We discuss the problem of estimating a general (mixed) qubit state. We give
the optimal guess that can be inferred from any given set of measurements. For
collective measurements and for a large number $N$ of copies, we show that the
error in the estimation goes as 1/N. For local measurements we focus on the
simpler case of states lying on the equatorial plane of the Bloch sphere. We
show that standard tomographic techniques lead to an error proportional to
$1/N^{1/4}$, while with our optimal data processing it is proportional to
$1/N^{3/4}$.Comment: 4 pages, 1 figure, minor style changes, refs. adde

A. Peres

A. Rodriguez

D.F.V. James

D.G. Fischer

E. Bagan

G. Vidal

G.M. D’Ariano

J.I. Cirac

J.I. Latorre

K. Banaszek

K.R.W. Jones

M. Baig

M. Huebner

M. Keyl

N. Gisin

R. Derka

R. Josza

R. Muñoz-Tapia

R.D. Gill

R.T. Thew

S. Massar

Th. Hannemann

Z. Hradil

English

arXiv

We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can be inferred from any given set of measurements. For collective measurements and for a large number N of copies, we show that the error in the estimation varies as 1/N. For local measurements, we focus on the simpler case of states lying on the equatorial plane of the Bloch sphere. We show that the error using plain tomography varies as 1/N1/4, while our approach leads to an error proportional to 1/N3/4

