion of the minimization problem [12, 15]. More recently,the analysis of the feasibility, the form of the general so-lution and a method to compute it has been derived forfinite dimensional systems [16] and a question naturallyarises on how the approximate solution compares to theexact one, and in which regimes it could be convenient toemploy the former. Indeed, our analysis is motivated bytwo relevant properties of the approximate solution: onthe one hand it is given in a closed form which is moreconvenient for applications and, on the other hand, itmay be applied to a larger class of a priori states, includ-ing those described by a density operator not having fullrank.This paper focuses on qubit systems in situationswhere only the average of a single observable can be ac-cessed. We consider the use of mKE for estimation ofstates and for the characterization of weak Hamiltoni-ans, and compare the two solutions in terms of state fi-delity and operator distance respectively. We find thatthe approximate solution is generally close to the exactone unless the initial state is near an eigenstate of themeasured observable. Our results thus provide a rigor-ous justification for the use of the approximate solutionwhenever the above condition does not occur.The paper is structured as follows. In Sec. II we reviewthe mKE principle for a qubit system where a single ob-servable is measured, and present both the approximateand the exact solutions to the mKE estimation problem.In Sec. III a systematic comparison between the approx-imate and the exact solution is performed in terms offidelity, and the role of initial purity is discussed. In Sec.IV we address estimation of weak Hamiltonians by mKEand compare the approximate and the exact solutions interms of operator trace distance. Sec. V closes the paperwith some concluding remarks. II. THE MINIMUM KULLBACK ENTROPYPRINCIPLE
The quantum Kullback (Umegaki’s) relative entropybetween two quantum states is defined as [17–20]: K ( ρ | τ ) = Tr [ ρ (log ρ − log τ )] . (1)As for its classical counterpart, the Kullback-Leiber di-vergence, it can be demonstrated that 0 ≤ K ( ρ | τ ) < ∞ when it is definite, i.e. when the support of the first statein the Hilbert space is contained in that of the second one. In particular, K ( ρ | τ ) = 0 iff ρ ≡ τ . This quantity, thoughnot defining a proper metric in the Hilbert space (it isnot simmetric in its arguments), has been widely used indifferent fields of quantum information [20–25] because ofits additivity properties and statistical meaning in statediscrimination.Let us now consider a quantum system initially pre-pared in the state τ that, after some kind of evolution,unitary or not, is now in the final state ρ . In this casewe have some prior information that we can regard as abias towards τ . Furthermore, when some observables aremeasured, the information achieved (e.g. their mean val-ues or their full probability distributions) gives some con-straints about the state. The mKE principle, states thatthe best estimate for the state ρ is then the density ma-trix that satisfies the constraints and, at the same time,is somehow closer to the initial state τ , i.e. minimize thequantum Kullback entropy, given the constraints. ThemKE principle allows one to take into account the avail-able prior information as well as the new evidence comingfrom the data, while not introducing any other kind ofspurious or unwanted piece of information.The minimization can be done by Lagrange multipli-ers. If the constraint is given by the mean value of theobservable A (and by the normalization), then the quan-tity that should be minimized is: F ( ρ, λ , λ ) = K ( ρ | τ ) + λ (Tr [ ρ ] − λ (Tr [ ρA ] − (cid:104) A (cid:105) ) , (2) λ k being the Lagrange multipliers. Two approaches havebeen developed to solve this mKE problem. The firstis approximate and leads to analytic solutions in severalcases, e.g. when the final state is close to the initial one[12, 26]. More recently, the general feasibility of thisestimation problem and an exact method has been de-veloped, valid when the quantum system under investi-gation is finite dimensional [16]. Having at disposal anexact solution allows us to assess the approximate oneand to individuate the situations where it may safely ap-ply instead of the exact one. In the following we aregoing to systematically compare the two solutions for aqubit system subjected to the measurement of a singleobservable. A. General Solution