Bagán Capella, Emili

Baig i Aleu, Marià,

Muñoz Tapia, Ramon

Rodríguez González, Alberto

American Physical Society

Diposit Digital de Documents de la UAB

RAPID COMMUNICATIONSPHYSICAL REVIEW A 69, 010304~R! ~2004!Collective versus local measurements in a qubit mixed-state estimationE. Bagan,1 M. Baig,1 R. Mun˜oz-Tapia,1 and A. Rodriguez1,21Grup de Fı´sica Teo`rica and IFAE, Facultat de Cie`ncies, Edifici Cn, Universitat Auto`noma de Barcelona,08193 Bellaterra (Barcelona), Spain2Area de Fı´sica Teo´rica, Facultad de Ciencias, Universidad de Salamanca, 37008 Salamanca, Spain~Received 17 September 2003; published 23 January 2004!We discuss the problem of estimating a qubit mixed state. We give the optimal estimation that can beinferred from any given set of measurements. For collective measurements and for a large number N of copies,we show that the error in the estimation varies as 1/N . For local measurements, we focus on the simpler caseof states lying on the equatorial plane of the Bloch sphere. We show that the error using plain tomographyvaries as 1/N1/4, while our approach leads to an error proportional to 1/N3/4.DOI: 10.1103/PhysRevA.69.010304 PACS number~s!: 03.67.2a, 03.65.WjI. INTRODUCTIONKnowing the state of a system is of paramount importancein quantum information. Quantum measurements provideonly a partial knowledge of a state. Such a state can only bereasonably reconstructed if a large number N of identicallyprepared copies of the system is available. Since the seminalwork of Holevo @1#, there has been a lot of research on thissubject. Most of the quantitative analyses have mainly fo-cused on pure states, for which the optimal strategies @2–6#have been identified. They give the ultimate limits that canbe achieved in state reconstruction. However, they involvecollective measurements that, although very interesting fromthe theoretical point of view, are very resource consumingand very difficult to implement in a laboratory.In the real world pure states are very scarce and so mixed-state estimation is not just an academic issue. For instance, itis important to estimate the purity of a state, since this pa-rameter often determines its utility for performing quantuminformation tasks. In quantum tomography, a quorum of lo-cal observables is measured on a ~large! number of copies ofa state r . From the relative frequencies of the outcomes, onethen obtains an approximation or guess rg to the signal stater . However, the statistical deviations often yield unphysicalstates, e.g., tr rg.1. In this case, one can either discard theresults or use a maximum likelihood data analysis @7#. Usingthis analysis one infers the physical state that provides theclosest theoretical probabilities to the observed frequencies.Many variants of these techniques can be found in the litera-ture @7,8#, but there is a notorious lack of quantitative results~see though @9,10#!.The large numbers law ensures that with an infinite num-ber of copies (N→‘) and infinite measurements the statecould be exactly reconstructed by any sensible method. Inpractice, however, one has access only to limited resourcesand the crucial issue is to quantify the quality of the recon-struction procedure. This is the question we address here. Wefocus on qubit states and use the fidelity as figure of merit.We obtain the best estimate for any given measurement andcompute the analytical expressions of the average fidelity forboth collective and local ~von Neumann! measurements inthe asymptotic limit ~large N).1050-2947/2004/69~1!/010304~4!/$22.50 69 0103II. BURES METRICThe estimation procedure goes as follows. After we havemeasured on the N copies of the system, some result is ob-tained, which we symbolically denote by x. Note that xstands for both a single outcome of a collective measurementand a list of N outcomes ~one for each individual measure-ment! in local schemes. Based on x, an estimate for r can beguessed, rg(x). The fidelity is defined as @11#f 5@ tr Arg1/2~x !rrg1/2~x !#2. ~1!It determines the maximum distinguishability between r andrg(x) that can be achieved by any measurement @12#. Forqubits, Eq. ~1! readsf rW ,RW ~x !5 11rWRW ~x !1A12r2A12R~x !22 . ~2!Here r5urWu and R5uRW u, where rW and RW (x) are the Blochvectors of the states r and rg(x), respectively @r5(11rWsW )/2; sW 5(sx ,sy ,sz) are the standard Pauli matrices#.The fidelity can be viewed as a ‘‘distance’’ between twodensity matrices. The corresponding metric is usually knownas Bures metric. From the infinitesimal ‘‘distance’’ f (rW ,rW1drW) it is easy to obtain the volume element ~normalized tounity: *dr51)dr54pr2drA12r2dn , dn5sin ududf4p , ~3!where dn is the invariant measure in the two-sphere. Equa-tion ~3! is the natural uniform probability distribution func-tion, or the a priori probability distribution for a completelyunknown qubit state r . For our discussion below, we willalso need dr when the density matrices are known to lie in agreat circle of the Bloch sphere. It readsdr512prdrA12r2du . ~4!©2004 The American Physical Society04-1RAPID COMMUNICATIONSBAGAN et al. PHYSICAL REVIEW A 69, 010304~R! ~2004!Although in Sec. IV we use Eqs. ~3! and ~4!, our main resultsin the following section are independent of any particularchoice of the a priori distribution.III. FIDELITY AND OPTIMAL GUESSThe average fidelity, hereafter fidelity in short, is the meanvalue of Eq. ~1! over the a priori distribution and over allpossible outcomes x,F5(xE dr f rW ,RW ~x !p~xurW !, ~5!where p(xurW) is the conditional probability of obtaining out-come x if the signal state has Bloch vector rW . These prob-abilities are determined by the expectation values of positiveoperators O(x), such that (xO(x)5I, i.e., p(xurW)5tr@O(x)r# . Our aim is to maximize Eq. ~5!.For a given measurement scheme $O(x)%, there alwaysexists an optimal guess, as we now show. We first introducethe four-dimensional Euclidean vectorr5~A12r2,rW !. ~6!Note that rr85A12r2A12r821rWrW8 and uru5Arr51.With this, the average fidelity readsF5(xE dr 11rR~x !2 p~xurW !, ~7!where R(x) is defined in analogy to Eq. ~6!. Equation ~7! canbe cast asF512 112 (x V~x !R~x !, ~8!whereV~x !5E drrp~xurW !. ~9!Using the Schwarz inequality one obtains the upper boundF512 S 11(x uV~x !u D , ~10!which is saturated with the choiceR~x !5V~x !uV~x !u . ~11!Since the guess ~11! satisfies uR(x)u51 and its first compo-nent is non-negative, it always gives a physical state. In fact,for any set of measurements and any a priori distribution,Eq. ~11! is the best state that can be inferred and Eq. ~10! isthe maximum fidelity.As the number of copies of the system becomes asymp-totically large, any reasonable estimation scheme leads to aperfect reconstruction of the state, i.e., F→1. For a large but01030finite N, the relevant issue is knowing the rate at which theperfect estimation limit is attained. For pure states, it is wellknown that the best collective strategy yields F;121/N@F5121/(4N) for states on the equator of the Blochsphere# @2#. It has also been shown recently that thisasymptotic limit can be achieved with local measurements@6#. For mixed states much less is known. Most of the purestate results cannot be extrapolated to the mixed case, andsome others may look counterintuitive at first sight. Althoughthe space of mixed states seems to be larger, they are lessdistinguishable than pure states. The fidelity ~2! has a mini-mum value (12r)/2 which is never zero but for pure states(r51). Thus the average fidelity could, in principle, belarger than that of pure states alone. Note that any estimatedmixed state rg has some overlap with the signal state r . Tobe more concrete, imagine one does random guessing, with-out performing any measurement at all, i.e., p(xurW) is uni-form. Then, using Bures volume element ~3!, the averagefidelity is F rand51/218/(9p2), which is larger than the ran-dom value (F51/2) for pure states.IV. RESULTSA. Collective measurementsAs a first application of the results of the previous section,let us obtain the asymptotic behavior of the fidelity with theoptimal collective measurement scheme. The main resultsare contained in @9#, where an optimal ~and minimal! gener-alized measurement was obtained for qubit density matricesand generic isotropic probability distributions. However, nodefinite form for this distribution was assumed and no ex-plicit results were obtained. The conditional probabilitiesp(xurW), which can be read from @9#, carry two labels x5(k ,mW ). The discrete index k refers to the representations ofthe symmetric space spanned by $r ^ N% onto which the posi-tive operators of the measurement project, whereas the unitvector mW labels a continuous set of outcomes in the two-sphere @13#. We havep~k ,mW urW !5ckS 12r24 DN/22kS 11rWmW2 D2k, ~12!whereck5S NN/21k D ~2k11 !2N/21k11 . ~13!From Eqs. ~9! and ~10! we obtainF512 112 (k E dmuV~k ,mW !u. ~14!The sum in Eq. ~14! runs from k50 (k51/2) for N even~odd! to k5N/2, which coincides with the results in @9#.Taking advantage of the rotational invariance, the integrals inEq. ~14! can be evaluated exactly. The computation of theasymptotic limit is rather lengthy and will not be reproducedhere @14#. The final result is4-2RAPID COMMUNICATIONSCOLLECTIVE VERSUS LOCAL MEASUREMENTS IN A . . . PHYSICAL REVIEW A 69, 010304~R! ~2004!F512S 34 1 43p D 1N 1 . ~15!Note that this fidelity is only slightly worse than that of purestates: 3/414/(3p)51.17*1.The expression ~15! also gives us important informationabout the optimal fidelity when the a priori probability dis-tribution corresponds to states known to lie in the equatorplane of the Bloch sphere ~4!. Since in this situation we havemore information about the states, the fidelity cannot beworse than Eq. ~15!, i.e., the error, defined as E512F , mustsatisfy E<j/N , where j is a constant.B. Local measurementsLet us now tackle the problem of reconstructing a qubitstate r from local von Neumann measurements. They are themost interesting ones from the experimental point of viewand can be readily implemented in a laboratory. For simplic-ity, we will consider only states that are known to lie on theequator plane of the Bloch sphere ~4!. This is a nontrivialcase that can be relevant for quantum optics ~e.g., for polar-ization states of photons!. The techniques we used are essen-tially contained in @5,6#. Full details of the calculations willbe presented elsewhere @14#; here only our main results willbe sketched.Consider N52N copies of the state r . Quantum statetomography tells us that von Neumann measurements alongtwo fixed orthogonal directions x and y are sufficient to re-construct the state. After the measurements, we obtain a setof outcomes 11 and 21 with relative frequencies a i and12a i , respectively (i5x ,y). This occurs with probabilityp~aW urW !5)iS NNa iD S 11ri2 DNa iS 12ri2 DN(12a i). ~16!In the most plain quantum tomographic approach the guess isgiven byRT~aW !5~A12R2,R cos g ,R sin g!,R cos g52ax21, R sin g52ay21. ~17!In many instances, however, the statistical fluctuations pro-duce an unphysical guess ~the square root term becomesimaginary!. If one discards these cases, the asymptotic be-havior of the fidelity can be shown to be F512jT /N1/41 , where jT is a constant. Although plain tomographyyields a perfect reconstruction of the state in the asymptoticlimit, it is much worse than the optimal collective scheme@note the power 1/4 of N as compared to the power 1 in Eq.