It is possible to show that, for finite-dimensionalHilbert spaces, the minimum of the Eq. (2) exists, isunique and continuous with respect to the data [16]. Af-ter a suitable reduction of the problem to a subspace thatensures that the solution is full rank, and assuming that τ is full-rank on the same subspace, the optimal solutionof the problem, when the only outcome of the measure-ment is the mean value of observable A , is the followingone: ρ ( λ , λ ) = e log τ − I − λ X − λ X (3)where λ , λ are Lagrange multipliers, and X , X arethe operators obtained from I and A through the Gram-Schmidt orthogonalization process. Notice that if τ is notfull rank the above formula does not return a valid den-sity operator. In order to evaluate the Lagrange multi-pliers, the constraints of normalization Tr[ ρ ( λ , λ )] = 1and mean value of A , Tr[ ρ ( λ , λ ) A ] = (cid:104) A (cid:105) should beimposed. B. Approximate solution
Assuming that the evolution is not leading the sys-tem too far away from its initial preparation, we can findan approximate solution to the mKE estimation problemupon writing the infinitesimal increment of the densityoperator. More explicitly: in the Hilbert space of statis-tical operators, one considers an infinitesimal incrementof the operator ρ correspoding to the increment of anarbitrary parameter λ . Upon assuming that incrementsof the density operator are evaluated according to theFisher metric it is possible to introduce the differentialequation [26]: dρdλ = − { ρ, A − (cid:104) A (cid:105)} (4)where { , } is the anticommutator. As already men-tioned, the same equation can be obtained from Eq. (2),when the final state ρ is close to the initial state τ , accord-ing to the Fisher metric. In turn, the state ρ obtainedby integration of Eq. (4) is the approximate solution ofthe mKE problem with λ playing the role of a Lagrangemultiplier.This work focuses on statistical operator in the qubitspace, when the initial state is given and the only infor-mation obtained from measurement is the mean value ofthe observable A . The solution of the previous equationfor this case (notice that it is also correct for spaces withdimension larger than two) is: ρ ( λ ) = e − Aλ/ τ e − Aλ/ Tr [ τ e − Aλ ] (5)where λ can be found using the constraint Tr [ Aρ ] = (cid:104) A (cid:105) .As mentioned above, the approximate solution ρ ( λ ) maybe computed also if τ is not full rank. In addition, wenotice that ρ ( λ ) has the same rank of the a priori state τ . Since the approximate solution has been derived as-suming that evolved state is close to the initial one [26],one may expect that ρ ( λ ) obtained from Eq. (5) is nottoo far away from the a priori state τ . As we will show inthe following, this is basically true in the case of nearlypure initial states. Otherwise, when the initial state isappreciably mixed, the approximate solution can, in fact,be far away from the initial preparation. On the otherhand, also in these cases the approximate solution is closeto the exact one. In other words, having at disposal the exact solution allows us to assess the approximate onealso outisde the assumptions made to derive it, and toextend the regimes where it may be safely employed. III. COMPARISON BETWEEN THE EXACTAND THE APPROXIMATE MKE ESTIMATES
In this section the two solutions are compared, throughthe use of the fidelity, in order to establish whether, andin which conditions, the approximate solution can be con-sidered as a good replacement for the exact one. Thecomparison is made for qubits systems.In particular, we address situations where a single ob-servable A is measured. The most general qubit observ-able may be written as A = a I + a · σ , where a and a = ( a , a , a ) are real parameters and σ = ( σ , σ , σ ) denote the Pauli matrices vector. With-out loss of generality it is always possible to perform arotation and a scaling in order to rewrite the observableas U † A U = A = α I + σ (6)i.e. as a function of a single real parameter α .Once the rotation is made, we rewrite the general stateof the qubit in the new reference as τ = 11 + (cid:15) ( | ψ (cid:105) (cid:104) ψ | + (cid:15) (cid:12)(cid:12) ψ ⊥ (cid:11) (cid:10) ψ ⊥ (cid:12)(cid:12) ) , where | ψ (cid:105) = cos θ | (cid:105) + e iφ sin θ | (cid:105) is the generic pure state and (cid:12)(cid:12) ψ ⊥ (cid:11) its orthogonal com-plement. The parameter (cid:15) depends on the purity µ [ τ ] ofthe initial state τ , we have: µ [ τ ] = 1 + (cid:15) (1 + (cid:15) ) , with µ ∈ [1 / , θ ∈ [0 , π ], φ ∈ [0 , π ) and µ ∈ [1 / , α ∈ R and (cid:104) σ (cid:105) ∈ [ − ,