~15!#. One may suspect that the cause of this behavior is thenumber of copies discarded, but we now show that it is notentirely so.Within the maximum likelihood framework @7# all avail-able data are used. If R<1, the guess is the tomographicone: RML5RT @see Eq. ~17!#, andRML5~0,cos~F!,sin~F!! ~18!01030if R.1, where F is the solution of the equation cos(2F)5R cos(g1F). In the asymptotic limit one can expand thisequation as a power series in (R21), F5g2(R21)cot 2g1. In fact only the first term is necessary forour calculation. After some effort one getsF512jMLN3/41 , jML5G~1/4!325/49p2.0.2256. ~19!Notice the significant increase in the rate at which the fidelityapproaches unity as compared to plain tomography.Finally, we have computed the fidelity for the optimalguess ~10!. Here again, all available data are used to producea reconstruction of the state. In Fig. 1 we compare the opti-mal guess and maximum likelihood methods for up to N520 copies of a state. It is clear that the optimal guess strat-egy always performs better. We have also obtained theasymptotic limit. The fidelity reads in this caseF512jON3/41 , jO.0.1708. ~20!The analytical expression of jO is rather involved and will begiven elsewhere @14#. Notice that this fidelity approachesunity at a rate similar to the maximum likelihood one ~19!,but the coefficient of the first correction is lower (jO,jML), as it should be. The most important parameter is theexponent of the 1/N term in Eq. ~20!. It shows that there is agap in the quality of the reconstruction process between fixedlocal measurements and optimal collective schemes @recallEq. ~15!#. One may argue that we have not exploited classi-cal communication, i.e., we have not designed each indi-vidual measurement according to the outcomes of the previ-ous ones. We have also explored this possibility numericallyand observed that the fidelity is almost identical to that ob-tained from the optimal guess with measurements along twofixed orthogonal directions. Therefore, we are led to conjec-ture that an error rate E;1/N3/4 is the lowest that can beachieved using any local scheme involving von NeumannFIG. 1. Average fidelities in terms of the number of copies. Thediamonds ~crosses! correspond to the maximum likelihood ~opti-mal! guess.4-3RAPID COMMUNICATIONSBAGAN et al. PHYSICAL REVIEW A 69, 010304~R! ~2004!measurements. It is unclear whether or not this conjecturemay apply to generalized measurements ~local positive op-erator valued measures!.V. CONCLUSIONSWe have obtained the optimal reconstruction of a generalqubit state for any given set of measurements and have illus-trated our results in some interesting cases. We have com-puted the asymptotic expression of the fidelity for the opti-mal collective scheme. For local measurements we haveconsidered the simpler but important case of states lying onthe equator plane of the Bloch sphere. We have shown thatthe performance of plain tomography is very poor, with anerror that varies as E;1/N1/4 for large N. We have shownthat maximum likelihood does provide a much better estima-tion: E;1/N3/4. Using the same data, the optimal guessanalysis gives the best reconstruction of the signal state. De-spite this improvement, the asymptotic behavior of the fidel-ity does not saturate the optimal collective bound. This is in01030contrast to pure state estimation, where local measurementscan perform optimally in the asymptotic regime. Althoughwe have mainly focused on measurements along fixed or-thogonal directions, we have also analyzed the most generallocal strategy, in which one is entitled to change these direc-tions after each individual measurement. Our results stronglysuggest that for mixed states the asymptotic behavior of theoptimal collective schemes cannot be attained by any localstrategy.ACKNOWLEDGMENTSWe thank A. Acin, G. M. D’Ariano, and C. Macchiavellofor useful conversations. We acknowledge financial supportfrom MCyT Project Nos. BFM2002-02588 and BFM2002-02609, CIRIT Project No. SGR-00185, and QUPRODISworking group EEC Contract No. IST-2001-38877. R.M.T.thanks the Benasque Center for Science for hospitality.A.R.G. acknowledges the hospitality of the IFAE and espe-cially of the GIQ.@1# A.S. Holevo, Probabilistic and Statistical Aspects of QuantumTheory ~North-Holland, Amsterdam, 1982!.@2# S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 ~1995!; R.Derka, V. Buzek, and A.K. Ekert, ibid. 80, 1571 ~1998!.@3# K.R.W. Jones, Phys. Rev. A 50, 3682 ~1994!.@4# J.I. Latorre, P. Pascual, and R. Tarrach, Phys. Rev. Lett. 81,1351 ~1998!; N. Gisin and S. Popescu, ibid. 83, 432 ~1999!; E.Bagan et al., ibid. 85, 5230 ~2000!; Phys. 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Huebner, Phys. Lett. A 163, 239 ~1992!; R. Josza, J. Mod.Opt. 41, 2315 ~1994!.@12# C.A. Fuchs, Ph.D. dissertation, University of New Mexico,1995; e-print quant-ph/9601020.@13# In Ref. @9# the authors use a discrete set of directions instead ofthe continuous set considered here.@14# E. Bagan ~unpublished!.4-4

Collective versus local measurements in a qubit mixed-state estimation

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Collective vs local measurements in qubit mixed state estimation

Collective vs local measurements in qubit mixed state estimation

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