1] specify the measured observableand its mean values respectively: (cid:104) A (cid:105) = α + (cid:104) σ (cid:105) . Sincewe are dealing with qubit measurements (which have twopossible outcomes) the knowledge of the mean value isequivalent to that of the full distribution.Once we fix both τ and A , the approximate and exactsolutions can be evaluated using, respectively, Eq. (5)and Eq. (3). The approximate solution in Eq. (5) has ananalytic form, which is independent of α , and is given by[12] ρ ( λ ) = e − λσ τ e − λσ Tr[ τ e − λσ ] , (7)where λ is determined by solving the equationTr[ ρ ( λ ) σ ] = (cid:104) σ (cid:105) . The analytic form of the Bloch vector r = ( r , r , r ) of ρ ( λ ) = ( I + r · σ ) is given by: r = t Z ; r = t Z ; r = (cid:104) σ (cid:105) where t = ( t , t , t ) is the Bloch vector of the initialstate τ and Z = cosh λ − t sinh λ (see Appendix A for theexplicit expression in terms of the parameters θ , φ , and (cid:15) ). The corresponding value of the Lagrange multiplieris λ = arctanh t − (cid:104) σ (cid:105) − (cid:104) σ (cid:105) t . (8)For what concerns the optimal solution, the first La-grange multiplier λ is evaluated using the trace normal-ization for the state ρ , while the second one λ is eval-uated upon exploiting the constraint of the mean value.The equation for the last constraint is transcendental,and numerical methods are needed in order to find λ .When the values of the two solutions are found, for fixed τ and A , it is possible to compare them using the qubitfidelity: F ( ρ , ρ ) = Tr [ ρ ρ ] + (cid:112) − µ [ ρ ] (cid:112) − µ [ ρ ] (9)where µ [ ρ k ] is the purity of the state ρ k , ρ is the ap-proximate solution and ρ the exact one.In order to assess the reliability of the approximatesolution we have evaluated the fidelity between the ap-proximate and the exact solution as a function of the fiveparameters involved in the estimation problem. Our firstresult is that the fidelity does not depend on the angle φ ,i.e. the two solutions (approximate and exact) show thesame functional dependence on such parameter. Besides,the approximate and the exact mKE estimate, as well asthe fidelity, do not depend on the parameter α .The relevant parameters to assess the approximate so-lution are thus the angle polar θ and the purity µ [ τ ] ofthe initial state and the result of the measurement (cid:104) σ (cid:105) .The fidelity is also symmetric with respect to the trans-formations θ → π − θ and (cid:104) σ (cid:105) → −(cid:104) σ (cid:105) . Notice thatfinding the exact mKE solution requires the use of nu-merical methods, which pose a upper bound to the initialpurity µ [ τ ] (cid:46) − − , above which the solution becomesnumerically unstable.As we will see in the following, the fidelity shows differ-ent behaviors, depending on the purity µ [ τ ] of the initialstate τ . Before going to a detailed comparison, we noticethat if the initial state τ commutes with the measuredobservable A , i.e. [ τ, A ] = 0, then the two solutions co-incide as it is apparent by inspecting Eq. (3) and Eq.(5). A. Fidelity for highly mixed initial states
When the purity µ of the initial state takes values be-tween 1 / . FIG. 1: (Color online) Fidelity between the exact and theapproximate mKE estimate as a function of the parameter θ of the initial state and of the outcome of the measurement (cid:104) σ (cid:105) . The plots are for two fixed values of the initial purity: µ = 0 .
55 (top left) and µ = 0 . θ = π/ θ =5 π/
12 (black dashed), θ = π/ µ . As it is apparent from the plots, the fidelity betweenthe two solutions is extremely close to unit for a largerange of values of (cid:104) σ (cid:105) around (cid:104) σ (cid:105) = 0, whereas for val-ues of (cid:104) σ (cid:105) near ± θ andthe global minimum is achieved for θ = π/
2. It is worthnoticing that the values of these minima corresponds tofidelity always larger than F = 0 . τ tends to 1 /
2, i.e. the ini-tial state approaches τ = I /
2, the two solutions coincidesfor all values of θ and (cid:104) σ (cid:105) , as it may readily seen fromEq. (3) and Eq. (5). This corresponds to the case of asystem initially in a maximally mixed state and for whichthe measurement is not providing any additional infor-mation. The minimum of the fidelity between the twoestimates is observed for µ (cid:39) . FIG. 2: (Color online) Fidelity between the exact and theapproximate mKE estimate for µ = 0 . µ = 0 . θ for (cid:104) σ (cid:105) = 0 and for different (close to unit)values of purity: µ = 1 − − (solid red line), µ = 1 − − (green dashed), µ = 1 − − (blue dotted). The lower rightpanel shows the minimum of fidelity for for (cid:104) σ (cid:105) = 0 (solid redline), (cid:104) σ (cid:105) = ± . (cid:104) σ (cid:105) = ± . B. Fidelity for nearly pure initial states
Let us now analyze the situation in which the initialstate τ is closer to a pure state, with µ ∈ [0 . , θ around π/ |(cid:104) σ (cid:105)| → θ close to 0 or π and for (cid:104) σ (cid:105) = 0. Thesephenomena are illustrated in the upper panels of Fig. 2,where we report the behavior of fidelity as a function of θ and (cid:104) σ (cid:105) for two values of the initial purity. The plot for µ = 0 . θ and (cid:104) σ (cid:105) , while their fidelity starts to differ from unit when θ is near 0 or π . The discrepancy becomes more and moreappreciable as far as (cid:104) σ (cid:105) approaches 0. Another infor-mation extracted from these plots is that the range of θ values where the fidelity is appreciably smaller than onetends to shrink and move towards zero while the purityincreases. In other words, when µ →
1, and thereforethe initial state τ is pure, the minimum of fidelity stayson θ = 0, π , while for all the other values of θ and (cid:104) σ (cid:105) the exact and the approximate mKE solutions coincide.When the purity of the initial state decreases (still beinglarger than a threshold value, say 0.9), we find a neigh-borhood of θ = 0 (and π ) where the two solutions aredifferent. Furthermore, in this interval of θ values, the fidelity decreases towards a minimum, which approachesto 1 / µ goes to unit.The lower right panel of Fig. 2 shows the behaviorof the global minimum as a function of purity. Noticethat for a nearly pure initial state, the values taken bythe fidelity in the minimum may be far from one. Thisbehavior may be understood as follows: let us considerthe point of minimum fidelity, i.e. θ = 0 (or π ) and (cid:104) σ (cid:105) = 0, for µ →
1. This corresponds to assume theinitial state τ to be one of the pure, | (cid:105) or | (cid:105) , eigenstatesof σ . On the other hand, if the measured mean value of σ is zero this suggest that the state ρ is somehow mixed.In fact, the exact mKE estimate is the completely mixedstate ρ (cid:39) I /
2. On the contrary, the approximate solutionis the pure state | φ (cid:105) = 1 / √ | (cid:105) + | (cid:105) ). In other words,the very form of the approximate solution tends to keepthe purity of the initial state unchanged, as it is apparentfrom Eq. (5) when we consider τ = | ψ (cid:105) (cid:104) ψ | . FIG. 3: (Color online) The ratio, Z , between the two fidelities F ( ρ apx , τ ) and F ( ρ exa , τ ), as a function of θ and (cid:104) σ (cid:105) , fordifferent values of initial purity µ . In the first graphic wehave Z for µ = 0 .
7. Then, from top to bottom, from left toright, we find graphics for µ = 0 . µ = 0 . µ = 0 .
99. Itis evident that Z is always greater (or equal) than one, whichmeans that ρ apx is, in general, closer to τ than ρ exa . Actually, the approximate method appears to force thesolution to be closer to the initial state than the exact so-lution of Eq. (3) does, in agreement with the assumptionsused for its derivation. More precisely, the approximatesolution is closer, in terms of fidelity, to the initial statethan the exact one, for any values of µ , θ and (cid:104) σ (cid:105) . Thisphenomenon is illustrated in Fig. 3, where we report theratio Z = F ( ρ apx , τ ) /F ( ρ exa , τ ) between the fidelities ofthe two solution to the initial state. As it is apparentfrom the plots, we have Z ≥ C. The purity of the two solutions
As we have seen in the previous Sections, when theinitial state shows high purity the approximate solutionstends to preserve such purity irrespective of the resultsof the measurements, whereas this is not the case for theexact solution. Since this phenomenon represents the un-derlying reason of the behavior of fidelity reported in theprevious Section, here we provide a more detailed studyof the purity of the exact and approximate solutions asfunctions of the purity µ of the initial state τ . FIG. 4: (Color online) Purities of the approximate and exactsolutions of mKE estimation. The first three plots show thepurity of the approximate solutions as a function of the exactone for the initial purity in the ranges µ = 0 . − . µ = 0 . − . µ = 0 . − . θ ∈ [0 , π ] and µ in the given range, and simulatingrandom measurements with (cid:104) σ (cid:105) ∈ [ − , R µ as function of initial purity µ . Herethe grey points are obtained by randomly selecting the initialstates in the full range of θ and µ and (cid:104) σ (cid:105) ∈ [ − , θ in the range θ ∈ [ π/ − π/ , π/ π/
10 and |(cid:104) σ (cid:105)| ∈ [0 . , θ ∈ [ − . , .
01] or around π and (cid:104) σ (cid:105) ∈ [ − . , . The first three panels of Fig. 4 report the purity ofthe approximate solution µ apx as a function of the purityof the exact one µ exa for randomly generated values of θ ∈ [0 , π ] and (cid:104) σ (cid:105) ∈ [ − , µ of the initial state is randomly sampled in a fixed range: µ ∈ [0 . , .
6] in the upper left plot, µ ∈ [0 . , .
8] in theupper right plot, µ ∈ [0 . ,
1] in the lower left plot. Theupper left plot shows that for highly mixed initial states,the purities of the two solutions are close each other. Forintermediate values of the initial purity we see a mixedbehavior, whereas for the nearly pure initial states ofthe lower left plots the approximate solution tends topreserve their purities, such that µ apx is larger than µ exa for most of the values of θ and (cid:104) σ (cid:105) .The plot in the lower right panel of Fig. 4 shows theratio R µ = µ exa /µ apx as a function of the initial purity µ for a randomly chosen values of θ and (cid:104) σ (cid:105) . In orderto fully appreciate the content of this plot, let us firstconsider the case R µ >
1, i.e. µ exa > µ apx . This ratioachieves its maximum in the region µ ∈ [0 . , .
85] andthe same behavior may be recognized in the second plotof Fig. 4, where, for high values µ exa and µ apx , we seethat µ exa > µ apx . Notice that, for initial purity µ (cid:39) . θ around π/ (cid:104) σ (cid:105) close to ± θ around π/ (cid:104) σ (cid:105) near ± R µ < R µ , which achieve its minimum for pureinitial states. Again, this is due to the fact that the ap-proximate solution tends to keep the purity of the initialstate unchanged, while the exact one does not. Besides,this behavior may be recognized also in the third plot ofFig. 4. For initial high purities, the purities of the ap-proximate and the exact solutions are different only forvalues of θ close to 0 or π and (cid:104) σ (cid:105) . This may be alsoseen by randomly sampling points in that area, whichcorresponds to the red points of the last plot of Fig. 4. IV. COMPARISON BETWEEN THE EXACTAND THE APPROXIMATE MKE ESTIMATIONOF WEAK HAMILTONIANS
As mentioned above, the mKE principle is an usefultool to estimate the state of a system which has a biastoward a given state and when some information comingfrom measurements on the final state are known. There-fore, this principle may be naturally applied to the esti-mation of a weak Hamiltonian, which drives the evolutionof a system in the neighborhood of the initial state.Suppose that a qubit system is described by the initialstate τ , and it evolves according to the Hamiltonian H .The state after this evolution is given by ρ = e − iH τ e iH . (10)The Hamiltonian can be represented by the vector h =( h , h , h ) in the Pauli basis H = (cid:88) j =1 h j σ j . We assume to know the initial state and want to estimatethe Hamiltonian using the information coming from themeasurement of a single observable A after the evolution.Upon using the mKE principle to estimate the outputstate and expanding Eq. (10) to the first order in theHamiltonian strenght (in agreement with the hypothe-sis of weak interaction), the estimated coefficients of theHamiltonian are obtained [12]: h = τ × r | τ | (11)where τ and r are, respectively, the Bloch vectors ofthe initial state τ = 1 / I + τ · σ ) and of the final one ρ = 1 / I + r · σ ).Since the mKE estimate for the output state may beobtained using either the exact method or the approxi-mate one we have two possible estimates for the Hamilto-nian operators, which will be denoted by H exa and H apx .As a matter of fact, in both cases the coefficients are ob-tained from Eq. (11) and thus the difference between thetwo Hamiltonians is due to the difference between theexact and approximate mKE estimates for the states. Inother words, comparing the two Hamiltonians provide amethod to compare the two solutions of the mKE princi-ple in terms of their use as a probe, rather than in termsof their closeness in the Hilbert space. FIG. 5: (Color online) Comparison between exact and approx-imate solution for the mKE estimation of weak hamiltonians.On the left side, the trace distance D between the H exa andthe H apx is plotted for a purity of the initial state of µ = 0 . D is displayed for µ = 0 . In order to compare the two Hamiltonians we employthe trace distance, that is: D = 12 Tr [ | H exa − H apx | ]where | B | is the modulus operator of B , i.e. | B | = √ B † B . We found that this quantity does not dependon the phase φ of the initial state and shows the samesymmetries of the fidelity between the two solutions. In the following we present a brief analysis of the behaviorof D .When the initial state τ is highly mixed the differencebetween H exa and H apx is small, and reaches a maximalvalue for |(cid:104) σ (cid:105)| → θ = π/ / π (see Fig.5). This behavior is similar to the one of the fidelity formixed state, but instead of having a maximal differencefor θ = π/
2, here a minimal difference is found. This isdue to different estimates obtained for the componentsof the Bloch vectors which define the two mKE solu-tions. In fact, for θ = π/ |(cid:104) σ (cid:105)| →
1, the first twocomponents (here referred to as σ and σ ) of the Blochvectors are small but quite different for the exact andthe approximate mKE estimates. The last componentis anyway equal to one for both the Bloch vectors. Thefidelity is able to point this difference out, which how-ever is not affecting the estimation of the Hamiltonians,since the coefficients of the Hamiltonian focus only onthe larger component of the Bloch vectors. In turn, thetrace distance between the two estimated Hamiltoniansdo not detect a difference between the approximate andthe exact method. The behavior of the distance betweenthe Hamiltonians for µ ∈ [1 / , .
7] is analogous, thoughfor increasing µ another maxima appear, in the same wayminima appear for the fidelity.Let us now address nearly pure initial states: as it isapparent from the comparison of Figs.5 and 6 for increas-ing µ we see a transition in the behavior of the trace dis-tance between Hamiltonians: new maxima appear, for θ → , π and (cid:104) σ (cid:105) = 0. and their value increases for µ →
1. Overall, the behavior of the trace distance D fornearly pure initial states is analogous to that of the fi-delity in the same regime, and all the observations madein that case holds. FIG. 6: (Color online) Comparison between exact and approx-imate solution for the mKE estimation of weak hamiltonians.On the left side, the trace distance D between the H exa andthe H apx is plotted for a purity of the initial state of µ = 0 . D is displayed for µ = 0 . V. CONCLUSIONS
In this paper, we have considered mKE state estima-tion for qubits when the average of a single observable isavailable. In particular, a detailed comparison betweenthe approximate and exact solution of the mKE estima-tion problem has been performed, with the goal of findingthe regimes where the approximate solution may be ef-fectively employed. In this case, the advantage is thatthe approximate solution is given in a closed-form, andit may applied to a larger class of a priori states, includ-ing those described by a density operator not having notfull rank.In order to compare the two solutions we have analyzedin details the behavior of fidelity between the two esti-mated states as a function of the parameters of the initialstates and of the outcome of the measurement. Our re-sults show that the most striking difference concerns thepurity of the estimated states, with the approximate solu-tion that tends to preserve the purity of the initial state,while the exact one does not, being more sensitive tothe information coming from the measurement outcome.Moreover, we find that in terms of fidelity the approxi-mate solution is closer to the initial state than the exactone for the whole range of parameters. We have also addressed mKE principle as a tools toestimate weak Hamiltonians and compared the perfor-mances of the two solutions for this specific task. Em-ploying the trace distance to compare the estimatedHamiltonians, we found results that confirm those ob-tained analyzing fidelity.Overall, our analysis shows that approximate solutionsto mKE estimation problems may be effectively employedto replace the exact ones unless the initial state is closeto an eigenstate of the measured observable. In turns,this provides a rigorous justification for the use of theapproximate solution whenever the above condition doesnot occur.
Acknowledgments
This work has been supported by MIUR (project FIRB“LiCHIS” - RBFR10YQ3H) and the University of Padua(project “QuantumFuture”).
Appendix A: Bloch vector of the approximate solution in Eq. (7)
Upon solving the equation Tr[ ρ ( λ ) σ ] = (cid:104) σ (cid:105) , we obtain an analytic form for the Lagrange multiplier λ and, inturn, for the Bloch vector of the approximate solution of Eq. (7) r = ( (cid:15) −
1) sin θ cos φ ( (cid:104) σ (cid:105) ( (cid:15) −
1) cos θ + (cid:15) + 1) (cid:113) − ( (cid:104) σ (cid:105) ( (cid:15) +1)+( (cid:15) −
1) cos θ ) ( (cid:104) σ (cid:105) ( (cid:15) −
1) cos θ + (cid:15) +1) ( (cid:15) − cos θ − ( (cid:15) + 1) r = − ( (cid:15) −
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1) cos θ + (cid:15) + 1) (cid:113) − ( (cid:104) σ (cid:105) ( (cid:15) +1)+( (cid:15) −
1) cos θ ) ( (cid:104) σ (cid:105) ( (cid:15) −
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